Lectures in Micro Meteorology

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1 Downloaded from orbt.dtu.dk on: Sep, 018 Lectures n Mcro Meteorology Larsen, Søren Ejlng Publcaton date: 015 Document Verson Publsher's PDF, also known as Verson of record Lnk back to DTU Orbt Ctaton (APA): Larsen, S. E. (015). Lectures n Mcro Meteorology. DTU Wnd Energy. (DTU Wnd Energy E; No. 0075). General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research. You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam.

2 Lectures n Mcro Meteorology Søren E Larsen 0 015

3 Lectures n MIcro Meteorology Report DTU-Wndenergy-E By Søren E Larsen Copyrght: Reproducton of ths publcaton n whole or n part must nclude the customary bblographc ctaton, ncludng author attrbuton, report ttle, etc. Cover photo: [Text] Publshed by: Department of Wnd Energy, Frederksborgvej 399 Request report from: ISSN: [ ] (electronc verson) ISBN: (electronc verson) ISSN: ISBN: [ ] (prnted verson) [ ] (prnted verson)

4 Preface Ths report contans the notes from my lectures on Mcro scale meteorology at the Geophyscs Department of the Nels Bohr Insttute of Copenhagen Unversty. In the perod , I was responsble for ths course at the Unversty. At the start of the course, I decded that the text books avalable n meteorology at that tme dd not nclude enough of the specal flavor of mcro meteorology that characterzed the work of the meteorology group at Rsø (presently of the Insttute of wnd energy of the Dansh Techncal Unversty). Ths work was focused on Boundary layer flows and turbulence and was often amed at applcatons lke wnd energy, wnd loads, dsperson and deposton, ar-sea exchange and ar-land exchange, as well as flow response to surface nhomogenety. The course, dmensoned to 60 hours, was generally structured n the frst year, based on copes of papers and copes of the overheads used for presentaton. But t gradually flled out n the followng years, as wth power ponts and the typed manuscrpts consttutng ths reports. Most wrtng was fnalzed wthn the frst 10 years of the course, meanng most references are somewhat dated by now, although I have not ressted addng more recent work, f ongong projects made t easy. In the course I have tred to present the detals of the basc materal, tryng to avod the well known sentence of It s easly seen. But I have been less thorough and pedagogcal, when presentng the more llustratve materal. The orgnal report ncludes pages of course materal, drectly coped from other people s publcatons, used durng the lectures. Therefore ths report s an nternal report only. In the present report these coped pages have been removed n respect for the rghts of the orgnal authors. DTU-Wnd Energy, Søren Ejlng Larsen Lectures n Mcro Meteorology

5 Content Summary Introducton Concepts, scales of moton and statstcal tools Basc equatons Mean flow, turbulence and closures The Ekman boundary layers The atmospherc surface boundary layer. Monn-Obuchov scalng Near surface vscous layers, roughness lengths: z 0 and z 0T, nterfacal exchange Scalng n the atmospherc boundary layer Horzontally heterogeneous boundary layers Dsperson plumes from chmneys Boundary layer clmate, radaton, surface energy balance Wnd clmate, wnd energy, wnd loads Instruments, measurements and data References Acknowledgements Lectures n Mcro Meteorology

6 Summary The report contans the authors lecture notes to courses n mcro scale meteorology and atmospherc turbulence. The course has amed to touch upon both the basc theores and the specal formulaton amng at applcaton. The structure of the course can be seen from the Content. It has a chapter on concepts and statstcs, two chapters on the basc flud dynamcs and turbulence closure. Three chapters on varous atmospherc scalng laws, Ekman layer, surface layer and total boundary layer. It ncludes one chapter on the roughness and the surface atmosphere exchange, one chapter on heterogeneous boundary layers, and one on atmospherc dffuson and turbulence. It has two chapters on boundary layer clmatology nvolvng radaton, temperature and wnd ssues, and fnally a chapter dscussng measurement and nstrumental problems. The report has been loosely edted compared to the orgnal notes. [Text - The followng lne contans a secton break - do not delete] Lectures n Mcro Meteorology 5

7 1. Introducton Characterstcs of the atmospherc planetary boundary layer (ABL), also called the planetary boundary layer (PBL), are of drect mportance for much human actvty and well beng, because humans bascally lve wthn the PBL. Hence, we bascally derve our wnd energy from wnds n the PBL, and most of our ar polluton s dspersed, deposted and chemcally transformed wthn the PBL. The mportance stems as well from atmospherc energy and water cycles ssues, because the fluxes of momentum, heat, and water vapour between the atmosphere and the surfaces of the earth all pass through the PBL, beng carred and modfed by mxng processes here. Snce these mxng processes mostly owe ther effcency to the mechansms of boundary layer turbulence, a proper quanttatve descrpton of the turbulence processes becomes essental for a satsfyng descrpton of the fluxes between the surface and the atmosphere. Descrpton of the structure of the flow, relevant scalar felds, turbulence and flux through the atmospherc boundary layers necesstates that almost all types of the flows, that occur there, must be consdered. For these objectves, there are very few combnatons of characterstc boundary layer condtons that are not of sgnfcant mportance, at least for some parts of the globe. 6 Lectures n Mcro Meteorology

8 . Concepts, scales of moton and statstcal tools Addtonally to the synoptc weather patterns, the meteorology of the PBL s strongly nfluenced by the surface characterstcs and turbulence structure. Therefore, we wll n ths ntroductory secton shortly summarse qualtatve aspects the dfferent processes nfluencng the PBL condtons addtonally to a more detaled dscusson of the statstcal methods used n general and whch we shall use throughout the text to descrbe the characterstcs of the PBL. Both wth respect to mean characterstcs, varablty and fluxes the PBL s domnated by turbulent moton. Therefore t s approprate frstly to consder what we should understand wth turbulence, a subject that has flled many pages n the scentfc lterature. Here we just notce that motons of systems that can be descrbed by the nonlnear flud equatons tend to show strongly varyng stochastc components, the turbulence, as well as more smooth and predctable characterstcs. Turbulence can occur on many scales of moton and be descrbed by as ether two-dmensonal moton or three-dmensonal moton. In the PBL, the wnd speed as well as temperature and humdty, and ndeed all atmospherc varables, show ths stochastc behavour on all spatal and temporal scales of varaton. In fgures.1, ths s llustrated by a measured tme seres of the wnd speed observed through dfferent tme wndows. The followng fgures.-6 all llustrate dfferent processes and scales of varablty wthn the PBL. Whle the moton n the PBL can vary on vrtually all scales, the processes wthn the PBL that create so called PBL turbulence occur most on tme scales of the order of and less than one hour, wth assocated spatal scales. Ths PBL turbulence s three dmensonal and therefore can carry most of the vertcal fluxes that s essental for the couplng between the atmosphere and the surface. On these tme scales the man mechansm for producng turbulence s the vertcal gradent of the mean wnd. In fgure. we show typcal vertcal varatons of wnd speed, humdty, and temperature between ther surface values and values at the top and above the PBL. Temperature and humdty can both ncrease and decrease wth heght, dependng on whether ther surface values or values n the free atmosphere are the larger. However, the wnd speed wll always ncrease wth heght from zero at the ground to ts value n the free atmosphere just above the PBL. The vertcal wnd shear gves rse to overturnng of the ar, producng the turbulence (Tennekes and Lumley, 198). Ths provdes a formdable mechansm for carryng the vertcal fluxes compared to the molecular transport mechansm that would have been an alternatve. For example, a temperature gradent of K across the lowest 10 meter heght wth a wnd speed at 5 m/s gve rse to a heat flux of about 0.5 mk/s (or 600 W/m ). If the flux had to be carred by molecular dffuson only, the result would be mk/s (or 5 mw/m ) only. The temperature structure of the PBL strongly nfluences the turbulence producton through ts nfluence on the densty of the ar. If the ar s warmer and thereby lghter close to the ground, t wll enhance the producton; f t s cooler at the ground the producton wll be reduced. To a lesser extent the humdty has smlar, although smaller effect because also admxture of water vapour changes the densty of the ar. Lectures n Mcro Meteorology 7

9 Fgure..1 Upper fgure: Wnd speed measured 30 meter above flat homogeneous terran n Denmark from Troen and Petersen (1989). The data were obtaned from a one-year tme seres recorded wth 16-Hz resoluton. Each graph shows the measured wnd speed over the tme perod ndcated. The number of data ponts n each graph s 100; each averaged over 1/100 of the tme perod ndcated. The vertcal axs s wnd speed, 0-0 meter/sec. Lower fgure: Smlar plot from Jensen and Busch (198) here based on 1000 ponts per plot. The lack of fne structure n the 10 sec plot s due to the onset of vscous dsspaton at the hghest frequences here, see dscusson of spectra n the end of ths secton. 8 Lectures n Mcro Meteorology

10 Fgure.. Characterstc heght varatons (profles) of the mean values of the wnd speed,u temperature,t and humdty, q from the ground to the top of the PBL, ndcated by h. Also shown by the arrows on the u-profle s the overturnng of the flow nduced by the vertcal velocty gradent. The profles are shown for the followng characterstc stuatons: a) Thermally unstable, e.g. a sunny day. b) thermally stable, e.g. a clear sky nght, and c) Thermally neutral, e.g. a hgh-wnd overcast stuaton. The sze and magntude of the turbulent eddes are ndcted by the rotatonal moton ndcated on the fgure. Instead of temperature, T, one wll n meteorology typcally use the potental temperature, θ, see secton 3.(larsen,1993) Based on the above dscusson we can now specfy the planetary boundary layer as beng the layer through whch the atmospherc varables change between ther values n the free atmosphere and ther values at the surface, the transton beng mostly controlled by turbulent moton and mxng. The structure and character of the turbulence wll be dfferent for the dfferent thermal condtons, as s shown below n fgure.3, llustratng the turbulence structure for thermally unstable and stable condtons. Fgure.3 Depcton of the turbulence structure for unstable and stable atmospherc boundary layers. From Wyngaard (1990) Lectures n Mcro Meteorology 9

11 Above we have dscussed the PBL as t was horzontally homogenous, meanng that only the vertcal varaton was mportant. Next we shall llustrate the horzontal structure of a PBL as n fgure.4 and ts durnal varaton n fgure.5. Fgure.4 Horzontalvarablty n a PBL. The PBL s seen as consttuted by a number of overlappng and nteractng nternal boundary layers, reflectng the dfferent surface characterstcs. We shall later dscuss ths n detal. The dfferent characterstcs of the PBL, depcted above, all operate at dfferent tmescales and spatal scales. The durnal cycle s of obvous mportance for the chance of regmes seen n fgures. and.5. The heght of the PBL and the terran features are seen to mpose dfferent length scales n fgures.3 and.4, and overall wll the tme and spatal scales of the synoptc flow be seen n the PBL. Fgure.5. Durnal varaton of a PBL, from Stull (1991). The fgure depcts the rse of the PBL heght wth surface heatng at sun rse, and the subsequent rse of the nght tme PBL after sun set by radatonal coolng. Obvously overcast condtons wll modfy ths turn of event, see fgure.. In fgures. and.3 the producton of turbulence s envsoned as a swrlng moton nduced by the shear, and modfed by the temperature structure. A whorl consttutes a volume of localzed vortcty, whch we shall denote an eddy. Ths pcture of turbulence, as a soup of 10 Lectures n Mcro Meteorology

12 ntertwnng spaghett-lke eddes, has been very useful n the study of turbulence n spte of ts extreme smplcty. Each eddy can be assocated wth a sze, or spatal scale, and a tme or tmescale. When eddes have been produced they wll reman coherent for some tme, creatng ther own smaller scale shear. By the same process as for the mean shear ths eddy shear wll create smaller eddes transferrng the knetc energy to smaller and smaller eddes, untl t s turned nto heat by the vscocty of the ar. Smlarly many other atmospherc processes can be charactersed by ther tme and spatal scales, descrbng roughly the tme they typcally takes and ther extent when they occur. In Fg..6 s depcted a number of characterstc processes and ther characterstc spatal and temporal scales, rangng from weather systems, wth a tmescales of about one week and a spatal scale of about 1000 km, and down to the smallest turbulence eddes, called dsspaton range, where the flud eddes have such small scales that they are dsrupted by vscosty and the knetc energy s turned nto heat. The characterstc scales here are about fve ms and fve mllmetres. Fg..6 s n a way a space-tme scale representaton of the atmospherc motons that are presented as tme sgnals n Fg..1. Fgure.6. Tme and space scales for the processes nfluencng the flow n the atmospherc boundary layer (Busch et al., 1979). The three moton categores of smallest scale, all belong to the category of three dmensonal atmospherc turbulence that are of key mportance for the structure of the atmospherc boundary layer, and wll be descrbed more n the text. For comparson are shown as well characterstc spatal scales for aspects of the wnd power technology Lectures n Mcro Meteorology 11

13 From Fgure.6 reflects that there seems to be a rough relaton between the spatal and the tme scales of the dfferent moton elements, meanng that large scale moton elements are assocated wth large tme scales and smaller scales motons are assocated smaller tme scales. From the fgure s seen that there s proportonalty between the tme scales and the spatal scales of a moton type correspondng to about 1 m/s For the three dmensonal atmospherc PBL turbulence, the above rough proportonalty s formulated more precsely by Taylor s hypothess of frozen turbulence, whch state that the turbulence feld does change very lttle, whles t s blowng past the observer by the mean speed. Hence, what s observed as a tme change over the tme τ corresponds n realty to a spatal change along the drecton of the mean flow of the length l, such that: (.0) l = U τ, where U s the mean speed. Ths smple formulaton works remarkably well, and means that for ths type of turbulence one s really measurng the spatal varaton (along the mean wnd drecton) by measurng the tme varaton from a statonary meteorology mast. In vew of the above dscusson we shall next consder a few of the statstcal tools used to descrbe the atmospherc boundary layer turbulence. The coordnate system: We have two vectors n the system, the poston, r[m], and the velocty U(m/s) r = (x, x, x ) = ( x, y, z ). U = (u, u, u ) = ( u, v, w). (.1) Here the numbered varables are usual when formulatng the governng equatons n tensor form. The other varables are often used when the equatons are wrtten n ther component forms. We have as well a number of scalar varables: The ndependent varable tme: t (sec). Temperature: T [K]. The ar densty: ρ [kg/m3]. Water vapour: ρ w [kg/m3], water vapour mxng raton: q = ρ w /ρ. Concentratons of admxture: C [kg/m3], mxng rato for C: c = C/ρ. We shall consder an atmospherc varable. It can be any of those we have defned, U, T, q, C, c. For generalty, we use the varables s and e. We consder the varables as stochastc Both varables can be any of the varables above s, e, u 1, u, u 3, T, q, C, c. The varables s, e are functons of space and tme: s = s (x, t), where refers to the three coordnate numbers: 1,, 3. The smplest statstcal operator s the average or the mean value. We typcally operate wth three types of averages: Ensemble averages, tme averages and spatal averages. Averages are typcally denoted by an over-bar, or a bracket, lke <s>, or by captal letters. 1 Lectures n Mcro Meteorology

14 The coordnate system x 3 z s always vertcal!! x 3 z u 3 w x y u v T, C, q t x 1 x u 1 u Fgure.7. Commonly used coordnate systems and varables, wth a practcal example for near surface turbulent velocty and temperature. Notce that the average vertcal and lateral velocty are both close to zero, but smaller scale velocty turbulence s three dmensonal. W s close to zero, because vertcal mean values and slow fluctuatons cannot exst close to the ground. V s close to zero, because the horzontal coordnate system s algned wth the x along the mean wnd. The Ensemble Average: We magne that our varable, s = s (x, t), s part of an ensemble of representatons of the varable, s. Hence we wrte: s j = s j (x, t), where j s the ensemble ndex. N 1 (.) s( x,) t = (,) E s x t j N j Ensemble average s easest to magne for the case of a wnd tunnel smulaton, where one can restart the tunnels over and over agan, and ths way obtan an ensemble. For an atmospherc boundary layer one may magne that the boundary layer s started over and over wth smlar start condtons and smlar boundary condtons. Lectures n Mcro Meteorology 13

15 The use of ensemble averages s most convenent when dong mathematcs, e.g. takng the average of model output, where each output can be consdered one ensemble. Also, t s the average we use when averagng equatons to yeld equatons for the average varables. In practce one wll often use tme or space averages, ether because the sgnals avalable are functons of these parameters as e.g. for measurements of tme seres from meteorologcal towers, or because of the objectves of the study, as for example area averages often beng the goal of hydrologcal studes. The averagng procedures employed are lmted by the sgnals avalable but are also a matter of choce. As an example, we can take the sgnal n fgure.1 that would lend tself to tme averagng, snce t s a tme sgnal, but also to ensemble averagng usng ensembles of data from smlar days or hours. Tme average: Here, we average our varable, s = s (x, t), over a gven tme nterval T to obtan: T 1 (.3) sx (, t) = (, ) T sx t+ τ dτ T T Tme averages are especally convenent n connecton wth tme seres-obvously. Also all measurements are assocated wth tme (and space!) averagng, because of the fnte tme and space resoluton of all physcal nstruments. An example of a practcal tme averagng procedure s shown on fgure.8 below. Fgure.8. Tme seres of wnd speed and the formaton of the tme average. Spatal averages: Here, we average over a spatal nterval. Ths nterval can be a volume, an area and a lne average. 1 s( x, t) = (, ), (.4) s x + χ t dχ n. where the power n reflects the dmensonalty of the ntegral, n=1,, or 3. Spatal averages are much used when workng wth numercal models, where one knd of average s the average across a grd-element. From measurements, an area average s typcal what a satellte sees, snce t averages over the footprnt. A lne average would be what can be detected from arplane measurements and also from some knd of LIDAR or RADAR scatterng nstrumentaton. 14 Lectures n Mcro Meteorology

16 Ergode theorem: Assume that s (x, t) s statstcally statonary and homogeneous, and then we have (.5) lm s ( x, t ) = lm s ( x, t ) = lm s ( x, t ) = s = const.. T T N For such condtons we can therefore n the lmt change between the dfferent average values, snce they are all the same, but only when we can consder the varables statstcally statonary and homogeneous and when the seres are long enough n both tme and the relevant space dmensons. Statstcally statonary means that all statstcs of s s ndependent of t. Statstcally homogeneous means smlarly that all statstcs of s s ndependent of x. Note s (x, t) can be homogeneous n some of the dmensons, x, and not n others. For example n the boundary layer we have learned that the wnd speed ncreases wth heght. Therefore n general wnd speed s not homogeneous n the vertcal but t can well be consdered so along the horzontal axes, see fgure.. Whenever, we use sx (, t ) wthout ndcatons of whch knd of averagng that s appled, t should ether be obvous from the context or t wll mean ensemble average. Fluctuatons. For a stochastc varable, s, we defne the fluctuaton as the dfference between the s and ts mean value. The fluctuaton s typcally ndcated by s. (.6) s = s s Funny enough ths very smple equaton s honoured wth a name: The Reynolds Conventon. The mean value of the fluctuaton s zero: (.7) s = ( s s ) = s s = 0. N Snce s s a constant and the average of a constant s the same constant. Frequency dstrbutons, pdf s, (Co-) Varances, standard devatons, STD. Varances and standard devatons are measures of the magntude of the fluctuatons. The varance of fluctuatons s found by averagng the square of the fluctuaton: (.8) < s = ss = σ, 0 s where σ s s the so-called standard devaton. Alternatvely one can determne the varance as:. (.9) s = ( s s )( s s ) = s s. The covarance s found from two sgnals, s and e, as follows. (.10) s = e ( s s)( e e). Lectures n Mcro Meteorology 15

17 The magntude of the covarance depends on two features, the magntude of the two fluctuatons, s and e, and how well they correlate wth each other. To separate these two features one often study the normalse co-varance, called the correlaton. It s defned by: Fgure.9. The same tme seres as n fgure.8, but here wth the computaton of an estmate of the varance. Also shown s the frequency dstrbuton of the fluctuatons, s, used to descrbe the frequency of ampltudes of the fluctuatons around the mean value wth the varance.. The covarance s found from two sgnals, s and e, as follows. (.11) se =( s s)( e e). The magntude of the covarance depends on two features, the magntude of the two fluctuatons, s and e, and how well they correlate wth each other. To separate these two features one often study the normalse co-varance, called the correlaton. It s defned by: es (.1) 1 ρes = 1, σσ e s where the bounds on ρ es corresponds to the two stuatons that s and e are ether n perfect correlaton or n perfect counter-correlaton, that s s = e or s = -e. Notce, s / σ s = 1. Commutaton rules for Ensemble averagng and mathematcal operatons: Recall that the ensemble average, as defned n (.3), s just a sum: (.13) s 1 N u for N j, N j = 1 Ths means summaton and dfferencng commutes wth averagng! Dfferentaton and averagng: 16 Lectures n Mcro Meteorology

18 (.14) N N de 1 de j d 1 de = = e = j dx N dx dx N dx j= 1 j= 1 ; Smlarly for ntegraton: (.15) sdx dt = sdx dt. However, multplcaton and averagng do not commute: (.16) e s= ( e + e )( s+ s ) = e s+ es + se + se = e s+ se s e, snce the correlaton n general s dfferent from zero. Covarances nvolvng a velocty component can be nterpreted a transport along the drecton of the velocty component. Consder the covarance between the velocty component w and the concentraton, C. The wc obvously descrbes a transport of concentraton C through a plane perpendcular to the drecton of the w-component. (.17) Flux C = w C = ( w + w )( C + C ) = wc + wc + w C + w C = wc + w C. w It s seen that the transport across the surface perpendcular to w s composed of a flux gven by the mean speed tmes the mean concentraton plus a flux gven by the co-varance between the two fluctuatons. Therefore even f the mean w s zero there can be a flux. Ths s exactly how t s f we take w as the vertcal velocty. Close to the ground there can be no mean w, snce that would buld a postve or negatve pressure perturbaton close to the ground, whch would counteract the w wnd speed. On the other hand t s seen that f C /<C > <<1 then even a small<w> can gve rse to a flux, e.g. many trace gases lke CO. That the covarance can descrbe transport can be seen by breakng t down nto postve and negatve fluctuatons around the mean value, and by notng that the postve velocty perturbatons correspond to transport along the postve drecton of the w-wnd drecton, whle negatve w perturbatons correspond to transport along the negatve drecton of the w-drecton, see fgure.10. We now use the defnton of the ensemble average. (.18) wc = wc = wc wc wc wc j j N M M M M j Here the summaton s broken down nto subsets, correspondng to negatve and postve perturbatons on C and w as ndcated, N = M1 +M + M +M. The frst two terms correspond to transport of C to the rght hand sde of the fgure, ether by transportng postve perturbatons of C to the rght or transportng negatve perturbatons of C to the left. Mathematcally, these two terms are seen to contrbute postvely to the total co-varance. If there s a mean gradent of C(z) as shown n fgure.10, these two terms domnate the sum. The two last terms correspondngly lead to transport of C to the left, and contrbute negatvely to the co-varance. Lectures n Mcro Meteorology 17

19 The resultng flux s therefore determned by the balance between the frst and the second group of terms. w + >0 0 w <0 0 z <C(z)> Fgure.10. Transport by negatve and postve velocty fluctuatons. Postve excursons from the <C(z=0)> wll tend to be assocated wth w>0 and vce versa. In a broad sense the varance and covarance are used to descrbe the fluctuaton ntenstes and relaton between dfferent sgnals. Havng consdered the varances and co-varances one can consder hgher-order moments, and the dstrbuton functons of the sgnals to study dfferent aspects of ther behavour. However, snce t wll not be much used here, we shall proceed to the tools used to dentfy the scales of varaton. Seres statstcs. Above we have consdered statstc measures for stochastc tme and space varables. We have focused on measures measurng the ntensty and correlaton of the stochastc seres. Now we shall consder methods that also the memory aspects of the seres. Covarances and correlatons. Assume a tme seres, s (t). The auto-covarance functon s defned as: (.19) R ( t, τ) = s ( t) s ( t + τ). s If s (t) s statstcally statonary, the R s (t,τ) = R s (τ), because, by defnton, no statstcs can depend on t. For statonary condtons, we can wrte: (.0) R s ( τ) = R s ( t, τ). = s ( t ) s ( t + τ) = s ( t τ ) s ( t ) = R (, ) ( ). 1 1s s t τ = R τ 1 where we used the substtuton: t 1 = t +τ. Note further that: (.1) R s(0) = s = σ s 18 Lectures n Mcro Meteorology

20 The autocorrelaton functon, ρ s (τ), s obtaned by normalsng R s (τ) by σ s. It s seen that ρ s (τ) s an even functon n τ, and that ρ s (0) = 1. Fgure.11. Example of autocorrelaton functons, showng the defntons of the ntegral tme scale, and showng that Auto-correlaton functon can change sgn, but that ts value for zero lag s by defnton equal to one (Tennekes and Lumley, 197) For space seres we can smlarly defne an auto-covarance functon: (.) R ( x, χ ) = s ( x ) s ( x + χ ), s where subscrpt now refers to the three spatal coordnates. Correspondng to statonarty for tme seres, we have homogenety for space seres. Recall that for space seres we may have homogenety along some of the coordnates and not along others, e.g. the vertcal axs. For homogeneous space seres the auto-covarance functon s a functon of the ncrement, χ only, and t s an even functon n χ. We can also defne the autocorrelaton functon by normalsng wth seres varance. Fnally, snce we know that the atmospherc varables are functons of space and tme, we can defne: (.3) R ( x, t, χ, τ) = s ( x, t) s ( x + χ, t+ τ) = R ( χ, τ), s s where the last equalty sgn assumes that we have both statonarty and homogenety. Note that snce our basc varables are functon of space and tme, n the prncple, the spatal and temporal auto-covarance functons reman functon of the other coordnates. For example for a statonary tme seres of an atmospherc varable, s (x, t), we have: (.4) R ( x, t, τ) = s ( x, t) s ( x, t+ τ) = R ( x, τ), s s Correspondng to the auto-covarance functon we can also have cross-covarance functons, from the varable s and e. In general we have: Lectures n Mcro Meteorology 19

21 (.5) R ( x, t, χ, τ) = s ( x, t) e ( x + χ, t+ τ) = R ( χ, τ), se se Where, we agan have used sutable statonarty and homogenety crterons as needed. (.6) R ( x, t, τ) = s ( x, t) e ( x, t+ τ) = R ( x, τ), se se Agan, we can normalse wth es to obtan the cross-correlaton functon, ρ es (x, τ). Note that the cross correlaton functons are not necessarly even functons n nether χ nor τ. In terms of tme correlaton, ths s a consequence of: (.7) e () t s ( t + τ) e ( t + τ) s () t The correlaton functons are measures for the memory of the varables that are correlated and thereby also a measure of the memory of the processes behnd the varables. The correlatons tend towards zero for large lags, meanng that the correlated varables for such lag are ndependent of each other, whch s another word for that the memory for these lags has dsappeared. For auto-correlaton functons, one often uses the ntegral of the correlaton functon as a measure of the memory. The scale s called the ntegral scale, see fgure.11. We shall use the correlaton functon to study how well determned a gven tme average can be expected to be. Consder the defnton of the tme average: (.8) T 1 sx (, t) = sx (, t+ τ) dτ T T T For statonary condtons and for tme gong to nfnty the tme average approaches the true average followng the Ergode theorem, meanng that: (.9) sx (, t) sx ( ) fort T where the true mean value cannot be a functon of t due to statonarty. Droppng for the moment the space coordnates, x, we now consder the varance: (.30) δ = (() st s) T T Insertng nto the tme averagng ntegral we get: 0 Lectures n Mcro Meteorology

22 (.31) 1 1 δ τ τ τ τ T T T = ( s( t ) d s) ( s ( t ) d ) T + = T + T T T T T T 1 1 = s ( t t ) s ( t t ) dt dt Rs ( t t ) dt dt 4T + + = 4T T T T T From the appendx we obtan, not very easly: (.3) T 1 ξ δ ξ ξ T = (1 ) R( ) d. T T 0 Introducng the autocorrelaton functon ρ s (τ), and changng ξ toτ, ths expresson can be wrtten: (.33) T σ s τ δ = (1 ) ( ) d T ρ τ τ s T T 0 For T, δ T 0 as t should for a statonary tme seres. For T small, ρ s (τ) 1 for the whole ntegraton, the ntegral becomes the area of the trangle between (0, 1) and (T, 0), and δ T σ s. For T large, the correlaton functon, ρ s (T) 0 for whch reason we can ntegrate all the way to nfnty, and the ntegral becomes: (.34) σ s τ σ σ s s τ δ (1 ) ( ) d ( ) d ( ) d T ρ τ τ = s ρ τ τ ρ τ τ s s T T T T T σ σ s s ( T / ) 0 =, s T ( T / T ) s where the ntegral scale, T s, s gven by: (.35) T = ρ ( τ) dτ s s It should be noted that the ntegral scale s defned, n some references, as the ntegral from zero to nfnty, and s therefore only half the value obtaned from (.44). Ths ambguty s throughout the lterature, one just has to be observant. The result above equals the varance for the seres, s, dvded by an estmate of the number, N, of statstcally ndependent estmates of the tme average of s that can be made n the tme T, gven the ntegral tme scale, T s, for the autocorrelaton functon; N T/T s. As always smlar expressons and statements can be made for the homogeneous spatal seres, s (x ). Lectures n Mcro Meteorology 1

23 Practcal consderatons about averagng, statonarty and homogenety. Wthn the dealsed mathematcal world of ensemble averagng, there are few practcal problems to consder. If one moves to the other types of averagng, for example the tme averagng, one must consder the averagng tme from more practcal consderatons. One aspect s the statstcal uncertanty of the average. Here one can be guded by equatons lke (.33). However there are more qualtatve consderatons as well. As llustrated on Fgure.1 and.6 geophyscal tme seres fluctuate on all tme scales and at least on spatal scales less than km, meanng that the proper averagng tme s not obvous. When defnng an averagng tme, one defnes both the average values, fluctuatng on tme scales larger than the averagng tme, and the fluctuatons, fluctuatng at tmescales smaller that the averagng tme, see e.g. Fg..1. One sort of defnes whch flow varablty to call varaton of the mean values, and whch to call varaton of the fluctuatng values. Snce much of the studes n mcro scale meteorology are focused on relatons between average values and fluctuatons, one wshes to nclude all the processes, denoted boundary layer processes n the averagng. Comparng wth Fg..6, we see that ths corresponds approxmately to and averagng tme between 0 mnutes and two hours. Smultaneously, we wsh to nclude as much of the fluctuatons, contrbutng to the vertcal fluxes between the surface and the atmosphere through eq. (.16) n the fluctuatng part of the sgnal. In Fg..6, also the cumulus clouds are known to nvolve mportant vertcal wnd speeds. Hence one may be tempted to ncrease the averagng tme. However, the averagng tme could then get too close to the durnal varaton wthn the sgnal, and one could lose the statonarty approxmaton. Addtonally, experence shows that for averagng tmes longer than about one hour, the Taylor theory of frozen turbulence becomes less correct for three dmensonal turbulence. A further consderaton s that one wll prefer averagng tmes such that characterstcs of both the average flow and the turbulence are not too senstve to the accurately chosen tme. Here one wll often refer to the spectral language, where the frequences separatng average values and fluctuatons for averagng tme between 0 mn- 1 hour lay wthn the so called spectral gap n for example Fg..13. Ths means that small changes n the averagng tme wll not change the varance of the fluctuatons sgnfcantly. Fnally, an averagng tme longer than 30 mnutes wll smooth many transent phenomena of nterest, lke wnd gusts and frontal passages, whch wll be smoothed too much by the averagng. All consdered the normal averagng tme for meteorologcal statons conventonally has settled between 10 mnutes and one hour. Fourer and spectral analyss. Above we have consdered the correlaton analyss as a tool to study both correlatons (- and that means possble relaton between dfferent stochastc space or tme seres) and to study the memory or nerta n the processes behnd the data seres. We have attrbuted the word tme and space scales to dfferent processes. We shall now try to develop a more precse descrpton of scales through the use of Fourer analyss, where the gven seres are expanded nto snus and cosnes. Snce frequency and wavelength for these functons have a precse meanng, we wll be able to dscuss the tme and spatal scales n a more precse way. In a loose sense, we wrte a, say- tme seres, as a seres of sne and cosne functons of (ω t) wth dfferent ω. We obtan the Fourer spectrum of a tme seres by correlatng the seres wth cosne or sne functons of frequency, ω. The magntude of the correlaton for each frequency s a measure of the contrbuton to the ampltude of the tme seres from sne and cosne functons of frequency ω. The square of these correlatons s Lectures n Mcro Meteorology

24 denoted the spectrum and s a functon of frequency and measures the contrbuton from each frequency to the total varance of. A large spectral value for a gven frequency means that ths contrbute much to the varance, and vce versa for a small value. A basc aspect of Fourer analyss s that there exst pars. To a gven functon of tme and space there exst one and only one Fourer functon of frequency and wave numbers, provded certan condtons are fulflled. To prove ths mathematcally, one must formulate the condtons on the functons. The Fourer methods have been proven mathematcally for the followng types of functons (Lumley and Panofsky, 1964, Yaglom, 196): Perodc functons, Functons that can be ntegrated absolutely. Statstcally statonary/ homogeneous random functons. As usually, we shall start wth statonary tme seres to avod too much wrtng, we shall further assume that the mean value has been subtracted. A statonary random functon s (t) wth zero mean can be expanded nto another random functon, Z (ω), and back agan, by means of the Reman-Steltje Fourer ntegral. (.36) ωt s() t = e dz ( ω) s ωt 1 1 e Z ( ω) = s() t dt s π t dz(ω ) Z(ω + dω) Z(ω), meanng that f dz(ω ) s dfferentable, then dz(ω ) could be wrtten as some functon Y(ω)dω. Note, Z (ω) s a complex functon. Snce s (t) has a zero mean value t follows that so has Z (ω). Z (ω) further has δ-functon characterstcs: (.37) dz * ( ω ) dz( ω) = δω ( ω ) df( ω) dω = δω ( ω ) dω S( ω) dω, where the last transformaton demands that F (ω) s a dfferentable functon, as can mostly be assumed n our use. S s a real postve functon that s even n ω. When the two ω s are equal ther product s an absolute square, for whch reason ther result must be real and postve. S (ω) s called the power spectral densty, or shorter: the power spectrum. We have ntroduced * to ndcate complex conjugaton as s normal when multplyng two complex numbers. Now recall the defnton of the auto-covarance: (.38) * ωt * ω( t+ τ) ( τ) = () ( + τ) = ( ω) ( ω) s s s R s t s t e dz e dz t ( ω ω ) t + ωτ * ωτ s s s = e dz ( ω ) dz ( ω) = e S ( ω) dω Multplyng by e -ω τ and ntegratng over τ yelds: Lectures n Mcro Meteorology 3

25 (.39) ωτ ( ω ω ) τ R ( τ ) e dτ = S ( ω) dω e dτ = S ( ω) πδ ( ω ω ) dω = π S ( ω ). s s s s s It s seen that S (ω) and R (τ) s a Fourer transform par. As R (τ) s even n τ, S (ω) s even n ω. Lettng τ = 0, we get: (.40) s(0) = = s( ω) R s S dω Therefore the power spectrum descrbes the contrbuton to the varance from the dfferent frequences. Now we turn towards the stuaton wth two dfferent tme seres: (.41) ωt s() t = e dz ( ω) ωt e() t = e dz ( ω) s e As before the two stochastc seres have zero mean value. The cross-covarance s found from: (.4) * ωt * ω( t+ τ) es ( τ) = () ( + τ) = e ( ω) s ( ω) R e t s t e dz e dz t( ω ω) t+ ωτ * ωτ e s es = e dz ( ω ) dz ( ω) = e S ( ω) dω, where we have agan used the δ - functon behavour of the Fourer modes, correspondng to the equaton for the power spectrum: (.43) dz * ( ω ) dz ( ω) = δω ( ω ) dω df ( ω) = δω ( ω ) dω S ( ω) dω e s es es For the cross spectrum however, we must n general expect S es (ω) to be complex. As for the power spectrum and the auto-covarance functon, the cross-covarance and the cross-spectrum are Fourer transform pars. Ths s seen n a smlar way, by multplcaton of the equaton above wth e -ω τ, and ntegraton over frst τ and then ω. (.44) ωτ ( ω ω ) τ R ( τ ) e dτ = S ( ω) dω e dτ = S ( ω) πδ ( ω ω ) dω = π S ( ω ). es es es es Next we consder the cross-covarance and ts relaton to cross-spectra. R es (τ) s not necessarly even or odd nτ. However, we can generate an even and an odd part as: 4 Lectures n Mcro Meteorology

26 (.45) 1 1 R ( τ) = ( R ( τ) + R ( τ)) + ( R ( τ) R ( τ)) = E ( τ) + O ( τ), es es es es es es es Where E and O are the even and the odd part, respectvely. We see that (.46) E (0) = es, O (0) = 0. es es Insertng E and O n the Fourer Transform above yelds: (.47) 1 ωτ S ( ω) = e ( E ( τ) + O ( τ)) dτ es es es π 1 = (cos( ωτ ) E ( τ ) sn( ωτ ) O ( τ )) dτ = Co ( ω) + Q ( ω). es es es es π where we have used that e -ω τ =cos (ωτ) sn (ωτ). Co es (ω) s a real even functon of ω. It s called the Co-spectrum. It ntegrates to covarance between e and s. Q es (ω) s an odd functon magnary functon n ω. It ntegrates to zero. Ths can be seen by nsertng the Co- and the Quadrature spectrum for the cross-spectrum n the transform from S es (ω) to R es (τ), wth τ=0. (.48) * ωτ ωτ ( τ) = ( ) ( + τ) = ( ω) ω = ( ( ω) + ( ω)) ω, es es es es R e t s t e S d e Co Q d Generalsaton to spectra for many varables. Recall that we can consder meteorologcal varable as functon of three spatal r = (x 1,x,,x 3 ) and one tme varable, Faced wth ths we have optons when decdng on spectral or correlaton analyss. Ths can be exemplfed by the followng for example from spectral analyss: (.49) s( x,) t dz( k, ω) e or dz( k,) t e ( kx + ωt) ( kx ) k ω k ωt or dz( x, ω) e or dz( x,, t k, k ) e ω = k1, k 3 1 ( kx 11+ kx) To the dfferent analyses correspond dfferent power spectra, meanng that the varance of s s expanded nto the dfferent spectral descrptons: Lectures n Mcro Meteorology 5

27 s = S ( k, ω) dk dω or s () t = S (, t k ) dk s s k ω k or s ( x ) = S ( x, ω) dω or s ( x,) t = S ( x,) t dk dk s 3 s 3 1 ω k, k (.50) 1 R ( χ, τ) = s s R ( χ, τ) = k ( kχ+ ωτ ) kχ Ss( k, ω) e dkdω or Rs( χ,) t = Ss(, t k) e dk ω k ωτ kχ Ss( χω, ) e dω, or Rs( x3, χ,) t = Ss( x3, k1, k,) t e dk k ω Here the last lnes are seen to defne a cross-spectrum for the same sgnal measured at dfferent pont n tme and space. However, the descrpton n (.49) can easly be extended to cross correlaton between dfferent varables. Whch combnaton one should choose depends on how much one can stretch the arguments about statonarty and/or homogenety, snce these concepts have to be reasonably vald for the spectral/correlaton analyss method to be vald. Spectra, averages and statstcs. For smplcty, we consder a statonary tme seres wth zero mean value, s(t). Then from the defnton we have: ωt (.51) s() t = e dz s ( ω) The tme average s a before defned through: (.5) 1 st () T = st ( + τ) dτ T T T Insertng the Fourer expanson nto the averagng yelds: T 1 ω( t+ τ) s() t T = e dzs ( ) d T ω τ T (.53) T 1 ωτ ωt sn( ωt ) ωt = e dτ e dzs( ω) e dzs( ω) T = ωt T We see therefore that the tme averagng over tme T attenuates the frequency content at frequences larger than ω 1/T. Snce, we here have a seres we zero mean value, we can compute the varance of the tme averages around ts true mean value, denoted δ T n connecton wth the correlaton functons n (.9), by: 6 Lectures n Mcro Meteorology

28 (.54) sn( ωt) sn( ωt) δ ( ( ) ) ( ω) ( ω) ωt ωt * T = s t T = e dzs e dz s ωt ωt sn( ωt) sn( ω T) sn( ωt) = e dz ( ω) dz ( ω ) = ( ) S ( ω) dω ( ω ω ) t * s s s ωt ω T ωt ωω, Showng how the tme average approaches the true mean value for T, provded that the spectrum behaves well for low frequences. The varance between the raw sgnal, s(t), and ts tme average s estmated as: sn( ωt) ωt sn( ωt) (.55) ( st ( ) st ( ) T ) = ( (1 ) e dzs( ω)) = (1 ) Ss( ω) dω, ωt ωt whch shows that whle the tme average value st () T retans contrbutons of frequences less than 1/T, the varance around of the sgnal around the tme average manly reflect frequences larger than 1/T. Presentaton of Spectra. When plottng spectra one has to content wth that they often cover many decades both on the frequency (wave number) axs and along the ntensty axs. To compensate for ths one wll therefore try to plot logarthmcally to present the wde varety of scales n a representatve way. When dong ths t s further normal to multply the spectrum wth the frequency or wave number scales. Hereby, one can judge the relatve weght of the dfferent scales beng present. The dervaton below goes for the frequences (radans per sec, ω and Hz, f), but smlar relatons hold for the wave number or combned wave number- frequency spectra. (.56) ωs( ω) d(ln ω) = fs( f ) d(ln f ) = S( ω) dω = S( f ) df The bass for these transformatons s that the power spectrum s defned such that that t consttutes the contrbuton to the varance of the sgnal from an ncrement of the ndependent varables of the spectrum,.e. frequences and wave numbers. The followng fgures, , show the power spectrum of one years of wnd speed, measured at md-lattude. Frstly, the dfference n appearance between the Log-Ln and the Log-Log presentaton s obvous. In the Log-Ln presentaton the magntude of the dfference frequency bns provdes a good mpresson of the contrbuton to the total varablty from these bns, as can be seen from (.55) above. The log-log plot on the other hand present detals, not clearly present n the Log- Ln plot. Especally the Log-Log plot shows the hgh frequency part that s created by boundary layer three-dmensonal turbulence. The domnance of varance from the synoptc and the durnal varaton s clearly seen n the Log-Ln plot. In fgure.13 the spectrum s plotted versus the logarthm of the frequency, because of the many decades of frequency scales of nterest n geophyscal tme seres. Lectures n Mcro Meteorology 7

29 The strong ntensty of the spectrum between the annual and durnal-ntensty frequences derves from the moton of the weather systems across Denmark. Therefore, t can be dfferent n other parts of the world wth dfferent clmatology as are of course the ntenstes of the durnal and annual cycles. The contrbuton from the boundary layer turbulence descrbed above s represented by the small bump from about one hour and out. Around one hour s the famous gap between what n relaton to the boundary layer turbulence can be consdered as the ``mean flow'' and the three dmensonal turbulence. There has been some dscusson about the exstence of ths gap, because some convecton clouds actually create eddes wth about the tme scale of the gap, see fgure.6, and also snce the spectra so far used to llustrate ts exstence often have been composte from dfferent tme seres used to compute dfferent decades of the total spectrum, lke fgure.15. From the pont of vew of both modellng and measurement t s advantageous to use average values determned at tme and spatal correspondng to the spectral gap, because the absence of spectral ntensty here shows that only few ndependent processes create varablty n ths scale regon. Ths n turn means average values become better defned and that t also becomes smpler to decde f a partcular process must be parameterzed or explctly resolved by a numercal model. Fgure.13: The power spectrum of the one-year tme seres of wnd speed used n fgure.1 presented versus the logarthm of the frequency (Courtney and Troen, 1990; Troen and Petersen,1989) The annual frequency s not shown, snce only one year of data s used.in the lower fgure the prncpal tme scales are emphaszed. Fgure.14: The power spectrum of the one-year tme seres from fgure.13, but here the logarthm of the spectrum s presented versus the logarthm of the frequency. 8 Lectures n Mcro Meteorology

30 Fgure.15 presents the famous Van der Hoven Log-Ln spectrum, where the hgh frequency part s seen to be enhanced relatve to fgure.13. Ths spectrum s a composte; by that turbulence data from a storm event s glued to spectra from longer tme perods. It s famous and a lttle bt controversal because t gves people the wrong mpresson of the strength of the boundary layer turbulence relatve to other parts of the spectrum. Fgure.15. The Van der Hoven wnd spectrum from Brookhaven, NY, based on work by van der Hoven (1957), taken from Lumley and Panofsky(1964). The spectra presented n Fgures can be used to consder the choce of averagng tme n relaton to the scales of the dfferent atmospherc processes, also llustrated n Fgures.1 and.6. The exstence of a gap n the spectra around a tme scale of one hour can be used to nfer that the processes, takng place around that tme scale, are probably not very mportant for the total varance of the annual wnd sgnal. From Fgure.6 ths means that these processes, cumulus cloud convecton, breeze systems and PBL convecton, do not contrbute much to the annual varance at least not at tme perods around one hour. From (.53) s seen that the varablty of one hour mean values are domnated by a system of processes that s assocated wth tme scales larger than one hour. On the other hand, the sgnals assocated wth the fluctuatons around these mean values are assocated wth other processes wth tme scales less than one hour. If the gap n the power spectrum s pronounced, we have not only a tme scale separaton between the mean values and the fluctuatons, but also a separaton of the processes that have to be consdered, when the physcs of the two sgnals s to be understood. As dscussed above the exstence of the gap s not as pronounced as one could wsh for, based on these arguments, therefore we wll often have part of the same processes drectly affectng both mean values and fluctuatons around the mean values. Boundary Layer processes, eddes, scales and spectra. An mportant characterstc of the atmospherc boundary layer s that through t, the wnd speed s reduced from the free wnd speed aloft to zero at the bottom. As concluded earler we must start by consderng a horzontally homogeneous boundary layer, where turbulence s gong to transport momentum and everythng else between the surface and the top of the boundary layer on to the free atmosphere. Such a transport demands that there are w-fluctuatons avalable. For the power spectra we have seen ths manly occurs at scales at the hgh frequency hump of these spectra, where three-dmensonal turbulence occurs. Lectures n Mcro Meteorology 9

31 As we notce earler n ths note, the man producton mechansm s shear producton, as s llustrated on the next few fgures, taken from Tennekes and Lumley (197). The frst of these fgures shows that the mean wnd profle s unstable, contnuously sheddng eddes, Fgure.16. The mean speed at two levels on each sde of a gven levels tend to create a whrlng moton. Next fgure.17 shows how such a moton can extract momentum (and therefore also other varables) from a mean gradent, by movng flud elements from one level to another wth other mean characterstcs. Ths whrlng moton s assocated wth a rotatng flud element, whch we call an eddy. Fgure.16. Development of rotaton n a turbulent shear flow through overturnng of ar n ar parcels. (Tennekes and Lumley,197) Fnally, the thrd fgure,.18, shows how an eddy s stretched by the mean profle, thereby reducng ts radus. Ths stretchng also occurs by nteracton between dfferent eddes, settng up velocty gradents across each other. Asde from ths stretchng the velocty gradents of overlappng eddes force the eddes to shed smaller eddes correspondng to the processes assocated wth the mean gradents. Fgure.17. Transfer of momentum from one level to the next by a the whrlng moton derved n fgure.16 (Tennekes and Lumley, 197) The concept s that wnd shear s contnuously sheddng eddes, these eddes nteract wth the mean shear and each other to create ever smaller eddes. We talk about an energy cascade to smaller and smaller scale. As eddes grow smaller, the velocty gradents across them become strong enough for the molecular frcton to smooth out the moton. Ths smoothng out of moton removes varance from the wnd speed fluctuatons. It s called dsspaton and denoted by ε. 30 Lectures n Mcro Meteorology

32 Fgure.18. Stretchng of an eddy by the mean shear (Tennekes and Lumley,197) The largest of these shear produced eddes are produced wth scales, reflectng the scale, Λ, of the vertcal shear that creates them. Close to the surface, where the vertcal varaton of wnd speed s close to logarthmc, Λ z, the heght above the ground. Ths eddy producton wth subsequent cascade down to smaller sze and dsspaton by vscosty has been gven a poetc formulaton L.F. Rchardson paraphrasng a poem by Jonathan Swft. Bg whorls have lttle whorls, Ths feed on ther velocty; And lttle whorls have lesser whorls, and so on to vscosty. The orgnal So, Nat ralsts observe, a flea Hath smaller fleas that on hm prey And these have smaller fleas to bte em And so proceed to nfntum. The swrlng moton of eddes gves rse to the turbulent velocty fluctuatons, the velocty varablty at dfferent eddy szes are seen n a spectral analyss as the ntensty of the power spectrum at the assocated wave number. (.57) (,) k r u r t = e dz (,) t k (.58) * ( k,) ( k,) ( k,) j j S t = dz t dz t Where the subscrpts refer to velocty components 1, and 3. Lectures n Mcro Meteorology 31

33 When eddes are shed from the mean wnd profle, the drecton of the shear s of course mportant, but after a few steps n the eddy-eddy nteracton nvolved n the cascade, the eddes have lost sense of orentaton, and we say that the moton s sotropc, meanng that the flow statstcs s unchanged by rotaton n all drectons. Snce the flow cannot be truly sotropc, t s only sotropc for smaller scales, where the cascade has been actve n several steps. For sotropc turbulence we can defne power spectrum, E(k), beng only a functon of the length of the wave number vector, k, not ts orentaton. Below, E(k) s derved by ntegratng S (k)over all drectons of the k-vector, leavng only ts length as varable. (.59) Ek ( ) S( k) dk, kk= k (.60) 1 1 u u = ( u + u + u 1 3 ) = E ( k ) dk ( = S ( ) dk dk dk 1 3 ). k 0 Ek ( ) k k j (.61) S ( k ) = ( δ ), j j 4π k k We can separate the power spectrum nto three regons as shown on fgure.19, the producton range, wth k 1/Λ, where energy s extracted from the mean profle, a dsspaton range, where the flud moton s dsspated by vscosty, for k > η (ν 3 / ε) 1/4, whch for typcal atmospherc flows s about 1 mm. η Is called the Kolmogorov dsspaton scale and s a combnaton of vscocty and dsspaton as seen. In between there s a regon, where the spectrum depends only on the wave number and the dsspaton. Ths regon s called the nertal sub-range. Snce the spectrum descrbes wnd varance per wave-number ncrement, t has the dmenson: m 3 /sec. Dsspaton s destructon of varance by vscosty, hence t has the dmenson of varance per second, or m /s 3. Fnally, wave number has the dmenson of m -1. Dmensonal analyss then yelds: (.6) Ek ( ) = αε k /3 5/3 Whch s the famous Kolmogorov 5/3 law, where α s a unversal non-dmensonal constant, called the Kolmogorov constant, beng about 0.5. In Fgure.19, t s physcally realstc to expect sotropy only wthn the nertal range and the dsspaton range of E(k). To dstngush between such a flow and a truly sotropc flow (meanng that all scales of the flow s sotropc), we denote the boundary layer turbulence as a locally sotropc flow. The pseudo sotropc soup descrbed by E(k) s now advected wth the mean wnd speed, hereby defnng a coordnate system wth the x 1 -axs along the mean speed, u, the x 3 or z axs vertcal wth the wnd speed denoted w, and the other horzontal axs, x, wth wnd speed component, v, called lateral. 3 Lectures n Mcro Meteorology

34 Fgure.19. Schematcs presentaton of the energy spectrum for atmospherc boundary layer turbulence, defnng the producton range, wth scales reflectng the mean shear, dsspaton range and nertal range. (Tennekes and Lumley,197) The component spectra now depend on how we probe, E(k). A typcal way s that we use the wnd speed tself, meanng that we see all the components along the mean wnd speed from a statonary sensor. In (.63) γ ndcates the drecton of probng specfcally γ = 1 n (.64). γ (.63) S ( k ) = S ( k) dk j γ j γ (.64) S ( k ) = S ( k ) dk dk 1 j 1 j 3 Or, nsertng (.60) Ek ( ) k S k S k S dk dk dk dk (.65) 1 1 ( ) = ( ) = ( ) (1 ), u k = π k k 1 Ek ( ) k ( ) = ( ) = ( ) (1 ), v 1 1 k = 3 3 4π k k 1 Ek ( ) k3 ( ) = ( ) = ( ) (1 ) w k = 33 3 k, 3 4π k k 1 Ek ( ) k k j j 1 j 1 k j 3 3 4π k k S k S k S dk dk dk dk S k S k S dk dk dk d S ( k ) = S ( k ) = S ( ) dk dk = ( ) dk dk = 0, j Notce that the cross-spectrum between any two dfferent components s always zero, as t should be accordng to the assumpton of sotropy, reflected by the spectrum n (.60) From a statonary sensor, exposed to the wnd, we wll see a temporal sgnal varaton that corresponds to spatal varaton along the x 1 axs, followng Taylors hypothess of the turbulence as gven n (.0). Lectures n Mcro Meteorology 33

35 (.66) t = u x 1 where u s the mean speed, and s the sgnal ampltude varaton. A slghtly more general formulaton of Taylor s hypothess of frozen turbulence s that the advecton speed, u, s much larger than the speed of change wthn the turbulence soup passng by wth the wnd. Ths hypothess s surprsngly accurate for stuatons wth wnd larger than about m/s, for the three dmensonal turbulence charactersng the atmospherc boundary layer, as depcted n the power spectra, presented before, meanng for frequences larger than Hz and horzontal wave length smaller than say 10 km. From the equaton above, the relaton between the wave-number, k 1, wavelength, and the frequency s: (.67) k = π / λ = ω / u = π f / u 1 Probng of the turbulence feld, as descrbed by E(k), along the mean wnd speed, we can derve the component spectra. In the nertal sub-range one obtans, usng (.61) and (.64): (.68) /3 5/3 4 S ( k ) = αε k wthα = α = α, u where the α s are derved from the one unversal α of the sotropc spectrum above (.61). Also the scalar varables have nertal range forms smlar to the velocty component, for example for temperature: (.69) S ( k) = αε Nk, T 1/3 5/3 1 T 1 where N s the dsspaton rate of temperature varance, wth the dmenson K /sec. One can verfy ths wth same dmensonal analyss as for the velocty spectra. There are strong reasons to beleve that the Kolmogorov constant s the same for all scalars (Hll, 1989) wth α T beng about 0.8. We shall n the followng sectons see how ε and N can be estmated from the governng equatons. Overall the one- dmensonal power spectra, ks(k), therefore all tend to follow a bell shape, wth the /3 law consttutng the hgh frequency part (at least when the dsspaton range s not ncluded). Ths s llustrated on the followng fgure.0. One could add that t of course s possble to probe the boundary layer turbulence along other axes than the x 1, gven by the mean wnd speed, theoretcally or usng e.g. small arplanes or remote sensng. Dong so we fnd that the nertal ranges stll holds but that the α coeffcents changes, and that detals n the spectra changes as well, but the overall shapes reman the same. Indeed all power spectra tend to follow a bell shape, although co-spectra have somewhat dfferent form, as seen n Fgure.1 that ncludes co-spectra as well. The spectral range, depcted n Fgure.0 has been studed expermentally, qute ntensvely. Results from such measurements are exemplfed n Fgure.1 as analytcal forms of power spectra and co-spectra for the three wnd components and temperature based on measurements near the ground. The spectra are plotted versus the so called normalsed 34 Lectures n Mcro Meteorology

36 frequency, whch s seen to correspond to the horzontal wave number multpled by the heght z, accordng to the Taylor hypothess n (.66). In boundary layer theory use ths normalsed frequency collapses spectra from dfferent heghts nto the form shown on Fgure.1, see secton 6. Fgure.0. Prncpal sketch of the one dmensonal power spectrum as functon for κ 1, or alternatve as functon of frequency, transformed to wave number usng Taylors hypothess (.66), from Kamal and Fngan, 1994). The power spectra of Fgure.1 follows well the form depcted by Fgure.0, wth an nertal sub-range wth a Kolmogorov constant n accordng wth (.67) and (.68). The co-spectra, whch for sotropc turbulence should be zero, s seen to have a sgnfcant non-zero part, carryng the fluxes, as descrbed on p 11 and 1 n ths secton. At hgher frequences, all spectra approach local sotropy the power spectra approaches the predcted -/3 power law, whle the co-spectra approaches zero much faster n accordng wth the theory of approach to local sotropy. Lectures n Mcro Meteorology 35

37 Fgure.1. Power spectra and co-spectra for atmospherc neutral turbulence close to the ground, plotted versus the normalsed frequency n= fz/u zk 1 /π, see (.66), (Kamal et al, 197). The spectra are normalsed wth surface layer parameters, whch together wth the role of thermal stablty are descrbed closer n secton 6. The spectral formulatons and the Taylors hypothess are based on qute smple deas, manly of statstcal nature and have been found to be so broadly vald that they are extremely useful n both expermental and modellng work. The practcal lmtatons to the use of Taylor's hypothess show when there s too much varaton n the velocty relatve to the mean flow, ether due to turbulence or due to large vertcal wnd shear. Ths can nfluence the very low frequency, large-scale turbulence (Powell and Elderkn, 1974), and small-scale hgh frequency measurements (Wyngaard and Clfford, 1977; Mzuno and Panofsky, 1974, Larsen and Højstrup, 198). The lmtaton to the valdty of the nertal sub-range forms of the spectra s found when the assumpton behnd ther valdty breaks down, n the hgh-frequency end by the drect nfluence of the dsspaton and n the low-frequency end through the drect showng of the producton scales and the nearness of the surface (Tllman et al, 1994). The producton scales show up n Fgure.1 around the top of the bell shape of the power spectra, whch for shear produced turbulence scale wth vertcal wnd gradent, agan scalng wth the heght z, as dscussed on p. 4-5 n ths secton. The hgh-frequency lmt due to dsspaton s llustrated on Fgure.. Accordng to the dscusson n connecton wth Fgure.19, the dsspaton becomes mportant for the spectrum, 36 Lectures n Mcro Meteorology

38 strongly reducng the spectral ampltude, when kη 1, where η s the Kolmogorov dsspaton scale. fz/u Fgure.. Dfferent scaled spectra near the ground shown versus the normalsed frequency, fz/u (Larsen et al., 1980). 1 and shows dfferent power spectra lke, u, T, u, v, or w. The hatched area, 3, represents co-spectra, whle curve 4 shows a dsspaton spectrum, derved from a dfferentated sgnal as shown n secton 4, (4.57) : The -/3 slope of 1 and here become represented by a +1/3 slope, followed by ampltude reducton due to dsspaton.. For n <10-3, the velocty fluctuatons gradually looses ther 3D turbulence characterstcs and become manly horzontal fluctuatons and the normalsed frequency, n, loses ts relevance. Hence t s common now to present spectra as functon of ether wave number (m -1 ) or frequency (Hz). However, the power spectra here stll retan a knd of unversalty as seen n Fgure.3, showng the spectra from multple sources as functon of frequency between 10-6 and 10-3 Hz. Notced that the -5/3 of the nertal subrange, but not because of an nertal subrange here, descrbng the spectrum between the durnal cycle and 10-3 Hz. The spectra of Fgure.3 can be seen also n the one spectrum shown on Fgure.14. A weaker form for Taylor hypothess also governs the fluctuatons between the durnal cycle and 10-3 Hz n that the coeffcents for the smlar wave number spectrum can be found assumng advecton by the mean wnd of spatal fluctuatons. From the durnal cycle and down to lower frequences, the spectrum reflects the synoptc weather patterns, and no smple relaton between wave number and frequency exsts, as was also dscussed earler n ths secton, n connecton wth Fgures.6 and.14. The durnal cycle, showng clearly n Fgure.14 s n Fgure.3 more smeared out by the many data sets, wth dfferent measurng heghts. Notce also that Taylors hypothess has no relevance for the durnal cycle, beng an exclusvely a tme phenomenon. Lectures n Mcro Meteorology 37

39 Fgure.3. Composte plot of wnd speed spectra versus frequency (Larsén et al, 01), The unversal spectral functon, ftted to the data, s ndcated. APPENDIX A: Correlaton functon ntegral. In ths appendx we reduce the ntegral n (.30) (A.1) T T 1 1 T = ( s( t ) d s) ( s ( t ) d ) T + = T + T T δ τ τ τ τ T T T T 1 1 = s ( t t ) s ( t t ) dt dt Rs ( t t ) dt dt 4T + + = 4T T T T T To contnue we defne new coordnates, the so-called damond transformaton: (A.) t + t t t τ = ; σ =. and τ + σ τ σ t = ; t = (The followng detals are usually referred to n texts as: It s easly seen--) 38 Lectures n Mcro Meteorology

40 The absolute value of the functonal determnant for ths transformaton s: (A.3) t t 1 1 ( t, t ) τ σ = = = 1. ( τσ, ) t t 1 1 τ σ The ntegral now looks as: (A.4) T T t = T t = T 1 1 T = s s 4T = T T 4T t = Tt = T δ R ( t t ) dt dt R ( σ) dσdτ The new boundares on the other hand are complcated, and t s smpler to consder postve and negatve τ separately. For postve values of τ, the σ-ntegraton s bounded by: (A.5) T τ σ T + τ For negatve values of τ, the σ-ntegraton s bounded by: (A.6) T + τ σ T τ Fgure A.1. Integraton areas of (.34) wth the two sets of ndependent varables, t, t and (τ,σ). (Panofsky and Dutton, 1983)). Hence, we can contnue the ntegraton as: Lectures n Mcro Meteorology 39

41 (A.7) 1 δt = R ( ) s σ dσdτ = 4T 0 1 T+ τ T T τ = d R ( ) ( ) s d d Rs d 4T τ σ σ + τ σ σ = T T τ 0 T+ τ 1 = T t = T t = T t = Tt = T T T τ dτ R ( σ) dσ. 0 T + τ s The last ntegral s ntegrated by parts: (A.8) T T τ T T τ 1 1 T = s s T = 0 T = T+ τ 0 T+ τ δ dτ R ( σ) dσ dτ 1 F( τ); wth F( τ) R ( σ) dσ. Proceedng, we now obtan: (A.9) T T T δt = = d 1 F() F() d F(); T τ τ = τ τ ττ τ T T τ T τ wth F() τ = R ( σ) dσ. T + τ s The frst term s seen to be zero n both lmts. Hence we can wrte: (A.10) T 1 δt = d F( ) T τ τ τ τ 0 T 1 T τ ( T τ) ( T + τ) = d R ( ) ( ) ( ) s d Rs T Rs T T τ τ σ σ + τ τ 0 τ τ τ T + τ T T 1 1 = dτ τ { 0 Rs(T ) τ Rs( T + ) τ } = dτ τ R ( ); s T τ T T 0 0 Once more we substtute: ξ = T- τ, wth dξ= - dτ: (A.11) T 1 ξ δ ξ ξ T = (1 ) R( ) d. T T 0 Introducng the autocorrelaton functon ρ s (τ), and changng ξ to τ, ths expresson can be wrtten: (A.1) T σ s τ δ = (1 ) ( ) d T ρ τ τ s T T 0 40 Lectures n Mcro Meteorology

42 3. Basc equatons Our objectve s here to establsh useful equatons for our man atmospherc varables wth sutable smplfcatons. Varable: Pressure, p. Densty, ρ. Composton: here we shall dstngush between the densty of dry ar and ρ d and the densty of water vapour, ρ w..other trace consttuents are not dynamcally mportant, and wll consdered as passve tracers n the atmosphere. Temperature: T, and velocty; u = u 1, u, u 3. Complcatons: We are on a rotatng planet, meanng that we seek to nterpret the moton n a rotatng (acceleratng) coordnate system. Addtonal there s a varaton of p, ρ, T wth heght nduced by gravty. We start consderng 7 parameters, because we have two denstes, for water vapour and for dry ar. Equaton of state The deal gas law: (3.1) p= ρrt. Snce, both R and ρ depend on the composton, we decompose (3.1) nto ts partal pressure for dry ar and water vapour. R R 0 0 (3.) p= p + p = ρ T + ρ T, d w d w M M d w where R 0 s the unversal gas constant and M s the molecular weght for the gas consdered. Notce: ρ = ρ d + ρ w. We use the defnton for mxng rato, q ρ w /ρ. We can rearrange (3.) to: (3.3) R M = ρ (1 + ( 1) ) : M M 0 d p T q or d w (3.4) p= ρr T ; T T ( q). d v v Wrtng the equaton as above means, we treat the atmosphere, as t was dry ar only, wth respect to ρ, p and R. As a penalty we have to operate wth and artfcal temperature, T v, denoted the vrtual temperature. Note that typcally q << 0.1 kg/kg. Vertcal varaton of p and ρ: Consder frst a hydrostatc balance between the weght of a volume of ar and the pressure force on ths volume. Lectures n Mcro Meteorology 41

43 p (3.5) = ρ g, z where we have balanced the forces due to gravty and due to pressure. Insertng the equaton of state (1.1) and assumng an sotherm atmosphere, we can wrte, e.g.: p pg p (3.6) = z RT H where H s called the scale heght, typcally about 10 km. Wth g 9.81 m/s, T 88 K and R 87 J K -1 kg -1. Thereby H becomes 8.4 km. The two equatons of course allow for solutons also for more realstc varatons of T. Comparng wth atmospherc data on the heght varaton of pressure and temperature, we fnd that whle the measured pressure depends roughly as the exponental decay predcted by (3.6), the temperature varaton s very dfferent from the constant assumpton used, wth dfferent behavour n the dfferent layers of the atmosphere. The heght varaton of pressure and temperature also nfluences the dynamc stablty of the atmosphere. To see ths we shall move an ar parcel vertcally. Consder fgure 3.1, where we move an ar-parcel up- and down n an atmosphere wth vertcal structure ruled by the hydrostatc equaton between densty and pressure changes. As we move the ar parcel, t wll change t volume, V and thereby ts densty n response to the external pressure found at ts new poston. We now assume the moton of the ar parcel to be too fast for the parcel to exchange heat wth ts surroundngs on the way to a new poston. Ths means that the changes of the nternal parameters n the parcel, ρ, Τand p, are controlled by an adabatc process. In the followng we shall dstngush between parameters nternal to the parcel, denoted by subscrpt, and parameters pertanng to the surroundng atmosphere, wthout subscrpt. For our adabatc parcel parameters we have: (3.7) dp dt = ρ C p Also : ρ = RT p ; where C p s the heat capacty, or specfc heat, ~1010 (JK -1 kg -1 ) at constant pressure 4 Lectures n Mcro Meteorology

44 . p ρ 3,T 3,p 3 ρ,t,p p(z), ρ(z), T(z). p = ρ g, z ρ 1,T 1,p 1 Fgure 3.1 Adabatc vertcal moton of an ar parcel n a hydrostatc atmosphere. As the ar parcel s moved to a new poston t adapts to surroundng pressure wth the speed of sound (that s nstantaneously). If the ar parcel has moved dz the external pressure has changed n accordng wth the hydrostatc balance by: (3.8) dp = ρ gdz The nternal pressure has smlarly changed, ruled by the adabatc equaton: (3.9) dp = ρ C dt. p Assumng now that the ar parcel starts out from equlbrum between parcel varables and the surroundng varables: ρ ρ. As dscussed above dp = dp. Combnng the equatons for the external and the nternal changes, we obtan: dt g (3.10) = Γ 0.01 K / m = 1 K /100 m, dz C where we have defned the dry adabatc lapse rate, Γ. p Untl now, we have consdered only the changes n an ar parcel. Now consder an ar parcel movng up and down n three dfferent vertcal temperature gradents, beng n equlbrum wth the surroundngs n poston (a), and movng to the ponts (b) and (c). We assume the pressure to be the same for all three stuatons. The nternal parameters of the ar parcel wll always develop followng the dry adabatc lapse rate, as seen on the fgure. Lectures n Mcro Meteorology 43

45 . z b a T (z) =T 0 -Γz T 3 (z) c T 1 (z) T Th l f h l ll l d l Fgure 3. Ar parcel movng adabatcally up and down n an atmosphere wth dfferent thermal stratfcatons. If the ar parcel moves n the T (z)-atmosphere, t s seen to be n equlbrum also at the new heghts, b and c. If the ar parcel moves n the T 1 (z)-atmosphere: Startng at level c and arrvng at level (b) the ar parcel wll be warmer and therefore lghter than the surroundng ar. Arrvng at level (c) t wll correspondngly be colder and hence heaver than the surroundngs. Hence, all moton ntated wll contnue. If the ar parcel moves n the T 3 (z)-atmosphere, startng at level a, and arrvng at level (b) the ar parcel wll be colder and thereby heaver than the surroundngs. Arrvng at level (c) t wll be warmer than the surroundngs and hence lghter. Hence, an ntated moton wll be counteracted by the buoyancy forces. Based on ths mage, we call the T (z) - temperature stratfcaton for neutral, because t wll not nfluence ntated moton. The T 1 (z)-atmosphere s called unstable, because all ntated moton wll be amplfed. The T 3 (z)-atmosphere s called stable, because all ntated moton wll be damped by the temperature stratfcaton. Hence, such stratfcaton stablses exstng stuatons. Potental temperature The potental temperature,θ, s defned as the temperature an ar parcel acqures, when t s moved adabatcally to a reference pressure level, p 0. Snce much ar moton nvolves adabatc moton, we see that ar parcels wll retan ther potental temperature durng most motons, meanng that potental for many stuatons can be consdered a conservatve aspect of an ar parcel. From (3.7) we have: (3.11) dp dt = ρ C p Also : ρ = RT p ; 44 Lectures n Mcro Meteorology

46 Recallng that p always equalses nstantaneously wth the surroundngs, combnaton of these two equatons we get: (3.1) R dp dt R dp = = C p T C p p p T,p,ρ T = θ, p = p0 Fgure: 3.3. Defnton of potental temperature. We now follow the ar parcel down to the reference level: (3.13) θ T p0 dt R dp =. T C p p p or: (3.14) R 0 ln( ) ln( ), T Cp p or θ θ = T = p p R 0 C p ( ). Notce that the only parameter reflectng parcel parameters s θ θ because the ar parcel s n equlbrum wth the surroundngs before t s moved to level, p 0, and we assume contnuous pressure equlbraton. The equaton s called Posson s equaton. It s seen that every parcel of ar has a unque potental temperature that s conserved for dry adabatc moton. We say that θ can be called a conservatve quantty, because t does not change, when an ar parcel s moved around, as opposed to the real temperature that changes when the ar parcel s moved adabatcally up-or down, as dscussed above. Snce p and T are functons of heght z, so s θ, meanng that θ s a functon of the heght from whch we start movng an ar parcel down to the reference level. Dfferentatng (3.14) wth respect to z we obtan, usng the hydrostatc balance: p Lectures n Mcro Meteorology 45

47 (3.15) θ T p R p p p = ( ) ( ) = z z p C p z R R 0 Cp 0 Cp 1 0 T p p θ T R T p θ T R T pg = = = T z Cp p z T z Cp p RT θ T = +Γ. T z We assume that the reference level n the boundary layer s at the surface, z = 0, where then θ = T. Throughout the total boundary layer, we therefore have θ T 300 K. Hence, we see that θ T (3.16) +Γ; θ ( z) = T( z) +Γz. z z Returnng to the concept of stablty, we see that t can be charactersed two ways, usng the temperature or usng the potental temperature. For each temperature gradent, the characterstc of the atmosphere s shown n parenthess. T (3.17) = Γ ( neutral), or < Γ ( unstable), or > Γ ( stable). z or: θ (3.18) = 0( neutral), or < 0( unstable), or > 0( stable). z For a neutral dry boundary layer the temperature falls wth about 1 K/ 100 m. Note further that for adabatc moton, the potental temperature of an ar parcel s unque. We next turn to the complcatons from humdty. If water vapour s present the equatons we have to consder look as follows: dp p p (3.19) = ρc p( q); ρ = or ρ = and dp = ρgdz dt RT R( q) T v wth: T v = T(1+0.61q). Formally, (3.19) can be wrtten n the same way as the equatons for dry condtons, see (3.14): (3.0) Rd dp dt Rq ( ) dp dt = or alternatvely = C ( q) p T C ( q) p T p v p However, (3.0) cannot be ntegrated n a unque way anymore, because the ntegral now depends on the path to the reference level, and the varaton of humdty along that path. Only f we assume that q s a constant, can we ntegrate. 46 Lectures n Mcro Meteorology

48 If q s constant, we can choose to ntegrate wth respect to T or T v. If we do the frst, we arrve to that the atmosphere s neutral f: dt (3.1) = Γ( q), where Γ ( q) <Γ. dz For a typcal most atmosphere, one fnds that the temperature decreases wth about 0.5 K per 100 meter, not the dry adabatc decrease of 1 K/100m. Ths most adabatc varaton ncluded as well face changes between water and water vapour. If we ntegrate wth respect to T v, we arrve to that we can defne a vrtual potental temperature smlarly to the potental temperature. (3.) θ = θ( q) = T +Γ ( q) z v v Equatons of moton We shall now derve a sutable set of equatons for wnd, temperature and other scalars. We start by dervng Euler s equaton relatng partal and substantal dervatves. Euler s equaton We consder a volume V that moves through space, meanng that ts poston can be gven as (x (t), y (t), z (t)). Assume that we have a feld ϕ = ϕ(x,y,z,t). We wsh to determne the changes n ϕ wthn the volume as t moves through space. Ths means we consder ϕ(x (t), y (t), z (t), t). We change to wrte the coordnates as (x (t), t). We track the change as the volume moves from (x (t), t) to (x (t + δt), t + δt), where = 1,, 3. (3.3) δϕ = ϕ( x ( t+ δ t), t+ δ t) ϕ( x ( t), t). We use Taylor expanson: (3.4) ϕ ϕ ϕ ( δt) ϕ( x ( t+ δt), t+ δt) = ϕ( x ( t), t) + ( x ( t+ δt) x ( t)) + δt+ + + x t t, We now dvde by δt, and let t go to zero: (3.5) ϕ( x ( t+ δt), t+ δt) ϕ( x ( t), t) ( x ( t t) x ( t)) ϕ + δ ϕ ϕ ( δt) = δt x δt t t When lettng δt 0, we notce that: ϕ( x ( t+ δt), t+ δt) ϕ( x ( t), t) dϕ for δt 0; δt dt (3.6) ( x ( t+ δt) x ( t)) u for δt 0; δt where u p s the velocty of our volume. Hence, we arrve at Euler s equaton relatng total dervatves to partal dervatves: Lectures n Mcro Meteorology 47

49 d (3.7) ϕ ϕ u ϕ ϕ u ϕ v ϕ w ϕ = + = = ϕ + u ϕ; dt t x t x y z t Ths relaton wll be used extensvely n the followng and s shown here n ts dfferent forms. The contnuty equaton We express that n a statonary volume of dry ar n space, changes n densty can only happen through transport n and out of the volume. q y ρ 1 u 1 ρ u x 1 x Fgure 3.4. Densty change n a volume of ar, note the coped fgure operates n (x,y,z) and ncrements n these. Consder the change of densty due to the net flux n the u drecton: ρ( x x ) = ( u ρ u ρ ) t (3.8) (3.9) Lettng both t and δx go to zero yelds: (3.30) ρ ( u ρ u ρ ) = 1 1 t δ x ρ ( ρu) = t x Gong to 3D we now get: (3.31) ρ ( ρu ) ( ) ρ u ( ρ) + = 0 = + ρ + u t x t x x We can now use Euler s equaton: dϕ ϕ ϕ (3.3) = + u, dt t x and wrte an alternatve form of the mass balance, for a volume followng the flow: 48 Lectures n Mcro Meteorology

50 d ρ u (3.33) + ρ = 0 dt x Ths equaton can be nterpreted n terms of non-compressblty. The equaton shows that f: (3.34) 1 d ρ 0 ρ dt Then u (3.35) = 0 or u = 0. x Note, that the argument s somewhat weak. Because the assumpton s not consstent wth the result: (3.36) 1 d ρ ρ dt u << = 0 x A more thorough dervaton must utlse scale arguments about the possble rate of change for the two terms (Mahrt, 1986). The momentum equaton To derve the momentum equaton, we start wth Newton s second law for a flud element: du j (3.37) m = F wth = 1,,3 dt Where, m = mass, u = component of the velocty and F j s component of force, j. To proceed we have to dentfy the dfferent forces F j and also to descrbe the effect of havng a rotatng coordnate system, where the rotaton consdered s the rotaton of the Earth. Ths s done n the followng equaton, where we consder a unt volume, and hence denote the mass by, ρ: du p ρ = ρgδ ρω ε η u + F; 3 jk j k (3.38) dt x I II III IV Where, we have dentfed dfferent forces (one beng a vrtual force). Below we dscuss the term one by one: I: The gravtatonal force s n the vertcal, ndcated applcaton of the Kroneker- δ, whch s equal to 1 wth the two subscrpts beng equal and 0, f the subscrpts are dfferent. j II: The pressure force. Here we have already seen the vertcal pressure force, balancng gravty n hydrostatc equlbrum. Lectures n Mcro Meteorology 49

51 III: Corols force, whch s not a force, but appears because we consder the equaton n a movng frame of reference gven by the rotatng Earth. The term derves from Ω(η u), η s a unt vector parallel to Earth rotaton and Ω s the angular frequency of rotaton.ε jk s a cyclc operator, beng +1, -1 dependent on the order of, j and k and 0 f two of the ndces are equal. Term I ncludes as well a small component from the rotaton. When descrbng the moton n ts components, we must specfy the components of η n our chosen local coordnate system. Here we wll chose dfferent systems durng our excursons nto boundary layer flows. However they wll all have an x 1 and x that are horzontal, x 3 s vertcal, and we can therefore fnd the x 3 component of η to be equal to snϕ, where ϕ s the lattude. Fortunately the expressons for η 1 and η wll not be mportant, as we shall see. Snce, η 3 wll enter the mportant part of ths term we wll already now defne the so-called the Corols parameter: f c = Ω snϕ, whch depends on lattude. For md- lattude, f c s -1. At equator t s zero. x j The vscous stress τ j attacks a plane to x j n the drecton of x τ j X Fgure 3.5. Specfcaton of vscous or shearng stress,τ j, across a plane perpendcular to x j and n the drecton of x. IV: The frcton force F on an ar parcel s produced by the vscous force or shearng stress, τ j, across the flud element consdered n the fgure above. The stress s a tensor, τ j, where element,j attacks a plane perpendcular to x j n the drecton of x, see fgure. The stress s due to the movements and nteractons of molecules across the boundares of the flud element. The shearng stress s normally assumed to be proportonal to the velocty gradents: u u j (3.39) τ = µ ( + ), j x x whch s a general expresson assumng that the shearng stress s lnearly proportonal to the shear, when we assume ncompressblty. µ s denoted the dynamc vscosty. For the smplest stuaton wth a postve u gradent n the j-drecton and no u j gradent t s easy to descrbe what happens, from the molecular vewpont a postve the shearng stress n the -drecton results from a net downward transport of u - momentum by the random moton of the molecules. Because the u -momentum ncreases n the x j drecton, the molecules passng downward through a horzontal plane perpendcular to x j at any nstant carry more momentum than those j 50 Lectures n Mcro Meteorology

52 gong the opposte drecton. Thus there s a net transport n the negatve x j drecton. For smlar reasons also a u j gradent along the x drecton wll lead an ncrease τ j. Notce, the sgn of the shearng stress and the assocated momentum transport s opposte. Notce that τ j =τ j, meanng that the stress tensor has 6 ndependent elements. The shearng stress as gven above means that the flud above the plane act on the flud below the plane wth the gven a force per unt area gven by τ j. Correspondngly the flud below the plane acts on the flud above the plane wth -τ j, such that there s no net force on the plane. Next we turn to the vscous force on a dfferental volume, gven by the parcel n next fgure. Here the force on the upper plane s gven by τ j (x j1 ) δx δx k, whle the force actng on the parcel from below s gven by -τ j (x j ) δx δx k. δx x j The vscous stress τ j attacks a dfferental parcel. The volume of the parcel s δx δ x δ x k δx j τ j (x j1 ) δx k τ j (x j ) Fgure 3.6 Components of the vscous force attackng an ar parcel. The total force on the ar parcel, actng from τ j on the ar parcel s then found to be: ( τj ( xj1) τj ( xj)) Fδxδx δx = ( τ ( x 1) τ ( x )) δxδx = δxδx δx δ x j (3.40) or τ j Fj = for δ x j 0. x j j k j j j j k j k j Insertng the expresson for τ j and utlsng the ncompressble verson of the contnuty equaton, u / x = 0, we can smplfy the expresson for the forcng of the vscous forces n the momentum equaton: τ j u u j F = ( ( )) F = = µ + = j j x x x x j j j (3.41) u u j u = µ ( + ) = µ x x x x j j j Where, we have employed u / x = 0. Lectures n Mcro Meteorology 51

53 The momentum equaton now takes the form: (3.4) or dvdng wth ρ : (3.43) du p u ρ = ρgδ ρω ε η + µ ; dt x x 3 jk juk j du 1 p u = gδ Ω ε η + ν ; dt x x 3 ρ jk juk j where ν s called the knematc vscosty. For ar ν m s -1 = 0.15 cm /s. Mass balance for a scalar We denote the scalar concentraton by C, whch we can ether thnk upon as mass per volume [kgm -3 ], or as mxng rato [kg-c/kg-ar]. The change of C n a volume followng the wnd speed can be found from: dc FC (3.44) = + Sources Snks. ; dt x where F C s the flux of C along the drecton of x, see fgure, where the mass changes due to a gradent n F C across a volume s computed, ncludng an nternal net source, S C [ Cs -1 ]. F C1 δc F C X 1 X δx 1 = X X 1 Fgure 3.6. Change n concentraton of scalar C n a volume wth fluxes. (3.45) δcδxδx δx = ( F F ) δx δx δt + Sδxδx δx δt; j k C1 C j k j k S = Sources Snks; Dvdng by the dfferental volume and δt, one obtans 5 Lectures n Mcro Meteorology

54 (3.46) δc ( F F ) dc ( F ) S δt δx dt x C1 C C = + = + S forδ To proceed we need to descrbe the flux, F. Snce we follow the flow, the tme dervatve of the concentraton C must be substantal, and the flux n/out of our volume must be assocated wth the molecular moton. Also here we often assume the flux n a pont to go from hgh towards low concentraton. We can then ether say we assume the flux to be lnearly proportonal to the gradent of the concentraton, or consder t a Taylor expanson of the concentraton varaton across the pont, and then consder only frst order. C (3.47) FC = kc, x where k c s the molecular dffusvty of the substance C n the atmosphere. Ths type of assumpton about fluxes s used throughout physcs, especally for Flud Mechancs and molecular gas theory. It s not always true, and we shall n boundary layer descrptons see several cases where one has been forced to reformulate the theores because the assumpton does not hold. From molecular gas theory we have: (3.48) k a v, C C C where a s a coeffcent, and k C s assumed proportonal to the standard devaton of the veloctes of the C- molecules n the atmosphere, whle l C s the mean free path between the same molecules. Notce, the frst term wll change wth temperature, whle the last wll change wth pressure. The changes over temperature and pressure ranges of relevance for the atmospherc boundary layer are however so small that k C s consdered constant here. 0. (3.49) = + u = k + S = k + S dc C C C C C C dt t x x x x From the budget for a scalar, we can derve the equaton for the potental temperature: (3.50) d Q dt t x x C C x θ * θ LE 1 j u θ = + j = k θ T j j ρ p ρ p j, Where, k T s the thermal dffusvty, correspondng to k C above. k T s about m /s, about 1.44 larger than the knematc vscosty. Addtonally, we have n (3.50) ntroduced two specfc terms n the source, S. The frst wll be relevant, f water s present n the atmosphere as both lqud and vapour phases. E s the water vapour producton from evaporaton of lqud water wthn the volume,.e. from droplets. L s the coeffcent of evaporatve heat consumpton. C p s the heat capacty at constant pressure of the ar. The other source term that can be mportant n the atmospherc boundary layer s the dvergence of the net-radaton, Q j * n the j- drecton. Lectures n Mcro Meteorology 53

55 It should be noted that dervaton of the equaton for the potental temperature s farly complcated startng from conservaton of enthalpy. We avod ths dervaton here, but we note that the smplcty of the equaton demands that t s formulated n terms of potental temperature. Had we nstead derved an equaton for the temperature, T, we would have to nclude terms explanng how the temperature changed due to changes n pressure for vertcal transport, as consdered when dervng expressons for potental temperature above ( ). If only horzontal moton s consdered the equatons for T and θ become the same. In the sprt of keepng track of both lqud and gaseous water, we must keep track on both phases of the water: (3.51) ( mass of lqud) + ( mass of vapour) qt ql + q = mass of ar As q T s a scalar quantty ts budget equaton can be wrtten: (3.5) SqT = + = +, dt t x x ρ dqt qt qt q uj kq j j where S qt s the net source of total water n the ar, for example by water droplets beng njected nto the volume, and by evaporaton of such droplets. We wrte: S qt = S q + S ql, separatng S n a source term for vapour and for lqud. k q s the molecular dffusvty of water vapour n the ar. Notce, we magne that there s no dffuson of lqud water vapour the boundares of the dfferental volume. Hence we can wrte the equatons for the two phases of the water: (3.53) dq q q q q E = + uj = kq + + ; dt t x j x j ρ ρ dql ql q S L ql E = + u j =. dt t x ρ ρ In general we wll neglect these aspects of water and water vapor. Note S q and E wll often be combned. Smlar ssues arse when a speces, C, s not passve but chemcally/ or radoactvely actve, meanng that there are sources and snks, n the volume, ndependent of the turbulence transport. We shall not go further nto ths but refer to Lenschow and Delany (1987) and Krstensen et al (010) for studes nvolvng chemcal reactons, and Farall and Larsen (1984) and results nvolvng aerosols. j S, Summary of equatons and averagng consderatons. We have now establshed the necessary 8 equatons to handle our 8 unknowns. T, u 1, u, u 3, q, θ, ρ and p Equaton of state: (3.54) p = ρr T, wtht T ( q). d v v Relaton between T and θ: 54 Lectures n Mcro Meteorology

56 (3.55) θ = T +Γ z Contnuty equaton, n the ncompressble mode: u j (3.56) = 0; x j The momentum equaton: du u u 1 p u (3.57) = + uj = gδ3 Ω εjkηjuk + ν ; dt t x j ρ x x j Temperature: * d 1 j (3.58) θ θ LE Q u θ j k θ = + = T dt t x j x j ρc p ρc p x j Water vapour: dq q q q E (3.59) = + uj = kq + dt t x j x j ρ Snce we cannot solve these equatons drectly, we wsh to derve equatons for the mean quanttes, n keepng wth our dscusson n secton. In the equatons above we have used [ ] to ndcate the non-lnear terms,.e. terms where we have products or ratos of two varables. From the dscusson of the averagng procedures n chapter, we must expect complcatons for these terms, whle the other terms are lnear and wll behave ncely, when we average. To solve the equatons for the average varables, we must expect to have to consder also equatons for the fluctuatons. Therefore we now wsh to smplfy the equatons such that the average equatons can be obtaned more easly, whle the equatons for the fluctuatons are stll useful as well. The method we use for generatng equatons for the mean values s to nsert the varables, decomposed nto averages and fluctuatons, and then average the equatons (afterwards we derve equatons for the fluctuaton by subtractng the mean equatons from the equatons for the raw varables. (3.60) u = u + u ; θ = θ + θ ; ρ= ρ+ ρ ; p= p + p ; q= q+ q ; We neglect the lnearty problems assocated wth the E-terms n temperature and humdty equatons, and the radaton term n the temperature equaton. Thereafter, we have two types of non-lnearty, the one s assocated wth the advecton term n all equatons, whle the other s assocated wth the pressure term n the momentum equaton. We consder the last of these. (3.61) du 1 p u = gδ3 Ω εjkηjuk + ν ; dt ρ x x j Consder the frst two terms on the rght hand sde: 1 p (3.6) Y = δ3g ρ x Lectures n Mcro Meteorology 55

57 We wll now study the possblty for lnearsng Y for small values of the p and ρ, wthout removng too much nformaton for both average equatons and fluctuaton equatons, because we know from the start of the secton that the vertcal stratfcaton of p and ρ are very mportant for the flow characterstcs. Also, we shall see f p, and ρ can really be consdered small enough for lnearzaton. To the frst order n the fluctuatons we may wrte: (3.63) 1 ( p+ p ) 1 ρ ( p+ p ) Y = δ g δ g (1 ) = ρ + ρ ρ ρ x 3 3 x 1 ( p+ p ) ρ 1 ( p+ p ) = δ3 g +. ρ x ρ ρ x We wll now evaluate these terms for = 1, and = 3 separately,.e. the vertcal and the horzontal components separately. =3 1 ( p) 1 ( p ) ρ 1 ( p) ρ 1 ( p ) (3.64) Y3 = g + +. ρ z ρ z ρ ρ z ρ ρ z Now assume the hydrostatc equaton s at least approxmately vald for the mean values of p and ρ, and evaluatng the terms to frst order only (.e. removng the last term n (3.64). (3.65) 1 ( p) 1 ( p ) ρ 1 ( p) Y3 = g + ρ z ρ z ρ ρ z 1 ( p ) 1 ( p) ρ 1 ( p) = g + ρ z ρ z ρ ρ z 1 ( p+ p') ρ 1 p ρ = g(1 + ) = g(1 + ); ρ z ρ ρ z ρ As seen, frst order terms n the fluctuatons are the lowest order terms that remans, because the average terms cancels-at least approxmately due to the hydrostatc balance. Therefore, we should here keep the whole last term and not neglect ρ /ρ relatve to 1. = 1, Here we can safely neglect ρ /ρ relatve to 1. 1 ( p+ p ) ρ 1 ( p+ p ) 1 p (3.66) Y = δ3g (1 ) ; ρ + ρ x ρ ρ x ρ x For both these equatons, we must evaluate the mportance of the frst order terms, especally the term contanng the fluctuatng densty, ρ /ρ. Consder the equaton of state: 56 Lectures n Mcro Meteorology

58 (3.67) p ρ= RT d v or ln ρ= ln p lntv ln Rd. Dfferentaton yelds: (3.68) d ρ dp dtv ρ p T v = or = ρ p T ρ p T, Where, we have assocated dfferentals wth the fluctuatons, reflectng that the fluctuatons n the parameters here are small compared to the mean values. Exactly how small they are wll depend on the overall atmospherc condtons. However, the data ndcate: p 3 T v 3 (3.69) O( ) 10 ; O( ) p Tv Hence: ρ T v θ v (3.70), ρ Tv θv Where, we have ntroduced the vrtual potental temperature to generalse the result. If we assume the reference level for the potental temperature to be the surface t s seen that the mean values for the potental temperature and the vrtual potental temperature are very close to each other snce they only dffer wth Γ(q)z, and wth z small that term s small compared to 300K. The fluctuatons are the same snce Γ(q) z does not fluctuate statstcally. Note the order of magntude evaluaton of the mportance of the pressure and the temperature terms do not ndcate a large dfference, hence we can under specal crcumstances expect the pressure terms to become mportant. However, for now we conclude that the temperature term domnates the densty fluctuatons, and that the relatve densty fluctuatons generally must be consdered so small that they can be neglected compared to one. For = 1, we can therefore wrte: 1 ( p+ p ) 1 p (3.71) Y ; ρ x ρ x However, for =3, t s seen that the leadng terms are cancelled by the hydrostatc equaton, and the relatve densty fluctuatons become multpled by g, and therefore can be strong enough to nfluence the equatons for the fluctuatons. Therefore they 3 takes the form (3.65). Hence, we wrte the momentum equaton on the form: v v (3.7) du u u θ 1 p u (1 ) ; dt t x x x v = + uj = gδ3 Ω εjkηjuk + ν j θv ρ j I II III IV V VI In ths equaton we have removed one non-lnearty from the momentum equaton. We now proceed to nsert the Reynolds s conventon for fluctuatons and mean values and then take the mean value of the terms n the equaton. We proceed term by term: term I: Lectures n Mcro Meteorology 57

59 (3.73) Term II: (3.74) ( u + u ) ( u + u ) u = = ; t t t ( u + u ) u u u u u u ( uj + u j) = uj + uj + u j + u j = uj + u j x x x x x x x j j j j j j j u uu j u j u uu j = uj + ( u ) = uj + ; x x x x x j j j j j where we have repeatedly used the contnuty equaton n ts varous forms: uj uj u j (3.75) = 0; = 0; = 0; x x x j j j As s seen all the other terms n the momentum equaton are lnear n the varables. The procedure then proceeds as for the tme dervatve above. Term III: θ v (3.76) gδ3(1 ) = gδ3 Term IV (3.77) 1 ( p+ p ) 1 ( p+ p ) 1 p = = ρ x ρ x ρ x Term V: (3.78) Ω ε η u = Ω ε η u θ v jk j k jk j k Term VI: (3.79) u ( u + u ) u ν ν ν ; = = xj xj xj Therefore we can now wrte the equaton for the mean velocty components: (3.80) du u u 1 p u ; dt t x x x = + uj = gδ3 Ω εjkηjuk + ν uu j j ρ x j j Wth the momentum equatons processed, t s easy to summarse all the equatons for the mean varables (C s a scalar concentraton,.e. (3.49)): (3.81) u = u + u, = 1,, 3; T; θ = θ + θ ; ρ= ρ+ ρ ; p= p + p ; q= q+ q, C = C + C. Contnuty: u (3.8) = 0; Momentum: x 58 Lectures n Mcro Meteorology

60 (3.83) Potental temperature: (3.84) du u u 1 p u dt t x x x = + uj = gδ3 Ω εjkηjuk + ν uu j j ρ x j j d Q ; dt t x x x C x * θ θ θ θ LE 1 j = + uj = kt u jθ j j j ρ p ρc p j ; Water vapour: (3.85) Any Scalar: (3.86) uq dq q q q j E = + uj = kq + ; dt t x j x j x j ρ uc dc C C C j = + uj = kc + S; dt t x j x j x j It s seen that we have 9 varables and 7 equatons above. The remanng equatons are the equaton of state and the relaton between T and θ. Note that we consder E and Q* and S as externally gven varable. Also, we have neglected that strctly C p vares wth the humdty content of the ar. In general the equatons for the average varables have retaned the orgnal form for the raw varables, wth one mportant excepton that a number of correlaton terms nvolvng the fluctuatons enter nto the equatons. Ther orgn s the non-lnearty of the advecton terms. One result of ths s that we end up wth equaton wth more varables than we started wth for the raw varables. Ths s called the closure problem and s the theme of chapter 4. The physcal meanng of the correlaton terms s that they consttute a change n the average varable due to a dvergence of a turbulence flux. Recall that we have seen that a correlaton nvolvng a velocty component can be nterpreted as a flux along the lne of the velocty component. As notced above we have avoded gong deeper nto the S and E terms, ncludng breakng them up after Reynolds conventon. Lectures n Mcro Meteorology 59

61 4. Mean flow, turbulence and closures. In ths secton we shall, n more detals, consder the equatons for mean flow, temperature and humdty, as derved n Secton 3, and now loosely appealng to the standard condtons for atmospherc boundary layer, statonarty and horzontally homogenety. Ths wll lead to the so-called closure problem, whch n the prncple can be solved but only approxmately. The closure problem naturally leads to the need to understand the turbulence structure, whch leads to a dscusson of the equatons for the fluctuatons. Our varables can be summarsed: u= u+ u ; T; T = T( q); θ = θ + θ ; ρ= ρ+ ρ ; p= p+ p ; q= q+ q ; EQ, *; v We have used the Reynolds s conventon. Also we nclude E, Q*, the net-source of water vapour from water droplets, and the net-radaton. For smplcty we have neglected the general scalar varable, C, snce q and θ follow smlar relatons. For the varables we have, n secton 3, derved the followng equatons for the mean moton and the mean temperature and humdty. The equaton of state, and the T- θ relaton. (4.1) p = ρr T, wtht T ( q), θ = T +Γ ( q) z. Contnuty: d v v u x (4.) 0; Momentum: (4.3) 3 Potental temperature: (4.4) = du u u 1 p u = + uj = gδ Ω εjkηjuk + ( ν uu j ); ; dt t x ρ x x x j j j dθ θ θ θ LE 1 Q = + uj = ( kt u jθ ) dt t x x x ρc ρc x j j j p p j Water vapour: dq q q q E (4.5) = + u j = ( k q uq j ) + ; dt t x x x ρ j j j * j ; In general the equatons for the average varables have retaned the orgnal form for the raw varables, wth the mportant excepton that a number of correlaton terms nvolvng the fluctuatons enter nto the equatons. Ther orgn s the non-lnearty of the advecton terms. The physcal meanng of the correlaton terms s that they consttute a change n the average varable due to a flux dvergence of a turbulence flux. For the velocty-velocty correlaton we even retan the name stress and operate wth turbulence stress n parallel to shear stress for the molecular term. Recall that we have seen that a correlaton nvolvng a velocty component can be nterpreted as a flux along the lne of the velocty component. Also, here we have wrtten the correlaton terms n such a way that they are comparable to the terms derved from the molecular fluxes of momentum, heat and humdty. From the equatons t s apparent also how 60 Lectures n Mcro Meteorology

62 the gradents of both the turbulent and the molecular fluxes result n changes of the mean varables. Another aspect of the correlatons terms s that our generaton of equatons for the mean varables have resulted n many more varables, f we understand every correlaton term as a new varables, and countng only the<u u j > terms, we obtan addtonally 1 new varables: 3 varables for the heat flux and the humdty flux respectvely, and 6 varables from the turbulence stress, where we have reduced the 9 elements n the tensor to 6 ndependent elements due to symmetry n the subscrpt numbers. Snce we have got no more equatons we have to descrbe these new terms such that the mean value equatons can be solved. The smplest method s to prescrbe them. Ths typcal means that they, n parts of the boundary layer, are prescrbed to be constant, or behavng n nce predctable ways; some may even be zero. Ths s the methodology we use when formulatng smlarty theores, to whch we shall return. The next smplest method s to defne a turbulent dffuson coeffcent that relates the turbulent fluxes to gradents n the mean varables, just as s done for molecular gas theory. Recall that we for ncompressble condtons could wrte the molecular stress as: u u j (4.6) τ j = ρν ( + ) xj x Whle for a scalar lke temperature and humdty, we found C (4.7) F = kc, x where both ν and k C could be estmated from: (4.8) ν k a v, C C C wth v c and λ c beng characterstc velocty and length scales for the problem, typcally the standard devaton of the speed of the relevant molecules and ther mean free path respectvely. The success of ths approach from molecular gas theory has nspred to ts use for turbulence flow as well, that s u u j (4.9) uu j ( = τ j ) = Kj ( + ); x x uc = K (4.10) ; Followng the molecular gas theory, we can now start huntng for a turbulent velocty scale and a turbulent length scale to estmate the turbulent dffusvtes, K j and K C. The characterstc turbulent length scale s called a mxng length, and the theores developed to derve the dffusvtes are called mxng length theores. We can elucdate the characterstc scales from a smple argument. Consder the fgure, we have already used n chapter, wth an eddy moton n a shear flow. C j C x. Lectures n Mcro Meteorology 61

63 Fgure 4.1. Transport of momentum by swrlng turbulent moton between levels wth low and hgh wnd speed (Tennekes and Lumley, 197). Assume that the eddy moton takes a flud partcle from x = 0 to x n a tme t. If the flud partcle does not lose ts momentum n the transport t wll arrve at x wth a momentum defct gven by: M = ρ( u1( x, t) u1(0,0)) = (4.11) = ρ( u ( x ) u (0)) + ρ( u ( x, t) u (0,0)) ρ( u1( x) u1(0)) The mean velocty gradent can be consdered approxmately lnear. Hence: u1 (4.1) M ρ x x The volume of flud, per unt area per unt tme that at tme t passes n the x drecton, s u (t). Hence the nstantaneous momentum defct transported to pont x s gven by: u1 (4.13) M u ρ xu. x The average transport of momentum defct per unt area per unt tme transported n the x drecton s then gven by: u1 (4.14) τ1 = M u ρxu. x Here u and x are two random varables (u = dx /dt). If the ar parcel dd not contnuously exchange momentum wth ts surroundngs, u would reman constant and <x u> would contnue to ncrease wth tme. Ths s not realstc; nstead we expect the correlaton between the speed of the parcel, u, and ts poston to decrease to zero as tmes ncreases. Hence we can say that the parcel has delvered all t momentum defct to the surroundng when the ntegral scale of the cross-covarance, <u x >, has been reached. We could stop here and say that ths cross-covarance can be nterpreted at a velocty scale tmes a length scale. However, t s possble to elaborate a bt further by followng the decay of the correlaton between velocty and poston along the path of movng parcel as tme ncreases: (4.15) t u () t x () t = u () t u ( t ) dt = u () t u ( t ) dt = 0 0 t u ρlu( τ) dτ ut L, 0 = where we have assumed the tme to be large enough for the ntegral to be approxmated by the ntegral scale. The ntegral s conducted along the path of the transport of the flud elements, t 6 Lectures n Mcro Meteorology

64 and t s called a Lagrangan path ntegral that s followng the flud elements. Notce that we have assumed u (t) to be statonary and therefore have made the usual substtuton: t-t = τ. We shall return to ths type of ntegrals when dscussng plume dsperson. Here we notce that we can estmate the turbulent dffusvty for momentum as: (4.16) K = ux ut = u ( u T) u, j L L L where t s seen that we get a velocty scale that equals the standard devaton of u, whle the mxng length s derved from the Lagrangan tme scale and the standard devaton of u. All the above arguments could be repeated for a scalar gradent, where the mean gradent of the scalar would shove up n the expresson but the K tself would depend only on velocty statstcs. We notce that a dffusvty approxmaton to the turbulent transport does not always work well, and that one of the smplest cases where t does not work s the unstable mxed boundary layer, where the gradents are zero n the mddle of the layer, whle the fluxes are large. It s seen from (4.8) that to estmate the turbulence dffusvty, one has to estmate two scales of relevance for the turbulence dffuson, a velocty scale and a length scale. In numercal modellng ths s not n general smple to specfy these two scales. Therefore, one wll often use a slghtly dfferent approxmaton for the turbulent transport. From (4.1), we let: ' u1 (4.17) u c, x where we approxmate the turbulence standard devaton to be proportonal to the absolute value of the mean gradent. Based on such arguments, an often used parameterzaton of the turbulence transport n (4.9) s u u j u u j u u j (4.18) uu j ( = τ j ) = Kj ( + ) = c ( + ) ( + ), xj x xj x xj x and smlar for scalar dffuson. Notce that the stress and the momentum transport are defned to have opposte sgn. As seen we now have to estmate only one length scale to close the equatons, because the mean gradents are resolved varables. As an alternatve to parameterse the unknown correlaton terms n ( ), as sketched above, one can contnue the equaton- buldng for these terms. The procedure nvolves subtracton of the mean equatons from the basc equatons: (4.19) ( u + u ) = t u = t u = t Hereafter, we wrte: Lectures n Mcro Meteorology 63

65 (4.0) u j u = u j( ) t + u u j = u ( ) t. ( uu j) = u j( ) + u ( ) t Equatons are generated smlarly for the tme dervatves of heat flux and humdty fluxes and for the varances of temperature and water vapour mxng rato. Hereby, one can generate equatons for all the turbulence correlaton terms n the mean value equatons. There are two problems: the most mportant s that the new turbulence equatons wll nvolve 3 rd order turbulence terms, and a number of other and new turbulence terms, lke the followng: uu θ, uu q, u θq; u θθ ; pu / ρ; j j Such terms wll agan have to be closed, mostly employng length scales and dffusvty formulatons, for example: (4.1) u θ u jθ uu jθ = K( + ), x x pu ' '/ / wth K Λ u Iu. j uu j ρ = u Λ x Ths means that we must estmate a turbulent dffusvty from a velocty and a tme or length scale. Notce that the second order moments n ths expanson should not be parametersed because they are solved explctly n the equatons that now nclude both mean values and second order moments. Alternatvely, one can contnue to wrte equatons for thrd order turbulence correlatons, proceedng as we dd for the second order correlatons. However, such equaton wll result n new unknown fourth order correlatons, whch wll have to be closed, agan mostly by a mxng length dffusvty assumpton. As the smplest example, we shall consder the equatons for second order correlatons, as was sketched n (4.0) above. Followng Stull (1988), we start to smplfy the equatons. To see the mplcatons, start wth our average momentum equaton, from (4.3): du u u 1 p u = + uj = gδ Ω εjkηjuk + ( ν uu j ); dt t x ρ x x x (4.) 3 j j j We now wrte the three component equatons, but consder steady horzontal mean flow, that and further assume that the turbulence cross-co-varance terms depend essentally only on the vertcal coordnate. These condtons consttute a far approxmaton for the atmospherc boundary layer, and most of the examples of atmospherc boundary layer models presented 64 Lectures n Mcro Meteorology

66 here and n the followng chapters nclude these approxmatons (recall: ε jk s a cyclc operator, beng +1, -1 dependent on the order of, j and k and 0 f two of the ndces are equal), η are the components of Ω n the local coordnate system, x : (4.3) du u u 1 p u = + u = gδ Ω ε η u + ( ν uu ); dt t x x x x u 1 j 3 jk j k j j ρ j j : du u u u 1 p u = + u + u = Ω η u + Ω η u + ( ν uu j ); dt t x x x x x or ρ 1 j j du u u u 1 p u = + u + u = + fu + ( ν dt t x x x x C 1 ρ 1 3 x3 Correspondngly u : uu ). du u u u 1 p u = + u + u = f u + ( ν uu ). dt t x x x x x 1 C ρ Where f c = Ωsnφ, from chapter 3. f c s denoted the Corols parameter beng about sec -1. Correspondngly we get for the vertcal equaton: du u u 1 p u = + uj = gδ3 Ω εjkηjuk + ( ν uu j ); dt t x j ρ x x j x j u : (4.4) 3 du3 u3 u3 u3 1 p u3 = + u1 + u = g Ω( η1u ηu1) + ( ν uu 3 3); dt t x1 x ρ x3 x3 x3 or : 1 p 0= g Ω( ηu η u) uu. ρ x x3 It s seen that, for what could be called typcal atmospherc boundary layer condtons, the two horzontal equatons provde useful nformaton about the relaton between mean flows and turbulence, and about how the Corols force mxes the components. The vertcal equaton s seen to reflect a forcng that comes from devatons between the atmosphere and the hydrostatc equaton, the heght change of the varance of u 3 and the mxng n of the horzontal components by the Corols parameter. The term s seen to depend on how we arrange the horzontal coordnate system. In a way the vertcal equaton just tells us the necessary balance of mean terms to keep the mean vertcal velocty equal to zero. Therefore, we follow Stull to avod these uncertantes by usng ε j3, whch as seen takes the Corols parameter out of the vertcal equaton. Lectures n Mcro Meteorology 65

67 We now proceed wth the momentum equaton for the raw varables (3.7), correspondng to the average equaton n (4.). Followng the prescrpton from above, we have to nsert the Reynolds s conventon for the averages and fluctuatons. (4.5) du u u θ 1 p u dt t x x x v = + u j = gδ3(1 ) + fcεj3u j + ν or j θv ρ j ( u u') ( u u') θ v 1 ( p+ p ) ( u u ) + u j + uj = gδ3 + fcεj3 u j + u j + ν t xj θv ρ x xj ( ') (1 ) ( ) ; we expand (4.5) u u' u u' u u' + + u j + u j + uj ' + uj ' = t t xj xj xj xj (4.6) θ v 1 p 1 p u u = gδ31+ gδ3 + fcεj3u j + fcεj3u j + ν + ν θ ρ x ρ x x x v j j Next we follow the prescrpton of (4.0) and subtract the mean value equatons as gven n (4.4) and (4.5) to retan the equaton for the fluctuatons: (4.7) u' u' u u' + u j + uj ' + uj ' = t x x x j j j u u =+ gδ + f ε u + ν + θ v 1 p u í 3 C j3 j θv ρ x xj xj j Next we use (4.0), but we lmt ourselves to estmaton of the equatons for the turbulence varances. Hence we multply (4.7) by u, to obtan equatons for u, and subsequently average to obtan equatons for the <u >. Remember Ensten s summaton notaton for repeated ndces. Notce that we wll n the followng often use the notaton: e=< u u >for the total turbulence varance or energy, snce 0.5ρ e s the turbulent knetc energy of the flow. (4.8) u u u u = uj uu j u j t xj xj xj u θ u p u =+ gδ + f ε uu + νu + u u u v í 3 C j3 j θv ρ x xj xj j The left hand sde s obvous, Thereafter, we handle the two last terms on the rght hand sde, the last term s obvously zero, whle the second last tme can be reduced by takng the second dervatve of the varance or: 66 Lectures n Mcro Meteorology

68 (4.9) u u u u = + x j x j xj ν ν ν. Here, we normally argue that <u > s a farly slowly varyng functon for statonary and horzontally homogenous turbulence. Hence we shall neglect ths term on the left hand sde. The last term on the rght hand sde s the one, we wsh to reduce, and t can be consdered approxmately equal to mnus the lead term on the rght hand sde. Ths term emphaszes the small scale turbulence and s related to the dsspaton of turbulent wnd, ε, we dscussed n secton. See more n appendx 4A. The last term on the left hand sde can be converted, usng the ncompressblty of (3.35): (4.30) u u uu u uu u í j j j j = u = xj xj xj xj Fnally, we can wrte the pressure wnd correlaton as (4.31) u p 1 up p u = ρ x ρ x ρ x Hence, we can wrte (4.8) as: (4.3) u u + u j = t x j u u θ v pu p u uu j u = uu j + gδ3 + + fcεj3uu j ν xj θv ρ x ρ x x j x j We can ether sum over the repeated subscrpt, 1,,3, as we should accordng to the summaton rule, and as we already do wth subscrpt j n (4.30), or we could consder the equatons one by one as u 1 = u,u =v,u 3 =w. We start wth the frst approach, to obtan and equaton for the total turbulence varance, e=<u >: u u u u θ v pu uu j u + uj = uu (4.33) j + gδ3 ν ; t xj xj θv ρ x x j x j I II III IV V VI VII In (4.33), we have lost the second pressure correlaton term, because of ncompressblty, and the f C -correlaton term, because summaton over and j and the sgn propertes of the ε j3 - tensor. The last term can now be dentfed as twce the dsspaton of wnd varance as dscussed n secton and further consdered n appendx 4A. In ths term, summaton over both and j s mpled: Lectures n Mcro Meteorology 67

69 (4.34) u ε ν x j In (4.33) the left hand sde represent the change of total velocty varance wth tme (I), and due to advecton (II). The rght hand sde contans terms descrbng local producton and destructon. In understandng these terms, t s a good dea to magne the statonary horzontally homogenous stuaton as dscussed n secton, wth manly vertcal mean gradents, and wth <u 1 u 3 > generally negatve. Term III s producton of varance from the mean shear, as we have dscussed n secton. Note the domnant mean shear n the boundary layer s vertcal. Term IV s producton or destructon of varance from the mean temperature flux, or postve durng day, and negatve durng nght. Term V descrbes how pressure transports and re-dstrbute velocty varance across the volume. Term VI represents the varance transport by the turbulence velocty fluctuatons. Term VII equal twce the velocty dsspaton as descrbed by (4.34) We shall now shortly return to the component nterpretaton of (4.3) u u + u j = t x j (4.35) u u θ v pu p u uu j u = uu j + gδ3 + + fcεj3uu j ν xj θv ρ x ρ x x j x j III IV V Va VI VIa VII In (4.35) we now consder the three velocty components, one by one, =1,,3 correspondng to u,v,w, and now dscuss the ndvdual terms The left hand sde contnues to descrbe the change n u varance by tme and by advecton. Notce n case of statonarty and horzontal homogenety these terms are small or even zero, f the concepts are strctly appled. Assume further that the mean wnd s along the u component, and that only vertcal mean gradents are sgnfcant, then the varance producton n III exst only for the u=u 1 component. The producton/loss term of IV on the other hand apples to w=u 3 component only. The v=u component receve ts varance from the transfer terms V, Va, VI and VIa. Va s denoted the pressure redstrbuton term, precsely because t acts to transfer varance between components, amng to return an sotropc relaton between the 3 velocty components. VIa s generally consdered unmportant, beng small due to the Corols parameter. The dsspaton s actve n all components through VII that can wth some approxmaton be consdered twce the dsspaton of the ndvdual components. We have now derved the prognostc equatons for the velocty varance n (4.33) and (4.35), recallng that these were just the smplest of the equatons for the second order moments depcted n (4.0). We stll mss the covarance terms. Stll (4.33, 4.35) llustrate the terms that appear, and whch wll have to be parametersed, f a second order closure for turbulence s mplemented, the terms nvolvng pressure correlatons, thrd order moments and dsspaton, shown as V, Va, VI and VII n (4.35). 68 Lectures n Mcro Meteorology

70 Fnally, we consder the temperature and humdty turbulence equatons, whch do not have the complcatons of pressure terms, Corols terms, and beng vector quanttes. From (3.50, 3.53) we wrte the equatons smlar to (4.5) for velocty. * dθ θ θ θ LE 1 Q j = + uj = kt dt t x j x j ρc p ρc x p j (4.36) * * ( θ + θ') ( θ + θ') ( θ + θ') LE ( + E') 1 ( Qj + Q j ) + ( uj + u j) = kq t x x ρc ρc x j j p p j For humdty the equatons are qute smlar (4.37) dq q q q E = + uj = kt + dt t x j x j q + q q + q q + q L E+ E ( ) ( ) ( ) ( ') + ( uj + u j) = kq + t xj xj ρ ρ Comparng wth (4.5), for velocty t s seen that the equaton for temperature and humdty can be handled smlarly replacng u by q or θ, and neglectng the f c and p and the buoyancy terms, and ncludng the E and Q* terms. We obtan: (4.38) * u jθ E Qj + u j = θ uj kt L j j j j ρ ρc p j θ θ θ θ θ θ t x x x x x q q q uq j q qe + uj = qu j kq + t xj xj x j x j ρ I II III VI VII VIIIa VIIIa Clearly equatons for all passve scalars would end up smlarly, where terms VIII and VIIIa would be smlar source terms. Terms VII n (4.38) represent the dsspatons of the scalars temperature and specfc humdty, often denoted: (4.39) θ q N = εθ = κt ; εq = kq, x j x j In the followng chapters, we shall dscuss the behavour of the dfferent terms wthn the atmospherc boundary layer. However, here we shall consder the closure of numercal models n the lght of the turbulence terms consdered above. Closure consderatons: Consder the modellng as beng amed on establshng some knd of averages of the meteorologcal varables u, T, p as functon of space and tme wthn the boundary layer, usng the equatons n (4.1) to (4.5). Lectures n Mcro Meteorology 69

71 To estmate the mean values as gven by these equatons have to be closed by relatng the unknown hgher moments, contanng the turbulence correlatons, to the lower moments beng resolved. To smplfy the dscusson we shall mostly focus on the velocty terms. Here we have llustrated an mportant method based on the assumpton of a characterstc scales, at the very least a length scale or a tme scale, as s llustrated n equatons (4.6) and (4.7) or (4.9) and (4.10). Ths method s called frst order closure, because the second order moments are estmated from frst order terms and estmated parameters, lke dffusvty and length scales. The next step n complcatons s to generate equaton for the second order moments as llustrated above, and as dscussed ths method produces a number of new terms nvolvng thrd velocty correlaton, pressure-velocty correlatons and dsspaton terms, that now all have to be related to mean values and second order terms that are resolved drectly. Remember, that we have derved the varance equatons, as the smplest examples of the full set of covarances: <u u j >, <u θ >, < u q >. The closure wll typcally be based on length scales, dffusvtes and dmensonal analyss, as llustrated n equaton (4.1). Typcally one wll have: Down-gradent dffuson wth velocty and length scales. Return to sotropy, typcally used for pressure terms. Dsspaton s put proportonal to turbulence ntensty n sutable powers dvded by a sutable length scale. The process of generatng equatons for hgher order turbulence than the second order can be contnued n the prncple by the same method as already outlned n (4.19 and 4.0). Advantages are: The most mportant and vsble features of the flow are descrbed by the mean values and the low order turbulence moments. Therefore, the soluton should become better f these moments are descrbed by exact solutons, whle the closure s pushed to hgher order moments. The mean values and the low order moments carres the most characterstc features of the flow, e.g. ncompressble boundary layer shear flow. The nteracton and processes controllng the hgher order moments can be argued to be ndependent of the processes generatng the turbulence. They are more result of the nteractve processes between dfferent aspects of the turbulence when t has been generated. Hence, one can use data from dfferent areas of turbulence flow studes, when tryng to parameterse hgher order moments. Dsadvantages are: An ever-ncreasng number of varables and equatons. The smple ncrease n equatons and unknowns for the frst the closure types, only for the velocty correlaton s shown n the next equaton. The total number of equatons and unknowns grow even faster snce terms nvolvng pressure and dsspatons have to be nvolved, also f one s not consderng temperature humdty radaton etc. (4.40) du uu j 1. Moment closure : = + ; Equatons :3; New unknowns : 6; dt x j duu j uu ju k. Moment closure : = + ; Equatons : 9; New Unknowns :18; dt x k duu ju k uu ju ku m 3. Moment closure : = + ; Equatons : 7; New unknowns : 54 dt x k 70 Lectures n Mcro Meteorology

72 To go to hgher moments does not really ntroduce novel deas of closure. It s stll based on these relatvely smple deas already used extensvely. The conceptual models behnd most of the closure assumptons however become less and less transparent as the moment consdered ncreases, smultaneously data become more and more sparse. And, although the local K-dffuson works well qute often, we know that t does not always work. We shall not go nto detaled lstngs detaled local model wth dfferent types of closure. If you are nterested, please refer to the lterature startng wth examples and references n Stull (1988). Creatve closure methods. Mxed order closures. Whle the hgher order closure s complcated n terms of equatons and closures, t s straghtforward conceptually, but cumbersome, and often n- transparent. In many model systems one tres to lmt the number of equatons by retanng only some of the hgher order moments, e.g. one can operate use frst order closure, wth K-dffusvtes, but nstead of prescrbng these dffusvtes one can have let them depend on the turbulence characterstcs, lke for a temperature - wnd system, one can have K = K( e, <θ >). The equatons for e and <θ > then have to be retaned n smplfed versons, see (4.33) and (4.38), where e s the total varance that then have to be modelled. New equatons. A successful example s to use the dsspaton and the varance to generate dffusvty: ( cε 5e) (4.41) K Ths s called the energy-dsspaton closure, and has proven extremely successful both for atmospherc and laboratory flow, wth flow separaton, and s extensvely used wth CFD (Computatonal Flud Dynamcs) modellng. A drawback s that one has to develop an equaton for the dsspaton that s farly complcated and a bt artfcal. It s shown here. Its dervaton s qute lengthy and cumbersome (Arpac and Larsen, 1984). (4.4) ε ε u ε g u ε ε = c uu + c w θ c, t e x e x e ε ε1 j ε ε3 j θ where the parameters c ε are unversal parameters, or almost so. Actually they are dfferent from tunnel flows to atmospherc flows. Length scales can be approxmated by: 3/ e (4.43) Λ A smple Gaussan closure, uses the Gaussan approxmaton that relates fourth order moments to second order moments. ε (4.44) xyzw = xy zw+ xz yw + xw yz Ths can of course be used teratvely. For a Gaussan process the odd moments are zero. Hence the method cannot be used for odd moments. They are usually let equal to zero or determned somehow else, when the Gaussan approxmaton s ntroduced. The Gaussan Lectures n Mcro Meteorology 71

73 approxmaton s a good approxmaton, though not extremely good, for the even moments of the turbulence veloctes. Other Types of closure Spectral dffusvty: One useful closure s to descrbe the turbulence dffusvty n terms of wave numbers. Ths comes especally useful when the soluton of the models s partly or fully performed n Fourer space and for scalars. Addtonally the method addresses the problem that a smple K-dffuson descrpton does not always work, not even for very well known phenomenon as vertcal dffuson n the unstable atmospherc boundary layer, where the vertcal gradent s close to zero, but the assocate fluxes are large, see fgure 4.3, or the near feld dsperson from chmneys, as dscussed n secton 10. The method can be llustrated by consderng the flux as functon of wave numbers: C dc uc uc = KC ; = ; (4.45) x dt x dc C C = KC KC ; dt x x x Ths s a dffuson equaton. We have assumed that K C s constant n x. We now consder the concentraton feld as functon of wave number, a natural extenson of the equaton, also from the pont of vew of solvng the equaton, because dfferentaton becomes multplcaton by wave number n Fourer space. kx (4.46) Cx ( ) = C( k) e dk Fourer transformng the above we get: dc( k) (4.47) = KCkC ( k); dt A natural extenson of the above then becomes to let K C become a functon of k or just k. dc( k) (4.48) ( ) = kk ( ); C kck dt The physcal concept behnd ths equaton s that the dffusvty should be segregated accordng to eddy sze as well. The exact formulaton depends on where f one keeps some spatal varables as spatal varables and only Fourer transform n the others. It s seen that the concept, n a physcally reasonable way wll repar one of the defcences of the smple K- theory, the unstable boundary layer, snce the K C (k 1/z ) corresponds to the transport of the large eddes n the boundary layer and wll multply unto the correspondng gradent across the total boundary layer, represented as k z C(k z, k 1,k ), wth z= z after Fourer transform. K(k) s called the spectral dffusvty. It was orgnally ntroduced for dsperson computatons of ozone n the stratosphere, where Ozone s often dstrbuted wth large volumes wthout concentraton gradents followed by sharp gradents, much lke we see at the top of the atmospherc boundary layer Berkowcz and Prahm, 1979). It s worth pontng out that a K(k ) cannot be recovered as smple multplcatve K(x ) by back transformng (Troen et al,1980). 7 Lectures n Mcro Meteorology

74 Fgure 4.3 Characterstcs of the unstable atmospherc boundary layer (Kamal and Fngan, 1994): Translent turbulence theory: Ths theory formulates, for dscreet varables, the dea that the mxng s performed by eddes of many scales, as such t s well adapted to computer smulatons from ts brth. Consder a stuaton wth only vertcal mxng. The volume consdered s broken down nto N sub-volumes. The mxng s then consdered as the concentraton s updated n the next tme step, by havng contrbutons from each of the N sub-volumes wth a weght gven by the translent matrx, c j (t, t). (4.49) N C( t+ t) = c( t, tc ) ( t); j j j= 1 The formulaton ensures that sub-volumes spaced far apart can nteract as well by large eddes, as can sub-volumes closer to each other, through small eddes. provded one has the translent matrx well calbrated to the strength of the dfferent eddes. Ths calbraton has been done n the lterature by varous authors, see e.g. Stull (1991) C j c j c j C Fgure 4.4. The transfer between sub-volumes n translent turbulence transfer. Notce that the translent turbulence theory wll handle the unstable gradent-less profles wth ease as well. Large Eddy Smulaton (LES). In the above we have dscussed ensemble average models, and the parametersatons necessary solve them. These models have been qute successful n descrbng the turbulence Lectures n Mcro Meteorology 73

75 moments and n development of the mean flow. It has further been easy to compare wth data that farly easly can delver data on such moments, at least to not too hgh order. They have not been good n provdng a descrpton of how turbulence looks, snce the averagng tme/space mostly correspondng to the ensemble average has to be long enough to have all the turbulence ncluded n the fluctuatng part of the sgnal. Another approach s the volume average models that have become farly wde spread worth the computer developments as research tools. Consder the raw momentum equaton : (4.50) du u uu 1 p u ; dt t x x x j = + = gδ3 Ω εjkηjuk + ν j ρ j where we have neglected temperature effects and used the ncompressble contnuty equaton. Ths n the prncple s a volume-ntegrated model, snce we have ntegrated over a volume large enough to nclude molecular effects only through ther ntegrated effects on the flud moton; we consder say volumes of the order of 1 µm 3. If we could solve the equaton and absorb the data, we would get the full space-tme dependent flow feld. Ths s clearly not possble, but say that we feel, we could use such space-tme dependent solutons for all eddes larger than, say meters. Our experence from the basc equaton and from the ensemble average models tells us that we would have to expect to be able to parameterse not the molecular effect but also the effects of the eddes smaller than meters on the part of the flow we resolve drectly. In spectra lke the below, we wll then be able to decde drectly whch scales to resolve explctly n the model, and whch scales to parameterse, by makng the cut along the k-axs. Fgure 4.5. Representatons of the power spectrum for wnd speed n the atmospherc boundary layer from Kamal and Fnngan(1994). LES s formulated not on bass of ensemble averages but on spatal averages that can be wrtten generally as: (4.51) [ ] η( x,) t = G ( x x ) η ( x,) t dx Volume Ths formula just says that we obtan the spatal average, [η], by averagng η over a certan volume, wth a weghtng functon, that can be a smply the volume, but also can be more complcated to facltate further computatons. If all flow varables vansh at the boundares, the spatal averagng commutes wth the dfferental operators: 74 Lectures n Mcro Meteorology

76 G (x -x ) 1/( ) - x - x Fgure 4.6. One Dmensonal representaton of a smple spatal averagng wndow.. η( x,) t = x (4.5) G( x x ) η( x, t) dx ( Gη) G( x x ) η( x, t) dx = Boundares = x x Volume Volume 0 + G( x x ) η( x,) t dx G( x x ) η( x,) t dx = = η( x,) t x x x Volume Volume [ ] The smplest averagng functon s a block average as shown on the fgure 4.6. It s nstructve to compare the less formulaton of the equatons wth the ensemble mean formulaton. Consder the momentum equaton for u. The ensemble mean can be wrtten: (4.53) du u ( uu + uu ) 1 p u ; dt t x x x j j = + = gδ3 Ω εjkηjuk + ν j ρ j Where we have kept the advecton terms together and as usual used the ncompressble contnuty equaton. If all flow varables vansh on the boundares, the averagng procedure commutes wth the dfferental operatons n the raw equaton and we have a LES formulaton that looks smlar to the mean value equaton: d 1 u u uu j g 3 p jk j u k u dt t x j ρ x x j (4.54) [ ] = [ ] + = δ [ ] Ω ε η [ ] + ν [ ] ; The two equatons look very much the same, but we somehow must determne and expand the term [u u j ]. To a start we have: =, (4.55) u u [ u ] Lectures n Mcro Meteorology 75

77 where the fluctuatons are defned much smlar to the Reynolds equaton for ensemble averages and fluctuatons. A dfference and complcaton s that both u and [u ] now must be consdered random varables. Another dfference s that we do know that u occurs on a smaller scale than does u, as we have seen n the earler dscussons about spectra and scales. For the smple block ntegraton over wdth as shown n the fgure, we have the spectral expanson of the varable, η, compare dscussons n secton (.51)-(.55). kx η( x,) t = e dz( k,); t (4.56) η () t = S ( k,) t dk 1 k( x x ) kx ( x,) t = e dz( k,) (,) 3 t dx = e dz k t 8 k [ η ] sn k ( x,) t ( ) S ( k,) t dk ; [ η ] η = η k sn k sn k kx η = η [ η] = (1 ) e dz( k, t) k sn k η = (1 ) Sη ( k,) t dk; k Because both u and [u ] must be consdered random varables the expanson of [u u j ] becomes less smple: (4.57) [ uu ] = [[ u][ u ]] + [ u [ u ]] + [ u [ u]] + [ uu ] j j j j j Notce that we cannot assume the two mddle terms to average to zero, as s the case for ensemble averagng. We now defne the stress as: (4.58) τ = [ uu ] [ u ][ u ] = [ u [ u ]] + [ u [ u ]] + [ u u ] [ u ][ u ] + [ [ u ][ u ]]; j j j j j j j j Ths leads to the volume averaged momentum equaton can be wrtten: 1 + j = δ3 Ω εjkηj k + ν + τj t xj ρ x x j xj (4.59) [ u ] [ u ][ u ] g [ p] [ u ] [ u ] Ths equaton s now close to the average equaton for ensemble means, and can be solved by the same means, but the τ-term has to be understood and parametersed. Ths s pretty cumbersome, but has been done, and the LES modellng s a well-establshed tool n the effort to understand boundary layer turbulence. The most useful parametersatons look as we are used to from the ensemble average models: [ u ] u [ ] [ ] (4.60) ( j u u u ) ( j u τ ) ( j j = K + c + + ) x x x x x x j j j Indeed wth sutable formulatons for the spatal averagng and for the scale λ (Llly, 1966) the LES system offers a determnstc soluton for the Naver- Stokes equatons for realstc turbulent flows (Ladyzhenskaya, 1969), meanng that not only the statstcs but also ndvdual structure development n LES s meanngful.. 76 Lectures n Mcro Meteorology

78 The LES computes exactly as the acronym say the temporal and spatal development of the large eddes, wth full spatal and temporal pcture, wth a spatal resoluton gven by the spatal averagng, and the temporal resoluton gven by the tme step. It has been found to work well for unstable condtons where the large eddes are especally characterstc. It has also been developed for stable condtons, where the large eddes are smaller and not so characterstc. Onset of gravty waves has been found a problem here. It has been useful for computaton of turbulence statstcs, because t provdes the full felds (wth the resolutons gven by the models). If one consders our standard pctures of eddes n the boundary layer, one should therefore magne that these eddes move as n move. Fgure 4.7. Characterstc boundary layer flows. Note the LES modellng, presents the tme varyng Large eddes part of the flow (Wyngaard 1990, Kamal and Fnngan, 1994). Appendx 4A: Spectral descrpton of the vscous dsspaton In the dervaton of the energy equaton n Stull (4.3.1b-d), we derved an expresson for the vscous dsspaton, whch we ntally only had dscussed n terms of how t work to smooth out the turbulent flud moton, or otherwse formulated, removed varance or energy from the turbulence. From (4.9) we have: (4A.1) ν u u u u. x ν j x ν = j x j In the dscusson of the two terms we can argue that the frst term s the second dervatve of the varance of u. Snce the varance only vares slowly n space and tme n the boundary layer ths term s much smaller that the second term, whch s the vscous dsspaton. That the two terms are dfferent can be best llustrated usng the three dmensonal scalar spectrum, ntroduced n the dscusson of scales and spectra. The velocty feld can be Fourer expanded (see dscusson n secton ) kr (4A.) u(,) r t = e dz( k,) t Lectures n Mcro Meteorology 77

79 (4A.3) * ( k,) ( k,) ( k,) j j S t = dz t dz t For the sotropc turbulence approxmaton we can defne the scalar three-dmensonal power spectrum, E(k), beng only a functon of the length of the wave number vector, k, not ts orentaton. (4A.5) Ek ( ) S( k) dk, kk= k (4A.6) = = uu ( u u u ) E ( k ) dk Hence n spectral form, the frst rght hand sde element n (4.49) s: (4A.7) u ν E( k) dk ; = ν xj xj whch s just shows that ths term s qute small because the varance changes lttle for homogeneous turbulence. To derve the second term we start by: (4A.8) (,) k r u r t = k e dz (,). t j x k From ths follows: (4A.9) ν j k r k r * ( u (,)) r t = ν( k e dz (,))( t k e dz (,)) t j j x k k j k k ν j = ν k S ( ) d = k E( k) dk. Relatve to the varance, the vscous dsspaton term s seen to have much less contrbutons from the small wave numbers and much larger contrbutons from the large wave number regons of the spectrum. We recall the descrpton of the three wave number regons of E(k), from secton Lectures n Mcro Meteorology

80 Fgure 4A.1. Schematc energy, or varance, spectrum for the atmospherc boundary layer, shown wth the three characterstc spectral regons (Tennekes and Lumley,197). We can separate the power spectrum nto three regons as shown on the next fgure, the producton range, wth k 1/Λ, where energy s extracted from the mean profle, a dsspaton range, where the flud moton s dsspated by vscosty, for k > η (ν 3 / ε) 1/4, whch for typcal atmospherc flows s about 1 mm. η s called the Kolmogorov dsspaton scale and s a combnaton of vscosty and dsspaton as seen. In between there s a regon, where the spectrum depends only on the wave number and the dsspaton. Ths regon s called the nertal sub-range. Snce the spectrum descrbes wnd varance per wave-number ncrement, t has the dmenson: m 3 /sec. Dsspaton s destructon of varance by vscosty; hence t has the dmenson of varance per second, or m /s 3. Fnally, for the nertal sub-range the spectrum for the dsspaton looks lke: /3 + 1/3 (4A.10) Ε ( k) = kek ( ) = αε k To model the enhanced destructon of turbulence n the dsspaton range for k>η, the nertal subrange form of (4.57) s multpled by an exponental form that generally s represented as (e.g. Larsen and Højstrup,198) /3 + 1/3 4/3 (4A.11) Ε ( k) = kek ( ) = αε k exp( nk ( η) ), whch s the Kolmogorov dsspaton spectrum wth n beng a constant of order one. A dsspaton spectrum s plotted together wth other spectra n Fgure. It s obvous from ths descrpton that the dsspaton takes manly place at hgh wave numbers, whch s n accordng wth the here presented spectral pcture. Also the dscusson shows that that the second term n (A.1) s much larger and of completely dfferent nature from the frst term. Lectures n Mcro Meteorology 79

81 5. The Ekman boundary layers For convenence we repeat the mean value equatons: du u u 1 p u = + u = gδ Ω ε η u + ( ν uu ); j 3 jk j k j dt t x ρ x x x j j j (5.1) dθ θ θ θ LE 1 Q = + u = ( k u θ ) j T j dt t x x x ρc ρc x j j j p p j dq q q q E = + u = ( k uq ) + ; j q j dt t x x x ρ j j j * j Neglectng molecular terms, radaton term and E-terms, we fnd for horzontally homogeneous statonary flow wth zero vertcal mean velocty. The three momentum equatons. (5.) 1 p 0 = + f u ( uu ) ρ x C x3 1 p 0 = f u + ( uu ). ρ x C 1 3 x3 1 p 0= g Ω( ηu η u) uu. ρ x The scalar equatons : x3 0 = ( u θ );0 = ( uq ); x x3 For such smple condtons t s seen that the scalar fluxes are constant wth heght. At the top of the boundary layer the turbulence terms dsappear, by defnton, hence we see that the pressure terms and the Corols terms must balance, gvng rse to the so called Geostrophc wnd, gven by: 1 p 1 p 1 p 1 p (5.3) G = ( U, U ) = ( u, v ) = (, ) = (, ); 1G G g g f ρ x f ρ x f ρ y f ρ x C C 1 C C where t s ndcated that we can use several notatons for the same varable. See Fgure Lectures n Mcro Meteorology

82 Fgure 5.1. Schematcs of the Geostrophc balance between pressure gradent force and Corols force, denoted PGF and CF. We start consderng only the two horzontal equatons, whch we can now wrte: 0 = f ( u U ) ( uu ) C G 1 3 x3 (5.4), 0 = f ( u U ) ( uu ). C 1 1G 3 x These two equatons descrbe the so-called Ekman spral wnd profle. Dervaton of the spral: We have the two equatons for the horzontal wnd components. 0 = fc( u UG) ( uu 1 3) x3 (5.5) 0 = fc( u1 U1 G) ( uu 3). x 3 3 Fgure 5.. Wthn the boundary layer the wnd s seen as a balance between the pressure, the Corols and the frctonal forces, denoted CF and F respectvely. So far, we have not consdered the drectons of the horzontal axes. We now fx them such that x 1 s along the Geostrophc wnd. Changng also to the (x,y,z) and (u,v,w) form for the varables, the equatons take the form: Lectures n Mcro Meteorology 81

83 (5.6) 0 = fcv ( uw ) x 3 0 = fc ( u G) ( vw ). x 3 For smplcty, we now use a gradent dffuson for the turbulence terms, wth constant dffusvty. In later sessons we shall consder more realstc closure schemes.: u u fcv = ( uw ) = K = K z z z z (5.7) v v fc ( u G) = ( vw ) = K = K. z z z z Combnng to the complex varable, and multplyng the second equaton by, we get: W = u + v; (5.8) v fc ( u G) = K z Addng the two last equatons yelds, snce 1/(-) = : W 1 (5.9) K = f ( ) ( ) ( ) C u G + fcv = fc u + v G = fc W G z or: W (5.10) K f ( ) 0, C W G = z Ths s a smple second order dfferental equaton wth constant coeffcents. Hence, the soluton can be wrtten: W G= Aexp((1 + Z ) ) + Bexp( (1 + Z ) ); (5.11) 1 Z = z/ h ; h = ( K / f ) ; = u fcv K z E E C where we have ntroduced the Ekman heght scale, h E. The numercal value of the Corols parameter s nterestng only f one wants to work on the Southern Hemsphere. The boundary condtons are: (5.1) W = 0 for z = 0; W G for z ; The last of these means that A = 0. The frst then means that B = -G. Hence, the soluton of W looks as: Z (5.13) W G = Ge (cos( Z) sn( Z) sgnf C ). Gong back to the two components, we get: 8 Lectures n Mcro Meteorology

84 (5.14) u = G e Z = G e z h Z z/ he (1 cos( )) (1 cos( / E )); v = Ge Z sgnf = Ge z h sgnf Z z/ he sn( ) sn( / E) C. Snce, n ths coordnate system, u s along the Geostrophc wnd, we can fnd the angle between the surface wnd and the Geostrophc wnd, θ. (5.15) dv Z Z v ( cos( ) sn( )) tan lm lm dz e Z e Z sgnfc θ = = = lm = sgnfc ; z 0 z 0 z 0 Z Z u du e cos( Z) + e sn( Z) dz The result θ 45 s not correct, for neutral homogeneous condton t s closer to 0. But some results hold: Lookng along the vectors, the surface wnd s to the left of the Geostrophc wnd, and further the angle s ndependent on the sze of f C. Under all crcumstances, t vares much wth other condtons lke stablty, baroclnty and other types of heterogenety. Below, we show a three dmensonal dagram of the turnng of Ekman spral on the Northern Hemsphere. Fgure 5.3. Ekman spral on the Northern Hemsphere (Larsen and Jensen, 1983), showng the pressure gradent force, P, the frctonal force, F, and the Corols force, C, from the top of the boundary layer and down towards the surface. Next we consder a number of specal aspects of the above dervaton. The turbulence dffusvty: Above the Ekman heght, h E, was defned from the Corols parameter and the turbulence dffusvty. We know h E typcally to be of the order of 500 meter n the atmosphere, hence we can deduce the magntude of K. (5.16) ( / ) E = C = C E 15. h K f or K f h m s Comparng wth the correspondng knematc molecular dffusvty, ν m s -1, t seen that turbulence dffuson s about a mllon tmes more effcent than molecular dffuson for the same gradent. Therefore, t s often justfed to neglect the molecular terms n the equatons compared to the turbulence term. Compare the mean value equatons for the boundary layer, Lectures n Mcro Meteorology 83

85 where neglect of the molecular parts of the flux dvergence terms wll almost always be justfed by the comparson above. : du u u 1 p u = + uj = gδ3 Ω εjkηjuk + ( ν uu j ); dt t x ρ x x x (5.17) j j j dθ θ θ θ LE = + uj = ( kt u jθ ), dt t x x x ρc j j j p dq q q q E = + uj = ( kq uq j ) + ; dt t x x x ρ j j j Surface characterstcs of the Ekman layer Above we have seen that the u and v parts of the Ekman layer mean wnd speed profles converge toward same values for z 0. We shall now consder n more detals the behavour of the Ekman layer close to the surface. We start wth the total stress: (5.18) 1 = + xz yz ρu τ ( τ τ ) ; It s seen that the two components of the stress can be consdered as vectors, attackng a horzontal plane along the x and y drectons, respectvely. Ths means that we can combne these two components as vectors. Note further, that we have ntroduced a scale velocty for the turbulence, u, denoted the frcton velocty. The stress components are computed as: τ xz u τ (5.19) ; yz v = uw = K = vw = K ; ρ z ρ z. We can now fnd the total surface stress as defned above, from the Ekman profle and the relatons above. We use subscrpt 0 to denote values at the surface. : 1 1 u v (5.0) u*0 = (( uw ) 0 + ( vw ) 0 ) = (( K ) 0 + ( K ) 0 ), z z The wnd profles were dfferentated when we determned the angle between the Geostrophc wnd and the surface wnd (5.15). Usng these results we get (5.1) u = GhE fc *0 / Ths shows the relaton between the surface stress, the Geostrophc wnd and characterstcs of the Ekman profle, namely, f C and h E. 84 Lectures n Mcro Meteorology

86 When studyng the behavour closest to the surface t s most convenent to rotate the coordnate system, such the x axs s along the surface mean wnd, u, rather than along the Geostrophc wnd. Agan we have z, w beng vertcal and y, v beng the horzontal component perpendcular to x, u. y G x s Ekman Sprall y s θ x G Fgure 5.4. Relaton between coordnate systems algned wth the Geostrophc wnd and wth the surface wnd. - The general formulaton for the Ekman spral was: 0 = fc( v VG) ( uw ) z (5.) 0 = fc( u UG) ( vw ) z In the coordnate system wth x along the mean wnd, <v> =0, because u s algned wth the mean wnd. Hence, the equaton reduces generally to: (5.3) ( uw ) = fcvg; z ( vw ) = fc( u UG); z Integratng from z = 0 to z, we obtan: (5.4) ( uw ) = ( uw ) f V z= ( uw ) + f V z; z 0 C G 0 C G z ( v w ) = ( v w ) + f ( U u ( z )) dz f ( U u ( z )) dz, z 0 C G C G 0 0 z where we have ntroduced that f C V G s negatve n ths coordnate system and that the v-w cross covarance s close to zero at the surface. Ths equaton s rather general wthout closure assumptons etc. In the surface wnd system, we would expect that at the surface all the surface stress was along the mean wnd carred by the u-w-cross-co-varance, whch would then equal u o, whle the v-w-cross-co-varance would be vary close to zero close to the ground. We can check f ths s consstent wth the Ekman soluton, by turnng the coordnate system, as gven n the fgure above. We get: Lectures n Mcro Meteorology 85

87 (5.5) u = u cosθ + v sn θ; s g g v = v cosθ u sn θ. s g g π θ. 4 where, we denote the wnd n the surface system by subscrpt s and the wnd n the Geostrophc system by subscrpt g. From the Ekman spral equatons we now get: (5.6) du du dv uw = K = ( K cosθ + K sn θ) = s dz dz dz s g g KG Z Z Z Z = (cos θ( e cos( Z) + e sn( Z)) + sn θ(( e cos( Z) e sn( Z))) = h E 1 Z = fchg E (( e cos( Z)) fchg E / = u*0 forz 0. smlarly dv dv du vw = K = ( K cosθ K sn θ) = s dz dz dz E s g g KG Z Z Z Z = (cos θ( e cos( Z) e sn( Z)) sn θ(( e cos( Z) + e sn( Z))) = h 1 Z = fcheg(( e sn( Z)) 0 for z 0. Where, we have utlzed the varable Z = z/h E, and also used the results from above relatng the Ekman heght and the dffusvty, K. The sgn of the Corols parameter s neglected here. It s seen that n the surface wnd also the Ekman soluton yelds that the total surface stress s carred by the u-w cross-co-varance, whle the v-w-cross co-varance at the ground s zero. We notce that the earler ntroduced surface stress scale, u o, equals (-<u w >) o ½ n ths coordnate system. By frst order expanson, the equatons for the surface stress can now be wrtten n terms of heght varaton for u. Ths makes t somewhat easer to evaluate the mportance of the heght varaton, because we can now consder the relatve varaton of u / u o : (5.7) ( uw) z ( uw) 0 fcvg z; or: u z u o fcvg z; or = + = + f V z V z z u u (1 ) u (1 ) u (1 ); h C G G z o o o u o Gh E E ( v w ) u f ( ) dz u (1 ) z z UG u ( z ) u( z ') dz z = o C = o u u 0 o o U 0 G he 86 Lectures n Mcro Meteorology

88 In the equaton for u, we have gradually ntroduced approxmatons from the Ekman spral, frst through the expresson for the total surface stress, second assumng θ 45. For z/h E less than about 0% we see that u s wthn 10% of u o and t reduces approxmately lnearly. In the same z- nterval <v w > ncreases from about zero and up slghtly slower than proportonally to z/h E. We can further see that although the stress may vary across the layer, the surface value u o wll characterze not only the magntude but also the changes across the layer, and the changes wll be small relatve to u o. Ths layer closest to the surface s called the surface layer. It s characterzed by that all fluxes can be consdered constant wth heght, snce we saw n the begnnng of ths secton that the scalar fluxes were ndependent of heght. Although we have used the smple Ekman spral somewhat to derve ths concluson, we can also see that the exstence of the surface layer s more general than the Ekman soluton for the whole boundary layer. We shall return qute ntensvely to ths layer. In the coordnate system wth the surface wnd along the x-axs, the Ekman spral looks as seen below. Notce, the heght of the boundary layer has been approxmated by u o /f C. Fgure 5.5. The Ekman spral seen n a coordnate system algned wth the surface mean wnd. Notce that the boundary that z s normalzed wth u /f c rather than h E. Ocean current Ekman spral. Before leavng the Ekman layers, we shall see how an atmospherc Ekman layer above an ocean surface drves and ocean Ekman layer below, snce ths was really the way t was frst derve by Ekman n 1905, confrmng observatons of drftng ce by Nansen a few years earler, In the ocean we can neglect the Geostrophc wnd from the equaton, whch then looks lke: 0 = fcv ( uw ) z (5.8) 0 = fcu ( vw ) z Notce, we keep the same drecton of the z-axs. Ocean depths s then for z -. The equaton s reformulated wth dffusvty, whch we for water gve the subscrpt w, K w. Lectures n Mcro Meteorology 87

89 u 0 = fcv + Kw (5.9) z v 0 = fcu Kw ; z As for ar we defne a complex varable, W, and as for the ar we derve the equaton to be: W u + v; (5.30) W fc W = 0. z K Followng the procedure from the atmosphere we have: W= Aexp((1 + Z ) ) + Bexp( (1 + Z ) ); (5.31) 1 Z = z/ h ; h = ( K / f ) ; Ew Ew w C w where the water Ekman depth has got a subscrpt w, lke the dffusvty. Snce, W 0 for z -, t follows that B=0. We can now wrte the component equatons from W. (5.3) Z W = ( A + A ) e ( cosz + sn Z); r Z u = e ( A cosz A sn Z); r Z v = e ( A cosz + A sn Z); r The boundary condtons at the water surface derve from contnuty of stress at the surface. We formulate the stress contnuty as: u u 1 ρwkw = ρaka = ρaghe fc; z 0 z 0 (5.33) v v 1 ρwkw = ρaka = ρaghe fc, z 0 z 0 where the surface stress values on the atmospherc sde was derved earler. Dfferentatng on the ocean sde we get. u u Z 1 Z = = e ( ArcosZ A sn Z) = z Z z he Z Z e (5.34) {( ArcoZ A sn Z) + ( Ar sn Z A cos Z)} he 1 ( Ar A ) for Z 0, and smlar for v h E 88 Lectures n Mcro Meteorology

90 (5.35) Ths solves to: (5.36) u K 1 ρ K = ρ ( A A ) = ρ Gh f ; w w w w r a E C z 0 hew v K 1 ρ K = ρ ( A + A ) = ρ Gh f, w w w w r a E C z 0 hew A = 0; A r 1 1 ρaghe fc ρaghe fc ρah = = = K 1 ρ h ρ fh w h where we have used expressed Kw n terms of h E and f C. w w Ew c Ew Ew ρw hew E G; Hence, we arrve at: (5.37) ρ h u z = Ge z h a E z/ hew ( ) cos( / Ew ), ρwhew ρ h v z = Ge z h a E z/ hew ( ) sn( / Ew ); ρwhew Frst of all the equaton shows that the surface current runs along the Geostrophc wnd, not the surface wnd or for that matter the surface stress. It was to explan observatons of ce floe ndcatng somethng lke that, that Ekman dd hs dervaton. As the depth ncrease the current veers toward rght. Notce, that we have to derve h Ew from other consderatons, or for that matter K w, just as we n the atmosphere used observatons for h E. We would expect the depth of the Ekman layer to be assocated wth the strength of the turbulence mxng. Due to the stress contnuty over the surface we have that u w u a (ρ a /ρ w ) ½. Takng u as ndcator of the turbulence strength we would assume that a smlar rato would relate h Ew and h E, and that h Ew h E (ρ a /ρ w ) ½, whch s approxmately true. Then we further see that the surface current relates to the Geostrophc wnd through the same rato, beng roughly 1/30. The followng fgure llustrates the characterstcs of the two Ekman sprals. Here one could also notce that the surface stress n the atmosphere s along the surface wnd. The surface stress at the ocean surface s 45º of the surface current. Lectures n Mcro Meteorology 89

91 Fgure 5.7. Coupled ocean atmosphere Ekman sprals (Stull, 1991) Appendx 5A. The Baroclnc Boundary Layer and the Thermal Wnd. The pcture of the boundary layer can be modfed by many knds of heterogenety and mproved parameterzatons. A smple modfcaton arses when one consders the mportance of horzontal temperature gradents, as wll often be present n realty. So far, we have only allowed for horzontal gradents of pressure to drve the flow. To feed n the nformaton about the changng horzontal temperature, we use the so-far unused equaton for the vertcal component. 1 p 0 = g Ω( ηu η u) uu (5A.1) ρ x3 x3 Dfferentatng ths wth respect to the horzontal coordnates, = 1, we obtan. 1 p 1 p 0 = ( g Ω( ηu η u) uu ) =. (5A.) x ρ x3 x3 x ρ x3 Assume now that ρ vares horzontally due to the horzontal temperature varaton. Dfferentatng by parts, we get: 90 Lectures n Mcro Meteorology

92 (5A.3) 1 p p 1 1 p = + = x ρ x x x ρ ρ x x ( ρg) ρ 1 1 p = + ρ = x x x ρ ρ ρ 3 g T 1 p 1 ρ 1 p = + + = T x x ρ x ρ x ρ x 3 3 g T 1 p 1 1 p = + = 0; T x x ρ x H ρ x 3 where, frst we have used the hydrostatc equaton n the frst term and reversed the order of dfferentaton n the second term. The horzontal varaton of ρ s assumed domnated by the temperature varaton as usual. Thereafter, we have multpled and dvded by ρ n the second term and dfferentated by parts, and fnally used the hydrostatc balance agan, ntroducng the scale heght of the atmosphere, H. Now expressng the horzontal pressure gradents as the Geostrophc wnd components, we obtan, for = 1, : 1 g T ( fu ) fu =, C G C G x H T x 3 1 (5A.4) 1 g T ( fu ) + fu = ; C 1G C G x H T x 3 We wll here neglect the second term, because we at most ntegrate to the Ekman- h E, beng of the order of 3-5% of the scale heght. Hence, we can derve a vertcally varyng Geostrophc wnd due to a horzontal temperature gradent:, g T (5A.5) U1 G U1 G0 x3= U1 G0 A x3; ft C x g T UG UG0 + x3 = UG0 + A1 x3, ft x C 1 where the ntegraton constants are the surface Geostrophc wnd. One can now nsert these expressons for the Geostrophc wnds nto the expresson for the Ekman spral equatons. (5A.6) u 0 = fc( u U Ax ) K x G = + + fc u1 U1 G0 Ax 3 K x 3 0 ( ). 1 u Lectures n Mcro Meteorology 91

93 Solutons are show on the next fgure, where the arrows show dfferent baroclnc forcng. In the solutons presented the x 1 axs s placed along the unperturbed Geostrophc wnd. Fgure 5A.1 Behavor of the Ekman spral for four examples of baroclnc forcng all wth the same magntude, ndcated wth the unt-length arrow. The drecton of the arrow refers to the sgn of the A 1 and A terms n the defnton of the thermal wnd that s to the sgns of the two horzontal temperature dervatves n the defnton of the A-terms.. From the top to the bottom the sgn of the forcng s: (A 1, A ) = (0,-1 ), (0, +1), (+1, 0), (-1,0). We note that the modfcaton of the Geostrophc wnd, and thereby also the wnd tself, s largest at the top. Comparng wth the defnton the resultng change s seen to be qualtatve reasonable. For example, n the lowest fgure A 1 = -1 and A = 0, meanng that we must expect lttle or no changes n u1, whle U G and u wll be modfed n negatve drecton (NO Jensen, personal communcaton). 9 Lectures n Mcro Meteorology

94 All examples have the same strength of baroclnc forcng, but the drecton of the arrows shows the sgn of A-terms of the correctons. Vertcal arrows, means only A (A reflect the horzontal temperature gradent n the x drecton) s dfferent from zero, wth sgn gven by the drecton of the arrow, smlarly for horzontal arrows. In the fgure we have chosen the magntude of the two A-terms to be A m/s/m, correspondng to a horzontal temperature gradent of about 0,05 C/km. From the fgure, we can conclude that the shape of spral s ndeed senstve to the thermal wnd. However, we see as well that the angle between the Geostrophc wnd and the surface wnd shows lttle senstvty to ths knd of forcng. Ths concludes our dscusson of the atmospherc Ekman boundary layer. For a horzontally homogeneous statonary boundary layer, we have seen that use of constant turbulence dffusvty allow us to derve the so-called Ekman spral soluton for the varaton of the mean wnd wth heght. We have estmated the dffusvty needed to be n accordng wth data and found t to be about one mllon tmes larger than ts molecular brother, the cnematc vscosty. We have been able to nclude horzontal varaton of the mean temperature and found that ths does modfy the form of the Ekman spral consderably. Indeed, although the Ekman spral s found n the data, t s found comparatvely rarely, ndcatng that boundary layers, smple enough to justfy the used smplfcatons, are somewhat rare. As expected the assumpton of a constant K-value s too smple, as we wll see a heght dependent K fts better the data. Also thermal stablty can be bult n to K. Now however, the equatons have to be solved numercally. Notce, that n our soluton of the Ekman spral, we have not avoded the closure problem. We have had to specfy K from other nformaton than what s n the mean value equatons themselves. We use nformaton about the h E, the Ekman heght, whch was based on measurements. Fnally we have dentfed a surface layer wthn the meters closest to the surface. In ths layer one could wth good approxmaton assume that all fluxes were constant wth heght. Lectures n Mcro Meteorology 93

95 6. The atmospherc surface boundary layer. Monn-Obuchov scalng In the dscusson of the Ekman layer, we have seen that the lowest part of the layer s charactersed by havng fluxes that vares lttle or not at all wth heght. Ths means that for horzontally homogeneous statonary flow, the lowest part of the boundary can be characterzed usng the surface fluxes plus maybe a few more varables. The use of dmensonal analyss and normalsng varables wth characterstc parameters has and long record wthn flud dynamcs. It generally nvolves some guesswork as to what are the mportant parameters, some expermental data to use as gudance and some theory as to how one should go about t, and fnally and very mportant expermental valdaton of the derved formulatons. In the earler chapters, we have seen smple dmensonal analyss, when the turbulence dffusvtes were establshed, as beng product of a characterstc velocty parameter and a characterstc length- scale. (6.1) K α v, where v s the relevant velocty scale, s a relevant length scale, and α s a dmensonless numercal constant, whch should be of order one, f the most relevant characterstc parameters have been chosen. As we found for the turbulence dffusvtes, there wll often be several dfferent sets of possble characterstc parameters. The expermental verfcaton s very mportant. The am of the followng analyss s to descrbe as well as possble the mean values, ther vertcal gradents, and the turbulence characterstcs for a statonary horzontally homogeneous atmospherc surface boundary layer. Such a surface boundary layer can be sad to be the smplest non-trval boundary layer one can magne. We shall use the equatons for mean values and the turbulence varance to map the problem. A general verson of the mean value equatons was wrtten on our way to the Ekman spral. 94 Lectures n Mcro Meteorology

96 (6.) The three momentum equatons. 1 p 0 = + f u ( uu ) ρ x 3 3 C x3 1 p 0 = f u + ( uu ). ρ x C 1 3 x3 1 p 0 = g Ω( ηu η u) uu. ρ x The scalar equatons : 0 = ( u 3θ ) x 0 = ( uq 3 ); x x3 Introducng the geostrophc wnd and turnng the coordnate system along the surface mean wnd, and changng to x,y,z coordnates we obtan: (6.3) The three momentum equatons. ( uw ) = fcvg; z ( vw ) = fc( u UG);. z 1 p 0 = g + Ωηyu w ρ z z The scalar equatons : 0 = ( w θ ) z 0 = ( wq ); z We can use the frst two of these equatons to derve the rate of change of the surface stress wth heght, usng the result to defne the surface layer, as the layer through whch the surface stress dd not change too much wth heght, through the followng equaton (remember V G s negatve when f c s postve). Lectures n Mcro Meteorology 95

97 (6.4) ( uw) z ( uw) 0 fcvg z; or: u z u o fcvg z; or f V z V z z u u (1 ) u (1 ) u (1 ); h C G G z o o o u o Gh E E ( v w ) u f ( ) dz u (1 ) ; or = + = + z z UG u( z ) u( z ) dz z o C = o u u 0 o o U 0 G he z and v w for << h u z u o ( ) z / u o 0 1. E In the last part of (6.4) we have used the results and notaton from secton 5 about the Ekman spral, thr Ekman heght, h E, and the frcton velocty, u., and that t close to the surface changes slowly wth heght from ts surface value, u o. Other fluxes of mportance are <w θ > and <w q > that control the fluxes of heat and humdty through the layer. As we have specfed the layer, we have <v > = 0, hence the Corols parameter s unlkely to be mportant, because t controls the cross talk between the equatons for the two horzontal veloctes. From the equatons, t looks, as the Corols parameter s mportant only for the rate of change of the stress through the surface layer. There s no obvous use of the equaton for the mean vertcal velocty, snce we have assumed that to be zero. All consdered we fnd lttle addtonal use for the mean value equatons for our (strongly) smplfed surface layer flow. We now proceed to the varance equatons derved n our effort to evaluate the dfferent closure possbltes for the equatons n Mcro Scale Meteorology 4. We use the notaton: e= u, as the total fluctuatng turbulence varance. The general equatons look as follows, from secton 4. (6.5) 1 e 1 e gu ( θ ) u 1 ( ue ) 1 ( u p ) + u = ε; t x x x x 3 v j j j uu j j θv j j ρ j θ + = ε 1 θ 1 θ θ 1 ( u j ) uj θ u j t xj xj xj 1 q 1 q q 1 t x x x ( uq j ) + uj = qu j εq j j j θ We now mpose the statonarty and horzontal homogenety demands of our smplfed surface layer, and turn the coordnate system along the mean surface wnd. 96 Lectures n Mcro Meteorology

98 (6.6) 1 de gw ( θ v ) u 1 ( we ) 1 ( wp ) = 0 = uw ε; dt θ z z ρ z v = 0= θ w ε dt z z 1 dθ θ 1 ( wθ ) = 0= qw ε q dt z z B P T T D 1 dq q 1 ( w q ) t p θ The budget equatons for the turbulence varance as derved n secton 4. They are here smplfed to the condtons n a smple statonary horzontally homogeneous surface boundary layer. The terms are denoted: B for Buoyancy, P for Producton, T t for flux dvergence of turbulence flux of varance, T p for flux dvergence of pressure nduced transport of varance, and D for Dsspaton. Also for these equatons we note that the Corols parameter does not enter, meanng that t s unmportant for our problem. However, the equatons nclude relatons between many profle quanttes and turbulence quanttes. We shall now show examples on how the dmensonal analyss can be used to relate the dfferent quanttes to each other for smple subsets of the equatons above. Frst, we wll consder only the velocty profle wthout temperature effects. If we neglect most of the terms n the frst equaton t takes the form: (6.7a) u u ε = z * 0 The dsspaton just follows the local energy producton n the frst term. The equaton underlnes also the nteracton between u and the velocty profle. Hence t argues for that u should be the velocty scale for the gradent. The profle must obvously also depend on z. Assumng that the wnd profles depends only on u * and z, we can normalse (6.7a) to obtan: (6.7b) z u zε 1 = =, u z u κ 3 * * where κ s a dmensonless unversal constant called the v Karman constant. It s a unversal constant, because t can only depend on u * and z from our assumpton, and no dmensonless argument can be made from these two varables. κ s expermentally found to be around 0.4. It s seen that a consequence of (6.7b) s that the wnd speed vares logarthmcally wth the heght, z. u (6.7c) uz ( ) u0 = ln( z/ z0), κ Lectures n Mcro Meteorology 97

99 Where the roughness length s ntroduced as an ntegraton constant, and as the heght where the wnd speed becomes equal to u 0. At the surface we normally take u 0 ~0. To formulate ths dervaton n more formal way, one can use the dmensonal analyss, denoted Bucknhams P theorem, descrbed by Jensen and Busch (198) n the appendx. Fnally, we can refer to that the resultng logarthmc behavour of the wnd above the surface s one of the oldest and best establshed results from wnd tunnels, where t s denoted the law of the wall. Also for the near surface atmosphere, the logarthmc behavour of wnd profles s well known. Havng found a soluton to the wnd profle, we can now consder how the 3 varance equatons wll behave when beng normalsed by proper parameters. From the frst results above, the constants and parameters n the equatons we defne the followng characterstc parameters to make the terms non-dmensonal. Notce, we obvously need a humdty scale and a temperature scale, snce we nclude both temperature and humdty equatons. (6.8) u uw ; uθ θ w ; uθ θ w ; uq qw ; * * * * * v v * * g/ θ ; z; Note that θ v enters only n the Buoyancy term. We have converted the humdty flux and the temperature flux to a water vapour and a temperature scale, usng u. The mnus sgn has a ratonal explanaton for u (that the momentum transport s downwards), for the other parameters the reason s just hstorcal, and to make the producton terms postve. We can now make the varance equatons non-dmensonal, usng the parameters above. (6.9a) 1 de gw ( θ v ) u 1 ( we ) 1 ( wp ) kz = 0 = uw ε; dt θ z z ρ z u v dθ θ w θ kz = 0 = θ w εθ, dt z z u θ 1 1 ( ) d q q w q kz = 0 = qw εq, dt z z uq 1 1 ( ) B P T T D t p 3 Multplcaton of the equatons above by the factors at the rght hand sde makes all terms n the equatons non-dmensonal. Note, we have followed tradtons by ncludng the non-dmensonal von Karman constant, k n the normalsaton group. We wrte the resultng equatons as: 98 Lectures n Mcro Meteorology

100 (6.9b) kz 1 de z z z z z z z = 0 = + ϕ ( ) ϕ ( ) ϕ ( ) ϕ ( ); z z ( ) ( ) L L 3 m Tt Tp ε u dt L L L L L L L kz d z z z z 1 θ = 0 = ϕ ( ) ϕ ( ) ϕ ( ), z ( ) L θ Tθ εθ u θ dt L L L L kz d q z z z z 1 = 0 = ϕ ( ) ϕ ( ) ϕ ( ), z ( ) L B P T T D q Tq ε q u q dt L L L L t p Below the ndvdual functons are derved and dscussed. The buoyancy term B, z/l, where L s the so-called Monn-Obuchov stablty length scale: (6.10) kz gw ( θ ) z z z = = u u u v 3 3 θ L v θv θv kg ( w θ ) v kg θ v It s seen that L ± for neutral condton, meanng for vanshng heat flux. Neutral condtons then means that z/l = 0. L s negatve for unstable condtons and postve for stable condtons. The producton terms, P, are derved as: (6.11) z kz u kz u ϕm ( ) = ( uw ) = ; 3 L u z u z z kz θ kz θ ϕθ ( ) = ( θ w ) = L u θ z θ z z kz q kz q ϕ ( ) = ( qw ) = q L uq z q z, where we assume that v. Karman's constant s the same for all varables. The flux dvergence for turbulence transport of varance, T t : Lectures n Mcro Meteorology 99

101 (6.1) kz1 ( we ) k ( we ) z z = z ϕ ( ); z ( ) L 3 3 Tt u z z u L L kz 1 ( wθ ) k ( wθ ) z z = z ϕ Tθ uθ z z uθ L z L ( ) L ( ); kz 1 ( wq ) k ( wq ) z z = z ϕ ( ); Tq uq z z uq L z ( ) L L z Note : z = ; z L z ( ) L The dvergence of the pressure transport term, T p, s done smlarly: kz1 ( wp ) k( wp ) z z (6.13) = z ϕ ( ); 3 3 Tp u z z u L z L ( ) L Fnally the dsspaton terms, D: (6.14) z kzε z kzε z kzε ϕ ( ) ; ϕ ( ) ; ϕ ( ) ; θ q ε = 3 εθ = εq = L u L u θ L uq Optmstcally, we have wrtten all the ϕ- functons as functon of z/l and only z/l. How relable s ths? In the appendx, the dmenson analyss above has been carred out for ϕ m (z/l) based on the momentum equaton, where the flux dvergence terms have been neglected. Therefore, we would expect also the smlar analyss to hold for ϕ h (z/l) and ϕ q (z/l), and therefore also for all the dsspaton terms. Addtonally, we see that z/l s the only free varable n the momentum equaton, ndcatng that at least other types dependences n the ϕ- functons must cancel, when these functons are added as prescrbed by the equaton or at least be of less mportance. However there are ndcatons that the flux dvergence terms do depend also on other parameters (Ellot, 197), manly the boundary layer heght, so that the M.O. scalng can be consdered an approxmaton only, although a farly good approxmaton. Ths s underlned also from the fact that the v Karman constant, a constant wthn the framework of the Monn-Obuchov smlarty, emprcally seems to show a tendency towards a week dependency on external parameters, lke the surface roughness number, z 0 u /ν (Larsen, 1993), but the statstcs s uncertan. Under all crcumstances, we noted above that the results of a dmensonal analyss must be consdered lke a hnt about where to look, rather than a truth derved from frst prncple, and expermental valdaton wll always be necessary. An approach based on the Bucknhams PI theorem for dmensonal analyss of the surface layer turbulence s presented n Jensen & Busch (198), see appendx 6A. Here s llustrated that t s sometme successful n explanng the expermental data, sometme not. The later result s usually taken as ndcaton of that one has somehow guessed wrongly for the mportant varables. Not all ϕ- functons have been equally well studed. Most comprehensve work has been conducted on the mean-profle expressons, followed by the dsspaton terms. The flux 100 Lectures n Mcro Meteorology

102 dvergence terms have been relatvely lttle studed, especally the one nvolvng pressure, mostly because of the large dffcultes n measurng the pressure fluctuatons properly. In a followng secton, we shall see that the Monn-Obuchov scaled surface layer s but one scalng regon n the atmospherc boundary layer. It can be consdered correct n an asymptotc way when the nfluence of the other scales have vanshed. Untl then we shall just consder t an useful approxmaton. The expermental campagns started n the mddle of the 1960, where one got nstruments that could resolve all of the three velocty components, the so-called sonc anemometer thermometer. The next fgure 6.1 shows the set-up of the frst of these. The setup on the fgure shows how one could measure the mean temperatures and velocty gradent. The soncs provded fluxes and turbulence varances. Addtonally, fast response hot-wre anemometers could provde the wnd dsspaton. Also other nstruments were operated as seen on the fgure. As seen the soncs operated at three levels, meanng the assumpton of constant flux wth heght could be valdated. Further t was possble to estmate many of the ϕ- functons at 3 levels, ϕ m (z/l), ϕ h (z/l), ϕ ε (z/l), and ϕ Tt (z/l). By plottng these functons versus z/l t was possble to evaluate f the functons could be consdered functon of z/l only, or they stratfed accordng to the dfferent measurng heghts. Humdty was not measured. Also varances and spectra were evaluated as functon of z/l. Bascally, the Monn-Obuchov smlarty was found to provde good approxmatons for these data and data from subsequent experment. The followng two fgures show data from such experments, ths tme conducted n Sweden. Fgure 6.1. Dagram of the expermental set-up durng the Kansas 1968 surface layer experment (Izum, 1971). The frst comprehensve expermental test of the Monn-Obuchov smlarty hypotheses (Bush et al, 1973, drawng:c. Kamal personal communcaton). Lectures n Mcro Meteorology 101

103 We shall now proceed to dscuss detals and consequences of some of the M.O. formulatons.: The stablty The stablty s descrbed by the parameter combnaton: z/l, wth z/l = 0 denotng neutral. From the defnton of L t s seen that t s a measure of the relatve strength of the mpacts on the turbulence from the dynamc part, through u, and from the thermal part through θ. The fact that all the stablty s descrbed by the rato between z and L means however that stablty s both a functon of the relatve strength between the thermal and the dynamc forces n the atmosphere and of the measurng heght, meanng that rrespectve of the overall atmospherc condtons, close to the ground the stablty s neutral. Profle Expressons. We shall now consder the forms for the mean profle for ϕ m (z/l), ϕ θ (z/l), ϕ q (z/l). The three functons all looks pretty smlar. The typcal form s llustrated on the fgure below. The functonal forms of the ϕ- functons can be summarsed (Panofsky and Dutton, 1989, Högström, 1990): (6.15) z z z z ϕ( > 0) ; ϕ( < 0) (1 α ) L L L L 1 1 ϕm: n, α 15. ϕθ, ϕq: n, α 10 4 n ϕ (z/l) z/l 1,0 z/l Fgure 6.. Characterstc behavour of the profle functons wthn the Monn-Obuchov surface layer smlarty. The expresson for the dsspaton functons do qualtatvely follow the producton functons (ϕ m z/l), ϕ θ (z/l) and ϕ q (z/l), as ndeed they must f the flux dvergence terms n the equatons are small. However, the exact match s debated, see next fgure, where estmates of all functons are shown for the momentum balance s shown. 10 Lectures n Mcro Meteorology

104 Fgure 6.3. Expermentally determned Monn Obuchov smlarty functons from Högström (1990) as functons of (z-d)/l, where d s the zero level for z, called the dsplacement heght, see secton 7.The terms, P,B, Tp,Tt, and D are defned n Equaton 6.9.The top fgure shows unstable values, whle the lower fgure shows stable values. Lectures n Mcro Meteorology 103

105 From the ϕ- functons above, we can calculate the profle expressons. As example we chose the profles of wnd and potental temperature. (6.16) z kz u ϕm ( ) = L u z z 1 ( ) u ϕ u du ( ) dz ( L ) dz ; u z z m z 1 = ϕm = kz L k z z 0 z0 z0 smlarly z kz θ ϕθ ( ) = L θ z θ θ z z z 1 ϕ ( ) θ θ z θ 1 dθ = ϕ ( ) dz ( L θ = ) dz ; kz L k z z 0 z0t z0t We use that u(z 0 ) = 0, and θ(z 0T ) = θ 0, that s some estmate of the surface temperature, and ϕ m (z/l= 0) = ϕ h (z/l= 0) = 1. Ths s really how the v. Karman constant has been chosen. Hence we get: (6.17) u z z z u z = ψ + ψ k z L L 0 ( ) (ln( ) ( ) ( )) 0 z z 1 ( ) z ϕ z m 0 ψ( ) ψ( ) = ( L ) dz L L z z 0 wth and (6.18) θ z z θ z θ ψ ψ ψ 0T ( ) 0 = (ln( ) θ( ) + θ( )) k z0t L L z 1 ϕ ( ) z L dz z z θ 0T θ( ) ψθ( ) = ( ) L L z z For unstable condton the ψ - functons are complcated and postve, see e.g. Stull(1991), whle for stable condtons the ntegraton s trval: z z z (6.19) ψθ ( ) = ψ( ) = 5 L L L The values of ψ for z 0 / L and z 0T /L are often neglected, and ψ(0) = 0. We are now able to show the prncpal behavour of the profles for stable, neutral and unstable condtons. It should be ponted out that the temperature gradent and the θ changes sgn together. 0T z wth 104 Lectures n Mcro Meteorology

106 The qualtatve behavour of the profles are shown on the next fgure, and t s clear how these functons do descrbe the lowest part of the characterstc profles throughout the boundary layer as shown earler Ln(z) z/l = 0, z/l< 0 z/l > 0 Ln(z 0 ) or Ln(z 0T ) u( z) θ ( z) θ0 θ Fgure 6.4.Characterstc profles for the surface boundary layer for stable, neutral and unstable condtons.. Fgure 6.5 Typcal boundary layer profles of wnd speed, U, temperature, T, and water vapour mxng rato, q: a)daytme unstable condtons. Part of the profles s well mxed,.e. no heght varaton. b) Nght tme stable condtons. Stronger heght varaton of profles. c) Thermally neutral condtons. The profles are logarthmc n a heght nterval. Lectures n Mcro Meteorology 105

107 Some aspects of the surface layer. The Rchardson number, R, s used throughout flud dynamcs to characterse the thermal stablty of a flud and the nfluence of thermal propertes on the turbulence. There are two dfferent Rchardson numbers, the flux Rchardson number, defned from gradents and fluxes, and the gradent Rchardson number, defned from the rato between the temperature gradent and the wnd speed gradent. We start our story wth the smplfed energy equaton used earler: (6.0) 1 de gw ( θ v ) u 1 ( we ) 1 ( wp ) 0 uw ε; dt θ z z ρ z v 1 de u g( w θ v ) u 1 ( we ) 1 ( wp ) u 0 ( uw )(1 /( uw ) ( + + ε) /( uw )); dt z θ z z ρ z z v From ths equaton we defne the R Flux (6.1) g( w θ v ) u RFlux = /( u w ); θ z v R Flux s seen to descrbe the thermal nfluence on the turbulence structure. If R Flux >0, the thermal propertes act to dampen the turbulence, and f R Flux s larger than about 1, the turbulence cannot exst. One talks about a crtcal value for R Flux. Usng a turbulent dffusvty, we can derve a gradent Rchardson number. (6.) R Flux θv θv K g ( w θ v ) g z g = = = z R θv u θv u u θv u uw K ( ) z z z z Snce we are not entrely certan about these dffusvtes, we consder the two R s as dfferent. The gradent R can be used to characterse the stablty of the surface layer as an alternatve to z/l. Indeed t was used before z/l, because t only demands that one can measure mean value gradents, whch s an easer and older technque than the measurement of the turbulence fluxes, necessary to derve the Monn Obuchov length. The relaton between R and z/l s found from: 106 Lectures n Mcro Meteorology

108 (6.3) R θ θv kz θ v v g z g z θv kz z ϕθ ( z / L) = = = θ u L ( m ( z/ L)) v u θv u kz ϕ ( ) ( )( )( ) z z u kz. Wth the forms of the smlarty expressons (6.15) gven above we see that for unstable condtons, R z/l. Smlarly for the stable condtons, the smlarty functons ndcate that R approaches a constant for z/l, correspondng to the so called crtcal R, where the contnuous turbulence des out,as descrbed above. Ths result s stll debated, though. Other researchers clam that R does not go to a constant. See llustraton n fgure 6.6, ndcatng that at least for ths experment the R goes to a constant. Fgure 6.6. The dependence of the gradent Rchardson number on stablty parameter, ζ = z/l,estmated n Busnger et al. (197). So far we have used constant turbulence dffusvtes to close equatons. Now, we wll try to derve behavour of K that s consstent wth the surface layer formulatons. We llustrate for momentum and a passve scalar lke humdty. u (6.4) uw = Km ; qw = K z q q z Lectures n Mcro Meteorology 107

109 Insertng now the smlarty scales and expressons we obtan: (6.5) From ths we get expresson for K m and K q : u q u = Km ϕm( z/ L); q u = Kq ϕq( z/ L); kz kz (6.6) K = ku z / ϕ ( z / L), K = ku z / ϕ ( z / L) m m q q As seen we have found a velocty scale, u, a numercal coeffcent, the v. Karman constant. The mxng length scale s seen to be equal to the heght z, dvded by the smlarty functons. The ϕ-functons ncrease lnearly wth z/l for stable condton, whle they are slghtly smaller than 1 for unstable condtons. Ths means that for unstable condton the mxng length s larger than the measurng heght, whle t for stable condtons s smaller than z. Indeed, for stable condtons: z z z (6.7) 0. L for ϕ( z/ L) = z L 1+ 5 L Typcally dffusvtes follow the surface layer formulatons for small z. For greater heghts n the boundary layer proper, the dffusvtes are assumed to become constant or dmnsh wth heght. One example s: (6.8) / ϕm( / ) exp( α C g / ); K = ku z z L f V z u where α s a coeffcent. If we Taylor expand the exponental t s seen that ths expresson s seen to be consstent wth the formulatons for the heght varaton of u, whch were derved earler n (6.4): fcvg z (6.9) u z u o( 1 ); u o lnz K(z) Typcal heght varaton of K(z) n the boundary layer. Fgure 6.7 Characterstc behavour of K(z) through the atmospherc boundary layer from (6.30) In secton 5, we have seen that a typcal K consstent wth the Ekman layer dervaton s of the order of 15 m /s. The surface layer value s consderably less, takng z = 10 m and u * equal to 0.5 m/s. Hence from (6.6): K m /s. On the other hand the surface layer K ncreases wth 108 Lectures n Mcro Meteorology

110 heght on to the top of the surface layer, beng from m above ground. Hence from ths we must expect the surface layer to look qute dfferent from ts predcted behavour n the secton about the Ekman spral n secton 5. Varances and Spectra. The varances and the spectra are obvously related, snce the varances derve from ntegraton of the spectra. The varances can be made non dmensonal wth approprate scalng parameters, snce we have ntroduced scalng parameters for each basc varable, wnd, temperature and humdty. The properly scaled data can be plotted as functons of z/l. The varablty of such functons wth z/l s a test of the applcablty of Monn-Obuchov smlarty functon for these varables. Generally, t s found to work for some of the varances, but not for all, reflectng some lmtatons to the surface layer concepts. As an example of a varance estmates, where surface layer scalng works well, we refer to σ w, the standard devaton of the vertcal velocty, and σ θ, the standard devaton of temperature. Followng the scalng phlosophy presented out above, we would scale or normalse these two wth u and θ, and expect the scaled varables to be functon of z/l σw z σθ z (6.30) = fw ( ) and = fθ ( ) u L θ L It s seen that Monn-Obuchov smlarty does ndeed work well for many of these. It does not work well for horzontal velocty components, as we shall see n Secton 8. Smlar experence has been found for the spectra. The scalng used s derved from a combnaton of the Taylor hypothess, the Monn-Obuchov smlarty, and the Kolmogorov laws for nertal range turbulence. We consder the nertal spectra for temperature and wnds. (6.31) ks ( k) = αε k ; /3 /3 1 u 1 u 1 ks ( k) = αε εk ; 1/3 /3 1 T 1 T θ 1 Insertng the smlarty functons for dsspatons, we obtan. ks 1 u( k1) αu /3 = ( kz /3 /3 1 ) ; u ϕε kvk (6.3) ks 1 T( k1) αt /3 = ( kz 1/3 /3 1 ) ; θϕ ε ϕεθ kvk where we summarse the expressons for the dsspaton functons, wth v. Karmans constant denoted k vk here, because we here use k 1 as horzontal wave numbers: Lectures n Mcro Meteorology 109

111 (6.33) z k zε z k zε ϕε ( ) = ; ϕ ( ) = ; L vk vk θ 3 εθ u L u θ The above shows that f we measure the spectra as functon of wave number, the nertal subranges for several measurng heghts would fall on the top f normalsed as shown. If we measure tme sgnals and analyse for frequency spectra then the Taylor s hypothess s used to wrte the k 1 z varable as a normalsed frequency n: (6.34) n f [ Hz] z kz z ; u π λ 1 = = = Warnng! : Before around 1980 n was called f, and f was called n. One has to check n older papers. We recall that n Secton1, the Taylor hypothess was formulated as: (6.35) t = u x 1 whch means that ω = u k 1. We recall as well k 1 S(k 1 ) = ns(n). The result of these consderatons s that f we normalse the power spectra as ndcated by (6.3), and plot the normalsed spectra versus the normalsed frequency n, the resultng plots wll unversal, and ndependent of wnd speed, measurng heght and stablty. Ths scalng has only been derved for the nertal subrange by the above scalng arguments, but t s generally used also a larger range and s emprcally found to work over a much larger frequency range, f one allows for addtonal dependency on the stablty, z/l. As for the varances t s generally found that some of the spectra and some parts of the spectra can sad to follow better these M. -O- smlarty formulatons, than other parts. Examples are shown on the followng fgure, Fgure 6.8, whch show all the man spectra and co-spectra n the surface layer, scaled wth dsspaton expressons for near neutral condtons, as descrbed above. As addtonal stablty dependency, Fgure 6.9 shows the varaton of the peak frequency wth stablty, showng a tendency for the spectra to move to larger n, or smaller scales, followng (6.34). 110 Lectures n Mcro Meteorology

112 . Fgure 6.8. Normalsed spectra, accordng to (6.31), for near neutral condton plotted versus the normalsed frequency, n (Kamal et al, 197) Fgure 6.9 Varaton of the normalsed peak frequency, n m of the spectra n Fgure 6.8 wth stablty. The absence of curves for unstable u and v-spectra s an ndcaton of that these spectra do not follow the Monn-Obuchov smlarty well for unstable condtons as we shall see n secton 8. Note that the co-spectra n Fgure 6.8 are normalsed by the total correspondng fluxes. Hence, the co-spectra show the contrbuton to the respectve fluxes from the dfferent frequency ntervals. Asde from the low frequency part of the power spectra for the horzontal velocty components, u and v, n unstable surface boundary layer (see Fgure 8.10), also other spectral characterstcs Lectures n Mcro Meteorology 111

113 obtaned wthn the atmospherc surface layer show devatons from the M.O. smlarty llustrated n Fgure 6.8 and 6.9. An example s shown n Fgure Ths fgure llustrates that not all sgnal varablty wthn the near surface layer can be charactersed as surface layer turbulence n the sense dscussed n secton as that not all varablty wthn the boundary layer s created from boundary layer processes. In Fgure 6.10 the low frequency fluctuatons reflect the low frequency spectra dscussed n secton, notably the meso-scale spectra presented n Fgure.3. Fgure Stable spectra measured at 46m above terran from Larsen et al (1990). The rght hand fgure shows 10 such spectra of the v-component. Here, the hgher frequences shows the spectral varaton consstent wth u and z/l as dscussed here, whle the lower frequences show no such varaton, and are bascally ndependent of the surface layer scalng and reflect the meso-cale spectra presented n Fgure.3. The rght hand fgure shows the average of the three velocty components, u,v, and w for the 10 runs presented on the former fgures. It s seen that only the w-spectrum s neglgble for the meso- scale frequency range, and ndeed only the w- component spectrum can be explaned fully by the surface layer scalng, as llustrated n Fgure 6.8 and 6.9. Just as we have consdered varances and co-varance for varables measured at the one pont, we may also consder correlaton between varables at dfferent ponts of space, varous knds of cross-varances and cross spectra nvolvng data from dfferent ponts n space. Especally we shall study correlaton between veloctes at dfferent spatal ponts, whch are mportant for wnd load modellng. Su( χ, ω) = Cou( χ, ω) + Qu( χ, ω) (6.36) Coh ( χ, ω) = S ( χ, ω) / S ( x, ω) S ( x + χ, ω) u u u u Ths frst of the equaton descrbes the cross spectrum of a velocty component at pont x and x+χ. In the second equaton one has smplfed by takng the absolute square of the cross 11 Lectures n Mcro Meteorology

114 spectrum and normalsed by the smlar one pont power spectrum establshed as geometrc mean of the spectra n the two ponts consdered. Ths s denoted the Coherence. Consder the correlaton expresson from (.49). (6.37) R u ( kχ+ ωτ ) Su( k, ) ( k1u e dkd ω (, ) = ) χ τ ωδω ω = k k S ( k ) e u ( kχ+ Uk1τ) dk Where we have ntroduced the δ-functon, defned n secton, to account for Taylor hypothess of frozen turbulence, as we have used throughout the spectral descrpton above, wth the mean wnd, U, along the x 1 -axs. The cross spectrum from (6.3) can now be computed from: 1 ωτ 1 ( kχ+ Uk1τ ωτ ) (6.38) Su(, ) = Ru(, ) e d Su( k) e dkd π = π Here we use another δ-functon (6.39) χ ω χ τ τ τ k 1 ( ku ωτ ) δ ( ku 1 ωτ ) = e dτ π Hence we arrve to an expresson for the two pont cross-correlaton spectrum χ ω 1/ ) ( 3 3) (6.40) (, ) ωχ U (,, ) k χ + S k χ u e Su k k e dk dk U = kk 3 ω 3 3 Whch shows a cyclc varaton between the Co and the Q part essentally controlled by the exp(ωχ 1 /U) or the phase delay. From (6.36) we now obtan the coherence as follows, usng horzontal homogenety n (6.3): (6.41) u( χ, ω) u( χ, ω) / u( ω) Coh = S S = ω ( kχ+ k 3χ3) Su(, k, k3) e dkdk3 /( Su(, k, k3) dkdk3) U U kk 3 kk 3 = ω, Based on ths very smplfed pcture we conclude that dsplacement along the mean wnd drecton wll nfluence the phase not the coherence that wll reman equal to unty. Only dsplacement perpendcular to the mean wnd wll reduce the coherence because the numerator wll be smaller than enumerator n (6.37). The model can be expanded assumng sotropy spectral relatons, as gven n secton. Ths s done n Krstensen and Jensen (1979), wth good results. However, we can already pont to many lmtatons to the valdty for ths modellng, as also ponted out by Krstensen and Jensen (1979) and Krstensen et al (1981): We know that there wll be a reducton of correlaton also along the mean wnd drecton. For many scales, relevant flow felds are not sotropc, especally n the z-drecton where t s not even homogeneous. Therefore, we shall cte some of the emprcal expressons e.g. Davenport (1961), whch are based on the smple exponental model below, wth the coeffcent a beng specfed for dfferent dsplacement and ambent condtons. Lectures n Mcro Meteorology 113

115 (6.4) Coh( f ) exp( af χ / U ) Where the coeffcent a s dependent on the separaton and drecton of the separaton χ, longtudnal vertcal or lateral and also depends on stablty, reflectng the ncreased larger scale contrbuton to the spectrum for ncreased nstablty. However, the stablty effects seem not as clear as for many of the other parameters studed n ths secton. Addtonally, we must expect to have to nclude the measurement heght, z, because the turbulence depends on the measurement heght. The frequency s now n Hz. The followng estmates of the a-factor for lateral and vertcal dsplacement, ndcated by y and z respectvely, but not ncludng stablty effects, are from Panofsky and Dutton (1984), but many other formulatons exst, e.g. Krstensen et al (1981) (6.43) a a a y z x χy χy for 4 z z χ z z σ w 60 U The values ndcate that the coherence for pure vertcal and the pure lateral dsplacement behave rather smlarly for close to neutral condtons, the lmtng value s ndcated for a y only though. The longtudnal coherence reflects eddy decay tme relatve to advecton tme. Addtonal to dsplacement drecton, we can also consder the three dfferent wnd components of the turbulence, stablty has been shortly dscussed. Fnally should be mentoned that the expressons n (6.38) are found from 3D surface layer turbulence. Also the low frequency parts of Fgure 6.10 gve rse to coherence that has now been parametersed, and behave qute dfferently from the boundary layer turbulence normally consdered n these notes (Vncent et al, 01) We now leave the dscusson of the atmospherc surface boundary layer. The next fgure shows the boundary layer, as we understand t now. We have especally studed the surface layer, and the scalng expressons that apply wthn ths layer. When we say that they apply wthn the surface layer, t rases the queston of what we expect to be descrbable wthn the surface layer. The short answer s local varables, meanng varables that can be defned wthn ths layer only. We have seen the varables, representng fluctuatons, can be consdered local, vertcal gradents and horzontal gradents lkewse. Mean values cannot be consdered local, n the sense that they can be determned from wthn the surface layer only. To determne wnd speed and temperature we have to ntegrate the gradents from a surface value and up, forcng us to ntroduce z 0 and z 0T, both parameters, layng outsde the turbulence boundary layer. Wth these lmtatons we have found the formulatons to work reasonably well wthn the surface boundary layer of the atmosphere. Some the expressons that can be derved do not work, and often t s argued that nfluence from nearby layers s penetratng nto the surface layer. Recall that the dea of dstnct layers s really n contradcton wth another dea, the deas of the boundary layer beng mmersed nto a soup of eddes of all szes and orentatons. 114 Lectures n Mcro Meteorology

116 Ln (z) h Boundary Layer Proper h s 0.1h h sl? Surface Boundary Layer Vscous nterfacal layer surface Fgure Schematc drawng of the atmospherc boundary layer wth sub-layers. The surface boundary layer s seen between h sl and hs. Between the surface boundary layer and the surface we have a vscous nterfacal layer, where also the molecular dffusvtes are mportant, because the turbulence s nhbted by the nearness of the ground. Above the surface layer s the boundary layer proper, where also the turnng of the wnd and the structure at the exchanges wth the free atmosphere are mportant. In the followng secton we shall consder the Vscous nterfacal layer. Appendx 6A. Dmensonal analyss. The Buckngham PI-theorem. In the man text, we have smplfed the momentum equaton to be vald for the statstcal statonary and horzontally homogenous surface boundary layer. Defnng sutable characterstc scales we could argue for the functonal form of relevant dmensonless meteorologcal quanttes. Here, we shall ntroduce a more formal method of dmensonal analyss leadng to smlar expressons, the so called Buckngham P theorem, whch we have summarzed from Jensen and Busch (198). Assume that a physcal system s descrbed by a number of physcal quanttes Q1, Q, Q3, ----, Qn., through the equaton: (6A.1) FQ ( 1, Q,, Q n ) = 0 Equaton (6A.1) s dmensonally homogeneous, meanng that that t s ndependent of the dmensonal unts chosen. The n physcal quanttes Q, are assumed to nvolve r ndependent Lectures n Mcro Meteorology 115

117 physcal unts, lke tme, length, mass etc. Now the P theorem states that (6A.1) s equvalent to another functon: (6A.) f ( π1, π,, πn r) = 0. Here π 1, π, π 3,---, π n-r are ndependent dmensonless power products of the Q quanttes. Note that that lettng the functon equal to zero, s to be understood such that the functon value remans constant for all allowable values of the parameters. Therefore, the soluton of (6A.) means fndng constrans between the parameters to allow f (-) to reman constant. The P theorem allow one to fnd the mnmum number of dmensonless parameters needed for a complete descrpton of the physcal problem, descrbed by (6A.1). Normally the P theorem cannot provde nformaton about the functonal form of the soluton to (6.A.1). It must be determned by other means, manly through measurements. As example we apply the P theorem to the smplest possble verson the surface layer formulaton, then neutral wnd relatons n (6.7a), where we agan assume that the wnd profle must depend on the frcton velocty and the heght. However, we neglect the dsspaton from (6.7a), because t assumes addtonally that the there s local balance between producton and dsspaton of knetc energy. Hence, we can wrte (6A.1) as: u (6A.3) F(, u, z) = 0. z Here we have 3 Q quanttes, dmensons (tme and length). Hence we need only one dmensonless parameter n the f-functon, we obtan: z u (6A.4) f ( ) = 0 u z Ths n turn n mples the valdty of (6.7b), neglectng the dsspaton, and leadng to the logarthmc profle. z u 1 (6A.5) = u z k Note that just n man text of ths secton, the mportant assumpton s the number of parameters that descrbe the physcal problem. In ths sprt we now contnue by ncludng the effect of buoyancy effects u g (6A.6) F(, u, zw, ' θ ', ) = 0. z T0 Now we have three basc dmensons, length, tme and temperature and fve Q-quanttes, meanng that we need two π-quanttes, e.g.: z u z g z (6A.7) π1 = ; π = w' θ' =, 3 u z u T0 L Where we have rentroduced the Monn-Obuchov stablty length L., from the man text of ths secton. Hence the f-functon looks as follows 116 Lectures n Mcro Meteorology

118 z u z (6A.8) f (, ) = 0, u z L Ths mples that: (6A.9) kz u m( z = ϕ ) ; u z L Where the v.karman constant, k has been ntroduced from (6A.5) and also ntroduced the stablty functon for the wnd profle, we found n the man text of the secton. The dmensonal analyss has been found very useful n meteorology, usng dmensonal argument for relatons between quanttes. However, t does not always work properly, often because one neglects parameters n the analyss that should have been nvolved. Lectures n Mcro Meteorology 117

119 7. Near surface vscous layers, roughness lengths: z 0 and z 0T, nterfacal exchange. Behavour of profles and flow close to the ground. The behavour of the velocty profle very close to the ground s llustrated below. Fgure 7.1. The logarthmc and the actual wnd profle near the ground. δ s the lower heght where the logarthmc profle apples, z 0 s the roughness length. The x- ponts show actual profle values. The profle on the fgure s a standard: (7.1) u z u( z) = ln( ); k z δ s the lower heght of valdty for the logarthmc profle. Note that z 0 s below δ, so z 0 s the extrapolated heght where the logarthmc wnd profle attans the true value of the wnd at the ground, namely zero. As a rule of thumb, δ 10z 0. Below δ the wnd s nfluenced by the ndvdual roughness elements as well molecular vscosty. If the surface s flat vscosty domnates we can estmate δ from the flux dvergence terms n the average momentum equaton, e.g. from (3.83), where the two last terms read: (7.) u uu j u ν = ( ν uw ), x x z z j where we, as usual, have assumed gradents only n the z-drecton. We have earler argued that the molecular term could be dscarded due to the much larger turbulence term. However, close to the ground the profle gradent becomes large enough for the molecular term to domnate. Insertng the surface layer expressons and nsertng δ as the heght where the molecular an turbulence terms are equal, the equaton above reads: (7.3) j u u ν ( ν uw ) = ( ν + u ) δ z z z kδ ku Snce ν 0.15 cm s -1, and u s typcally > 10 cm/s, δ s seen to be around 0.1 mm, so the above consderatons are vald only when roughness elements are below that heght Lectures n Mcro Meteorology

120 To see what happens n the more general stuaton, we take the non-averaged momentum equaton for the u-component, e.g. (3.7), usng ncompressblty to rewrte the advecton term. (7.4) u uu j 1 p u + + ν = t x ρ x x j j 0. Introducng explctly the components, u, v and w, and rearrangng slghtly, we obtan: 1 (7.5) u u u u + ( u + p ν ) + ( uv ν ) + ( uw ν ) = 0 t x ρ x y y z z Notce, we stll do not average. We now ntegrate the equaton from the true surface z = η(x, y, t). The tme, t, s ncluded because many surfaces move n tme, wth water surface wave felds as the most mportant example. The ntegraton wll be carred out from z = η to z = h, where h s well nto the turbulent surface layer wth M.O. smlarty, logarthmc profles etc., see fgure 7.. Fgure 7.. The actual surface, η(x, (y), t), the scale heght δ, and a heght wthn the turbulent surface layer, h. The ntegraton looks as follows: h u h 1 u h u u h (7.6) dz + ( u + p ν ) dz + ( uv ν ) dz + ( uw ν ) t x ρ x y y z η = 0 η η η We use Lebnz rule to move the dfferentaton outsde the ntegraton to facltate nterpretaton: h h h φ η h φ η φdz = dz φη ( ) + φ( h) = dz φη ( ), x x x x x x η η η Where we have used that h s a constant heght. Hence (7.6) takes the form: x Lectures n Mcro Meteorology 119

121 (7.7) h h h u 1 u u u h dz ( u p ν ) dz ( uv ν ) dz ( uw ν ) η t x ρ x y y z η η η = h h 1 u 1 u ( ν ) ( ν ) η ρ ρ η η η = udz + uη + u + p dz + u + p + t t x x x x η h u u η u u + ( uv ν ) dz + ([ uv] ν ) η + ( uw ν ) h ( [ uw] ν ) η = 0 y y y y z z η The terms n square brackets are for later use. Next we consder the no-slp condtons between the ar and the surface, η(x,y,t), meanng that an ar partcle at the surface remans there. The speed of a surface partcle can be wrtten: dη η η η (7.8) wη = = + uη + vη = 0; dt t x y Multplyng ths equaton wth u, t s seen we can elmnate the terms wthn the square bracket n the ntegrated momentum equaton. The equaton now takes the form: (7.9) h h 1 u 1 u η udz + ( u + p ν ) dz + ( p ν ) η + t x ρ x ρ x x η h η u u η u u + ( uv ν ) dz ν ) η + ( uw ν ) h + ν ) η = 0 y y y y z z η We now takes the mean value and assume, as usual, statonarty and horzontal homogenety for mean quanttes, n the turbulent surface layer, where all ntegrals get the most of ther contrbutons, collectng all terms wthout ntegraton frst: (7.10) h p η u u η u η u udz + ( ) η + ν( ) + ( uw ν ) t ρ x z x x y y z η h h u 1 u ( uv ν ) dz ( u p ν ) dz = y y x ρ x η η h or: (7.11) p η u u η u η wu = u = + ν ρ x z x x y y ) h h ( ) η ( ); 10 Lectures n Mcro Meteorology

122 The two last ntegrals above wll be close to zero n the turbulent surface layer, where they get most of ther contrbutons. The frst s for the lateral velocty, when the coordnate system s algned wth the mean wnd n the u-drecton. The second term s smply the balance between advecton and pressure forces n the mean wnd drecton. As usual, when we are n the turbulent surface layer we can neglect molecular terms. The result can then be formulated as, that the turbulent momentum flux n the turbulent surface layer, n the near surface layer s taken over by a mxture of pressure forces on terran features, called form drag and ncluded n the frst term, and molecular transport. For the molecular terms the frst are usually consdered the most mportant. The two last are normally neglected except n wave growth theores where they are consdered mportant. Next we study how the scalar varables are transported towards the surface through the vscous nterfacal layer. We have a smlar profle expresson n the turbulent surface layer: (7.1) 0 θ z θ θ = ln( ); k z We have a smlar fgure for temperature as fgure 7.1 for wnd speed: 0T Fgure7.3. The temperature profle close to the ground. s the lower heght of valdty for the logarthmc profle, shown as a straght lne.z 0T s the roughness length for temperature. Correspondng to the momentum equaton startng from (3.58) and neglectng the humdty term: (7.13) θ u jθ θ + νθ = t x x j 0 j The equaton s smlar to the momentum equaton for the u component, whch we used to study the surface stress. The only and mportant dfference s the absence of the pressure term. Hence we can fnd the result drectly, wthout redong the dervaton, we just dd. (We also here use η(x,y,t) as the actual surface). θ θ η θ η (7.14) wθ) h = νθ( ) z x x y y η Lectures n Mcro Meteorology 11

123 Ths means that close to the surface heat and all other passve scalars must be transported by the molecular transport only, as opposed to surface stress that could use pressure as well n the form of form drag. Hence, for many surfaces scalar transport s fundamentally dfferent from the momentum transport close to the surface. Roughness Formulatons and Parameters The purst atttude s that the z 0 parameters are smply ntegraton parameters, when Monn- Obuchov profles are ntegrated. It s a heght where the varable attans ts surface value (zero for wnd speed). From another pont of vew t s a characterstc of the surface, beng responsble for the atmospheres frcton and general couplng and exchange wth the surface. Determnaton of z 0 s then a feld were knowledge about the roughness of specfc surfaces has been determned mostly expermentally but also wth some theory. Below we cte such formulas for specfc surfaces. Surfaces wth characterstc surface elements The roughness s derved from the densty and shape of the roughness elements: (7.15) z0 0.5 h S/ A, A>> S Here h s the heght of the ndvdual roughness elements. S s the crosswnd area of a roughness element, and A s the average surface area for each roughness element (Lettau, 1969). Dense vegetaton, canopes, cereal felds etc. One of the smplest formulas here s: (7.16) 0 1 z = ( h d); d h; 3 3 Where h agan s the heght of the canopy, d s the so-called dsplacement heght. The dsplacement heght s generally ntroduced over denser vegetaton, as llustrated below. Here some of the canopy heght contrbutes to a change n surface level, rather than to roughness. Fgure 7.4 Illustraton of roughness elements havng both a roughness length and a dsplacement heght, here for a forest canopy (Stull, 1991) 1 Lectures n Mcro Meteorology

124 Theoretcal one can estmate the dsplacement heght as the level where the frctonal forces attacks the surface. Expermentally one has to ft two parameters from a logarthmc expresson nstead of just one, z 0. u z d) (7.17) uz ( ) = ln( ); k z0 The equaton shows consensus about how to ntroduce d nto the logarthmc profle equaton. As seen from the forest pcture, Fg 7.4, above the exact profle becomes qute dfferent from the turbulent surface layer profle. Ths s llustrated also on the next fgure showng the profle from just above and down through a cereal canopy. Fgure 7.5. Wnd profles above and through a cereal canopy, not beng logarthmc, but rather by an exponental varaton around the crop heght. Roughness of water surfaces. Over the ocean the roughness elements are assocated wth the wnd drven surface waves. Observng that the waves were generated by the wnd force on the surface, proportonal to u, and restraned by gravty, Charnock(1984) proposed a smple relaton for z 0 : ν u (7.18) z0 = β + α u g The frst term s later ntroduced for low wnd stuatons, wth few waves and wave roughness elements, where molecular vscocty becomes responsble for the momentum transport close to the surface, see equaton (7.3). The coeffcent, β, s usually taken as about 0.1. The coeffcent α s found to be very small and qute varable between 10 - and , dependent on ste and stuaton. Many more complcated expressons, but the scatter of the data, around both (7.18) and the more complex expressons, makes expermental evaluaton dffcult, see appendx 7A. The fact that the numercal coeffcent s so small s normally taken as ndcaton of that all parameters have not been nclude. (Hansen and Larsen,1997). The water roughness s usually qute small, mostly less than a few tens of a mllmetre. Indeed water s one of the least rough surfaces n nature. As seen t also changes wth wnd speed, a Lectures n Mcro Meteorology 13

125 thng that actually happens for many other surfaces, lke a blowng sand surface, a vegetated surfaces, where the ar does not just exchange momentum but also energy wth the surface, by movng part of the surface around, waves for the water, sand partcles for sandy surfaces and leaves and branches for vegetaton. See also Appendx 7A. Seasonal and other changes of roughness. Many surfaces show changes of roughness wth season because the roughness of vegetated surfaces change wth the seasonal changes of the vegetaton or due to aspects lke snow and ce cover n the wnter. Below we show a fgure of the seasonal change of the roughness for typcal Dansh felds. Fgure7.6. Seasonal varaton of the roughness length, z0, measured for three agrcultural felds,, 3 and 4 n Western Denmark (Semprevva et al, 1989) - Roughness (m) Month Landscape 1998 Landscape 1999 Sprng barley I Sprng barley II Wnter barley Wheat Grass Maze Beets Model wth hedges Fgure 7.7. The fgure shows the seasonal roughness varaton as measured from several small masts and one tall mast n the same area. Also the modeled area roughness s reported. The seasonal varaton of all data and models s clearly seen.(hasager et al., 003) The measured roughness n fgure 7.6 and 7.7 are obtaned from small masts each placed n a feld wth a characterstc crop. The measured landscape roughness n Fgure 7.7 s obtaned from a tall mast. The mportance of hedges for the landscape roughness s emphaszed. Addtonally, many types of vegetaton show z 0 to be a functon of wnd speed, a lttle lke water because vegetaton s moved by the wnd, and therefore energy s suppled to the vegetated surface just as for the wavy surface of water or wnd farm, see below. Also stablty nfluences 14 Lectures n Mcro Meteorology

126 z 0, beng smaller for stable condtons, due to nhbted nteracton between the atmosphere and the surface for strong stablty, see Zlltnkevch et al. ( 009). Roughness of a large wnd farms Fg Dervaton of roughness of a very large wnd farm (Frandsen, 199). As the Fgure 7.8 ndcates, Frandsen (199) fnds an expresson by matchng an outer logarthmc profle that reflects the roughness of the wnd farm at the wnd turbne hub heght, h, to an nner profle that reflects the background roughness: (7.19) U( z) 1 z = ln( ) z h u κ z wf 0wf U( z) 1 z = ln( ) z < h u κ z Where subscrpt wf denotes wnd farm. The relaton between the two frcton veloctes are derved from the areal thrust of the wnd farm as: (7.0) 0 ρu = ρu + ρcu( h), wf t, The thrust coeffcent for the wnd farm (per unt area) s found by dvdng the thrust of the ndvdual wnd turbnes, T, wth the surface area for each turbne. 1 π t= T/ xx 1 = ρcu T ( h) πd / xx 1 ρcu t ( h) 4 (7.1). πct x1 x ct = ; s1 =, s = 8ss D D 1 Here C T s the thrust coeffcent for the ndvdual wnd turbnes, and D the dameter of the area, swept by the blades, when the turbne works. C T s zero for non-workng turbnes, t then growths fast wth wnd speed to about one, and gradually decrease as 7(m/s) /U(h) to about 0. for 15 m/s, for characterstc wnd turbnes (Frandsen, 007). Here, we neglect the contrbutons from the statonary structure of the wnd turbne. Lectures n Mcro Meteorology 15

127 Matchng the wnd speeds of (7.19) n the hub heght, and solvng (7.0) and (7.1) for z 0wf one obtans: (7.) z0wf = h exp( ) h exp( ), c + ( κ / (ln( h/ z ) c t κ 0 κ t where the approxmaton holds, f the background roughness s small compared to z 0wf. Characterstc values can therefore be found from (7.). Assumng h=100m and s= 7, we fnd that, for ncreasng wnd speed z 0wf starts beng equal to z 0, to ncrease fast wth wnd speed to about 1.1 m for U (h) =7 m/s, thereafter to decrease to about 0. m for U(h)= 15 m/s. Notce, we are talkng about a fully workng wnd farm. If the turbnes do not work, the roughness predcted by (7.) goes back to the background roughness. At present expermental evdence s obvously dffcult both because the heght of the turbne structures actually exceeds the surface boundary layer, and also because the very large wnd farm s not yet constructed. Consensus tables and fgures summarsng the roughness of landscapes. The expermental and theoretcal knowledge accumulated s often collected nto fgures and tables. We show some below, where some are derved from the assessment of roughness from terran nspecton, whch s an essental part of evaluatng the wnd power resource for a gven locaton. One should recall the purst vew, that the roughness length s a constant of ntegraton for a logarthmc wnd profle. However, as dscussed above, t can also be related to the physcs of the processes n the nterfacal layer. However, roughness length scales for larger features, lke buldngs, forests and wnd farms that are obvously penetratng through the whole turbulent surface layer, where the logarthmc profle s supposed to exst, as well as for whole terran types stll rases nterpretaton questons. Indeed, for such terran forms, the roughness seem to be stablty dependent, reflectng that the roughness elements now reach nto the turbulent surface boundary a layer (Zltnkevch et al,006). In spte of these complcatons, the roughness length s obvously a relevant feature also here, at least n a qualtatve sense, snce the terrans n queston can be sad to exert a frcton wth the atmosphere. Here, t s mostly be used n connecton wth numercal models that need an expresson for the surface drag, also for these types of terran. Asde from the ncluson of very large terran features, relatve to an nterfacal layer, you wll notce that many roughness aspects of a terran are only partally, f at all, ncluded, as ndeed the roughness length s a measure of the frcton between the atmosphere and the ground. However, t s also a practcal parameter, whch must be deductble wthout too much montorng and complcaton Here we lst a few of the complcatons: Fgures 7.7 and 7.8 llustrate the dependency of roughness length on type of surface vegetaton, and ts changes. Some of these changes are natural, but for agrcultural felds, farmer s decsons on growng and harvestng perods are also mportant. We have seen that the roughness of water and wnd farms depends strongly on the wnd (for the wnd farm, f t operates), so t does for sand (f t s dry) and to some extent for all vegetaton that can be moved by the wnd. 16 Lectures n Mcro Meteorology

128 All landscapes can become snow covered and all low and flat landscapes covered wth water. These aspects have a strong season and locaton dependency, but wll also depend on random factors lke the weather. Fgure 7.9 Terran wth roughness correspondng to roughness class 0, water areas wth z m. The class comprses seas, fjords and lakes. (Troen and Petersen, 1980) Fgure 7.10 Terran wth roughness correspondng to roughness class1, open areas wth a few wnd breaks, z m The terran appears to be very open and flat or gently undulatng. Sngle farms and stands of trees and bushes (Troen and Petersen, 1980). Lectures n Mcro Meteorology 17

129 Fgure Terran wth roughness correspondng to roughness class, farm lands wth wnd breaks, the mean separaton of whch exceeds 1000 m, and some scattered buld up areas, z m. The terran s charactersed by large open areas between many wnd breaks, gvng the landscape and open appearance. The terran may be flat or gently undulatng. There are many trees and buldngs. (Troen and Petersen, 1980). Fgure 7.1 Example of terran wth roughness correspondng to roughness class 3, urban dstrcts, forests, and farm lands wth many wnd breaks,, z m. The farm land s charactersed by many closely spaced wnd breaks, the average dstance beng a few hundred metres. Forest and urban areas also belong to ths class. (Troen and Petersen, 1980) 18 Lectures n Mcro Meteorology

130 Fgure Schematcs of terran types, roughness classes, and z 0 values. (Troen and Petersen, 1980). Lectures n Mcro Meteorology 19

131 Fgure Schematcs of terran types, and z 0 values. Note some roughness varaton for smlar terran s ndcated.( Stull,1991). 130 Lectures n Mcro Meteorology

132 Scalar Roughness, z 0T, z 0q etc. There are two dfferences between the study of scalars and velocty n the vscous nterfacal layer. It s smpler because we know that fluxes have to be carred as molecular transport. It s more complcated expermentally, because the scalar roughness s the heght, where the profle attans the surface value of the scalar, meanng the way one determnes the surface value now becomes mportant for the value of the roughness. For velocty t s smpler because we know that the surface value of the velocty s zero or close to zero. We shall now reconsder the fgure 7.1, we used above, ths tme for the temperature profle. Fgure7.15. Temperature profle close to the ground. s the lower heght of valdty for the logarthmc profle, shown as a straght lne n the logarthmc plot. The x-ponts show actual profle values. We consder the heght to be both n the turbulent layer and n the dffusve layer. We can then establsh an equaton for the flux-gradent relatonshp n both layers. (7.3) θ H θ θ0 = ln( ) = ln( ); k z0t ku z0t H = ν ( θ θ )/ ; Equatng the temperature gradents n the two equatons we get: (7.4) θ 0 z ku = + = 0 ln( ) ln( ) ln( ), z0t z0 z0t νθ where we, somewhat artfcally, have ntroduced z 0. We can now formulate an expresson for z 0T /z 0. (7.5) (7.6) z ku = 0 ln( ) ln( ) kx T z0t ν θ z0 0T kx T ku = e wth kx T = 0 ν θ z0 z z, ln( ) We have no smple way of solvng for. Instead we try dfferent and reasonable formulatons for, and see that they result n the same knd of dfferences between z 0 and z 0T. 1) ν/u. Ths s the smooth surface transton we derved n the begnnng of ths secton. It s the heght where the strength of turbulence flux equals that of the molecular dffusvty. For ths assumpton we fnd the followng expresson for X T : Lectures n Mcro Meteorology 131

133 (7.7) X T wth ν 1 ν 1 = ln( ) Pr + ln( Re 0 ) ; ν k uz k θ ν 0 Pr ; Re 0 = ; νθ ν where we have used the defnton of Pr = Prandl Number, and Re 0 = the roughness Reynolds Number. The Prandtl Number s a specal case of the Smdt Number for a general scalar, Sc = ν/ν d, where ν d s the Brownan dffusvty of ths partcular scalar n ar. Insertng the expresson for X T, the expresson for the rato between z 0T and z 0 becomes. (7.8) z 0T z 0 0 uz = e = e Re kxt k Pr 1 0 We now consder another assumpton for. ) αz 0, that s s proportonal to z 0. Insertng nto the expresson for X T, we get: (7.9) ν uz = α ln( α) = αpr Re ln( α). 0 kx T k k νθ ν kx T kα Pr Re αe (7.30) 0 z 0T z 0 = e = Although the resultng equatons are dfferent, they both yeld that z 0T /z 0 dmnshes wth growng Pr Number and wth growng Roughness Reynolds Number. To see the actual dfferences, we wll have to go to data, such as s shown on the next fgure, whch shows result from Brutsaert (198) on X T. Typcally, we have condtons wth Reynolds Number between 0.5 and 00. The lowest curve on the fgure corresponds to Pr = 1.0, whle Pr for the lower atmosphere s about 0.7. Hence, we have to go a bt below the lowest curve. If we choose X T 5, then we fnd that the rato: z0t kx (7.31) T = e = e ; z0 whch s known as the e - -law. These consderatons apply not well for surfaces made from fbrous materals, because for such surfaces the z 0T and z 0 wll also depend on dfferent surface propertes, an aspect that s not ncluded n the analyss above, but also here ther rato s found often to be about e -. Ths law seems also to apply approxmately for many of the other scalar roughness. 0 For other scalars than temperature, we change Pr to Sc, the Smdt number, and for some the Brownan dffusvty s very dfferent from the vscocty, gvng rse to Sc beng very dfferent from the above value around one. Consder molecules and aerosols as spheres, the Brownan dffusvty can be derved from Enstens formula (Chapman and Cowlng, 1970): kt (7.3) ν d = 3πνρd Here, k s the Bolzmann constant, and we have used subscrpt d to ndcate any partcle, wth dameter d, molecular or larger. For a typcal molecule n ar, we have d= m, wth ν ν d. Aerosols typcally wth d= m wth ν p cm /s and Sc rangng from For such small dffusvtes partcles have obvous dffcultes ever reachng the surface, whch s consstent wth that z 0d beng close to zero n (7.8 and 7.30). Indeed one has found the 13 Lectures n Mcro Meteorology

134 modelled surface flux to be too small, whch has led to modfcatons of the models of the near surface layer to nvolve other processes than Brownan transport, to allow suffcent partcle flux through the layer to descrbe expermentally found partcle flux to the surface. Addtonally also the aerosol dynamcs s expanded to nclude mpacton, gravty and slp between the ar and the aerosols (Lss and Slnn, 1983, Farall and Larsen, 1984). However, here we shall not go further nto ths subject. X T Fgure The fgure shows expermental data on ln(z 0 /z 0T )/k = X T n (7.) versus the roughness Reynolds number Re 0 = u * z 0 /ν for dfferent Prandtl and Smdt numbers. Note that the reasonng above, appled to heat flux, apply to all scalars, for whch reason the Schmdt numbers are cted as well. The fgure s taken from Brutsaert (198). The sold curves correspond to X T = 7.3Re 0 1/4 Sc 1/. The Smdt number s the rato between the vscocty,ν, and the relevant dffusvty; ν/ν d for the Smdt number and ν/ν θ for the Prandtl number. Fgure Schematcs of the dfferent sub-layers wthn the atmospherc boundary layer. We are now about leave the consderatons about the vscous atmospherc nterfacal layer. As seen n the last fgure, we have been able to update the fgure Mcro Scale 6 about the dfferent layers wth new nformaton. In the next secton we shall revst the total boundary layer, Lectures n Mcro Meteorology 133

135 especally the part that s above the turbulence surface layer. Ths s the part, for whch, we derved the Ekman spral n secton 5. Deeper nto the nterfacal layers. Before leavng the nterfacal layers, we shall take a short dscusson of ther general characterstcs. They are the layers, where the atmosphere handshakes to the non-atmospherc layer below, through at least one boundary condton. In the dscussons above, we have used the surface values of wnd speed, temperature and humdty as examples. We have seen that only for wnd speed over sold land were the boundary condton so smple, u 0 = 0 that there was an effectve decouplng between the atmosphere and the sol below wth respect to speed. For all other atmospherc varables, there s no such decouplng, because ther near surface value n the ar wll have relaton to ther near surface value n the medum consttutng the surface. In general the surface condton ether must be measured or determned by modellng systems determnng the varables below or at the surface. These systems wll then as upper boundary condtons have the lower boundary values for the atmospherc system. Hence, the two systems have become coupled, as they are n the physcal world, of-course. Examples: For wnd speed over water or other movng surfaces, the wnd speed s not zero at the bottom, and the surface stress nfluences the condtons wthn and on the surface, whch wll agan nfluence the surface stress. For forecastng of condtons above and n the ocean the exchange across the nterface s so mportant for the outcome of the forecast that coupled models now domnates the felds, see also appendx A. The temperature of a surface s typcally forced by the radaton balance at the surface, ncomng solar radaton and outgong longer wave radatons, and the heat fluxes (sensble plus latent) to the atmosphere and down n the ground, see secton 11. The humdty at the surface wll be a complcated functon of the water budget wthn the sol and the water flux above the sol. Only over free water surfaces humdty becomes smple, because the water vapour pressure at the water surface wll be the saturated water vapour pressure at the temperature of the water surface. Just as the heat balance at the surface wll be nfluenced by evaporaton and water vapour fluxes, also the water vapour budget wll be strongly nfluenced by the heat budget. From a modellng pont of vew, both surface values and surface fluxes of many varables are nether very relevant nor well defned. They are not well defned because many surfaces are not well defned on closer nspecton, lke e.g. vegetated surfaces. They are not relevant, because the relevant flux at the surface takes place between a reservor n the ar and a reservor below the surface. Examples here are trace gas exchange across the water nterface, whch s parametersed on the dfference between the water concentraton and the ar concentraton, both specfed well away from the nterface. Also gas fluxes between vegetaton and the atmosphere wll often be controlled by the concentraton dfference wthn the vegetaton and n the atmosphere, coupled through a stomata resstance. For both cases the exact values at the 134 Lectures n Mcro Meteorology

136 surface become rrelevant, because the atmosphere and the layer below s coupled now through the boundary flux only. Resstance modellng A useful concept for such consderatons s the resstance-models that allow us to handle such surface flux consderatons. In the next fgure, we have an upper and a lower layer, where the concentraton of a substance s controlled by a dfferental equaton DC DC (7.33) = F( C,), and = G( C,) Dt Dt The soluton of these equatons nvolve among others the boundary values and fluxes Fgure Illustraton of exchange between to volumes, one above and one below the surface, The fluxes between the two layers are determned under the assumpton of statonarty and horzontal homogenety, meanng that the vertcal flux s constant through the layers. Hence we have: (7.34) F Fs = Fv = Fbs = F +, where F +- s the flux across the nterface. For F s we fnd, for example usng K-theory, whch we developed for surface layer turbulence: (7.35) C ku z C ku zm 1 Fs = K = ( Ch C ) ( C ); s δ h C s δ z ϕ ( z/ L) z ϕ ( z/ L) h δ r C C s a where we have defned a resstance for fluxes through the turbulence surface layer, r a, subscrpt a stands for aerodynamc. Smlarly we get for the flux through the vscous nterfacal layer: C D 1 (7.36) Fv = D = ( Cδ C0+ ) ( Cδ C0+ ) z δ rδ As of now we have presented no model about the fluxes at the nterface and below the surface, they may be trough stomata or roots or nvolvng chemcal bndngs or smlar strange pathways. Lectures n Mcro Meteorology 135

137 Whatever they are, we assume that the processes carryng the fluxes allow us to cast them n the resstance frame, hence: 1 1 (7.37) F = ( C0 C0 ), Fbs = ( C0 Cd ). + r + r The soluton of ths system s easly seen to be: (7.38) rf = ( r + r + r + r ) F = ( C C ); + a δ + bs hs d bs Ths system shows analogy wth Ohm s law, wth the flux as a current and the concentraton dfferences as voltages. The total resstance of a flux path s seen to equal the sum of resstances, smlarly a parallel pathway for the fluxes, can be computed wth a parallel resstance that wll combne wth the total lke parallel resstances n electrc network. Fnally, the fact that all the fluxes are equal means that the concentraton gradents wll be large, where the resstance s large, and vce versa. A lmtaton to the resstance concept s need for statonarty for the system to work that s for the flux to be constant through all layers. Hence, when the transport tme along a flux path way become too long, one must change to dfferental equatons n the dfferent layers and the modellng becomes more complex. The last fgure shows the system of layers usually employed when estmatng fluxes across the ar sea-surface, usng a resstance formulaton: Fgure Schematcs of a full resstance model for the atmosphere ocean exchange. All the fluxes shown on the fgure are to be modelled by the domnant processes at each layer Lectures n Mcro Meteorology

138 Dscusson. In ths secton we have dscussed the characterstc of the surface boundary condtons for the atmospherc boundary layers. We have seen that they typcally are specfed n terms of roughness parameters and surface values. Velocty has the smplest surface value and roughness characterstcs, n that the wnd speed s zero at the ground due to the no-slp condton. We have seen that n general the surface characterstcs nvolve the momentum transport to the surface by Brownan dffuson and through flow separaton around roughness elements. Hence, there s a mnmum roughness, as gven by the molecular term n (7.19). Asde from ths, at the smplest the roughness can be descrbed as a landscape parameter, as s presented by some of the characterstc landscape drawngs and the schematc summary of landscape and roughness. In the next approxmaton, the roughness may be expected to depend on wnd speed, strongest for water surfaces, but also for dry sand and vegetaton. Characterstc seasonal changes must be expected as well, reflectng seasonal dfferences n the vegetaton (leaves and growth of agrcultural crops) and ce and snow cover (of obvous mportance, but not dscussed n the secton). Indeed, the roughness changes wll often be seasonal, but obvously n a detaled analyss they are assocated wth physcal changes of the surface characterstcs that n average (but not always and not only) follows the changng season. Indeed, as we ponted out, when roughness elements reach nto the turbulent boundary layer, the roughness may even depend on stablty, because the eddes encountered are now modfed by stablty. Close to the surface the transport of scalars can utlse only the molecular dffuson, as opposed to momentum transport. We have accounted for ths through a roughness Reynolds number dependency. The scalar roughness s shown to be velocty dependent as well, both through the wnd roughness and n ther relaton to ths roughness, nvolvng the roughness Reynolds number. They furthermore has an uncertanty related to that the surface value of the scalars s not forced to zero by the no-slp condton, as for velocty, but must be determned ndependently from measurements or modellng. Addtonally, t should be ponted out that t often s not possble to wrte a functonal dependency between z 0 and the scalar roughness, because they can dependent on dfferent features of the surface and nvolve addtonal processes n the flow. In spte of these uncertantes the scalar roughness for many substances s often found, followng a roughness Reynolds number relaton, gvng about a tenth of the wnd roughness, when ths roughness nvolves the flow separaton around roughness elements. If modellng of the surface fluxes and values are chosen, one must consder also the condtons n- and below the surface as presented n ths secton through the resstance modellng, and n secton 11 about the energy balance at the surface. Lectures n Mcro Meteorology 137

139 Appendx 7A. Characterzaton of a water surface. The marne ABL has qute dstnct features compared to the land ABL, reflectng the specal characterstcs of the water both dynamcally and thermally. Therefore, we shall shortly summarze these characterstcs n ths appendx. The water s sem transparent meanng that the radatonal heatng and coolng s dstrbuted downwards. Addtonally the water s very effcent n redstrbutng heat vertcally. The surface waves and crculaton systems, lke the Langmur Cells, combned wth turbulence gve rse to extensve mxng. Addtonally, when heated from above, the surface water evaporates, and t wll start snkng; now beng heaver, because t retans the salt from the evaporated water. If the surface water cools, t also becomes heaver due to the coolng and snks. All ths gve rse to an ntense mxng n typcally the upper 10 meter of the ocean. In the heat exchange wth the atmosphere the water therefore consttute a very large heat reservor that only can change ts temperature slowly, and addtonally has ts own heatng and coolng from the ocean currents. Indeed when an ar mass moves over an ocean for enough tme, t ends up at the temperature of the ocean surface. For these reasons the homogeneous marne ABL s always close to neutral. The durnal radaton cycle shows very lttle nfluence on the water surface temperature, although t can be measured, but typcal ampltudes are less than a few tenths of a degree (Pena et al, 008). The annual radaton cycle on the other hand has sgnfcant nfluence on the sea temperature, because they nvolve enough heat and tme to change both the temperature and the depth of the mxed layer. However, stable and unstable condtons happens over the ocean as well on shorter tmescale,, but they are mostly transtonal, assocated wth ar masses movng across water surface wth a dfferent temperature, ether comng from a nearby land or assocated wth movng weather systems. We shall return to these phenomena when comng to the nhomogenous and nstatonary ABLs n secton 9. The sea s also an obvous source of water vapor, ndeed over the ocean, q 0, the surface value of the water vapour mxng raton, s derved from the saturated pressure at the surface temperature. The ocean s also a source of lqud water n the form of sea spray convertng to marne aerosols. In wnter tme the spray s the source of cng on shps and offshore structures. The roughness elements over water mostly take the form of small steep waves of a wave length of around 5 cm, although momentum can be transferred also by larger scale breakng waves. The ocean surface s depcted n Fgure A1. Snce the roughness s assocated wth the waves and the waves are generated by the wnd and modfed by gravty, Charnock (1955) proposed that the roughness should depend on u and g. A slghtly updated verson of the roughness for water looks as follow: ν z 0.11 ( c/ u, ) u = + α (7A.1) 0 u g Whch we have taken from (7.19), now wth an estmate of the coeffcent β, and the Charnock coeffcent,α, now beng a functon of the phase speed of the domnant wave c, and the frcton velocty, u, snce the roughness elements wll be mowng wth the phase speed of the domnant waves n the drecton of the wnd. The term, c/ u, s denoted the wave age, because c ncreases wth the duraton of the actng wnd. The Charnock coeffcent,α, s varyng between 0.01 and 0.07, beng smallest for md-ocean mature waves wth large phase speed. A typcal value for regonal seas s 0.015, α can be a functon of other parameters as well: e.g. bottom topography, swells, both modfyng the waves and ther drecton of propagaton, and very hgh 138 Lectures n Mcro Meteorology

140 wnd (e.g. hurrcanes) results n foam covered waters that reduces β further (Makn, 1997). As seen n α (A1) may also depend on other varable, here denoted by -. Other varables may be a sutable expresson for the wave heght or wave steepness, ether overall or n certan frequency bands, a characterstc wave length for the waves, etc. We shall return to ths later. Fgure 7A.1. The wnd profle close to the water surface, wth the wave nduced vortcty and the small scale roughness element rdng on the larger scale waves, wth a phase speed c. Also the rotor movement of a water wave s shown. A short summary of the surface waves s now useful; The surface wave ampltudes η( x, t) s a sem perodc statstcal functon of space and tme. Smlar to turbulence, one can consder wave spectra of ether wave numbers or frequences, connected through the dsperson relaton. η( xt, ) acos( k x ωt) dsperson relaton : ω = gk(1 + γk / g) tanh( kd) phase speed : ω k g k g k c = = kk ω k kk The requres kd >> and γ k g << " " 1 / 1 (7A.) The wave ampltude depends on the tme or the fetch that has been avalable for the wnd acton on the water surface. Young waves have small wave lengths, large steepness, and small phase speeds. The waves are generally produced wth propagaton along the wnd drecton ± 5. Older waves are characterzed by larger wave length and lower frequences, wth larger phase speeds, larger ampltudes and lower steepness. Some waves propagate from afar, generated by other wnd felds. They are typcally of even longer wavelength and lower frequences.they are denoted swells. Waves can be dffracted by bottom topography. Fgure A llustrates typcal wave ampltude spectra as functon of frequency. The red one pertans to coastal areas, whle the black broken curve reflects more md-ocean condtons. Lectures n Mcro Meteorology 139

141 As dfference frequency regons of the spectra can propagate n dfferent drectons and wth dfferent phase velocty, one often uses the characterstcs at the peak, e.g. ω p, c p and S(ω p ) to characterze the whole spectrum, snce the spectrum s farly narrow wth a sharp peak. Fgure 7A.. Two characterstc spectral functons, The Person Moscwtch (PM) and the JONSWAP form. The PM form s generally used over the open sea, whle the JONSWAP form s used n coastal areas. (IEC6140-3, 009) The peak frequency s found to vary wth fetch as (Frank et al, 000). Alternatve to fetch one can use tme snce start of producton wth t=x/u 10. c p / u π xg = 35 U 10 1/3 (7A.3) Based on the above coeffcent we can now show the Charnock coeffcent,α, versus the wave age n Fgure A3. It should be emphaszed though that many wave felds are so rregular that the scatter around functonalty depcted n the fgure can be farly large. As seen even wth the selected data n the fgure the scatter s large. Much of the scatter s related to occurrence of swell that per defnton are not local and hence could not be expected to be scaled wth local varables, as s mpled by Fgure A3. The type of behavor, depcted n Fgure A3, s generally used for coastal regons wth short fetches for offshore wnds. The scatter mples that the uncertanty s hgh, and also other expressons exst, as mentoned n connecton wth (A1). An example s shown n the next formula, beng due to Taylor and Yelland (001), who as well dscusses many other formulatons: B z 0 H S = A (7A.4) HS LP Where H S, the sgnfcant wave heght s defned as the standard devaton of the waves multpled 4. L P s the wave length of the domnant waves, meanng for the waves at the spectral 140 Lectures n Mcro Meteorology

142 peak. Hence ther rato can be taken as the wave steepness. For the coeffcents we have A 7, B 4.5. Equaton (A4) can be used for vrtually all wave condtons, open sea and coastal areas, hgh wnds and low wnds, wth a scatter smlar to the one seen n Fgure (A3). One reason that so many forms exst s that the dfferent forms can be transformed to each other because surface waves are such smple varables that many characterstc are related, and all the dfferent forms have been ftted to much the same data sets. The form n (A4) for example wll reflect much the same dependence of fetch or duraton as do the form n Fgure (A3), because Hs, Lp wll depend on wnd speed, and fetc, reflectng the varaton of the wave spectrum, as also seen n (A3). Fgure 7A.3. The Charnock s coeffcent, here denote A c, s shown versus recprocal wave age, u /c. Typcal wave ages n nature s between 5 and 30, (Franck et al. 000). The data for u /c> 0.5 reflects water tunnel stuatons, where the wave feld s just beng formed by the wnd, and they are very steep and rough wth c beng small Here the roughness ncreases the more the wave spectrum-and the wave feld- bulds up wth those young waves. For u /c< 0.5 and decreasng, the phase speed of the waves ncreases untl the phase speed s of the order of the wnd speed, at what tme the waves can only extract lttle momentum from the wnd. The roughness therefore decreases u /c n ths regon. Fnally we should menton that n expermental efforts to determne the surface condton over the sea, one often operates wth the so-called drag coeffcent, defned from as u κ D10n = = + 10n U z = 10m 10n C a bu ln( ) z 0, (7A.5) C D10n s the drag coeffcent referred to a wnd at the heght of 10 m correspondng to n meanng neutral condtons, where one for standardzaton refers the measured the wnd speed to z = 10 m and neutral condtons, usng the Monn-Obuchov formulatons. In (A5) the second term consttutes the defnton, the thrd term apples the logarthmc wnd profle wth k beng the v Karman constant, and show the relaton to z 0. The fourth terms s a typcal expresson used, wth a and b 0.07, when the wnd speed s measured n m/s. C D10n s purely emprcal Lectures n Mcro Meteorology 141

143 but has the advantage of drectly relatng the surface stress to the wnd speed, wthout nvolvng z 0. Expermentally t s drect to estmate from estmates of u * and U(z). As seen from (A5) t s possble to derve z 0 from C D10n (Geerrnaert,1990). In spte of the functons shown n Fgure A3,and from (A4) the roughness of the sea surface remans one of the smallest, one can encounter n nature. Ths means that hgh wnd speeds wll be less effcent n forcng the stablty towards neutral over water than over land, although also over water frequency of neutral stablty ncreases wth wnd speed. Stll hgh wnd can be encountered assocated wth strongly stable flows over water, agan reflectng an nhomogenous stuaton where warm ar s advected over cold water, and the frcton almost dsappear. Agan we shall return to ths ssue, when dscussng nhomogenous boundary layers. Just as wnter snow can modfy the roughness of a land surface strongly, the wnter wll some part of the world cover the water wth ce, and the roughness now wll depend on the characterstcs of the ce surface, rangng from extremely low for smooth sold ce, to qute rough for pack ce. The small z 0 also means that the turbulence typcally s lower over the water than over land, reflected also n a lower ABL heght over water than over land. Addtonally small z 0 means that the z 0T and z 0q are close to z 0 for low wnd speeds, wth a small roughness Reynolds numbersee (7.6, 7.6) and start devatng only for rough pack ce or larger wnd speeds, wth rough sea.. 8. Scalng n the atmospherc boundary layer. Asymptotc Scalng We start wth the fgure from last secton about the regons n the atmospherc boundary layer. 14 Lectures n Mcro Meteorology

144 Ln (z) h Boundary Layer Proper h 0.1h s h sl 10 z 0 z 0T 0.1 z 0 surface Surface Boundary Layer Vscous nterfacal layer Fgure 8.1. The sub-layers of the atmospherc boundary layer, and ther approxmate heghts of separaton. We consder the two Layers, the surface boundary layer and the boundary layer proper, above. The coordnate system s algned wth the mean surface wnd, and we start consderng neutral condtons only. z0 << z hs << h u 1 z (8.1) = ln( ); u k z0 v = 0. u For an upper part of the atmospherc boundary layer we have, the velocty defect profle, estmatng the devaton between the Geostrophc wnd and the actual wnd, from the top of the boundary layer and down to a heght where the roughness becomes an mportant parameter: (8.) z << z h 0 ; u u g 1 = f vw, z 1 v vg = uw ; f z C C As to the nterval of valdty, we have learned that the logarthmc law s vald for z about 10z 0 and up to h s. For (8.) we just know t to be vald for z = h down to agan a heght much larger than z 0, but not as close to z 0 as the surface layer formulaton, because the formulaton does not contan z 0. The last equatons, we can wrte as: Lectures n Mcro Meteorology 143

145 (8.3) z << z h; 0 u u u vw vw z ( ), g = = = F u fc z u ( z/ h) u h v v u uw uw z ( ); g = = = G u fc z u ( z/ h) u h u wth h f C It s seen that we can formally wrte the equatons as functon of z/h only. Based on the above, we argue that t seems that we can wrte the wnd profle as: z z h u ug u = F( ξ ), (8.4) v vg u = G( ξ ); wth u ξ = z / h = z / fc and z0 << z hs << h (8.5) u 1 z = ln( ) = f ( η ), wthη = z / z0. u k z0 v u = 0. Hence, we have assumed that we have two heght ntervals, one upper where the proper heght varable s ξ, and one lower, where t s η. We now assume that there s an nterval for z for z 0 <<z<<h, where both expressons are vald. Ths s formulated that both expressons yeld the same normalsed wnd speed gradent n ths heght nterval. From the upper layer equaton we have: (8.6) u F ξ u F = u =, z ξ z h ξ z u z F F or = = = ξ u z h ξ ξ For the lower layer: (8.7) u f η u f = u =, z η z z0 η z u z f f or : = = η u z z 0 η η Hence we have: 0 << ; 144 Lectures n Mcro Meteorology

146 (8.8) z u f = η = ξ F for z0 << z << h u z η Ths s also formulated as that the two branches of the velocty profle are matched asymptotcally for ξ 0 and η smultaneously. From (8.8) we can argue that these two functons can only be equal to each other f they are constant, snce they do depend on two dfferent varables, ξ, and η. Choosng ths constant as the v. Karman constant, we get: (8.9) η f 1 1 = f ( ) ln b wth z / z0 ; for z0 z h η k η = k η + η = << << (8.10) ξ F 1 1 = F( ) (ln A), wth z / h.; for z0 z h ξ k ξ = k ξ + ξ = << << where A and b are two constants of ntegraton. ξ We have now determned the two functons, f (η) and F (ξ ) n ther overlappng nterval of valdty. That s, recallng the defnton of the functons: (8.11) u 1 z = f ( η) = ln u k z0 ; for z0 << z hs << h and (8.1) u ug 1 z = F( ξ ) = (ln + A) ; for z0 << z << h u k h Here we have chosen the ntegraton constant, b = 0, n accordng wth our knowledge about the logarthmc profle. The ntegraton constant A s determned from data to approxmately. An nterestng possblty s to subtract the two equatons to yeld: ug 1 h (8.13) = (ln A ) ; u k z0 In the prncple t s only vald n the matchng nterval, where u can be wrtten wth both expressons, but the dfference have no heght varaton, and t s seen to consttute a statement on the relaton between the surface stress and the Geostrophc wnd ncludng the boundary layer heght and the surface roughness. Correspondngly, one fnds for the v-component: v = 0; for z0 << z hs << h u (8.14) v vg = G( ξ ); for z0 << z << h u Agan, we take the dfference, and get: vg z B (8.15) = G( ) const., u h k Lectures n Mcro Meteorology 145

147 where we see that G n the consdered heght nterval must be a constant snce v g /u does not vary wth heght. The B value found from data s around 5. As usual the v.karman constant s ntroduced for later convenence. The above relatons between the surface stress and the Geostrophc wnd s called Resstance Laws drawng on an analogy between current and voltage, wth the surface stress beng the one, the Geostrophc wnd the other and the expresson then takng the role of resstance. The two components of the Resstance Law Equatons can be combned: u G = ug + vg ; h = ; f (8.16) C 1 kg h = ((ln A) + B ) ; u z0 vg h α = arctan( ) = arc tan( B /(ln A)); u z g Lettng B 5, A 0, u * 0 cm/s, z 0 10 cm and f C 10-4, we get tanα 0.5 and α 7. It s seen that α ncreases wth z 0 and that overall we fnd a somewhat more realstc α than from the smple Ekman soluton.. For non-neutral cases and extenson appears straght forward, usng the Monn-Obuchov length scale, L, combned wth the boundary layer heght, h, to create a stablty parameters: µ h/l, such that: A= A(µ) ; B = B(µ). Ths means also that α wll change wth stablty, n accordng wth known evdence. Fgure 8. llustrates the varaton of A and B wth µ. As apparent from the fgure the scatter becomes very large, when such an approach s appled. Typcal values and varatons of these functons are as llustrated, whch show as well that there s an enormous scatter on the estmates. Ths reflects probably, that for non-neutral condtons these functons nvolve a smplfcaton that s too lmtng to descrbe the realty reflected n the data. Indeed, t seems that a reason for much of the scatter s to be found from baroclnty, see secton 5, emphaszng the fact that on a boundary layer scale truly horzontally homogeneous condtons are rare, at least over land. Also, the formulaton of the resstance laws nvolves some arbtrarness, of whch the heght of the boundary layer s one of the more uncertan. The neutral boundary layer s typcally taken as proportonal to u */ f C, wth some arguments about the constant of proportonalty, whch s normally taken as 0.3, but coeffcents between 0.1 and 1.0 has been chosen n the lterature. Such choces wll of course nfluence the qualty of the ft, especally when several data sets are been used n the model evaluaton. For non-neutral boundary layers the u */ f C scalng s not suffcent, and several other models for the boundary layer heght s proposed, as we shall see later. Indeed, many of these models have that the boundary layer heght s not determned by smple scalng at all, but has to be descrbed by ts rate equaton Lectures n Mcro Meteorology

148 Fgure 8.. The stablty varaton of the resstance law coeffcent A and B versus µ h/l from Melgarejo and Deardorff (1974). To ths should be added the expermental uncertanty assocated wth the expermental determnaton of the surface flux and the roughness parameter. As we shall see later n secton 9, surface parameters determned have to be determned as averages over km. A parameter determnaton that s obvously both dffcult and not well defned. The farly complcated structure of the Resstance Law (8.16 ) has nduced many to try to develop smpler expresson, wthout sacrfcng the not very mpressve accuracy of (8.16). Here we can menton. From Jensen (1978) we get: (8.17) u / G 0.5 / ln( Ro) wth the Rosby number, Ro = G / z0 f C. For strongly unstable stuatons, Wyngaard et al. (1974) fnds: (8.18) u / G k / ln( L/ z0). Smlarly to resstance laws for wnd speed they have been defned as well for scalars, such that. θ ( h) θ0 qh ( ) q0 (8.19) = H( µ ); = Q( µ ); θ q Also, here the scatter between model and data s tremendous. To the uncertantes mentoned above, uncertanty on determnaton of θ ο and q 0 should be added, see secton 7. Lectures n Mcro Meteorology 147

149 The resstance laws, derved above, were obtaned by subtracton of the surface layer profles form the general velocty defect profles n the asymptotc matchng regon, where both profle expressons were supposed to be vald smultaneously. The velocty defect profles allow us as well to derve estmates of the profle expressons throughout the boundary layer as functon of ξ= z/h. Several authors have ftted profle data from tall masts to the velocty defect curve. Here we cte data of v.ulden and Holtslag (1980), based on data from several 00-meter towers combned wth extensve pressure maps analyss, provdng the Geostrophc wnd. We repeat the velocty defect expresson: (8.0) z0 << z h; u ug vw = = F( ξ ), u ξ u v vg uw = = G( ξ ); u ξ u u wth ξ = z / h = z / f We know that F(ξ) and G(ξ) have to be consstent wth the expressons derved above for ξ 0 n the overlappng regon wth the surface layer expressons. Also we have that F(ξ) and G(ξ) 0 for ξ 1, where the velocty becomes the Geostrophc wnd. Addtonal ntegral constrans on F and G can be derved, notng that: v w = 0, u w = u for ξ 0. (8.1) v w = 0, u w = 0 for ξ 1. Integratng (8.0) wth respect to ξ from 0 to 1, and usng (8.1) we obtan: (8.) 1 1 F( ξ ) dξ = 0, G( ξ ) dξ = 1, 0 0 V.Ulden and Holtslag obtan the followng curves and expressons: C Fgure 8.3 The two non-dmensonal velocty defect profle. The functons ψ v, ψ u correspond to the functons G and F n the dscusson above. The pont symbols correspond to dfferent data (v. Ulden and Holtslag, 1980). 148 Lectures n Mcro Meteorology

150 The curves n Fgure 8.3 are descrbed through the followng functons: (8.3) u ug 1 ξ = F( ξ) = ψu ( ξ) = (ln ξ + A+ aξ)(1 ) u k c v vg 1 ξ = G = B+ b u k c ( ξ) ( ξ)(1 ) ; The coeffcents are ftted wth c 0.3, a 15, and b -8 (based on A=1.9 and B =4,7). The second order term s obvously made to force the functons to zero at the heght of the boundary layer. Note, the above fgure and equatons show estmates of the F and G functons for the entre nterval of valdty, not just the overlappng nterval wth the surface layer profle As noted above, the boundary layer heght s only one thrd of the scale heght, when the boundary layer heght s understood as the heght, where the velocty defect becomes zero. The Geostrophc wnd components are as derved n the resstance law. A smpler expresson s obtaned from Larsen, et al (198), forcng the profles towards the drag law formulaton at ξ 1. Introducng the resstance laws formulatons for u g and v g, from (8.13) and (8.15) we obtan: u z z z z u = (ln A + a (1 ( ))), k z0 h h h (8.4) u z γ v = B ( ) ; k h whch as seen converts to the drag law f z = h. Ths s of course not consstent wth that the Geostrophc lmt should be reached for z = 0.3h as n Fgure 8.3, whch s uncertan anyhow. Wth A=, B=5, a = 10, equaton (8.4) predcts that u ncreases more wth heght that n the logarthmc layer, above the surface layer, correspondng to (8.3), whle v essentally ncreases lnearly wthn the boundary layer. The power γ (beng around one) can be used to regulate the behavour of the equaton for z approachng h. Choosng A and B from Fgure 8., and lettng a depend on L (the Monn-Obuchov stablty length), from secton 6, we can estmate the profles through the boundary layer also for non-neutral condtons. In the orgnal Larsen et al (198) a=0 and γ = 1 were used, and t should be emphaszed that the behavour of the profles wthn the general boundary layer s presently far from beng resolved. However, the results presented n Fgure 8.3, as gven by (8.3), wth a faster ncrease than logarthmc of u(z) s also found by Grynng et al.(007), who usng a dfferent theoretcal approach and data from several tall land masts, also determnes the wnd profles over a wde range of thermal stabltes, see Fgure 8.3a. Lectures n Mcro Meteorology 149

151 Fgure 8.3a. Wnd profles n the boundary layer, from Grynng et al (007). Data compared wth models. The model of Grynng et al s an extenson of the surface boundary layer wnd profle, as dscussed n secton 6. The profle s derved from: du u = ; = + + dz κ LSBL LMBL LUBL, α u = u (1 z/ h) ; L = z/ ϕ( z/ L) 0 where the surface length scale z s extended wth length scales pertanng to the Surface Boundary Layer, the Mddle Boundary Layer and the Upper Boundary Layer, and u change wth heght through the boundary layer. The dervaton of expressons for the three length scales can be found n Grynng et al (007). Specfcally for L SBL one can compare wth (6.3) n secton 6. SBL Thus, the wnd profle throughout the entre neutral boundary layer s slghtly more complcated and uncertan than for the surface boundary layer. Another mportant dfference may be that, whle a large fracton of the wnd stuatons n the surface layer can be charactersed by the neutral wnd profle, the neutral boundary layer profle characterses comparatvely fewer stuatons at the top of the boundary layer. Thermal structures of both the boundary layer and the ar above are expected becomes more mportant at these greater heghts. Recall for example that the buoyancy term n the Monn-Obuchov formulaton s gven as z/l, and hence ncreases lnearly wth heght. Wth ths n mnd we shall now summarse the scalng formulatons pertanng to the whole boundary layer and nclude specfcally the thermal structures. Summary of scalng laws for the boundary layer. We now try to summarse the total famly of scalng laws that are n use wthn the atmospherc boundary layer. 150 Lectures n Mcro Meteorology

152 We notce that we have seen three length scales n our dscussons, so far: The Monn-Obuchov stablty length scale, L, and the boundary layer heght, h or as t s often denoted for unstable condtons, z I, and fnally the measurng heght, z that s the vertcal characterstc heght n the boundary layer.. Also, when we have dscussed eddy szes wthn the boundary layer, we have notced that the largest eddes scale wth the boundary layer heght. Further, we have notced that at each measurng heght eddes are produced wth scales of the order of the measurng heght. For unstable condtons the large eddy structure s partcular clear. Fgure 8.4. Structure of the unstable atmospherc boundary layer. The boundary layer sze eddes are partcularly clear for these condtons (Wyngaard, 1990). Below we repeat as well the structure of the stable boundary layer, whch as seen s qute dfferent from the unstable boundary layer, wth the boundary layer eddes beng much less obvous, but stll wth an dentfable boundary layer heght. Fgure 8.5 Structure of the stable atmospherc boundary layer. The boundary layer sze eddes are here suppressed by the buoyancy. Turbulent mxng s relatvely smaller than for unstable condtons (Wyngaard, 1990). Fnally, we have dscussed qute ntensvely the structure of the neutral boundary layer, wth ts Ekman spral, resstance laws and ts boundary layer heght. In Secton 6, we dscussed the surface layer and the Monn-Obuchov scalng as well as the general scalng methodology. Especally we used the equatons for turbulence varance to Lectures n Mcro Meteorology 151

153 dentfy relevant scales. Here, we repeat the varance equatons for the wnd and the temperature. For smplcty, we wll neglect the humdty and other passve scalars n the present dscusson (8.5) (8,6) 1 de gw ( θ v ) u 1 ( we ) 1 ( wp ) = 0 = uw ε; dt θ z z ρ z v kz 1 de z z z z z z z = 0 = + ϕ ( ) ( ) ( ) ( ); 3 m ϕtt ϕtp ϕε u dt L L L z L L z L L ( ) ( ) L L 1 dθ θ 1 ( w θ ) = 0 = θ w εθ, dt z z kz 1 dθ z z z z = 0 = ϕ ( ) ( ) ( ), θ ϕtθ ϕεθ u θ dt L L z L L ( ) L Based on these equatons we derved the followng mportant scales for the turbulence surface boundary layer: (8,7) u uw; uθ θ w; uθ θ w; g/ θ ; z; L; * * * * * v We now ask whch addtonal parameters, we should nclude descrbng not only the surface boundary layer, but the whole boundary layer between the top at heght, h, and to the bottom of the surface boundary layer. As n Secton 6, the wsh s to descrbe local varables wthn the boundary layer by means of the chosen scalng parameters. By local varables, we mean varables that can be derved solely wthn the boundares of the layer, such as horzontal and vertcal gradents of mean values and fluctuatons, and the fluctuatons themselves. An example of varables that cannot be derved wth the boundares of the layers s the mean values lke <u>, <θ> and smlar quanttes that all take knowledge about the condtons at the surface or above the top of the boundary layer or both. From our earler dscussons, we see that we at least have to add the followng addtonal parameters: The Corols parameter, the boundary layer heght and parameters descrbng the fluxes through the top of the boundary layer from the atmosphere aloft. (8.8) f,,,, C h uw vw θ w h h h Addtonally, we would expect the need to specfy external parameters outsde the boundary layer, such as the surface values z 0, z 0T, u 0 and θ 0, wth addton of the smlar values at the top, G and θ h (z), snce we know that the background temperature gradent above the boundary can be mportant. As n Secton 6, the usefulness of the scalng formulatons dmnshes f the number of relevant scales to a problem s too large. The soluton becomes too complcated and, general. v Hence, there has been an effort to dentfy sub-sectons of the boundary layer, where smpler and therefore more useful scalng laws apply. Snce, we have three length scales for the boundary layer, t customary to organse the scalng laws usng pars of ratos between the scales, z/h and z/l, or h/l and z/h. Below s shown a 15 Lectures n Mcro Meteorology

154 schematcs based on z/l and z/h, because t s the smplest, and allow us to buld drectly on the surface boundary layer scalng laws, we start the dscusson here. Fgure 8.6. Scalng regons for the stable and unstable boundary layers from Olesen et al. (1984).The lnes and ponts ndcated n the surface layer part of the dagram refer to data sets from the Kansas and Mnnesota experments used to formulate and evaluate much of the scalng laws, dscussed here (Olesen et al, 1984). As seen the boundary layers are separated nto stable and unstable boundary layers, wth neutral n between. For z/h less than about 0.1h we fnd surface layers, where the man scalng parameters are: u, θ q and z/l. For stable condton we have seen that as z/l ncreases, z dependency tend to dsappear from many of the scalng laws. Therefore the regme s called z-less. As example, we consder the velocty gradent for stable condtons: (8.9) z kz u z ϕm ( ) = = 1+ 5 L u z L u u z u z = (1 + 5 ) 5 for z kz L kl L From the equatons we see that the velocty gradent becomes ndependent of z (z-less) for large z/l. Ths then goes for all the other varables, snce all the ϕ -functons, we know, has a smlar form for stable condtons, and ends up beng proportonal to z/l for large L. For the z-less regmes and further up nto the poorly understood regon one often tres a local scalng system, where the approach s to use local fluxes as scalng parameters, rather than surface fluxes as done as part of the Monn-Obuchov smlarty system. The argument s that the surface fluxes cannot be relevant snce z-less means that the vertcal flow gradents do not feel the dstance to surface, hence the local fluxes become more relevant. Local flux means that the fluxes have to be estmated n the same heght as the mean gradents. Ths local flux works as follows, as s llustrated on for the gradents of temperature and wnd speed: (8.30) uw = u ; θ = θ w / u z z z z z Wth these fluxes we can now normalse the gradents, ether usng the boundary layer heght, h, or a local Monn-Obuchov length scale,λ defned through the local fluxes n (8.30). Examples Lectures n Mcro Meteorology 153

155 of the scalng employed are shown n the next equaton. As seen we can employ Λ and h several ways n a scalng formulaton. kz u z kh θ z (8.31) = fu ( ); or = fθ ( ); u z Λ θ z h z Whch of the scalng approaches one tres depends on f one s closest to the top of the boundary layer or one s closer the heghts, where Monn-Obuchov apples. To make the system useful one has for both formulatons to specfy how the local fluxes vary wth heght. From measurements, one fnds the followng approxmatons: (8.3) z u z z w θ z = (1 ) ; = (1 ) u h w θ h 0 0 α1 z α where α1= 3/4 and α =1 seem to gve the best ft. Formally t s seen that wth (8.3) the surface scalng and the local scalng formulatons used n (8.31) can be related to each other. Although expressons lke the above wll work for some parameters throughout the stable boundary layer, the regon above the surface layer s stll less well understood. The regon s charactersed by very small vertcal fluxes, wth turbulence appearng as ntermttent bursts, rather than an ongong contnuous process. Addtonally the sgnal fluctuatons are often of nternal gravty wave type rather than turbulence type. The characterstcs of varances and spectra n the stable boundary layer can to a large extent be smoothly extrapolated from the surface layer, where the peak frequency for the power spectra move to larger normalsed frequences for ncreasng z/l, as ndcated by Fgure 6.9, and the formulas for the varances n (6.31) reman largely vald. Before we move onto the unstable sde of the dagram, we shall make some general consderatons about the heght of the boundary layer. As we have dscussed for the neutral boundary layer a best estmate of ts heght seem to be: (8.33) 1 u h = f 3 C For not too stable boundary layers the followng regresson formulaton offers a far approxmaton: (8.34) h CuL ( / f C ) 0.5, where L s the Monn-Obuchov stablty length scale, and C s found between 0.7 and 0.4. Ths boundary layer heght s seen to be very close to a geometrc mean between the neutral heght and the Monn Obuchov stablty length scale. It s seen that heghts less than 100 metres s qute realstc. There s however a prncpal aspect. For the horzontally homogeneous statonary boundary layer that grows nto a lkewse thermally neutral free atmosphere, there are not many scales to choose from, when one constructs a boundary layer heght. Bascally the boundary layer heght s the upper lmt for mpact of the frcton aganst the ground and for the assocated turbulence generaton. Here u * /f C appears as relevant parameters, and the coeffcent n front s also of the order of one. The only other velocty avalable for the expresson would be the Geostrophc wnd, whch obvously yeld much too hgh a boundary layer heght. 154 Lectures n Mcro Meteorology

156 If the thermal forcng s mportant, whch t s by defnton for stable and unstable boundary layer, then the growth of the boundary layer wll obvously be nfluenced both by the thermal propertes of the boundary layer and of the layers aloft. The thermal propertes of the boundary layer wll obvously depend on the durnal cycle as n Fgure 8.7, whch was shown before. For such boundary layers, t s more correct to magne that the heght s determned by ts own rate equaton, descrbng local changes as well as advecton. dh h h = + u = w + w + w dt t x (8.35) j s e ls, where the three vertcal velocty scales reflect rse n boundary layer heght due to local surface fluxes, w s, due to local fluxes through the top of the boundary layer from entranment (mxng across the boundary layer heght) processes, w e, and due to large scale processes such as subsdence, w sl. Such formulatons are often used for determnaton of h. For the unstable boundary layer there s an especally smple formulaton descrbng the ncrease of the boundary layer heght as a functon of the ncomng solar heatng of ground. We shall derve ths equaton later. For now we wll assume that the structure of the boundary layer below the boundary layer heght can be derved from scalng laws, ncludng among other parameters the boundary layer heght, even when the boundary cannot be consdered statonary anymore, because the boundary layer heght s determned from a rate of change equaton. j Fgure 8.7. Durnal varaton of the boundary layer from Stull(1991). We now cross over to the unstable sde of the dagram remanng n Fgure 8.6, startng n the surface layer. We see two layers, a shear domnated layer and convectve matchng layer, also called the layer of free convecton. The defnton of these relates to the turbulence wnd varance equaton, as we derved earler. It s shown below n both before and after scalng accordng to the Monn-Obuchov smlarty. Lectures n Mcro Meteorology 155

157 (8.36) 1 de gw ( θ v ) u 1 ( we ) 1 ( wp ) = 0 = uw ε; dt θ z z ρ z v kz 1 de z z z z z z z = 0 = + ϕ ( ) ( ) ( ) ( ); 3 m ϕtt ϕtp ϕε u dt L L L z L L z L L ( ) ( ) L L The two producton terms, the buoyancy producton, becomng z/l, and the shear producton, becomng ϕ m (z/l). ϕ m (z/l) start out as 1 at neutral and then gradually decreases wth z/l. Therefore we can say that when - z/l ncreases to more than 1; the turbulence producton becomes domnated by buoyancy producton. The most clear cut example of such a layer, drven by heat flux only, s the well mxed unstable layer above the surface layer. As seen n Fgure 8.4 for the unstable boundary layer, there s no shear n any of the mean varables wthn ths layer. Snce shear s unmportant, so s the shear producton of turbulence, and the heat flux wll have a domnatng mportance. The actual heght, z s mportant, and the boundary layer heght wll be the domnatng heght scale lmtng the sze for the domnatng eddes. The scales of mportance are then: g (8.37) Q0 = w θ, z, h= zi, We note that our man varables wll stll have dmensons of velocty and temperature; hence we generate a velocty scale and a temperature scale from the above set. g 1/3 1/3 (8.38) w ( zq ) = u( z / κl) ; θ Q / w, 0 I 0 I ML 0 θ Where the relaton between w * and u * smply comes from the defnton of L. w * s seen to be much larger than u *. Typcal values for u * s m/s, whle w * s of the order of 1-5 m/s. In the mxed layer scalng varables scaled by w * and θ ML* wll be functon of z/z I. The next fgures show a few examples of such behavour. θ Fgure 8.8. Typcal varaton of the sensble heat flux through the unstable boundary layer. 156 Lectures n Mcro Meteorology

158 Fgure 8.9. Varaton of the level of turbulence fluctuatons of w and θ through the unstable boundary layer. The dfferent symbols refer to dfferent experments. The evaluaton of the mxed layer scalng laws proceed the same way as for the Monn- Obuchov smlarty, namely that data are plotted accordng to the predcted scalng law, and the valdty of the law s evaluated by the qualty of the ft between data and model. Note n Fgure 8.8 that the heat flux generally decreases throughout the mxed layer. It also changes sgn at the top. The sgn change at the top means that the heat flux goes from aloft nto the boundary layer there. Ths s consstent wth Fgure 8.4 showng the unstable boundary layer, where the temperature aloft s larger than the temperature n the boundary layer. The turbulent transport between the boundary layer and the atmosphere aloft s called entranment. The heght varaton of the heat flux shows that the mxed layer s not statonary. Recall that we found that for a truly statonary boundary layer the vertcal fluxes of all scalars were constant wth heght. For temperature we have: dθ w θ (8.39) = dt z The fact that the heat decreases wth heght means that the data have been obtaned when the temperature wthn the unstable boundary layer ncreases. Indeed, ths s also the perod when ths boundary layer s best defned, as we shall see below. However, frst we notce that below the mxed layer we fnd a matchng layer or a free convecton layer. Ths layer s charactersed by that t, lke the mxed layer s drven by the heat flux, and by convectve eddy moton. Unlke the mxed layer t s however not lmted by z I because we are down n the surface layer, where z<<z I. Alternatvely, ths regon can be consdered a matchng regon between the surface layer Monn-Obuchov scalng and the mxed Lectures n Mcro Meteorology 157

159 layer scalng, n that t s a regon were they should both be true. Examples on the scalng laws applyng n the free convecton regme are shown n Jensen & Busch (198). The matchng approach s explaned n detal n Panofsky (1978) The structure of the unstable boundary layer can be seen also from the pont of vew of spectra and eddes. In Secton 6, the power spectra of turbulence n the surface layer are descrbed, usng a combnaton of data, and Kolmogorov hypotheses and Monn-Obuchov hypotheses. Here t s shown that turbulence spectra by and large are well descrbed by the above set of hypotheses, but that the unstable horzontal components would not adapt to ths termnology. The reason s that the low frequences of the spectra for unstable condtons are more descrbed by the mxed layer scalng than by the surface layer scalng, even n the surface layer. The low frequency part of the spectra can be seen as footprnt of the large boundary layer sze eddes n Fgure 8.4. Ths s llustrated n the next fgure, where the unstable v-spectrum s descrbed as a supposton of a hgh frequency, small scale part that scales accordng to the formulatons n secton 6, and a low frequency part that scales wth the mxed layer scalng. n = fz/u Fgure 8.10: The unstable Spectrum of the horzontal lateral component composed by a hgher frequency component scalng by the neutral Monn-Obuchov smlarty functon from secton 6 and a lower frequency component scalng wth the mxed layer scalng (Højstrup, 198). In Fgure 8.10 the low frequency part of the spectrum scales wth w *, whle the hgher frequency part scales wth surface layer scalng. Hence the total spectrum can be formulated a sum of the two: nsuvw,, ( n) = Auvw,, ( n) w + Buvw,, ( n) u, (8.40) 1/3 fzi fz wth w* = ( ziqg / T ), n =, n =. u u Where the normalzed frequency for the mxed layer, n s defned for the mxed layer the same way as n s defned n the surface layer. The two forms A(n ) and B(n) represents spectral forms defned n the surface layer, as gven n secton 6, and the mxed layer respectvely. In Fgure 6.9, we saw that peak frequency the unstable horzontal spectra could not be descrbed as functons of z/l only, we now see that the reason s the low frequency part, and thereby the 158 Lectures n Mcro Meteorology

160 peak, s functon of two varable z/l and z I /L (through equatons 8.37 and 8.38). As our measurng heght moves up n the unstable boundary layer the A(n ) of the spectrum of (8.38) gradually domnates the B(n) part. On the other hand the A(n ) part does not dsappear when we move closer to the ground. The forms of (8.40) have been extended to neutral condtons by Højstrup et al (1990), see also Mann(1998) for overall dscusson of the neutral velocty spectra. To the spectral functons n (8.40) corresponds smlar varances of the turbulent velocty components as seen n Fgure 8.11, where the velocty varances are plotted versus z/z I and scaled wth w *. Notce the dfference for the w-varance plotted aganst a lnear heght scale n Fgure 8.9 and the plot of Fgure 8.11, where the heght scale s logarthmc. We see that the smple schematcs presented n Fgure 8.6 for some varables s countered by a behavor as seen n Fgure 8.10, where the scalng that apples depends on the orgn of the eddes encountered rather on the measurng heght. The behavor of the scalar spectra and varances, lke temperature and humdty are not as well establshed as for the velocty components. Fgure Data on model and data for varances of the three velocty components, scaled wth w * and plotted versus heght wth z I /L as parameter (Højstrup, 198). As a last comment to the dagram n Fgure 8.6, we notce that t ncludes as well a Near Neutral Upper Layer. Ths layer s charactersed by that -z/l<1 all the way to the top of the boundary layer. Hence both shear and buoyancy s mportant for the turbulence producton and both z I and L are mportant length scales charactersng the flow. Indeed ths s a layer where all the parameters we lsted above must be expected to be mportant. As a consequence the number of parameters s too large for scalng laws to be of help n understandng ths part of boundary layer. Top-Down/ Bottom-Up Scalng Here, we menton a scalng approach, rather than a scalng regme. It s the so-called Top- Down/Bottom-Up approach used for unstable stuatons. The approach s llustrated below for Lectures n Mcro Meteorology 159

161 temperature. We recall that a problem wth unstable boundary layers s that K-theory does not work n the mddle of the layer becomes the vertcal gradents of the varables s very small, due to ts well-mxed characterstcs. A way around ths problem s to break the profle down nto a sum of two profles, as ndcated, one profle startng above the boundary layer and decreasng down to the ground, an other startng at the surface and decreasng all the way to above the boundary layer. The real profle can then be generated as a weghted sum of the two. z I θ bu θ td θ Fgure 8.1. Breakng up the temperature gradent nto a top-down and a bottom-up profle. The two profles each follow ther own scalng laws. The bottom up scales wth the surface heat flux, t vares as 1-z/z I, whle the top-down profles scales wth the heat flux at the top that s heat entraned from above the boundary layer, and t vares wth z/z I. An advantage wth the approach s that one can nclude detals about the structure of the surface layer at the ground and of the entranment layer at the boundary layer top. The fluxes through the top of the boundary layer, here of substance C wll often be descrbed usng the entranment velocty, w e, and a mean gradent. (8.41) Flux( C) = we C h h Smple descrpton of the growth of the unstable atmospherc boundary layer. The growth of an unstable boundary layer s depcted on Fgure The stable nght tme profle s descrbed by the lapse rate γ, wth the surface temperature θ 0.(Remember all ar temperatures are potental temperatures.) As the sun heats up the ground, the ar temperature starts to ncrease and at a gven tme s θ d. at that tme the boundary layer heght s h. We assume the whole boundary layer to be well mxed,.e. at any tme wth constant potental temperature. From the fgure s seen that we have: (8.4) θ = θ + γh 0 The relaton between the change n boundary layer temperature and the flux dvergence s gve by: dθ (8.43) = 1 ( w θ w θ ) w θ (1 + A ) / h, 0 h 0 dt h 160 Lectures n Mcro Meteorology

162 where A s normally found to be about 0., but of course can vary wth many thngs, and be modelled more or less complcated. The A factor s an extremely smple form of the entranment formulaton mentoned above. Insertng the frst equaton n the second we get: dh 1 dh (8.44) h = = w θ (1 + A) / γ, 0 dt dt Denotng: (8.45) we can wrte: t θ 0, 0 Q( t) = w (1 + A) dt () (8.46) ht () Qt = γ If the heat flux s constant wth tme h(t) s seen to ncrease wth the square root of t. 1 z γ h Q h Q 0 θ 0 θ d θ Fgure Schematcs of the growth of an unstable boundary layer drven by the heat flux at the surface and at the top. The equaton gves a reasonable descrpton of the growth of the unstable boundary layer as long as the heat flux s ncreasng or constant wth tme. After the heat flux decreases and turns negatve n the end of the day the boundary layer structure changes, and the above equaton does not descrbe the stuaton anymore, compare Fgure 8.7 that shows how a so called resdual layers s slowly formng, consstng of decayng eddes. Indeed what happens at a somewhat more detaled level of descrpton s that the heat flux nto the boundary layer creates turbulence and the turbulence s responsble for the growth of the boundary layer, through turbulence dffuson. A more detaled model demands that the turbulence varance equatons are ncluded n the process descrpton. Such models have been developed and also gve a more comprehensve descrpton of the development of the unstable boundary layer, at the prse of complexty. Lectures n Mcro Meteorology 161

163 An example s presented below (Grynng and Batchvarova, 1990). (8.47) h Cu dh w' θ ' 0 + ws = (1 + Ah ) BκL γ( g/ θ) [(1 + Ah ) BκL] dt γ Where A, B and C are emprcal constants, wth A beng the one already used n (8.43), about 0., and B and C about.5 and 8 respectvely. In spte of ts larger complexty t s seen that (8.47) does not contan new parameters relatve to the ones already encountered. It s seen that for the growth of a pure mxed layer, u =0, (8.47) reduces to (8.44). It should fnally be noted that nether (8.44 nor 8.47) allow for a descrpton of the boundary layer heght and the boundary layer structure, when the surface heat flux starts decreasng n the md-afternoon, and the unstable boundary layer gradually loses ts characterstcs and starts gven room for the nght tme stable boundary layer. New developments for boundary layer scalng. In (8.44 and 8.47) knowledge about the lapse rate,γ s essental for descrbng the rse of the boundary layer h. One could therefore thnk that the thermal stratfcaton of the atmosphere above the could be mportant for the boundary layer heght under all crcumstances, especally snce t s known that that the atmosphere above the boundary layer s generally stable stratfed, and because t seems reasonable that ths stratfcaton must nfluence how easly the turbulent boundary layer can grow nto the atmosphere aloft not only for an unstable boundary layer, whch we have dscussed above. The parametersaton s often formulated n terms of the so called Brunt-Vasala frequency, N (radans/sec). g θ 1/ (8.48) N = ( ). θ z The scale N can be combned wth a velocty, for example u *, to yeld a length scale. The new scale s seen to be ndependent of the thermal stablty and wll modfy the profles predcted n ths secton also for neutral. From N one can generate a length scale as u * / N, whch s seen to have the same form as the boundary layer heght, h, n (8.3) and onwards. How these two scales nteract and nfluence the flow s not yet settled completely. Therefore, we shall here not contnue to elaborate ths new scale, but more pont out ts possble nfluence and refer to Esau and Zltnkevch (006). However the correcton s unlkely to be large for the normal condtons, and a fnal form for the correcton s not yet well establshed. Example of measurement program n the full boundary layer. In Chapter 6 about the Monn-Obuchov smlarty and the surface layer scalng, we showed the set-up of the Kansas 1968 experment to llustrate the consderatons necessary for settng up such a surface layer experments. Addtonally, the experment has been crucal for the formulaton and the valdaton of ths scalng. Smlarly, we show n Fgure 8.14, the set-up for the Mnnesota 1973 experment that was conducted by the same core-group, also here supplemented by other groups of scentsts. The Mnnesota 1973 experment was also crucal for formulaton and valdaton of many of the scalng laws, pertanng to the full boundary layer, as dscussed n ths chapter, just lke the Kansas experment was for the surface layer expressons n Chapter Lectures n Mcro Meteorology

164 Fgure The expermental confguraton for the Mnnesota 1973 experment ( Itzum and Changhey, 1976) for the whole boundary layer. The lower rght hand sde set-up corresponds to the Kansas experment, also denoted CRL n the nstrument descrpton. The MRU confguraton are all attached to the tether sonde balloon, and descrbed as MRU sensors. The experment has been central for our descrpton of the unstable and the neutral boundary layer as gven n ths secton. It was less useful for our understandng of the neutral condtons, because the tether sonde had problems for hgher wnd speeds. (Bush et al, Drawng : C Kamal personal communcaton) Dscusson: We have llustrated how scalng laws have been establshed for many aspects of the atmospherc boundary layer throughout the boundary and for many dverse boundary layer flows. Relatve to Fgure 8.1. we have now establshed laws and formulatons for the whole boundary layer. Occasonally we have been forced to accept that scalng laws could not be utlsed for certan phenomena or only for these phenomena wthn lmted parameter ntervals. Also we have been forced to relax some of the basc assumptons behnd the dervaton of the scalng laws or of the consstency between the dfferent applcatons of the same scalng formulaton. Below s a short summary. Lectures n Mcro Meteorology 163

165 1. The demand to statonarty and horzontal homogenety was relaxed from the start, where horzontal pressure gradents had to be accepted for havng a smple Barotropc Ekman spral.. Incluson of a Baroclnc Ekman spral meant that also horzontal nhomogeneous temperature felds would have to be accepted as a normal part of the atmospherc boundary layer. In secton 5 s seen that even very modest horzontal temperature gradents lead to sgnfcant devatons from the Barotropc Ekman profle. 3. The mxed layer scalng for the unstable boundary layer s very useful, but typcally the boundary layer heght and the mean temperature feld wll now be non-statonary and governed by ther own rate equaton. However, ths non-statonary boundary layer heght can be used n connecton wth mxed layer scalng, and the heat flux typcally follows a mxed layer scalng wth heght, havng a gradent reflectng the change n mean temperature. Also strongly stable layers are smlarly non-statonary, as the radatonal coolng gradually decreases the surface temperature. 4. For unstable condtons power spectra and varances of velocty follows a mxture of surface layer (Monn-Obuchov ) and mxed layer scalng, even well wthn heght ntervals, where surface layer formulatons govern most of the moton. The reason s that the effect of certan larger eddes can be felt n the surface layer, even when they orgnate n the mxed layer. 5. Applcaton of many scalng laws s based on the assumpton that some parameters are much less mportant than other parameters, and therefore can be neglected. In practse one wll often meet stuatons, where much less mportant has to be replaced by slghtly less mportant, for whch reason the less mportant parameters wll show up n the results as well, albet wth less mportance. 6. The necessty to nclude consderatons about horzontal nhomogenety and nstatonarty s much more mportant for the descrpton of the flows n the total boundary layer than for flows n the surface layer, as we have seen or remarked repeatedly n ths secton. After the next secton about nhomogeneous boundary layers, we wll be able to dscuss ths statement n more detals. 164 Lectures n Mcro Meteorology

166 9. Horzontally heterogeneous boundary layers. So far we have assumed that the boundary layers could be consdered horzontally homogeneous, by and large. We shall now leave that assumpton and dscuss how to handle heterogenety, a feld that s stll far from fully l developed. Consderng the natural landscape n the fgure 9.1, we obvously have to structure the type of nhomogenetes one can meet, and whch often wll have to be handled by dfferent models or concepts. A common concept s to consder the nfluence of a new surface characterstc n terms of a new nternal boundary layer growng n to the old one, and reflectng the new surface characterstcs. In prncple we expect the Geostrophc wnd and the general atmosphere over the boundary layer to be unperturbed by the nternal boundary layer. However, recall that from the resstance laws we know that the surface flux, the boundary layer heght and the Geostrophc wnd, wth all scalars as well, are nterconnected. Therefore, f an nternal boundary layer growths to become a new homogeneous boundary, whch wll happen f the fetch s large enough, some adjustment wll have to happen such that the new boundary layer s n equlbrum wth the free atmosphere, as depcted by the resstance laws. Fgure 9.1. Conceptual vson of a heterogeneous surface wth assocated development of Internal Boundary Layers. The fgure llustrates the response of the atmospherc boundary layer to that the ar encounters surfaces wth dfferent surface characterstcs. Internal boundary layers (IBL) start growng, and each surface have to be charactersed by ther respectve surface characterstcs such as terran elevaton, dfferent roughness and surface values of other varables under consderaton. Lectures n Mcro Meteorology 165

167 Below, we shall consder dfferent types of nternal boundary layers: 1) Change of surface roughness ) Change of surface heat flux. 3) Change of surface level, hlls and rdges, escarpments. 4) Change of turbulent spectrum for changng surface condtons. 5) Addtonally, we shall present a selecton of characterstc specal response to terran nhomogenous stuatons, wthout gong nto detaled dervatons. 6) In appendx 9.A we shall present a smple dervaton of the flow over low hlls, both to llustrate the physcal/mathematcal prncples appled, and to pont to practcal applcatons. Whenever the wnd blows over a new surface, theory and data agree that the surface flux changes qualtatvely as shown on fgure 9.. The man aspects are that the change n surface fluxes s strongest just at the transton, thereafter the flux relaxes towards a steady level after a few hundred meters. Ths steady value then slowly relaxes to the value t would have f the new surface contnued to nfnty and the nternal boundary layer became a new homogenous boundary layer. The fetch necessary for ths fnal relaxaton s usually estmated to be km, for neutral stablty, shorten for unstable and longer for stable condtons. Fgure 9.. Qualtatve presentaton of change n surface fluxes as the ar passes over surfaces wth dfferent surface characterstcs, S1---S5. The brute force approach to these nternal boundary layers s to use a numercal model. Also n the end meteorologcal numercal models must be able to handle nternal boundary layers. Therefore, t s nstructve to consder such an approach frst From Secton 3 the general equatons, we obtan the governng equatons for the mean flow n the followng form, where we have ntroduced the specal perturbaton pressure p n addton to the surface Geostrophc wnd as well the thermal wnd, or baroclnty, from secton 5, reflectng larger scale layer averaged horzontal temperature gradent, θ L. 166 Lectures n Mcro Meteorology

168 (9.1) The three momentum equatons. u u 1 p + uj = fc( vg v) ( uu j ) t x ρ x x j v v 1 p + u = + f ( u u) + ( vu ). t x x x j C g j j ρ j w w 1 p + uj = t x ρ x x j 3 wu. g θ g θ u = u x ; v = v + x L L g g0 3 g g0 3 fcθl x fcθl x1 The contnuty equaton : u x = 0; The scalar equatons : j θ θ q q + uj = ( u jθ ); + uj = ( u jq ); t x x t x x j j j j j j In these equatons we have already smplfed somewhat. However, we have ntroduced the perturbaton pressure, whch wll be essental to descrbe the flow around hlls and other changes n surface elevaton and wll appear n general when changng surface characterstcs ntroduces abrupt changes n wnd speed. They are assumed of such small scale that they do not contrbute drectly to the larger scale features, lke the Geostrophc wnd. Also, we have ntroduced the changes of the Geostrophc Wnd wth heght due to the larger scale horzontal temperature gradent, snce the possblty of baroclnty (see secton 5) must be consdered for heterogeneous stuatons. θ L means that the temperature must be layer averaged (0-x 3 ). Especally at coastal areas baroclnty can be mportant, due to the very dfferent heat capacty of a land and an ocean surface. To smplfy further we now assume that the change takes place along an nfnte lne parallel wth the y-axs. The wnd close to the surface s comng n along the x-drecton. Indeed, we do not need the last assumpton, because for an nfnte lne of change along the y-drecton, there can be no dervatves along that drecton. Also that the changes do not nvolve terran levels and that the pressure perturbaton can be neglected (For roughness changes that s true just a short dstance away from the lne of change). Further neglect baroclnty and assume statonarty, and that the vertcal flux dvergences are stll much larger than the horzontal ones. (Agan that can be shown to a good approxmaton). Lectures n Mcro Meteorology 167

169 (9.) The three momentum equatons. u u w + u = fc( vg v) ( u ) ( uw ). z x x z v v w + u = + fc( ug u) ( u' v ) ( vw '). z x x z w w w + u = uw ' w. z x x z The contnuty equaton : u w + = 0; x z The scalar equatons : θ θ w + u = ( w θ ) ( u' θ ) z x z x q q w + u = ( wq ) ( u' q ); z x z x where we have retaned only the most mportant vertcal and horzontal terms. As seen the equatons are not very complcated, and t s temptng to close them wth the same knd of turbulence dffusvty as for homogeneous condtons. However the turbulence closure cannot be assumed to be the same as for homogenous condtons, because t s known that the flux profle relatons dffer. In the horzontally homogeneous surface layer, we can wrte: ku z (9.3) K = ; ϕ For heterogeneous condtons K can be formulated the same way, but the ϕ functons behave dfferently, and for neutral condtons they are generally dfferent from 1.0 (Jensen and Busch, 198, Jensen et al, 1983). Substantal work has been nvested n formulatng and valdatng K-formulatons for such condtons. Models have been developed as well based on nd order closure and also 1.5 order closure. For the last methods a total turbulent varance equaton s carred along, and K s determned from the varance and the dsspaton, see e.g. Rao et al.(1974, 1974a). See also dscusson n secton 4. Such equatons have some success, wthout beng completely convncng, see Fgure 9.4. Here we shall not dwell nto ths approach, nstead we shall present a few smple models that expermentally have been found very successful. Therefore, they wll gve a better pcture of the characterstcs of dfferent nternal boundary layers, than a lengthy dscusson of the closure possbltes. They have been found as well to be essental for the development of the proper turbulence closure formulatons, because they provde background and comparson for these formulatons. The penalty of ths approach s that we end up wth a collecton of very dfferent models, that has been specalzed to extract the essental physcs for specfc types of nternal boundary layers wth as few as possble of mportant terms and characterstc parameters. 168 Lectures n Mcro Meteorology

170 Neutral nternal boundary layers orgnatng from an abrupt change of terran roughness, surface layer formulaton. The stuaton s depcted on fgure 9.3. The nfluence of the new surface s growng through turbulence dffuson n an nternal boundary layer. dh h (9.4) = u ( h) = A1σ w0 = Au 0, dt x where we assocate the growth rate wth one of the two veloctes assocated wth vertcal dffuson, w and u. Subscrpt 0 ndcates that we are consderng surface values. It s worth notng that for heterogeneous boundary layers, we know that the constant flux layer approxmaton s not vald. Hence, we must be careful about specfyng the heght at whch the turbulence quanttes are estmated. In the surface layer σ w and u are proportonal to each other, so for the tme beng, we settle on u. We can now wrte the growth rate as: (9.5) dh dx Au u ( h) 0 = = κ A. h ln( ) z where t s assumed that the logarthmc profle scales wth the surface stress, although we must assume that also n the surface layer the stress cannot be constant wth heght close a roughness transton. 0 Fgure 9.3. Descrpton of the structure of the nternal boundary, of heght h, due to a step change n terran roughness. Rule of thumb h~ 0.1 x. Equaton (9.5) ntegrates as follows: h 1 h dx = ln( ) dh, or : Aκ z z0 z0 0 (9.6) x x h h C( 1) = (ln( ) 1) + 1. z z z where all the 1 es are relevant only very close to the transton. C s an emprcal constant (= Aκ) approxmately equal to 0.9. Overall the formula depcts that h s growng somewhat slower than x, about as x 0.7. An often used smplstc approxmaton s h~ x/10. The physcal sgnfcance of the exact heght of the nternal boundary layer, as depcted above, s knd of dffcult to evaluate, but we can now determne the rato between the surface stresses Lectures n Mcro Meteorology 169

171 by matchng the up-stream and down-stream profle n h(x). The down-stream profle s determned by that t matches the up-stream value n h(x) and passes through (u,z) = (0, z 0 ), see fgure 9.3: (9.7) 1 1( ) ln( ) ln( ) k zo 1 k zo u h u u = = = u h u h = = h h z0 z0 ln( ) ln( ) ln z z z z M 1 1 ; h h h h ln( ) ln( ) ln( ) ln( ) z z z z The rato between the two surface stresses s a very well defned quantty to measure, and such measurements for short fetches show the model accurately to predct ths rato. From comparng wth data, one fnds C 0.9. An example of such data model nter-comparson s shown n the next fgure, where ths smple model s shown as well to perform better than a second order closure model by Rao et al. (1974), especally for the rough-to-smooth transton. In the fgure the open symbols correspond to the model descrbed here, wth two dfferent values of C. From the formula and the measurements t s seen that surface stress rato frst shows a large excurson wth a sgn gven the by logarthm of the z 0 -rato. For longer fetches the rate of change reduces strongly. The logarthm rato between the two roughness lengths s denoted M n (9.7), and t s as seen a ratonal way of descrbng the magntude of a roughness change. Fgure 9.4. Comparson between surface stress measurements and dfferent model predctons, the IBL model of (9.7), and the second order closure model of Rao et al, (1974 for smooth- torough and for rough-to- smooth transtons. The data (ponts) are from Bradley (1968. The sold curves are from Rao et al.(1974). The trangles and crcles correspond to C equal to 1.0 and 0.5 respectvely.(hedegaard and Larsen,1983). 170 Lectures n Mcro Meteorology

172 Addtonally, for long fetches the model does not work well n that the rato of surface stresses goes to 1 for x gong towards nfnty. For long fetches we expect the condtons to approach the condtons for horzontally homogeneous boundary layers, where the Geostrophc wnd, z 0 and u * are nterlnked through the resstance laws, as shown n (9.8). u G = ug + vg ; h = ; f (9.8) C 1 kg h = ((ln A) + B ) ; u z0 vg h α = arctan( ) = arc tan( B /(ln A)); u z We shall return to ths pont later when dscussng extensons of the model of (9.4). g 0 For measurement heght less than the heght of the nternal boundary layer, we can estmate the wnd speed rato between up-stream and down stream wnds. (9.9) u ( z) u u z z h z ln( ) ln( ) ln( ) z z z = = 1( ) u z h z 1 ln( ) ln( ) ln( ) z z z for z h. As seen ths rato s not as transparent as the surface stress rato, because t s a balance between two opposng ratos. However, gong back to the model fgure, t s seen that f z 0 > z 01 then u (z)< u 1 (z) wthn the entre nternal boundary layer, and vce versa. The velocty rato predcted by (9.9) s not n accordng wth data. The model of (9.4) yelds an excellent descrpton the rato between up-stream and down-stream surface stress. If one measures the wnd profle at any pont downstream the roughness change, one fnds a profle that s closer to the profle of fgure 9.5 below. Here we can see that the formula above wll work only for z< h, whle the up-stream profle prevals down to h 1 h/3. In between one must nterpolate. The general lesson of ths s that nternal boundary layers are pretty dealsed features, and ther actual heght wll often vary wth the parameters, one consders. Note, however that for ths case, they all vary proportonally to each other. Lectures n Mcro Meteorology 171

173 Fgure 9.5. Prncpal structure of the wnd profle wthn an nternal boundary layer. The outer and nner profles (thn lnes) are matched at z=h, to provde the surface stress. The nner profles n equlbrum wth the surface stress reach untl z= h. The outer profle remans vald down to z=h 1 ; between h and h the profles are nterpolated. The relaton between the heghts s h 1 h/3, h 0.1 h (Jensen and Peterson, 1977, Hunt and Smpson, 198). One of the mportant smplfcatons n the equaton for the growth of the nternal boundary layer of (9.6) s that t ncludes one z 0, only, but does not dstngush between the two dfferent z 0 s that appear n the problem, the upstream or the downstream value. Therefore, one has to consder ths problem outsde the model dervatons: The use of the formula comes wth the rule that one must use the largest z 0 to determne the growth rate of the IBL heght, the arguments beng that the larger z 0 creates most turbulence, and the growth of the nternal boundary layer s controlled by the turbulence of the one of two layers that s the most turbulent. Ths has ndeed been supported by data that shows that the success of the model depends on the use of ths addtonal rule see Fg The model descrbed above has been qute successful wthn ts somewhat narrow lmtatons. Ths could ndcate that the physcal prncples behnd the model are robust. Therefore, several efforts have been made to extend ts valdty. Extenson to large fetches for neutral boundary layers. From (9.7) s seen that the surface stress rato relaxes to 1 for fetch gong to nfnty. Ths s not consstent wth the resstance laws, as gven by (9.8), whch show that for larger fetches, two dfferent z 0 s must correspond to two dfferent values of surface stress for the same Geostrophc wnd. Generally the model behaves unrealstc for large fetches, wth respect to the surface stress rato, and because the nternal boundary layer heght ncreases wth no lmts as fetches goes to nfnty, and at least extend beyond valdty of the surface layer formulatons, beng basc to the model. An extended model, reparng some of the fallaces, s shown n fgure 9.6. The extended model s based on the same smple dea as used n (9.4) dh h (9.10) u = A1 σ w dt x 17 Lectures n Mcro Meteorology

174 Fgure 9.6. Growths of nternal boundary layers that growths towards the total boundary layer heght for long fetches, and where a surface stress rato s acheved that s consstent wth the predctons of the resstance laws as gven n (9.8) ( Larsen et al.,198, Semprevva et al, 1989).. But both σ w and u s now gven by expressons that are vald throughout the total boundary layer (Wyngaard, 198): (9.11) σw σw0 h (1 ) u0 u0 H u 0 h h u = (ln( ) ) κ z0 H u 0 wth : H = f where we have ntroduced the scale heght of the boundary layer, H, and the wnd profle throughout the boundary layer, as derved n Secton 8. As for the surface layer descrpton, equaton 9.10 wth (9.11) can farly easly be ntegrated to: h x z0 h h C 1 = ((ln 1) Y( )), (9.1) z0 1 h z0 H H h h wth : Y ( ). H H c, The resultng model follows the surface layer model for short fetches and converges to the resstance law predctons for large fetches, wth changes gong on out to fetches of about 100 km, but wth extremely small changes occurrng between 0 and 100km. It s seen that the model starts out as a surface layer model for small fetches, and then gradually changes and approach the large fetch solutons. The matchng relatons nvolve the profles of (9.11) rather than just the logarthmc profles. It s seen from Fg. 9.6 how the matchng characterstcs depend on f the IBL growth wthn the upwnd boundary layer or drectly aganst the free Lectures n Mcro Meteorology 173

175 atmosphere. In the latter case the cross-sobarc angle also starts approachng the angle for the new equlbrum boundary layer for x. See Fgure 9.7 The smplstc way of phrasng the models of nternal boundary layers (IBL) after a step change s to say that the heght of the IBL grows as between x/10 and x/100, dependent on the parameter used to characterze the IBL. Ths can be turned n two ways. When measurng from a mast at heght, z, t means that surface changes closer than about 10z cannot be seen n the data. On the other hand, f one wants to use the resstance laws n secton 8 wth a characterstc roughness together wth the wnd at z = 1 km, t means that ths roughness must be characterstc for an area wth a wdth between 10 and 100 km. Fgure 9.7. Behavour of the full boundary layer large fetch model, taken from Hedegaard and Larsen (1983) Extenson to changng vertcal velocty In the formulatons above, t has been assumed that the average vertcal velocty remans zero or nsgnfcant. Ths s not consstent wth (9.), where the contnuty equaton lnks changes n <w(x,z)> to changes n <u(x,z)>, and changes n <u(x,z)> are well establshed, and s one of the key outputs of the models dscussed. A smple approach ncludes <w> teratvely by consderng a modfcaton of (9.4). h (9.13) u( h) = Au 0 + w( h) x From ths equaton one can determne <u(x,z)> as descrbed above, followed by estmaton of <w(h,z)>. 174 Lectures n Mcro Meteorology

176 (9.14) u w uzx (, ) + = 0, or : w( h) = x z x Generally, the wnd speed s decreasng for smooth to rough transtons, and ncreasng for rough to smooth transtons. The correcton wll therefore ntroduce an asymmetry n growth rate for the nternal boundary layer n the two stuatons. Obvously t wll be most mportant for shorter fetches, where the change of u s strong and h s large enough for the ntegral n (9.14) to be sgnfcant. The contnuty equaton s routnely ncluded n numercal models of nternal boundary layers, to nclude the effect changes n vertcal velocty from the change n horzontal velocty. h z0 Extenson to non-neutral condtons Obvously, there are stuatons where the thermal stablty of the flow s of mportance for ts response to change of terran roughness. The same rule s extended to stuatons where stablty plays a role n one or the two of the boundary layers consdered. Let us reconsder the growth of the nternal boundary layer for non neutral condtons, correspondng to (9.10), we can somewhat smplstc and navely wrte: (9.15) dh dx Aσ A kfw( h/ L), u ( h) h h ln( ) ψ ( ) z L 1 w = = where we have added the stablty correctons to both the profle expresson and σ w /u *0. These functons are very dfferent for stable and unstable stuatons. The forms, defned n Secton 6, are: (9.16) z z z z z f for and f + for L L L L L 1/3 w( ) w( ) ( ) 0 The profle functon ψ(z/l) derves from ntegraton of the non-dmensonal shear, ϕ(z/l), as dscussed n Secton6. z z z z z 1/4 z (9.17) ϕ( ) 1+ 5 for 0, and ϕ( ) (1 16 ) for 0, L L L L L L From ths derves the ψ(z/l): z z z z z z (9.18) ψ ψ ϕ L L L L L L For unstable condtons, the form of ψ(z/l) s actually a very complcated functon. The approxmaton mentoned here s due to Jensen et al (1984). 1 ( ) 5 for 0, and ( ) ( ( )) 1 for 0 Consderng the functons above, t s seen that the heght of the IBL, h, wll grow faster for unstable condtons and slower for stable condtons. Indeed, neglectng the f w -functons, Jensen et al. (1984) constructs an approxmate soluton as: (9.19) h(ln( h / z0 ) 1 ψ( h )) ϕ( h ) Cx, L L 0 Lectures n Mcro Meteorology 175

177 whch s seen to become equal to (9.6) for neutral stablty. For stable and unstable condtons the growth of the IBL s depcted on the fgure below: Fg. 9.7a. Growth of nternal boundary layers for dfferent stabltes. Lower blue lne s stable, black s neutral and upper red lne s unstable Ths concept can further be extended by up-stream and down stream profles that ncludes dfferent stablty correctons as well as dfferences n roughness. As seen the ntegraton and matchng procedures loose the esthetcal smplcty of the pure neutral case, but can n the prncple be performed, and the approach does catch the man characterstcs of nfluence of both stablty and roughness for the development of nternal boundary layers, although there have never been measurements avalable for really comprehensve nter-comparson between data and theory. As for (9.6), equaton (9.19) s derved wthout specfyng whch surface characterstc, we must use, as we by defnton has two dfferent surfaces wth dfferent roughness and probably also dfferent stablty. Agan, we wll have to appeal to the concept of the strongest turbulence, meanng usng the z 0 and L that yeld the strongest turbulence. Here, the concept may be breakng down, because the two parameters may pont to n dfferent drectons, and may well each domnate at dfferent fetches, snce the IBL wll start neutral and become more nfluence by thermal forcng as h/l ncreases. Also another nconsstency n the model becomes apparent. For example, gven the defnton of the Monn-Obuchov length: (9.0), L u 3 / wθ we can see that t s only for very specal dstrbutons of the heat flux that one can magne an nternal boundary layer wth systematc changes of u * wth a constant characterstc L-value. As for the large fetch model, t s seen that for short fetches also the thermally nfluenced model starts out as a neutral - surface layer model. It also appears that the stablty and the full boundary layer models can be combned, by ncludng the proper parametersatons, although the qualty of such a combned model has not yet been tested. 176 Lectures n Mcro Meteorology

178 The growth of a thermal nternal boundary layer (TIBL) wthn a stable upstream boundary layer. When the heat flux s of domnatng mportance for the development of an nternal boundary layer, a dfferent smple model s often used to descrbe the development of the TIBL, the thermal nternal boundary layer. In ts smplest form t s an unstable nternal boundary layer that growth aganst a stable boundary layer. The concept s taken from the growth of the thermal mxed layer from the mornng stable layer over a homogeneous plane, as we dscussed n Secton 8. Indeed our whole treatment wll be buld of the formalsm developed here. We have changed the text slghtly. The man change of concept s that the temporal change n Secton 8 s now taken as a temporal change wth ar mass transport tme from the boundary between the upstream somewhat stable boundary layer and the TIBL, one can magne a coast lne wth a cold sea and a warm land or vce versa, see Fgure 9.9. The growth of an unstable boundary layer s depcted on Fgures 9.8 and 9.9. The stable upstream profle s denoted γ, wth the surface temperature θ 0. As the ar moves over the warm surface, t heats up, after a gven transport tme the temperature s θ d, at that tme the boundary layer heght s h. We assume the whole boundary layer to be well mxed,.e. at any tme the temperature and velocty are constant wth heght. From (8.40) - (8.45), and Fg. 9.8 s seen that we have: (9.1) θ = θ0 + γh The relaton between the change n boundary layer temperature and the flux dvergence across the nternal boundary layer s gven by, see (9.): dθ w' θ' 1 (9.) = ( w θ w θ 0 h ) w θ 0 (1 + A ) / h, dt z h where t s the flyng tme from the surface change to x = <u>t. A s normally found to be about 0., but of course, t can vary wth many thngs. Such varatons are seen to be easy to mplement nto the model. Insertng the frst equaton n the second, we obtan: (9.3) Denotng: (9.4) we can wrte: (9.5) dh 1 dh h = = w θ (1+ A) / γ. 0 dt dt t θ 0, 0 Q( t = x / u ) = w (1+ A) dt () ht ( x/ u) Qt = = γ 1 If the heat flux s constant wth tme h(x/<u>) s seen to ncrease wth the square root of x/</u>,. If the heat flux vares, ths can easly be taken nto (9.4). Lectures n Mcro Meteorology 177

179 Fgure 9.8. The prncple for growth of a TIBL due to heat nput from the top and bottom. The equaton gves a reasonable descrpton of the growth of the unstable nternal boundary layer, as long as the heat flux s not decreasng systematcally wth tme. It can be refned consderable ncludng detals of the temperature profle and decreasng heat nput, as well as the forcng by the wnd, also allowng for neutral or even unstable upstream condtons, through a proper ncluson of the dsspaton that mposed nternal lmtatons to the growth. In prncple the heatng can go on as long as the ar moves over the warm surface. As a lmtaton one can menton that ths model s dffcult to combne wth models that explctly descrbe the changes n the velocty profle, because the TIBL model s based on that velocty only plays the role of an advecton velocty, beng constant wth heght. The more advanced model presented n (8.47), Grynng ( 006) on the other hand ncludes the turbulence created by the shear profles, n spte of that the explct velocty ncluded n the model s a constant advecton velocty only. Also ths equaton can be turned to an equaton for growth of a TIBL. h Cu dh w' θ ' (9.6) 0 + ws =, (1 + Ah ) BκL γ( g/ θ) [(1 + Ah ) BκL] dt γ Now wth: dh h h h (9.7) = + u u dt t x x The model now descrbes the stuaton depcted on the Fgure 9.9 below. Smplfcatons may easly be derved: Lettng u * and L go to zero, recovers the smple soluton of (9.5). For L gong to nfnty we obtan an equaton for the growth of a neutral nternal boundary layer aganst a free atmosphere wth a lapse rate of γ : h ( g/ θ ) C h (9.8) u + 1/ γ 3 =, Bu Bu γ x where the second term n the parenthess s seen to correspond to the same growth rate as n (9.4), whch here s seen to correspond to the ntal growth, before the frst term n the parenthess starts domnatng. The frst term reflects the nfluence on the growth from the turbulence produced wthn the IBL, The soluton for the boundary layer heght, when t s domnated by the frst term n the parenthess s found as (Grynng (006): 3 1/3 6Bu t (9.9) ht ( = x/ u) =, γ( g / θ) whch s seen to depend on the stratfcaton aloft. 178 Lectures n Mcro Meteorology

180 Fgure 9.9. Schematc llustraton of the change of surface condtons for use of (9.6 and 9.7) Wnd Profles over a hll. When ar ascends a hll, streamlnes converge, and the ar accelerates. Typcal behavour of the phenomenon s gven n the fgures 9.10(a-c) below showng profles of mean speed and turbulence across a low hll, actually we talk about a two dmensonal hll, rather a rdge. The fgure also shows the pressure and wnd perturbatons around the hll, showng how a postve pressure perturbaton n front of the hll dmnsh the flow, whle a negatve perturbaton around the crest accelerates the flow, followed agan maybe by a deceleraton behnd the hll. The theory of ths behavour has mostly been developed for neutral condtons, although extensons to stable condton have been made. The key reference to the area s Jackson and Hunt (1975). Furthermore the slope must generally be consdered small, for the theory to apply, and one generally consders the flow n stream lne coordnates or smply terran followng coordnates. Fgure 9.10 (a-b) llustrate that for larger steepness flow separaton wth recrculaton may occur. In practse one can often go to larger steepness than theoretcally predcted wthout causng separaton. Jackson and Hunt (1975) separate the flow over the hll nto three layers. Layers above the heght L are undsturbed by the hll. L s a characterstc scale for the wdth of the hll typcally taken as the half of the wdth at half heght, see Fgure 9.10a,b,c, where also the heght of the hll, h, s gven. The layer below z= L s nfluenced by the pressure perturbatons around the hll, retardng the ar before and behnd the hll and acceleratng t the top. The layer closest to the ground s denoted the nner layer and has the heght,. In ths layer the flow s nfluenced by the hll surface through turbulence exchange. As usual the wnd s forced to zero at the ground. It s a regon wth large wnd shear. In the heght nterval between and L s the outer layer, whch s prmarly nfluenced by the hll from the pressure perturbatons. Notce the characterstc aspect of a pressure perturbaton, that t mpacts on the flow both up-stream and down-stream Lectures n Mcro Meteorology 179

181 of the change, as opposed to the nternal boundary layers from roughness and or heat flux changes that prmarly nfluences the downstream flow, and let the up-stream flow unchanged. Fgure 9.10a. Characterstcs of neutral flow across a two dmensonal hll, showng wnd, pressure and turbulence standard devatons (Hunt and Smpson, 198). Fgure 9.10b. Characterstcs of neutral flow across a two dmensonal hll, showng wnd, pressure and turbulence standard devatons (Hunt and Smpson, 198). Two depctons 180 Lectures n Mcro Meteorology

182 Fgure 9.10c Characterstcs of flow across a two dmensonal hll, wth defntons of flow regons and scales (Brtter et al, 1981). From arguments about turbulence dffuson, the thckness of the nner layer s found from: (9.30) ln( / z0 ) k L, where k as usual s the v. Karman constant, because profle relatons are nvolved n the argument. Ths result s seen to follow from the roughness change model of (9.6), f we consder the heght of the modfed velocty layer for a characterstc fetch equal to the wdth of the hll, L, compare eq. (9.6). In practce one fnds: 0.1 L. Below z, the logarthmc profle exsts n the usual way wth equlbrum between surface stress and wnd, although just as for the roughness change boundary layer one fnds a stronger horzontal varaton of stress, than for the horzontally homogeneous boundary layer. For the flow over the hll, one estmates a speed-up factor, usually denoted S(x,z). S x,z = u / u (9.31) ( ) 0 It vares across the hll and wth heght, as seen n fgures 9.10 and appendx 9A, and e.g Zeman and Jensen (1987), and n appendx 9A, beng slghtly negatve n front and behnd the hll and postve at the top. Wthn the nner layer, z<, t s however essentally constant wth heght. It s defned n (9.31), where Δu s the change of u from far upstream to the top of the hll at a fxed heght over the local terran (or rather fxed stream lne), normalsed by the upstream wnd, u 0. Wthn the nner layer at the top of the hll, t can be wrtten: ln ( L/ z0 ) 0 ln ( / z0 ) u h h (9.3) S = =, u L L where the last approxmaton s normally used. Lectures n Mcro Meteorology 181

183 As S s constant wth heght n the nner layer, the profle remans logarthmc, but wth enhanced shear gven by S. The maxmum speed up takes place at the top of the hll, at the top of the nner layer. There s a gradual varaton from the reduced wnd n front of the hll to the maxmum at the top and down to the wnd reducton behnd the hll. Ths s llustrated n Fgure 9a,b and n appendx 9.A, where we derve the flow over the hll from a smple lnear model, applyng Fourer decomposton of the hll. In the outer regon the shear decreases, and a smple way to fnd the wnd s to nterpolated between the enhanced wnd at heght,, and the unchanged wnd at heght L. Extenson of hll effects Below we show a fgure depctng the constructon of wnd profles above roughness changes, flow over hlls and flow over escarpments that can be treated as hlls, wth the outer scale, L, gven by the dstance between the terran change and the observaton pon, based on Jensen(1983). The fgure llustrates as well how one can combne roughness change and terran change effects as multplcatve, provded the modfcatons are small. It works by smply multplyng together, the effects on the flow from the ndvdual modfcatons. Fgure Dfferent types of nternal boundary layers and ther combnaton (Hedegaard and Larsen, 198) In the fgure IBL heghts are denoted δ, and gven by the relevant equatons n the text (9.6 and 9.30). The sold lne shows the profle at poston b over the terran. The theory, summarsed above, has been developed for not real a hll but rather a twodmensonal rdge perpendcular to a neutral wnd flow. As such t has been found to match data well, see Fgure 9.1 from the experments on the Askerven hll n Scotland, and the modellng by Zeman and Jensen(1987). Snce t s bascally a lnear theory, t apples as well for the opposte of a low rdge, a shallow valley. 18 Lectures n Mcro Meteorology

184 It has been found that t extrapolates qute easly to non-neutral stuaton, where the man effect has been that the up-stream profle has larger shear for stable condtons and smaller shear for unstable condtons. See llustraton of unstable, neutral and stable flow over a hll n Fgure 9.1. The formalsm has been found to combne well n a lnear fashon wth roughness change models, such that one can estmate the combned effect from each of the ndvdual effects by addng them, see Fg Fnally, the hll effect has been extended to descrbe flows over three-dmensonal terran of arbtrary elevaton varaton, or arbtrary topography, as long as ths topography varaton does not nvolve too much steepness. It s presented n Appendx 9A n detal, because the approach here llustrates very well the bascs physcs, and smultaneously, allow for and easy extenson to an arbtrary landscape. Combned wth a smlar roughness descrpton ths knd of development s the backbone of the development of the wnd resource evaluaton system at Rsø, the so called Wnd Atlas or WAsP. Fgure 9.1. Results from the Askerven Hll experment, as reported by Troen and Petersen (1980). The lower fgure, shows the relatve speed-up ratons perpendcular to the man axs of the rdge. The dots are measurements, open squares reflect a lnear models, whle two other models are ndcated by full and dashed lnes. Passve IBLs In passve IBL only surface values of passve tracer change values, t can be ether a change n concentraton or a change n surface flux. Ths means that both roughness and stablty s unchanged. The equatons for the changes can then be taken from (9.) takng only one of the scalar equatons, now denotng the passve scalar by c. cxz (, ) (9.33) uz ( ) = ( cw ) ( uc ' ) ( Kz ( ) cxz (, )), x z x z z where we have approxmated the flux equaton by a smple K-dffusvty approxmaton. Lectures n Mcro Meteorology 183

185 It s seen that most of the terms n (9.) have vanshed, snce we magne that we have only step n the surface values of c. The soluton of (9.33) can be based on normal surface layer formalsms (Grynng et al, 1983) wth (9.34) ( ) u (ln z ( z )), ( ) / c ( z uz = ψ K z = κuz ϕ ); κ z L L 0 The soluton of depends on our assumpton for the change n surface concentraton. If we magne that the changes take place as a pont source, the soluton becomes the soluton to the dsperson for a surface pont source. If we magne a surface step change along a lne perpendcular to the wnd speed, we can magne any such step-change, whch nvolve only c and nether of the other meteorologcal parameters. Actually, we can reverse the soluton to a queston about the nfluence of a surface source change at a gven heght and a gven dstance downstream, ths normally denoted the foot-prnt formalsms, used to attrbute measured flux or concentratons to gven areas up-stream. For good reasons the model n (9.33) corresponds to the descrpton of dsperson from a chmney that we derve n secton 10. We can now turn that dervaton around and ask where concentraton partcles measured by a sensor at a gven heght are ejected from the upstream surface. Ths s the opposte problem of where does the mass ejected from a source pont end up on the surface. The surface orgn (the foot prnt) of concentraton measured at a gven heght and locaton n the terran wll therefore to a large extent look lke the surface concentraton for a chmney wth the same heght as the sensor locaton and wth the opposte wnd drecton. Detaled treatment of the foot prnts has become an mportant ssue n the scentfc lterature, and the foot prnt depends on the terran, stablty and not least f fluxes or concentratons are dscussed. Foot prnts are as well of mportance, when fluxes are measured over heterogeneous terran, where t become mportant to document what part of the terran the fluxes refer to, see Horst and Wel (1983) for more detals. A typcal foot-prnt form s llustrated n Fgure It s seen also to be a typcal shape for the ground concentraton for the elevated chmney plumes n Secton 10. As expected areas too close to the mast cannot nfluence the sensor at the gven heght, snce the tracer wll not be able to dffuse to the sensor heght from these areas. On the other hand, tracers orgnatng from far away wll be hghly dluted when they reach the sensor. Unfortunately, even the very smple model of (9.33and 9.34) cannot be solved analytcally, but farly smple by a dgtal technque, Grynng et al.(1983). Flux per length unt Upwnd dstance Fgure Foot-prnt for a measured concentraton or flux at a gven heght. The dstance between the measurng pont at heght h and the start of the foot-prnt s found from σ z (x) = h, where σ z (x) s the vertcal spread of the plume from Secton 10.b 184 Lectures n Mcro Meteorology

186 Approaches to multple IBLs, brute force and blendng heghts. As seen n Fgure 1, the real atmospherc boundary layer can be seen as composed of a large multtude of dfferent and overlappng nternal boundary layers. Bascally, two approaches exst to handle ths: 1) applyng the ndvdual changes by the models and buld the resultng effects by multplyng the effects of the ndvdual models, ) usng the so-called blendng heght approach. The frst approach s used n the Wnd Atlas (Troen and Petersen, 1980), where especally the abrupt changes n surface stress near the terran change s very mportant to obtan the relable estmates of the wnd speed that s necessary to estmate the wnd power resource at a gven ste. Fgure 9.14a. Introducton of multple IBL s to descrbe the flow across a real landscape (Hedegaard and Larsen, 198). An alternatve approach s to ntroduce a blendng heght, where the effects of ndvdual bts of ndvdual of terran has been lost, or s blended together. The actual blendng heght formulaton of course wll have to dependent on the actual form and scales of terran features. An example s shown n Fgure 9.14 for an urban boundary layer. A rough estmate of the blendng heght can be obtaned from dffuson arguments, see secton 10: ' w (9.35) hb ( ) L c, u Where L c s the characterstc horzontal length scale for surface nhomogenety (Mahrt, 1996). Fgure 9.14b Indvdual IBL s and ther mergng at a blendng heght wthn the total planetary boundary layer (Grynng,006) Lectures n Mcro Meteorology 185

187 Above the blendng heght the profle flux relatons can be wrtten as: (9.36) eff u z z uz ( ) = ln( ) ψ ( ) eff eff κ z0 L Where the effectve values for z 0, u * and L have to be derved from detaled consderatons about the blendng heght. Here dfferent authors use dfferent blendng heghts. Wernga (1976) and Claussen (1991) refer a heght related to the heght of the roughness element, whle Mason (1988); Wood and Mason (1991) and Mahrt (1996) prefer a heght related to the horzontal scale for the roughness changes and the same upward dffuson rate as used n the step change model. The apparent smplcty of (9.36) s acheved at the prze of farly complcated sub-models descrbng how one can compute the effectve values of z 0, u* and L from the surface structure and the flow condtons (Hasager and Jensen, 1998). The dscusson above has been concentrated on the response of the boundary layer to nhomogeneous surface condtons. As a note of cauton, t should be ponted out that heterogenety n the boundary layer mght be due to other reasons as well. For example, slowly developng synoptc systems and cloud systems wll ntroduce nhomogenty that s orgnally ndependent of the surface condtons although the surface mght be nfluenced by for example clouds changng the radaton condtons at the surface., Turbulence changes wthn an IBL. Change of turbulent spectrum for changng surface condtons. Untl now we have dscussed the change n mean profles and the surface fluxes that scale the mean profles for nternal boundary layers. We shall now dscuss the changes n the turbulence varances and power spectra. Snce dfferent scales of the turbulence are modfed dfferently, t s most ratonal to consder the modfcatons of the turbulence spectra. Recall, we consdered and rejected usng the transport equatons (9.1) to derve the change n mean flow wthn the nternal boundary layer. Smlarly we shall avod usng the transport equatons for fluxes and varances, although, we have these equatons avalable from sectons 4 and 6, also they wll show qute complcated to use. Instead we shall use some smple sem-emprcal models based on spectra, whch buld convenently on to our smple IBL models for the mean flow and fluxes. As for the mean flow consderatons we start wth changes n surface roughness, and or heat flux. We wll consder the change n turbulence, and turbulence spectra. We notce that after a surface change, there wll be a local turbulence producton, u du/dz, wthn the new IBL that wll frst be close to the surface and then ncrease to greater heghts, and thereby larger wavelengths, as the IBL grows, because we have seen that ths shear producton manly njects energy at eddes wth a wavelength λ=z (= π/k 1, where k 1 s the wave number along the x- drecton, see dscusson n secton ). Ths local turbulence wll gradually replace the turbulence beng advected from upstream. Followng Højstrup (1981), Panofsky et al.(198), we consder the spectral modfcaton to follow a frst order system: (9.37) ds( k )/ dt = ( S ( k ) S( k ))/ τ ( k ) , 186 Lectures n Mcro Meteorology

188 S 1 (k 1 ) s the downstream equlbrum spectrum,τ(k 1 ) s an eddy lfetme that depends on the eddy scale, descrbed by k 1. For eddes n the nertal sub-range standard dmensonal reasonng results n: (9.38) τ( k ) ε k 1/3 /3 1 1 For eddes outsde the nertal sub-range, Panofsky proposes to use (9.39) τ( k ) ( kσ ) 1 1 Whle Højstrup (1981) proposes ( k 1 S(k 1 )) ½ nstead of σ u, when focused on unstable condtons, whle Mann(1994) assumes (9.38) for all condtons. The eddy lfetme s seen to ncrease wth eddy sze for both (9.38) and (9.39), as well as for the assumpton of Højstrup (1981). The man dfference between the expressons s the rate of ncrease wth eddy sze. Assumng a constant advecton velocty, U, wth x= Ut, equaton (9.36) s easly solved to: S ( k ) S( k ) = ( S ( k ) S ( k ))exp( x/ Uτ ( k )), (9.40) u 1 Where S 0 s the upstream spectrum that s also the spectrum above the nternal boundary layer, and x=0 s coordnate of the terran change, S 1 /S 0 s the rato between the equlbrum spectra upstream and downstream. We follow wth an llustraton from Højstrup(1981) n Fgure Equaton (9.40) s used to model the transton. Snce t s an unstable stuaton the modfcatons of u and U are modelled by used of (9.19), whle the equlbrum spectra are descrbed as gven n (8.40). To understand the behavour of the spectra n Fgure 9.15, we reformulate (9.40) and (9.38) n terms of the normalsed frequency, n= k 1 z/π =z/λ. (9.41) ns ( n) ns( n) = ( ns ( n) ns ( n))exp( x / Uτ ( n / z)) /3 /3 /3 (9.4) τ( n) ε z n From these equaton s seen that n a n-system, the hghest frequences are gettng modfed frst, and the frequency nterval that are modfed by the new surface becomes wder the lower s the measurng heght. Ths means, that the model s consstent wth the dea,that wthn the nner layer, see Fgure 9.5, the turbulence s n equlbrum wth the mean flow, as f the stuaton was horzontally homogeneous. Outsde the IBL, the spectrum s assumed to be equal to the upstream form, denoted ns 0 (n) n the equatons above, the model does not nclude any nterpolaton forms. The model also contans other strong smplfcatons, e.g. that the modfed spectrum form must be wthn the IBL, an argument meanng that z<h, n spte of that ths s not explctly formulated wthn the model tself, just as for the wnd speed predcton n (9.9). Addtonally, t s not qute clear how the spectrum behave for z larger that the nner layer heght and smaller that the matchng heght, for h <z<h n Fgure 9.5. Turbulence spectra for flow over hlls. Also for flow over hlls the eddy tme scale of (9.38) s mportant to characterse the turbulence modfcaton. As for the surface flux modfcaton above, we wll consder the spectra, because dfferent scales modfes dfferently. We shall follow Frank (1981), who has tred to summarse the work of several authors n terms of the spectral modfcaton through a transfer functon G(n= fz/u = z/λ), such that (9.43) Sa( n) = Ga( ns ) 0( n), Lectures n Mcro Meteorology 187

189 n= fz/u Fgure Model spectra compared to data for a water-land transton from (Højstrup, 1991). The fetch s 53 m from the transton, and the measurement heght s 1,,3 and 5 meter. The heght of the boundary layer, Z I =1000 m. At 5 meter only the very hgh frequency part of the spectrum are affected by the boundary layer change, whle at the bottom a wder frequency range s nfluenced., Where S 0 (n) s agan the upstream spectrum for horzontally homogenous condtons, S a the deformed spectrum above the hll, both measured at the same heght above the local ground level, or more precsely along the same trajectory.the forms are llustrated for the u- spectrum n Fgure The form of G(n) depends on the type of spectrum consdered and on the scales of dstorton as well as the (x,z) poston. For the largest scales, we have λ>> L n<< z/l, where L s the half wdth of the hll, defned n Fgures 9.10, and λ the wave length of the turbulence. For ths n-nterval, eddes can be consdered quas-statonary, whle they move across the hll. Hence the ampltude s amplfed by the same S amplfcaton one fnds for the mean wnd, see (9.31), therefore, for ths n- nterval : (9.44) G n = G n = + S G n, u( ) v( ) (1 ), w( ) Lectures n Mcro Meteorology

190 Compare Fgure The subscrpts on the G-functons refer to the velocty components. That G w s unty s based on measurements (Frank, 1996). It s understood that S(x,z) s a functon of both x and z. Close to the ground we fnd the nner layer (9.30) whch s charactersed by that the local profle s n equlbrum wth the local stress, meanng that we stll have a standard logarthmc profle, but wth a local stress. Here all the turbulence scales changes such that all turbulence spectra follow the standard forms, scaled by the enhanced stress, meanng that (9.45) ( ) = (1 + ), =,,. G n S uvw Rapd dstorton: For transport tme across the hll, t = L/U <<τ(k 1 ),where τ s the eddy lfe tme scale, defned n (9.38), the eddes s modfed by the changed shear. However, they have not tme to nteract wth each and start the non-lnear cascade, we dscussed n secton. They wll expand, contract and rotate n the dfferent drectons, often leadng to a decrease of u and enhanced w and v. As eddes are central to the concept, the lnearsed vortcty equaton (not dscussed n these notes) s often used for estmatng the effects on the velocty components. It can be used n a lnearsed verson because the eddy tme scale s exactly a measure of when the non-lnear effects of eddy-eddy nteracton becomes mportant. The computatons are usually based on sotropc upstream condtons, wth correcton for non-sotropy (Batchelor and Proudman, 1954, Hunt and Carruthers,1990, Zeman and Jensen, 1987, Brtter et al, 1981, Newley, 1985). Based on that the equatons are lnearsed Mann (1994) could extend the soluton technque to operate drectly on the spectra. The flght tme lmtaton for the rapd dstorton can be reduced to a normalzed frequency range for ther actvty can be deduced as follows from (9.38) 1/3 /3 (9.46) t = L/ U << τ ε k 1 Insertng standard expresson of the logarthmc profle, ε u 3 /kz, and k 1 =πn/z, equaton. (9.46) can be brought on the followng form: z 3/ (9.47) n<< nr z/ λr ( ln( z/ z0)), L Where n R s an upper lmt for rapd dstorton. Equaton (9.47) shows that the n-frequency nterval for rapd dstorton expand wth heght, beng charactersed by (9.48) z/ L<< n << nr. The lower lmt corresponds to the upper lmt for the quas-statonary regme, see (9.45). For n>n R the eddy tme scale s now small compared to the dstorton tme, non-lnear cascade effect start actng, and there s no bass for rapd dstorton anymore. Therefore the spectra are unmodfed. Also close to the ground, n the nner layer, there s no bass for rapd dstorton due to lower wnds and stronger shear (not ncluded n the rapd dstorton formulatons, and (9.44) or (9.45) controls the spectral modfcaton. For the rapd dstorton one fnd the followng approxmate forms for the G(n) functons (Newley, 1985,Frank, 1996): Lectures n Mcro Meteorology 189

191 (9.49) 4 Gu ( n) = µ 1 ( S) (1 1 / R) 35 G ( n) = Rµ ( R 1)(1+ 4 / 5 S), v G ( n) = Rµ ( R 1)(1+ 6 / 5 S) w 3 Where R = (σ u /σ w ) upstream, s called the ansotropy rato for the upstream condton. For truly sotropc turbulence, all components have the same varance and R=1.The µ are defned as: µ = (σ /σ upstream ), = u,v,w, for sotropc deformed turbulence. Townsend(1976) and Brtter et al (1981) fnds the followng smple varaton for the (unrealstc!) condtons of sotropc turbulence: For = u,v, w, µ = (σ /σ upstream ) = 1±4/5 S, where the + pertans to = w,v, whle the pertans to =u. The R-factor n (9.49) clearly shows the mportance of the non-sotropy of the upstream turbulence, as s ponted out by Zeman and Jensen (1987) from a nd order closure model. z/l < n < n R n = zf/u Fgure Deformaton of the longtudnal velocty spectrum above a hll. The fgure shows the transfer functon G(n) and the undstorted and dstorted spectra, both normalzed by the upstream u,as well three dstorton regmes, delmted by z/l and n R (Frank, 1996).Note that here and throughout ths hll turbulence L means that half-wdth of the hll. Also the curvature of the streamlnes s mportant for the dynamcs of the turbulence modfcaton. Over the hll summt the curvature s convex, resultng n a stablzng effect and a reducton of the w-component, ndeed Zeman and Jensen (1987) fnd model the modfcaton of the w-component due to streamlne curvature, r: σ U w U uw ' ' x r Cur Assumng that the curvature can be estmated from the hll curvature tself, 1/r -(d h/dx ) - H/L, one fnd a curvature effect, cu G w (n), to be multpled on to the G w s n (9.43). Cu σ u ' w' x u LH w Cu (9.50) G ( n) 1 H / L S G ( n) S. w w σ σ r σ L 0 Cur Lectures n Mcro Meteorology

192 Tltng modfcaton. Fnally the spectra are modfed, because the axes of the large quas-two dmensonal eddes wll not be tlted smlarly to the mean flow on the up-wnd of the hll, therefore there wll be a component of the upstream horzontal moton of these bg eddes n the drecton of w. The tlt correcton can be computed by a rotaton around the y-axs: u = u 0cos( α) + w 0sn( α) (9.51) w = u 0sn( α) + w 0cos( α) Where subscrpt 0 ndcates sgnal wth x n the true horzontal drecton, and α s the slope angle. There wll be a tlt effect for all components, but snce the large scale, low-frequency w- component s small, compared to u and v, the effects wll be largest for the w-spectrum and for the uw-co-spectrum., see Fgure Ths fgure llustrates the behavor of the spectral modfcaton accordng to the model dscussed above. Fgure Spectral modfcatons from Frank (1996). Upstream (full lne) and deformed power and co-spectra 10 m above the surface at the summt of a rdge. The parameters used n addtonal to the ones mentoned ar R=1.5 (9.46), L * s the Monn Obuchov scale n ths text. The tlt angle s denoted α. Addtonal coeffcents, as the ones specfyng the lmts for the dfferent dstorton regmes, are found n the paper. Note that the paper has reversed the f and n on the normalsed frequency axs. The resultng combned G a (n) functons n (9.43) thus depend on the detals of the flow, the hllstructure and the (x,z) coordnates of the measurng pont. However, we notce that most of the spectral modfcatons are presented n terms of S(x,z) and hence has a certan generalty. Frank (1996) compares the spectral model to result from a number of measurement campagns for hlls and escarpments and fnds, what one can call a moderately close match wth the Lectures n Mcro Meteorology 191

193 dfferent components of the G a (n) varyng n mportance wth parameters n queston and locaton (x,z) over terran. Dscusson of the turbulence models. Overall we can say that the models for turbulence for dfferent IBLs are qute smplfed, and work better for some quanttes than for other, The u- spectrum s reasonably well predcted, but the w- spectrum and the uw-co-spectrum show large varablty around the predcted values.. The model for the change n surface flux has the advantage of beng smple, whle the hll model s qute complex, nvolvng many dfferent components and processes. None of the models should be consdered accurate. They also nvolve a number of numercal coeffcents, whch we have neglected here, where we are focused on the physcal concepts only. The coeffcents can be found n the papers, referred to. As a techncal aspect should be ponted out, that the use of spectra as functon of wave numbers, and normalsed frequency strctly speakng s ncorrect, snce a the turbulence s horzontally nhomogeneous. However, a frequency spectrum can be obtaned. The argument about the senstvty to eddy sze s also reasonable and the upstream spectral structure can be consdered to follow standard boundary layer forms. Hence, t s convenent to use such a notaton also for ths nhomogeneous terran. For comparson wth the model uncertantes, we nclude the concluson from the revew of Panofsky et al.(198) about the spectral modfcatons for changng terran condtons: Change n roughness: Spectral denstes are affected only wthn the nternal boundary layer, and the nertal subrange spectra follow model spectra for homogenous condtons. For horzontal spectra only the hgh frequency adjust vrtually mmedately, low frequences adjust under the nfluence of the eddy lfetme (that n general nfluence all scale of the spectrum, but s domnated by the local stress producton for hgher frequences). Over hlls and escarpments. In the nner regon for the hll tops, the spectra follow and adjust n the same way to the surface stress. For the horzontal spectra the low frequency components are reduced. Above the nner layer rapd dstorton and other processes have to be appled. The vertcal velocty spectra follow generally well the homogenous forms because of the overall suppresson of the low frequency part. To the comparson of Panofsky et al (198) one could add that there wll often be a mxed hll and surface flux modfcaton nvolved wth many hll flows. Fnally, we can pont out that the terran effects on the wnd turbulence has been ncluded n the WAsP Engneerng program developed at DTU-WIND, based on the modellng by Mann (1994,1998) and (WAsP Engneerng, 008) Examples of dfferent thermally nfluenced IBL s The Polynya, represent probably the largest step changes n surface temperature and heat flux wthn the atmospherc boundary layer, see Fgure The Polynyas are persstent free water areas wthn the Arctc sea ce, occupyng only a small fracton of the total area of the arctc seas, but are very sgnfcant for the total heat exchange between the Arctc Ocean and the atmosphere. 19 Lectures n Mcro Meteorology

194 Fgure 9.18 Internal Boundary Layer over a Polynya, wth very ntense heat flux (Andreas, 1980). Notce the sharp C step change n surface temperature. The opposte of an unstable TIBL (Thermal Internal Boundary Layer) s the stable TIBL, often called SIBL, when warmer ar passes over a cold surface, most clearly demonstrated for a cold water surface. Stable TIBLs typcally use very long fetches to reach equlbrum, because the ar eventually wll be cooled by the water, but the surface heat flux, responsble for ths coolng, s very small, both because of the stablty and because of the small water roughness. Eventually at very long fetches, hundreds of klometres, the whole IBL s cooled to the water temperature, and the boundary layer becomes neutral. The orgnal water- ar temperature dfference s now concentrated n an elevated nverson. Fgure Stable nternal boundary layer (SIBL) over water for small and larger fetches (B. Lange, personal Communcaton) A short example of a SIBL n the Øresund regon s seen on the next fgure, where the predcted heat flux between ar and water as functon of fetch s presented (Melas, 1998). Lectures n Mcro Meteorology 193

195 p z = g ρ * land p z = g * ρ land Fgure 9.0. Characterzaton of the dmensonless heat flux n the growng SIBL, showng the dmensonless heat flux versus the dmensonless fetch over a cold a relatvely cold Øresund, see left hand fgure (Melas,1998) Importance of the thermal structure of the atmospherc boundary layer for flows over hlly terran. For moderately non neutral stratfcatons the thermal structure of the flow modfes the flow over the hll n Fgure 9.10 and 9.11, as seen n the fgure below. Fgure 9.1. Flow over a hll wth unstable (red), neutral(black) and stable(blue) upstream wnd profles. The upstream profle s shown to the left of the hll. The black curve reflects the neutral logarthmc profles. The sold blue lne shows a stable profle, whle the broken red lne reflects 194 Lectures n Mcro Meteorology

196 an unstable profle. The fgure llustrates why stable condtons results n larger speed ups on the top of the hll, than neutral, that n turn yelds larger speed up than unstable condtons. Recall that the wnd at heght L s unchanged, for whch reason the larger stable wnd gradent wll ncrease the speed up over the hll, f we follow the speed up constructon descrbed n connecton wth Fgures 9.10 to 9.1. For larger stable stratfcatons the thermal structure changes the flow consderably, and may determne f the flow wll be able to flow across a hll or wll have to flow around t. For flow over hll, wth heght h, one often uses the Froude number, F, to characterze the flow. The more stable the flow, the smaller s F. The hgher the wnd speed, U, the larger s F. U Froudenumber = F = Nh g ρ 1/ g θ 1/ N = ( ) = ( ) ρ z θ z Fgure 9.. The effect of the Froude number for flow above and around a hll, from emsson heght, h s, Reynolds number, Re, (Hunt et al, 1978). Lectures n Mcro Meteorology 195

197 Cold ar dranage s another typcal thermally drven flow, where the dense ar close to the surface runs lke water followng the terran elevatons. For larger and longer slopes one talks about catabatc wnds, but even short lopes and moderate fetches can result n small scale dranage flow, as llustrated from Rsø, where westerly flow at the top of a meteorologcal mast turns easterly ( downhll) close to the ground when the stablty, measured by a Rchardson Number, ncreases (Mahrt and Larsen, 1990). Fgure 9.3. Weak cold ar dranage flow for moderate slopes (Mahrt and Larsen, 1990).Terran and flow condtons. Addtonally, the lower fgure (Mahrt and Larsen, 198) shows that even for such weak dranage flows, the ntal phase may take the form of a surge wth a clear frontal structure The thermally controlled nternal boundary layers can change to dynamc systems wth closed crculatons, where the dfferental heatng of land and water gve rse to sea-breeze systems or land breeze systems wth return flows. A smlar forcng may be derved from the so-called Urban Heat Island, beng nduced by the hgher temperatures n urban centers, compared to the surroundng land. For breeze systems the dfferental heatng typcally reflects the durnal cycle. Such systems typcally have horzontal scales up to km, see Fgure 9.4b. If the scales ncrease to contnental scales, and they reflect the dfferental heatng and coolng on an annual tme scale, the systems are called monsoons. See llustraton below that could llustrate as well a breeze as a monsoon, dependent on the scale of the water and land surfaces nvolved. 196 Lectures n Mcro Meteorology

198 Fgure 9.4a. A sea breeze system under development. Notce the return flow, the mbedded TIBL, and a between the cool and the warm ar. Such fronts occur often between ar masses of dfferent characterstcs. (Wallace and Hobss, 006), see also the dranage surge llustraton, n Fgure 9.3. Fgure 9.4b. Schematc of meso-scale summer and wnter monsoon crculaton (Mller and Thomson, 1970). Fgure 9.5 llustrates a valley wnd system that oscllates between cold ar dranage and rsng ar flow along sun-heated slopes. Lectures n Mcro Meteorology 197

199 Fgure 9.5. Durnal cycle of valley wnds caused by heatng and coolng of the sdes of the valley (Hunt and Smpson, 198) A sea breeze and ts return flow can occasonally be seen at chmney plumes, as shown n the pcture below from Long Island off New York. Fgure 9.6. Change of flow drecton wth heght at a land-sea breeze stuaton, llustrated by smoke plumes released at two dfferent heghts. Fnally, we show a typcal expermental set-up to test or at least valdate the models for nternal boundary layers, here the DUDAMEX experment on a Dutch sland Schermonnkoog (Højstrup 198 Lectures n Mcro Meteorology

200 et al,198). The man nformaton here s that such experment by becomes more expensve than smlar experments to test the behavor of homogeneous boundary layers, because measurements has to be done at least n two dmensons, the vertcal and along the drecton perpendcular to the surface change, here a water front.. Fgure 9.7. Layout for studyng change of surface IBL at a coast, from (Højstrup et al, 198). Compared to the Kansas experment n Fgure 6.1, an addtonal horzontal dmenson has to be montored. Appendx 9.A. General lnear descrpton of neutral stablty flow over hlls and other orography. We consder a two dmensonal hll, a rdge, wth a constant velocty u 0 approachng the hll perpendcular to the rdge. It s the same hll (rdge) consdered n the man text of Secton 9, Fgure 9.10c. In ths appendx we wll consder such hlls as beng descrbed by snus and cosne functons. Hereby we wll see that that the solutons can be appled to arbtrary terran, as long as the aspect rato s low enough for lnear operatons to work. Addtonally, the detaled dervaton llustrates the smplest physcs that can be appled to derve the flow over a hll (Troen et al, 1990, Troen and Petersen, 1989).. Lectures n Mcro Meteorology 199

201 Fgure 9A.1. Smplfed flow over a two-dmensonal hll. Upstream velocty s assumed constant wth heght. The heght of the hll s h, and the wdth s L, measured at h/. From (3. 83) we have: (9A.1) u u 1 p u t x x x x + uj = gδ3 Ω εjkηjuk + ν uu j j ρ j j Assumng statonarty, neglectng molecular vscocty and neglectng the Corols force, (A1) reduces to: (9A.) 3 u 1 p u j = gδ uu j x ρ x x j j We now neglect the horzontal turbulence fluxes compared to the vertcal and wrtes: (9A.3) u 1 p 1 p u g uw g K u j = δ3 = δ3 xj ρ x z ρ x z Where we have used a constant dffusvty, K. The upstream condtons are defned by u j0 = (u 0, 0,0), and p=p 0 (z). The wnd u 0 s a constant and the pressure,. For these condtons, (A3) takes the form: u0 1 p0 u0 u0 = + K, or 0= 0 x ρ x z (9A.4) w0 1 p0 w0 1 p0 u0 = g + K, or 0= g x ρ z z ρ z Whch show that we assume the hydrostatc balance to be fulflled for the chosen upstream condtons, whle all other terms are zero. The hll perturbs the flow such that: (9A.5) p = p0+ p, uj = uj0+ u j, u j = ( u,0, w ) 00 Lectures n Mcro Meteorology

202 Note that we assume that the perturbed lateral velocty s zero, because out hll s a twodmensonal rdge along the y-drecton. Note further that the perturbed varables, u, p are dfferent from the turbulence fluctuatons that are statstcal functons of space and tme, whle the perturbed flow varables here are determnstc functons of the vertcal and the horzontal coordnates around the hll. Equaton (9A.3) can now be wrtten: (9A.6) ( u + u ) 1 ( p + p ) ( u u ) g K ( u u ) j0 j 0 j0 + j = δ3 j0 + j xj ρ x z Keepng only terms of the frst order n the perturbed parameters, we can wrte: u 1 p u u0 = + K, x ρ x z (9A.7) w 1 p w u0 = + K, x ρ z z Where we subtracted the zero terms, as gven by (9A.4), and neglected second order terms of the perturbed parameters. (9A.7) s seen to have three unknowns, p, u, w, but ncludes only two equatons. We ntroduce the contnuty equaton as the thrd equaton: j (9A.8) ( u0 + u ) ( v0 + v ) ( w0 + w ) + + = 0, or x y z u w + = 0. x z As seen, we have a system of partal dfferental equatons. To reduce t to a system of ordnary dfferental equatons, we ntroduce the Fourer transformed varables, performng the Fourer transforms n the x-drecton, where the solutons obvously must be well behaved, due to the lmted scale of the hll perturbaton. We ntroduce the Fourer transforms of the varables as follows: (9A.9) kx kx u ( x, z) = uˆ( k, z) e dk ; w ( x, z) = wˆ( k, z) e dk; 1 kx p ( x, z) = pˆ ( k, z) e dk ρ, wth correspondng reverse transformatons. Here k s the wave number along the x-drecton. Insertng (9A.9) n (9A.7) and (9A.8), we obtan the equaton for the transformed varables: uzk ˆ(, ) ku0uˆ(, z k) + kpˆ(, z k) K = 0, z pzk ˆ(, ) wzk ˆ(, ) (9A.10) ku0wˆ( z, k) + K = 0, dz z wzk ˆ(, ) kuˆ( z, k) + = 0. dz Lectures n Mcro Meteorology 01

203 To proceed, we can dfferentate the second of these equatons wth respect to z, and use ths and the last equaton to elmnate ŵ. Thereafter one dfferentates the frst of the equatons twce and uses ths to elmnate ˆp as well. The resultng equaton for û becomes: 4 uˆ ku0 uˆ ku0 (9A.11) ( k + ) + uˆ = 0. 4 z K z K We seek a soluton wth the heght varaton of the form: exp(-αz), where α s a complex coeffcent, because we expect the perturbaton to vansh for large z. Insertng ths n (A11), we obtan: 4 ku0 ku0 α ( k + ) α + k = 0, or (9A.1) K K ( ku Kα )( k α ) = 0 0 Wth the solutons: α = ±k, and α = ±(u 0 k /K) 1/ = ±(1+)(u 0 k/k) 1/, where we choose the postve k to be systematc. Hence, the soluton for û becomes: (9A.13) abs( k ) z u0 abs( k) uˆ( z, k) = ae a exp( (1 + ) z ), K We have chosen the mnus sgn n the exponental functons to ensure that the perturbatons dsappear for large z. There s one coeffcent, a, only, because û = 0 for z =0. Correspondng to (9A.13), we fnd the expresson for ŵ, agan usng equaton (9A.10) (9A.14) abs( k ) z u0abs( k) wˆ ( z, k) = we 1 + wexp( (1 + ) z ) K We now make the connecton to the Flow over Hlls n the man text of ths secton. Consder a cosne shaped hll, where now a wdth could be the wavelength, λ =π/ k. Fgure 9A.1 llustrate that t the length scale customary used to characterze hlls s, L corresponds to about λ/4 that can be taken as L 1/ k. In the man secton was defned as well the nner length scale,, whch here must be related to the second exponental, that s: (9A.15) K KL = u abs( k) u 0 0 To estmate K n a realstc way, we use the surface layer results K = κu z κu, where we have chosen the nner scale as a characterstc z. The nner scale equaton now develops as : (9A.16) κ u L ul L 1, or κ κ = = L, u u ln( z ) Where we have used the logarthmc profle for u 0, somewhat nconsstent wth the start assumpton of a constant u 0 n ths appendx. It s seen that we arrve to the same approxmate relaton between L and l, as n the man part of the secton (9.30). We now have an ntermedate result: 0 Lectures n Mcro Meteorology

204 (9A.17) Wth z z(1 + ) L uˆ( z, k) = ae ae z z(1 + ) L 1 wˆ (, z k) = we + we L = 1/abs(k) and l(k) κ /ln(l/z 0 )/abs(k). To solve for the three unknown parameters a,w 1,w, we consder the boundary condtons at the ground, whch smply states that the flow has to follow the terran at zero -level. h(x) u 0 + u w dh/dx = w /( u 0 + u ) dh dx Fgure 9A.. The flow followng the terran close to the surface. From fgure 9A. and (9A.5) s seen that dh dh wx (,0) = ( u + ux (,0)) u dx dx (9A.18) 0 0 Followng the procedures above, we ntroduce the Fourer transform of h(x) (9A.19) ˆ kx ( ) ( ) ; ˆ kx h x = h k e dk h( k) = h( x) e dx; Usng (9A.9), the boundary condton at (9A.18) can be brought on the form: wˆ ( k, z = 0) = u kh ˆ ( k) = w + w. (9A.0) 0 1 From (9A.17) we therefore obtan: (9A.1) z z(1 + ) L ˆ ˆ w(, z k) = w e + ( u kh() k w ) e From the contnuty equaton n (9A.6) we get: Lectures n Mcro Meteorology 03

205 (9A.) Dfferentatng (9A.1) we can now solve for w1, to fnd: khˆ( k ) u0 (9A.3) w1 = 1 (1 +L ) wz ˆ( = 0, k) kuˆ( z = 0, k) + = 0; Snce kuˆ( z = 0, k) = 0, dz wz ˆ( = 0, k) = 0. dz We use the contnuty equaton once more to fnd the remanng mssng coeffcent, a, n (9A.17). Assumng the z>l, we can focus on the e -z/l term n the varables (9A.4) wzk ˆ (, ) 1 khˆ ( k) u ku z k kae e or dz L 1 (1 + ) L z/ L 0 z/ L ˆ(, ) =, : hku ˆ( ) 0 (1 ) a = L + Hence we can now wrte: (9A.5) hku ˆ( ) L (1 + ) khˆ( k ) u wzk e e 1 L(1 + ) L(1 + ) z z(1 + ) 0 L uzk ˆ(, ) = ( e e ), z z(1 + ) 0 ˆ L (, ) = ( ) To use (9A.5) to see how the perturbaton would look for a real hll, we agan consder a cosne hll, defned by: π π (9A.6) hk ( ) = H( δ( k ) + δ( k+ ) L L π x Hence, hx ( ) = Hcos, and the perturbaton velocty s found from (9A.9, 9A.5, 9A.6): L (1 ) ˆ( ) z z + kx kx hku 0 (, ) ˆ L u k z = e u( k, z) dk = e ( e e ) dk L (1 + ) π π H(( δ( k ) + δ( k+ )) u0 z z(1 + ) kx (9A.7) L L L = e ( e e ) dk L (1 + ) π x uh 0 cos z z L L ( e e ), L Where we have neglected the n the (1+)- terms, whch would change the poston of the maxmum slghtly from the top of the cosne. From (9A.7) s seen that the relatve overspeedng s largest at the top of the hll and equal to H/L Ths s the same result as n (9.31) n the man part of ths secton. Here, however we see as well how the over-speedng vares across the hll, from -H/L to H/L. From (9A.5) s seen that the w-perturbaton vares not at a cosne, but as a snus functon, wth the maxmum/mnmum w at the mddle of the slope. 04 Lectures n Mcro Meteorology

206 The heght varaton s sketched n the next fgure, where for ncreasng z, frst e -z/ goes to zero, followed by that e -z/l goes to zero. z L uxz (, ) Fgure 9A.3.The heght varatons o f the perturbaton veloctes across hll. The total veloctes across the hll s derved by addng the perturbaton veloctes to the upstream velocty (u 0, 0,0). In the dervaton we have assumed a constant u 0. From the dervaton above t s seen that the solutons for the dfferent k-values are uncoupled. Hence, one can choose dfferent u 0 for dfferent k-values. For example one can mplement a logarthmc wnd profle by: u u( k) ln( Lk ( )/ z) κ (9A.8) 0 0 Further, t s seen to be t s relatvely smple to extend the two-dmensonal case dscussed above to a three dmensonal case, solvng the followng equatons: (9A.9) u v 1 p u u0 + v0 = + K, x y ρ x z v v 1 p v u0 + v0 = + K, x y ρ y z u v g K x y ρ z z w w 1 p w = + Wth the varables; ( u, v,0), and ( u, v, w ), together wth p and p, (9A.30) And the soluton method based on the two dmensonal Fourer transform: (9A.31) h( x, y) = hˆ ( k, k )exp( k x + k y) dk dk x y x y x y u ( x, y) = uˆ ( k, k, z)exp( k x + k y) dk dk x y x y x y Lectures n Mcro Meteorology 05

207 and smlarly for the other varables. From ths we see that the flow response s related to snusodal terran varatons, and snce all terran varatons through the Fourer transformaton can be consdered composed by snusodal varatons, the soluton to any arbtrary landscape and not only hlls can be solved by the above systems., provded the hlls are not too steep (n theory h/l<<1, n practce h/l<0.4). Hence the startng pont for modelng the flow response to a real terran s to establsh a dgtal verson of the elevaton h(x,y). The Two dmensonal Fourer expanson provdes the snusodal boundary condtons for solvng the 3D flow soluton followng the outlne above. In Fgure 1.10 such a map wth a supermposed grd system s shown. Note that we have not ncluded stablty effects n ths summary. They need addtonal consderatons, e.g. Dunkerley et al (003). 06 Lectures n Mcro Meteorology

208 10. Dsperson plumes from chmneys. In ths secton we shall focus on the prncples behnd dsperson from a chmney, both to see the governng prncples and to see what t can tell us about the nature of atmospherc turbulence. Turbulent dffuson from stacks Consder Fgure10.1 below; the chmney has a source strength, denoted Q [kg/sec.]. Normally one neglects the turbulent dffuson s the x-drecton, because the transport n ths drecton s domnated by the advecton by the mean speed. We consder dffuson n the z and y-drecton. Actually, we wll normally only consder dffuson n one drecton and assume the other one to be smlar. To smplfy further we assume the chmney heght to be large enough for us to use a constant dffusvty K = ku z ku h. z x C(x,y,z) h y U(z) K Fgure Dsperson from a chmney, wth coordnate system, concentraton feld, C, wnd speed U and dffusvty K. We shall consder the crosswnd ntegrated concentraton dstrbuton n the z-drecton: (10.1) c( x, z) = c( x, y, z) dy The mean concentraton, c(x,z) and the source strength Q s related through: (10.) Q = u c ( x, z) dz, meanng smply that materal comng out of the chmney dsperses but does not dsappear. Note, here s a flux entrely carred by the mean speed. To determne the concentraton dstrbuton, we can use the equaton for the development of the mean concentraton of a passve scalar, see for example. (9.3). As seen, t s bascally the same equaton for all passve scalars, temperature, humdty and trace concentratons. Lectures n Mcro Meteorology 07

209 (10.3) dc c c uc j = + u j = dt t x x j j We consder only the crosswnd ntegrated concentraton dstrbuton c(x,z) and follow the assumpton above that K can be consdered constant: cxz (, ) wc cxz (, ) cxz (, ) (10.4) u = = K = K. x z z z z Ths equaton s solved, usng Fourer transform along the z-drecton. We employ the transform par: kz 1 kz (10.5) cxz (, ) = cxke ˆ(, ) dk; cxk ˆ(, ) = cxze (, ) dz; Insertng these n the equaton above we obtan: π (10.6) cxk ˆ(, ) x k ( K/ ucxk ) ˆ(, ) 0; + = whch solves to: (10.7) cxk ˆ(, ) = ce 0 ( K/ u) k x We can now fnd c(x,z) by back transformng the above: (10.8) kz ( K / u ) k x kz 1 z π ( K/ ux ), c ( x, z) = cˆ ( x, k) e dk = c e e dk = c exp( ) 4( K/ ux ) where the last transformaton takes use of an ntegral table. Wth c 0 = 1, t s a normalsed Gaussan functon, meanng that ntegraton over z yelds the value c 0. As we saw above the concentraton and the source strength s related through: (10.9) Q = u c ( x, z) dz = u c0 c0 = Q / u We have now establshed the crosswnd ntegrated concentraton dstrbuton n the z-drecton as a Gaussan dstrbuton gven by Q 1 z (10.10) cxz (, ) = exp( ) u πσ ( x) σ z ( x), z wth the standard devaton gven by: σ = (10.11) z K x u 08 Lectures n Mcro Meteorology

210 It s seen that extenson to c(x,y,z) s trval, so we shall contnue to consder only the dstrbuton n the z-drecton. It s further possble to ntegrate equaton for more realstc K-profles wth some varaton along the z-drecton. For a full numercal ntegraton n accordng wth surface layer formulatons, see Grynng et al (1983). Lagrangan vew of the same problem. To determne the concentraton n the smoke plume from the Lagrangan vew, we release the smoke at the chmney top as a number of concentraton partcles. We thereafter sample the partcles as they arrve at a dstance downstream, and measure the number of packets sampled a gven dstance from the centrelne, N(z), as a measure of the concentraton dstrbuton, compare Fgure10.. Here, we assume that we send the partcles off wth a tme separaton large enough for the partcles to ht the samplng plane ndependently of each other. Ths means that the dstrbuton of N(z) also here becomes a Gaussan dstrbuton. Fgure 10.. Partcle trajectores for plume dsperson consderatons Denotng, the velocty of the partcles along ther trajectory for w (t), we can descrbe the poston of the partcles by: (10.1) t z() t = w ( t ) dt 0 We notce that w s a fluctuatng velocty wth zero average. b The varance of the dstrbuton of partcles after flght tme t, can be found as: (10.13) = 0 t z () t w ( t ) dt b The tme (of flght) evoluton of the varance can be found from: Lectures n Mcro Meteorology 09

211 (10.14) d dt t t d z t = z t z t = w t w t dt = w t w t dt () () () () ( ) () ( ) dt 0 0 t t t w ( t τ) w ( t) dτ RL( τ) dτ w ρl( τ) dτ = = = Here we have assumed that w(t) s statstcally homogenous and statonary, and we have ntroduced t = t-τ,the auto-covarance functon R and ts normalsed verson the autocorrelaton functon, ρ. Both functons have been gven the subscrpt L for Lagrangan, to remnd us that the velocty s measured followng a gven partcle. Notce that for a statstcally homogeneous and statonary feld, we have: (10.15) w L = w E, where subscrpt E refers to Euleran, meanng that the velocty s measured at a fxed pont. The reason that the two varance estmates are equal s that for a statonary and homogeneous feld all averagng procedures n space and tme wll approach the same Ensemble average value. We now wrte the descrpton of the spreadng of the plume as: t d (10.16) z() t = w L( ) d. dt σ ρ τ τ 0 The spreadng of a smoke plume wll therefore depend on the structure of the Lagrangan autocorrelaton functon. From the fgure below, we defne a near feld- dsperson for small travel tmes, where ρ 1, and a far feld dsperson for large travel tmes, when ρ 0. Near feld: The spreadng of the plume can now be wrtten: t d (10.17) () t = w ( ) d w t. or: (10.18) z L dt σ ρ τ τ 0 σ () t = w t ; σ () t = w t. z z Fgure Langrangan Autocorrelaton functon 10 Lectures n Mcro Meteorology

212 Far feld The spreadng of the plume can here be wrtten: t d (10.19) ( t) = w ( ) d w ( ) d = w T, z L L 0 0 dt σ ρ τ τ ρ τ τ where T s the ntegral scale for ρ L (t), that s the Lagrangan tme scale. We now get the standard devaton drectly: (10.0) σ () t = w T t, σ () t = w T t z z Comparng wth the results derved by the Euleran pcture, usng K-theory. (10.1) σ () t = K t, σ () t = K t, z z where we have used that t = x/u. The two methods are seen to yeld smlar results, for the far feld soluton, and even allow us a method to estmate a K, whch s consstent wth our normal way of understandng K. Comparng (10.1) wth (10.0) and (10.18) (10.) Far feld K u = T u and T : σw σw σw. Near feld K u = t u and t : σw σw σw. However, n the near feld the two ways of estmatng the plume dsperson do not agree, snce the Euleran estmate predcts that also close to the source the plume dsperses as the square root of the flght-tme, t ½, whle the Lagrangan method yelds that the plume dsperses proportonally to the flght-tme. Alternatvely one must have a near feld K becomes a parameter determned from the plume hstory (flght tme) not only of the turbulence structure, as for the far feld. Ths s contradctory to the concept of dffusvty. We know that the K-dffusvty closure can has problems, and the Lagrangan pcture s based on very smple physcal concepts (and s confrmed by measurements). Therefore, we beleve that the Lagrangan pcture reflects the truth, and the results show another example of the falure of K-theory. What happens physcally s that the spreadng of a plume n the begnnng s nfluenced of the dstance to the startng pont. Frst when the change n plume spreadng becomes ndependent of the condtons at the source, n the far-feld, we can approxmate the dsperson as K-dffuson. Near the source the dsperson processes are stll dependent on the release condtons at the source. The K-dffusvty has to be a functon of flud varable and not a functon of dstance to an arbtrary source. Relaton between Lagrangan and Euleran correlaton functons. It s not as smple to measure a Lagrangan correlaton functon as a Euleran correlaton functons. The relaton between the two types of correlaton functon have however been establshed both through measurements and through theoretcal consderaton. At ts smplest, one can say that they appears smlar n form and the Lagrangan ntegral tme scale s α tmes longer than the Euleran ntergral tme scale, wth α 4. Ths s n accordng wth that we would expect that the memory of the velocty fluctuatons would be larger for Lagrangan fluctuatons, Lectures n Mcro Meteorology 11

213 where one follows a specfc partcle, than for an Euleran measurements, where the sensor, -so to speak-, measures the velocty of new ar partcles passng through the sensor every tme t measures. The Lagrangan correlaton mage of the dsperson also means that the transton between near feld and far feld happens at dfferent travel tmes for the y-drecton than for the z-drecton, because ntegral scales of the correlaton functons for the two drectons are dfferent and behave dfferently wth heght. The reason for ths s that wthn the boundary layer the nearness of the ground suppress large scale sgnals n the w- components, as we have dscussed before n secton. On a more phlosophcal note, we notce that we here have been presented for another stuaton, where the K-dffusvty, whch s extensvely used n turbulence closures, s not always correct, compare secton 4. Practcal estmaton of atmospherc dsperson. In practse one wll translate the dsperson as functon of travel tme to dsperson as functon of dstance to the source, usng t =x/u. (10.3) σ ( x) = { z w, w T t = X x Far feld u w w t = x, Near feld u It s seen that we can wrte the spreadng of a plume as product of the dstance, x, and what we normally call the turbulence ntensty, and for the transton between near feld and far feld, we have the ntegral scale, X. The turbulence ntensty s descrbable by the smlarty functon, whch we have consdered n secton 6 and 8 and throughout the lterature. Several scalng-formulatons are possble, dependent on whch scalng regmes that s the most relevant for the stuaton under consderaton. (10.4) w w u u u u = f 1( z/ L) f( z/ L, Z0) or F1( z/ zi, zi / L) F( z/ zi, z/ L) F ( stablty, ) z In (10.4) the stablty s measured by several dfferent formulatons, both n surface layer formalsm and as mxed layer formulatons. The turbulence ntensty s a functon of stablty, however t s estmated n a partcular scalng regme, plus some boundary parameters as the roughness and the boundary layer heght. Generally ths functon ncreases wth ncreasng nstablty. The behavour of smoke plumes wth stablty s llustrated on Fgure 10.4 Smlar relatons pertan to the y-dsperson, as to the, here descrbed, z-dsperson. Descrpton of dsperson for dfferent stabltes s llustrated on the next fgure, whle the two fgures 1 Lectures n Mcro Meteorology

214 hereafter show practcal sgma -values for y and z, see fgure these values have trough the tme been derved based on an accumulaton of dsperson experments, theoretcal consderatons and experence wth the used values, much the same ways as the consensus roughness dagrams n secton 5 have been determned. Fgure 10.4 Chmney plumes for dfferent stabltes, ndcated by temperature stratfcaton and the standard (Meteorology and Atomc Energy, AEC, 1968) Lectures n Mcro Meteorology 13

215 Fgure Typcal behavour of plumes wdths as functon of dstance and atmospherc stablty. (Meteorology and Atomc Energy, AEC, 1968) 14 Lectures n Mcro Meteorology

216 Addtonal aspects of plume dsperson. It s worth pontng out that whle the plume grows as ndcated by the sgma curves dscussed above, for an elevated source one wll often have a dfferent experence at the ground, where the surface concentraton close to the chmney s zero, because the plume has not reached the ground yet. Ths happens approxmately, at a dstance, where σ z (x) has grown to become equal to the chmney heght, h. Around ths dstance one wll one experences a very fast growth of the surface concentraton wth x, much as llustrated on Fgure As llustrated n Fgures 10.5 and 10.6, σ z (x) s a strong functon of stablty. p g Upwnd dstance. Fgure Surface concentraton downstream of an elevated source. When σ z (x) grows from zero at the source to become equal to the chmney heght h, the ground concentraton reach a maxmum, followed by a slower dluton as the plume ncreases and ts concentraton therefore decreases. Modern modellng of ar polluton dsperson Modern modellng of ar-polluton dsperson wll normally employ numercal models, ether solvng ( ) smultaneously, or mportng meteorologcal felds, derved from meteorologcal models and data, and subsequently solvng (3.88). As dscussed n chapter 4, soluton of these equatons employs some knd of K-closure, and hence corresponds to the Euleran concept n ths chapter. When pont-sources lke sngle stacks are mportant, plumemodels, as the ones consdered here, are stll used, due to the need of gettng the near-feld dsperson correct. Even for large scale transport models the near-feld Lagrangan characterstcs of a pont sources can be mportant because of the mportance for the, mostly heat nduced, resultng plume heght, at whch the larger scale advecton takes place ( Brandt, 1998). See Fgure Lectures n Mcro Meteorology 15

217 Fgure Illustraton of a European wde ground modellng across Europe from extensve multple sources.(brandt, 1998) Modern plume models are normally formulated n terms of puffs, clouds of partcles, whch can then be advected by a changng mean wnd feld. Also combned numercal modellng that allow for mportant pont-sources, where near-feld dsperson characterstcs s then ncluded. Physcally, dspersons from puffs are derved consderng the change n dstance between pars of partcles, released nearly smultaneously at the source as opposed to the plume dsperson consdered here, where the dstance between sngle partcles from a centre lne s consdered. Ths dfference yelds a stronger scale dependency for puff dsperson than for the plume dsperson. Addtonally the wnds movng the ndvdual partcles are now correlated, makng the statstcal treatment more complcated. Therefore, we shall not dscuss puff dsperson here, where we have just touched upon dsperson, but we refer to Mkkelsen et al (1984, 1987). A concentraton pattern from a European wde dsperson experment from such models are llustrated on Fgure Spectral descrpton of plume dsperson. In the above dscussons we have used the Lagrangan correlaton functon to derve the plume dsperson. For llustraton we shall summarse the same dervaton usng Lagrangan spectra for the same dervaton. We shall use results from secton, equaton (.50) to (.5), wth only slght modfcatons. We can Fourer decompose the Lagrangan vertcal velocty, w (t) n (10.1) as follows: ωt (10.5) w'( t) = e dz w ( ω) Next we follow (10.1) for estmatng z(t): (10.6) t t ωτ ωt/ sn( ωt / ) w'( τ) dτ e dzw( ω) dτ t e dz ( ω) w ωt / 0 0, zt () = = = 16 Lectures n Mcro Meteorology

218 where we have performed the τ-ntegraton. Next we square z(t) and average to arrve to: (10.7) σ sn( ωt / ) S ( ω) w () t = zt () z*() t = wt dω z ωt /, w where * denotes complex conjugaton as usual, and we have followed the defntons and methodologes descrbed n secton, for creatng varances, see (.35) to (.53). As dscussed n secton the snc (ωt/) functon acts a low pass flter suppressng frequences ω >π/t. We can use ths to check the near feld and far feld solutons n (10.17) and (10.30). For the flght tme t very small the snc functon s seen to be unmportant, the normalsed spectrum ntegrates to 1, and we recover (10.18), for the near feld spreadng. For t large, the snc-flter reduces the spectrum to ts value at ω = 0, and (10.7) takes the form: (10.8) sn( ωt / ) S ( ω = 0) T π w () = z ω = = ωt / w π t σ t w t d w t w Tt, where we have recovered equaton (10.0). We have used (.38), for the relaton between the ntegral scale and S(0), wth the excepton of that (10.0) defnes T from ntegraton over the postve parts of the correlaton functon (0, ), whle we n secton defne t by ntegraton (-, ). The π/t derves from the ntegraton of the snc-functon. Euleran- Lagrangan spectral statstcs. We shall now change the Langrangan turbulence spectra used (10.7) to Euleran turbulence spectra. An advantage here s that we can more easly shft between frequency and wave number statstcs, usng Taylors Hypothess. Above we dscussed that the Lagrangan correlaton functon could be related to the Euleran correlaton through R L (τ) = R E (τ/α), wth α at ts smplest around 4. From ths we fnd the correspondng spectral relaton: (10.9) 1 α S ω = R τ e dτ R τ α e d τ α αs αω π = = π ωτ ( ωα )( τ / α ) ( ) ( ) ( / ) ( / ) ( ) L L E E Whch shows that the Lagran spectrum relatve to the Euleran one s larger at lower frequences and lower at hgher frequences, and s consstent wth a Lagrangan tme scale equals α tmes the Euleran one. We now use (10.7), emphaszng the dfference between Lagrangan and Euleran spectra: Lectures n Mcro Meteorology 17

219 (10.30) sn( ω ' t / ) S ( ω ') Lw σ () t = zt () z*() t = wt d ' z ω ω ' t / w sn( ω' t / ) S sn( ωt / α) Ew α ( ωα ' ) S ( ω) Ew = wt dω' = wt dω = ω' t / w ωt /α w = w sn( kx / α ) x α Ew U kx / w S ( k) dk Where we have used Taylor hypotheses n two forms: x = U t, and ω = U k. From the last equaton n (10.30), t s seen that the snc-functon determnes whch eddes that contrbute most to the dsperson, because such eddes are charactersed by wave numbers, k, or a wave lengths, λ, wth k x /α << π or λ x/α. Therefore, close to the source, wth x close to zero, the full w-spectrum contrbutes to the dffuson, but as the dstance to the source ncreases, the smaller eddes are cut out, and only larger and larger eddes contrbute. As x becomes large enough, the dsperson approach the far feld mode, as the spectrum approach S Ew (0). Ths s consstent wth the analyss n (10.) showng that the length scale assocated wth a K- dsperson of a plume, should be of the order of the standard devaton of w tmes the flght tme, n the near feld, untl t approaches the standard devaton of w tmes the Lagrangan tme scale, n the far feld. As we have dscussed n secton and 6, Euleran spectra contnue to smaller and smaller frequences and therefore larger and larger scales, compare Fgure 6.10 and,3, and especally for the horzontal wnd components, responsble for the horzontal plume dsperson, there s consderable energy at the lower frequences. Ths means that the smple hypothess of the spectrum convergng towards a S(0), reflectng one and only one characterstc tme scale, for small frequences does often not apply n practse, where the spectrum rather become modfed by new processes beng responsble for the varablty. Also for these practcal stuatons, equaton (10.30) contnues to apply, as long as we can apply the smple Lagrangan- Euleran transfer and also apply Taylor s hypothess. On the other hand, as the flght tme and dstance to the source, other aspects of the present model become less realstc, such as the assumed statonarty of wnd speed, wnd drecton and source strength. Therefore, one wll today use numercal codes, ether n the form of Euleran dffuson models or Lagrangan puff-models or combnatons of such models (Mkkelsen et al, 1987,Brandt, 1998). 18 Lectures n Mcro Meteorology

220 . Dsperson experments from an ndvdual source for dfferent atmospherc condtons, compare Fgure (T. Mkkelsen, personal communkaton) Lectures n Mcro Meteorology 19

221 11. Boundary layer clmate, radaton, surface energy balance. In the earler secton we have derved characterstcs of the boundary layer flow and turbulence n terms of ts response to dfferent forcng and boundary condtons. We wll understand boundary layer clmatology as descrpton of the characterstc weather phenomena wthn the boundary layer at a gven locaton, n response to ts pecular boundary condtons, and the forcng. Dfferent types of boundary condtons have been presented n these earler sectons, and we have seen how these condtons can modfy the local meteorologcal condtons. Therefore, we shall characterze the forcng by the larger scale meteorology, beng responsble for brngng dfferent ar masses across a locaton, and the radatonal forcng of local coolng and heatng of the ar. As also the large scale weather s drven by the radaton balance, the mportance of ths forcng s obvous. Hence, we start dscussng the of the radaton balance at the Earth s surface. Radaton and energy balance at the ground. Some defntons: All bodes wth temperature above zero Kelvn emt radaton. A black body emts maxmum possble radaton for ts temperature. The energy emtted from a body of temperature T 0 follows the Stephan Boltzmann law: (11.1) Energy emtted = ε σ T 0 4, Where T 0 s the surface temperature of the body, σ, the Stephan Boltzmann constant. σ = Wm - K -4 and ε 1, s denoted the emssvty and beng 1 for a black body and less for a normal (grey body). The wavelength dstrbuton of the emtted radaton follows a Planck curve, see Fg For ncreasng temperature the emsson ncreases strongly, and the Planck curve s shfted towards smaller wavelengths, followng Wen s Dsplacement Law: (11.) λ max = [mk]/ T 0, Fg Spectral dstrbuton of energy radatng from the Sun (T= 6000 K), at left hand sde, and Earth (T= 300 K), rght hand sde, consdered a black bodes. (Oke, 1985). When a radaton flux of wavelength λ ncdents on another body, the response of ths other body s characterzed by ts transmssvty, Ψ λ, ts reflectvty α λ and ts absorptvty, ζ λ = ε λ, where t can be shown that the absorptvty s the same as the emssvty (denoted Krchhoff s law). The relatons between these terms are 0 Lectures n Mcro Meteorology

222 (11.) Ψ λ + α λ + ζ λ = 1; Strctly speakng ths relaton s correct only for a specfc wavelength, but t holds approxmately for wavelength bands as well. For Solar radaton one talks about the reflectvty as α, and t s denoted the albedo. For opaque bodes (Ψ λ =0), the reflectvty and the emssvty are relates as: α λ = 1- ε λ. The dfference n wave length between solar emsson, beng of relatvely short waves, and the Earths emsson beng of relatvely long wavelength, results n the so-called green house effect, because the transmssvty of Earth s atmosphere s much larger for the solar radaton, than for the more long wave Earth radaton, see Fg. 11. Fg The flux of solar radaton, rradance, outsde the atmosphere and at the sea level. The much larger tramssvty n the vsble regon versus the lower transmssvty at the longer wavelength regon, the nfrared s clearly seen.(oke, 1985) Atmospherc radaton can be dscussed ether from the perspectve of what happens wth a specfc radaton, or how radaton enters nto the energy balance at a certan spot or level. We start consderng the Solar rradance,.e. the radaton flux reachng the top of the atmosphere. The solar constant, S 0, s the power (Wm - ) of the solar radaton reachng a sphere at Earth dstance from the Sun and outsde the Earth atmosphere. One has S 0 = 1370 Wm -. Fgure 11.3 shows such how the Solar rradance s dstrbuted wthn the atmosphere. The rradance outsde the atmosphere s taken as 100%. Note that 100% does not necessarly correspond to S 0, snce t depends on the solar heght at the locaton studed and the averagng tme employed. Fg Dstrbuton of the solar rradance as t reaches the Earth. It s seen that 30% s reflected back to space, 51 % s absorbed by the ground and 19% s absorbed by the atmosphere (Shodor,1996). Lectures n Mcro Meteorology 1

223 In the next fgure, Fg. 11.4, we consder the energy balance for certan parts of the Earth system. Fg Energy balance for varous Earth systems, the atmosphere, clouds, the surface. The unts are the 100% solar rradance arrvng at the top of the atmosphere n Fg As seen 51 % reach the surface and 19 % reach the atmosphere. The reflected 30% s neglected n ths fgure. Each system should have balance between gans and losses for the temperature to reman constant (Shodor,1996). Note n Fg that the Earth system lose as much energy by radaton as t receves from space, for the Earth to retan ts temperature. The wavelength of the Sun and Earth radaton are dfferent, but the energy s the same for the total Earth system to retan ts temperature. In Fg s shown the durnal cycle of the radaton balance and the resultng surface temperature. It s llustrated how the surface gans heat and temperature durng the day due to that the ncomng solar radaton domnates the energy losses of the surface. At nght the loss terms domnate and the surface temperature decreases, ensurng that the average of the durnal temperature varaton s approxmately zero. The varaton of the surface temperature wll be modfed also by advecton of ar wth dfferent temperatures, and by heatng and coolng from the ground. Also, the durnal temperature varaton n response to the ncomng radatonal heatng wll depend on the characterstcs of the surface, such as heat capacty and heat conductvty. Fg Durnal varaton of the surface energy balance and the assocated surface temperature (Shodor, 1996), Lectures n Mcro Meteorology

224 A more thorough dscusson of the energy balance demands that we consder the system of equatons controllng the balance. It s normally wrtten as: (11.3) Q* = K + K +L +L = Q H + Q E + ΔQ s. Q* s net radaton, and K s ncomng short wave radaton. K = α K s outgong short wave radaton, α s the surface albedo. L s ncomng long-wave radaton, and L s outgong long-wave radaton = εστ 4 + (1- ε)l, ε s the surface emssvty,and σ s the Stefan-Boltzmann constant. T s the surface temperature. Above, we notced that an often used relaton between s α λ = 1- ε λ, but ts applcaton assumes that the same wave length apples on both sde of the equaton. Here, we wll use the albedo, α, n connecton wth the short wave radaton and the surface emssvty n connecton wth the long wave radaton. Characterstc Values of albedo and emssvty for dfferent surfaces are shown n Table 11.1 below. Contnung wth the non-radatve terms of the energy balance, we have two turbulence flux terms and a heat conducton term. Q H = ρ C p <w T > = ρ C p u * θ *, the sensble heat flux, ρ s the ar densty, C p the heat capacty of ar at constant pressure, where the ar densty, ρ, s about 1 kgm -3 and Cp ~ 1010 J K -1 kg -1. The u * θ * -term reflects the Monn-Obuchov- scalng, dscussed n secton 6. Q E = ρ L v <w q > = ρ L v u * q *, the latent heat flux, L v the evaporaton/condensaton heat beng approxmately equal to.45 MJ kg -1, where we have shown the surface layer formulaton as well. Smlarly to u * θ *, u * q * s defned and dscussed n secton 6. The sensble heat flux obvously reflects turbulence heat flux between the surface and the ar, whle the latent heat flux reflects that the surface can regulate ts temperature by evaporatng or condensng water. ΔQ s s the heat storage per unt tme n the surface materal ~C dτ/dt, wth C beng the heat capacty of the surface, where C depends on the surface characterstcs. Here we obvously have to consder that the surface have a certan depth to dstrbute the heat nput or loss. Typcal values for C (JK -1 m -3 ) are shown n table 11. for characterstc surfaces. It s seen from the fgure that wet surfaces have larger heat capactes. Defnng the heat conductvty, λ, for the surface, we can trace conducton of ΔQ s down nto the surface materal, through ΔQ s = - λ (δt/δz), where ΔQ s (z<0) s now consdered a functon of z. Combnng ths wthn the heat conducton equaton, we obtan: (11.4) dt T 1 Q s 1 T λ T T = = = ( λ ) = κ dt t C z C z z C z z Where, κ = λ/c, s the thermal conductvty. Ths s the same equaton, we use to solve for a dsperson plume from a chmney n secton 10. Here we notce that the surface temperature on both annual and durnal frequences s related to a snusodal, and assume that the surface temperature can be wrtten as : T s = T av + δtsn (ωt), where T av s the average temperature and δt s the surface durnal temperature ampltude related to the radaton. Insertng and solvng (11.4), we fnd: (11.5) z ω/κ T T Te t z s av = + δ sn( ω ω / κ), Whch llustrates how surface modulaton of the temperature propagates downwards, wth an ampltude attenuaton and a phase change gven by the frequency of modulaton and the thermal conductvty. Equaton (11.5) s llustrated on Fgure Characterstc radatonal parameters are gven n table 11.1, 11. and Lectures n Mcro Meteorology 3

225 Table 11.1 reflectvty and emssvty for dfferent natural surfaces, ε corresponds to long wave radaton, whle α reflects shorter wave lengths.(oke, 1985). Table 11. Heat capacty for dfferent types of surfaces, [Jm -3 K -1 ],(Chrsten 004) Table 11.3 Thermal conductvty for dfferent types of surfaces (Chrsten, 004). From the dscussons above, t s obvous that the energy balance at the surface wll depend on the surface characterstc, and hence can be pretty dfferent for nearby but dfferent surfaces. 4 Lectures n Mcro Meteorology

226 Ths s llustrated n Fgure 11.6, showng the durnal varaton of some of the terms n the energy balance for a suburban and a rural Vancouver ste, as well as ther dfferences. Fgure 11.7 shows the annual varaton of the net radaton, Q*, at a rural ste n Denmark. Fg Durnal varaton of some of the terms n the energy balance for a suburban and a rural Vancouver ste, as well as ther dfferences. Notce the dfference between the latent heatlux of the two stes (Oke, 1985). The notaton s as defned above n the text. Fgure Annual varaton of Total daly net-radaton at an agrcultural ste n Denmark at 55degr North, The fgure shows both average values and extreme values for the perod consdered (Larsen and Jensen, 1983). A characterstc behavour of ground temperature profles for a land surface versus tme for a durnal varaton s shown on Fgure 11.8 below. Lectures n Mcro Meteorology 5

227 Fgure 11.8 Examples of the durnal varaton of ground temperature profles for a Pne forest n Germany (Chrsten, 004) The local clmate The local clmate of any stes wll depend on the larger scale clmate around the ste, and on the local energy balance, as dscussed above. Also other local features are mportant however, such as local terran, thermal and roughness characterstcs and wetness of the ste, whch wll nfluence the wnd, the energy balance and temperature and humdty statstcs. Larger free water surfaces are able to regulate ther temperature because of, addtonally to large heat capacty of water, t apples latent heat flux at the surface and enhanced vertcal mxng due to waves and turbulence n the nner volume. For salty water ths mxng s further enhanced because warm surface water evaporates, and hence becomes heaver due a hgher salt concentraton. On the other hand cold surface water snks, because t s heaver than the water below. Therefore, we fnd that durnal temperature varatons s smaller both on the surface and wth smaller penetraton nto the water mass than for a land surface. For lattude wth sgnfcant annual perods n the radaton budget (11.5) can stll be used now wth a sutable annual frequency. For ths frequency one generally fnds that the temperature changes at water surface and below s closer to those of a land surface. Further one fnds that the effectve thermal capacty and conductvty of the water wll be functons of current and wnd speed and buoyancy nfluencng both evaporaton and the nternal turbulent mxng n the water boundary layer, just as n the atmospherc boundary layer over the water, but wth the added complexty of the mportance of salnty for the buoyancy. Obtanng the clmate characterstcs. Due to the varety of forces behnd the clmate at a gven spot, t s mostly determned emprcally from multyear measurements of the relevant parameters at the pont of nterest. Therefore many basc clmate data are obtaned from observatons from a measurng staton, whch can be centred around a meteorologcal mast as seen n Fgure A staton, lke ths, must be mantaned and servced for several years to develop a local clmatology, formally referrng to the exact locaton of the mast. The strength of such a staton 6 Lectures n Mcro Meteorology

228 s that t obvously montors the clmate at a certan locaton, whle our ablty to extend the results to wder areas depends on the heterogenety of the surroundngs and on our understandng of the mcrometeorologcal characterstcs of the data and the ste. Some parameters are more local than other. We have seen that the wnd qute senstve to roughness, terran varaton and shelter effects. Surface temperature s beng senstve to the local energy balance, wll be senstve to slopes and orentaton of slopes etc, therefore one should always consder the localty of a gven record and of a ste, where the clmate has to be evaluated. Here t s often possble to draw from knowledge about boundary layer meteorology, such as presented n ths course. In the measurements depcted n Fgure there s an effort to cover a larger area wth many pont measurements, supported by arplanes and satelltes. Such a network of measurements are obvous both more expensve and dffcult to keep operatng but can be used for valdaton of modellng. However, also here the aggregaton of the ndvdual data sets have to nclude a substantal amount of the mcrometeorologcal understandng and modellng. Fgure A typcal meteorologcal mast nstrumented wth wnd, temperature and humdty measurements at several heghts. Such a mast wll typcally be supplemented wth measurements of pressure, precptaton, varous radaton measurements, and varous sol ground measurements, stuated on and around the mast structure (A typcal wnd Rsø, DTU- Wnd-confguraton). Lectures n Mcro Meteorology 7

229 Fgure Clmate measurng program set up to cover an extended area. Fnally, we turn to clmatology data derved from the so called reanalyss data, whch s based on applyng many years of the ntal meteorologcal felds used n weather forecast modellng or clmate modellng. Here bascally all the observatons, used to ntalze these models, are used to provde a homogenous clmate data set coverng all the grd pont n the models used. The obvous advantages wth ths method, s that t provdes homogenous data wth good area coverage, and also provde the clmate for a vast sute of meteorologcal parameters beng derved from the model. However, the clmate data are now seen through specfc numercal models, and the representatveness of the data has to be tested. Generally the dsadvantages are the farly coarse resoluton both n space and tme, see Fgure The boundary layer structure, especally the vertcal varatons of parameters are lmted due to the closure relatons used. Addtonally, the results wll often have to be translated to ponts or areas of nterest, by fner scale numercal models and/or analytcal models as presented n secton 9. Reanalyss data are often updates as the modellng mproves both wth respect to qualty and resoluton, Contnually updatng grdded data set representng the state of the Earth's atmosphere, ncorporatng observatons and numercal weather predcton output Fgure Reanalyss of the global pressure feld wth surface wnd. 8 Lectures n Mcro Meteorology

230 NCEP/NCAR reanalyss ( ) 1. Reanalyss data 1: 1948 present, about 50 km, 6 hrly. Reanalyss data : 1979 present, about 50 km, 6 hrly 3. Clmate Forecastng System Reanalyss: , about 40 km, 1 hrly & 6 hrly ECMWF reanalyss ( ) 1. ERA-15, , about 50 km, 6 hrly. ERA-40, , about 50 km, 6 hrly 3. ERA-Interm: 1979 present, about 1.5, 6 hrly Other reanalyss data: e.g. Japanese 5-yr reanalyss Fgurre Characterstcs for a number of reanalyss data set. The horzontal and temporal resoluton s ndcated. The vertcal resoluton amounts to 0-30 grd. Objectve of obtanng the boundary layer clmate. From the above s seen that the energy balance at the surface, together wth terran features as z 0 and orography, consttute the surface condtons for the boundary layer flows, smultaneously wth that the energy balance s nfluenced by the boundary layer turbulence through the turbulent fluxes. By boundary layer clmate, we wll understand the clmate characterstcs of the surface and the near surface atmosphere. Ths clmate s obvously strongly nfluenced by synoptc meteorology, beng responsble wnd speed and the advecton of ar masses brngng ther respectve humdty and temperature across the locaton, for whch the clmate s studed or establshed. Snce there s a strong randomness assocated wth weather, we wll n one way or the other have to buld a clmate descrpton on both average characterstcs and dstrbuton functons. Snce the radaton s very mportant for the local clmate, we must expect at least to be able to descrbe the durnal cycle and the annual cycle of characterstc parameters. These clmate characterstcs must be buld on measurements maybe ncludng reanalyss data and modelng. Snce the annual perod s mportant and the longest systematc cycle n the clmate system, an observaton seres of several annual cycles s necessary to obtan at least the mnmum statstcs coverng annual cycles. The standard meteorologcal clmate perod s 30 years, and there are several sem-perod phenomena wth longer perodctes than one year, as e.g. the EL Nno cycle and the North Atlantc Oscllaton. Ths may all argue for longer tme seres than a few years. Lectures n Mcro Meteorology 9

231 Example: ERA-40 parameter lst Fgure Reanalyss characterstcs: Example of the parameter avalablty from a reanalyss data set. For some uses one must emphasze not only mean characterstcs, durnal and annual cycles and multdmensonal dstrbuton functons of the characterstc parameters. Addtonally, one wll need specal attenton to the clmate of extreme values to prepare aganst hazardous stuatons. Below s cted what type of clmate nformaton that s needed for dfferent knds of actvtes: Landscape and urban plannng, farmng and forestry: Radaton balance, Dstrbutons of wnd and drecton, temperature, precptaton, water balance. Jont dstrbutons, durnal and annual varaton Constructon works: Dstrbutons of wnd and drecton, also extreme dstrbutons, precptaton. 30 Lectures n Mcro Meteorology

232 Wnd energy and wnd loads : Dstrbutons of wnd drecton and wnd speed, also extreme dstrbutons and spatal varablty. See dscusson n secton 1. Fg Smple wnd dstrbutons accumulated over 10 years, from WAsP (Mortensen et al, 199). Solar energy: Dstrbutons of ncomng solar radaton for several wavelength bands, durnal and annual varaton, cloud-cover. Polluton dsperson: From secton 10 s seen that : Dstrbutons of wnd speed, wnd drecton and thermal stablty, boundary layer heght wll allow us to compute to compute dsperson and deposton for know polluton sources, both n the form of annual averages and as extreme realsatons. Human outdoor establshments, lke outdoor restaurants, theatres etc. toursm: Radaton, temperature, precptaton (snow, ran), annual varaton. Fnally one should not forget data amng to smply to characterze scentfcally the local clmate and ts development. In the followng secton, we shall focus on the wnd clmate, not only the basc wnd clmate varables, but also derved hgher order varables. Although the descrpton there wll be on wnd, obvously the same methodology or smlar methodology can be used n connecton wth the other types of specfc clmatology, mentoned above. Lectures n Mcro Meteorology 31

233 1. Wnd clmate, wnd energy, wnd loads In ths secton we shall consder estmaton of the wnd clmate at a gven locaton, as a specal example of a boundary layer clmate and also wth reference to the mportant practcal aspects of wnd energy applcatons and wnd load estmaton. Estmatng the wnd clmate often nvolves two dfferent tasks: The procedure when data, measured and/or modelled are avalable, at the locaton of nterest, at a reasonable qualty and resoluton, and the procedures employed to transfer the needed nformaton to ths locaton, f t s avalable somewhere else, or f t s avalable wth a too course resoluton and systematc naccuraces from larger scale or meso- scale models. In a way ths secton could also be consdered a part of the clmate secton 11, but the clmate of stronger wnds has a number of characterstcs that justfes ths specal secton. One of these s that for stronger wnds one can often neglect the couplng between the wnd feld and other meteorologcal felds. The reason s that stronger wnds condtons typcally are close to neutral stablty condton, because a strong wnd the wnd shear wll force the Rchardson number (or alternatvely z/l) toward zero, for just moderately rough surfaces. That the wnd can be consdered alone, wthout all the other clmate parameters, of course smplfes both theory and observatons consderably and also smply the transfer of a wnd clmate from one ste to another. On the other hand the couplng between the wnd and the pressure n the governng equatons means that wnd tend to be more senstve than other atmospherc parameters to terran characterstcs, such features, such as hlls, valleys and roughness changes. Descrpton of the data needed and methodologes used. Bascally we wsh to descrbe the local wnd clmate. Therefore one needs a clmate record of the wnd, ths s normally understood as about 10 years of not too old data. Ths s shorter than the usual clmate perod, 30 years recommended by the World Meteorologcal Organsaton. The shorter record s chosen because, the real need the future clmate, not just a clmate record, meanng that one can utlze the many advantages of a shorter newer record (bascally newer and more homogeneous data closer to the present and thereby also the future). Although one mght get away wth recordng only wnd speed and drecton, especally f the record s establshed at the ste of nterest. It s never the less advantageous to nclude other observatons as well, such as pressure and temperature to obtan the ar densty, and temperature gradents to obtan thermal stablty, and also other parameters. Ths wll also facltate the use of the record at other stes than the one, where the record s obtaned. There s a need for both turbulence and mean wnd speeds. Here one can of course use all the raw data, but for smplcty one wll usually convert the raw data to dstrbuton functons. For mean wnds, for medum to strong wnd speeds, one has found that the Webull dstrbuton s mostly a sutable dstrbuton. For low wnds and for extreme hgh wnds one wll often use other dstrbutons. Hence for medum to hgh wnd speeds, the smplest complaton of wnd clmate data s the frequences of the wnd drectons, combned wth a Webull dstrbuton for each wnd drecton, see Fgure 1.1 and Lectures n Mcro Meteorology

234 Fgure 1.1. The Webull dstrbuton ftted to mean wnd measurements. Also the analytcal form of the functon wth ts two parameters s shown and compared wth llustratve data sets ( Mortensen et al, 199,Troen and Petersen, 1989). Fgure 1.. Drecton dstrbuton and wnd speed dstrbuton for one of the drecton (Mortensen et al, 199). As dscussed above the objectve of wnd records s mostly amed to estmate the knetc energy n the wnd, T= ½ ρu. The power delvered to any statonary object wll hereby be proportonal ut= ½ ρu 3, rrespectve of f the power s used to drve a sal boat or a wnd turbne or to generate surface waves on the water surface. Specfcally the power taken out by a wnd turbne s formulated as: (1.1) P(U) = 1 ρu 3 C p A, Lectures n Mcro Meteorology 33

235 C p s the power coeffcent of the turbne and A = πr s the area swept by the rotor wth radus R. C p s a desgn coeffcent for each wnd turbne and s provded by the manufacture. Typcally several dfferent C p exst for each turbne to optmze for dfferent condtons. Alternatvely one provdes the power curve P (U) for a gven ar densty. In Fg. 1.3 s shown a Webull dstrbuton and a power curve for a specfc turbne. Wth the wnd dstrbuton and the power curve gven, one can determne the expected average power producton from: (1.) < power > = d P(u)Wd(u)du, 0 where Wd(u) s the Webull dstrbuton for each drecton, and the summaton s taken over the drecton dstrbuton. Fg Examples of a Webull dstrbuton and a Power curve.( Mortensen et al, 199) The force from the wnd on structures erected nto the wnd, lke wnd turbnes, buldng brdges, trees etc., can be descrbed by a thrust coeffcent, C T, measures how effcent the knetc energy mpact on the structure, and the wndward area A. (1.3) T = 1 ρ u C T A The structures have to be desgned to wthstand the wnd force. The wnd forces can result n fatgue loads dervng from a combnaton of a wnd load for a gven wnd and the frequency of ths wnd. These loads are called fatgue loads. It s not extreme but strong and happenng often enough to be mportant for breakng the structure. The mportant wnd speeds here are medum to hgh wnds, and the wnd dstrbuton are typcally well descrbed by the Webull dstrbutons, but one mght have to nclude the turbulence loadng as well. Another mpact s assocated wth an extreme wnd event that damage or destroys structures. Here the engneers wll ask about the extreme wnd dstrbuton to desgn the structure to handle a gven extreme wnd, for example the wnd that n average occurs every 50 year or every 100 year. The Webull dstrbuton functons less satsfactorly for such rare wnds. One has to extrapolate observatons to events so rare that they are unlkely to be present n the data 34 Lectures n Mcro Meteorology

236 avalable, f the seres has duraton of only 10-0 years. The extrapolaton s helped by the fact that t has been shown mathematcally that most common realstc dstrbuton functons have extreme value tals that under very general condtons converge to one of three types of exponental dstrbutons of the Gumbel famly, provded one can assume statonarty (meanng no sgnfcant clmate change here!). Therefore dentfcaton of the Gumbel dstrbuton (Gumbel, 1958) for a gven data record serves as a ratonal bass for extrapolaton of the estmated dstrbuton to event so rare that they have not occurred n the obtaned record. Assume u to be the largest value of the wnd speed over some bass perod, T 0, selected wthn a longer seres of length T. The cumulatve Gumbel dstrbuton for such a seres of u, meanng probablty for that u < x s descrbed by: (1.4) G(x) = exp ( exp x β α ), Wth the correspondng probablty densty functon gven by: (1.5) g(x) = dg dx = 1 α x β x β exp( ) exp ( exp ) α α, Where β s the most probable value, whle α s an expresson for the standard devaton. Integratng g(x) one fnds that 0 (1.6) < x > = xg(x)dx = β + αγ., wth γ beng the Euler constant~ For a gven tme seres of length T, say 0 years, one can select a maxmum value for each subseres of length T 0, ths way to obtan N= T/T 0 estmates of the maxmum wnd, u, wthn the subseres. We now rank these u s accordng to magntude, from the largest to the smallest. The maxmum value of these u s, U 1, appears once durng the tme T, and T 0 /T= 1/N s hence an estmate of the probablty for that u U 1. We have that G(U 1 ) = 1- T 0 /T, because G(x) s the accumulated dstrbuton for u < x. The second largest U smlarly reflects the probablty T 0 /T for u U, and G(U ) = 1- T 0 /T and so on, for U j, one fnds that G(U ) = 1-jT 0 /T, and fnally for U N, G(U N ) = 1-NT 0 /T= 0, whch shows that the ndcated probablty estmaton s a bt rough. In practce one does use numbers slghtly dfferent for the here used ntegers to estmate relevant probabltes. The selected maxmum values U can now be plotted versus ther estmated probablty, n fgure 1.5 formulated n terms of the return perod, whch s the tme t n average wll take to encounter a maxmum U j for a subseres tme of T o. For j= 1, only one value has been encountered n 0 years, and the return perod s 0 years. For = 3, we have encountered 3 values larger or equal to U 3, and the return tme s 0/3~ 6.7 years, etc. T 0 /T, as shown on Fg For further nterpretaton we note from (1.4) that (1.7) G(U ) ~1 T 0 /T = exp( exp U β α )) Or: (1.8) U = β α ln ( ln 1 To ) T Lectures n Mcro Meteorology 35

237 Ths means that the maxmum values plot as a straght lne versus the double log to 1-T 0 /T. The parameters α and β can then be determned from ths straght lne. They reflect two aspects of the tme seres, the selected subseres length,t 0, and the processes controllng the wnd clmate. From (1.8), we can now extrapolated to larger tmes, T=T λ to obtan the largest wnd U Tλ that s lkely to occur just once n the tme T λ, correspondng to =1 n (1.8). (1.9) U Tλ = β α ln ln 1 To β + α ln T λ Tλ T0, Where we have used that T 0 /T λ << 1 n the approxmaton, and where α and β stll refer to the chosen subseres length, T 0, and physcal processes controllng the wnd clmate. An example of the process s shown on n Fg Fg.1.4.Establshng an extreme maxmum wnd as functon of the return tme, estmated from a 0 year seres of one year maxmum values followng the Gumball approach, and further extrapolaton to a return tme of 100 year (H. Jørgensen, personal communcaton). We have now estmated the extreme average wnd that n average happens once over a tme T λ, on bass of the extremes for subseres of length T 0. Obvously, ths approach to estmatng extreme values from gven seres s used for many other varables than the wnd, for example also wth respect to floodng both along rvers and along coast lnes. It should be understood that the present text s only an ntroducton to the Gumbell - statstcs. The methods for on how probabltes are ascrbed to the estmated subseres maxmum values are somewhat more complcated than the nteger ndex method used here. Also the error bounds on the extrapolated maxmum values should be and s ncluded n the techncal applcatons, but are not treated here. On the top of the extreme average wnd, estmated above, we wll often need also to estmate the extreme of shorter tme turbulence fluctuatons, whch can result n even larger short term wnds on the top of the mean wnd, and also gve rse to fatgue loads from repeated exposure 36 Lectures n Mcro Meteorology

238 of the structures. If not drectly measured the extreme wnd s usual estmated from ether boundary layer physcs, or from tme seres analyss. The frst of these methods are often used by operatonal meteorologst, who assumes that the extreme wnd at any pont develops through ntruson of an ar packets movng wth the Geostrophc wnd above the boundary layer down to the heght level of nterest. The turbulence extreme related to the tme seres analyss s related ether to the standard devaton of the turbulence record, compare Fgure 1.5, wth an extreme value taken as, u p (turbulence, extreme) ~ C( ) σ u, where C( ) can be taken as functon of varous condtons, or can be taken a constant, typcally -3. Note that snce turbulence s close to normally dstrbuted, C = means that about 4% of the varatons have hgher ampltude that the extreme turbulence estmate, whle C =3 means that only 0,3 % of the fluctuatons are larger than the extreme estmate. However, for a more thorough extreme analyss, we need to consder the extreme values of the turbulence measured by well specfed averagng tmes, and also to consder the Exceedance statstcs of such extreme values, for the fatgue loads studes. Exceedance statstcs and gusts The extreme exceedance statstcs s typcally assocated wth an averagng tme and an extreme value denoted a gust. A typcal averagng tme s 3 seconds, denoted 3 seconds gust, but also 1 second gusts are typcally appled. Fgure 1.5. Turbulence record wth a normal dstrbuton. We shall here follow (Rce, 1943,1944) through Krstensen (1989): We start wth our tme seres u(t), see Fgure 1.5, but we here nclude as well as the assocated seres ú = du/dt. Durng tme T the jont seres (u,ú) spend the tme dτ n the parameter nterval (u,u+du) and (ú,ú+dú): dτ= T P(u, ú)dudú, where P(u, ú) s the jont probablty dstrbuton for u and ú. The tme spend crossng the nterval (u,u+du) wth speed u s gven by: dt = du/ ú. Half of the tmes the nterval s crossed upwards, and durng the tme T, the number of upward crossngs can now be wrtten: dn= dτ/dt=½túp(u, ú)dú wth ú>0. From ths we get the number of tmes a level u s crossed wth an upward tendency: (1.10) N = T úp( u, ú) dú 0 Lectures n Mcro Meteorology 37

239 To proceed, we have to estmate P(u, ú). Snce u and ú can be consdered uncorrelated, due to the assumed statonarty of u(t) ( 0=<du /dt> = <u ú>), we wll follow the normal assumpton of P(u, ú) = P 1 (u) P (ú). Here P 1 (u) beng a Gaussan s usually a good approxmaton, see Fgure 1.6. For starters we can assume also ú to be Gaussan. Hence we can wrte: (1.11) 1 ( u u) ú Puú (, ) = exp πσ σ σ σ u ú u ú By ntegraton of (1.10) we now fnd: 1 σ ú ( u u) (1.1) Nu ( ) = T exp, πσu σ u Here we have ndcated that the number of exceedances, N, refers to the wnd speed u. Now we defne a gust as the u-value that s exceeded n average just once durng tme T. Lettng N=1 n (1.1), the Gust value u G s found from: (1.13) u σ Tσ G u ú G 1 = 1 = ln( ) u u πσ u 1/ The Total tme, Θ(u 0 ), for whch u(t) s larger than a gven u 0, can be derved from : (1.14) ( u ) T du úp( u, ú) dú Θ 0 = u0 0 Hence, the average duraton, T (u 0 ) of any excurson beyond u 0 can be determned from: σ σ u u (1.15) I ( u ) =Θ( u )/ N π 0 0 σ u u, ú 0 Where the last approxmaton demands that u 0 -<u> >> σ u, and n ths approxmaton shows the smple results that the duraton of the excurson s reversely proportonal to the devaton from the mean. The two last equatons of course apply for all large u(t) values ncludng u G n (1.13). Hence the Rce formalsm allows us, and the engneers, to understand not only the frequency of extreme events but also ther expected duraton, and hence allow for smulaton of dfferent load cases for constructons. The formulaton llustrates also the senstvty to the duraton T of the tme seres that we have studed. There s however another tme dependency, the averagng tme for the u(t), whch more comprehensvely should be descrbed as u(t, T, τ), where τ s the averagng tme for the sgnal, derved ether from the nstrumentaton or by post processng. In the equatons all u(t) and ú(t) are seen through such flters, and the varances defnng the dstrbuton functons are nfluenced by these flters. In secton t s descrbed how such flterng nfluences the varance, because the flters cut away varance from the relevant power spectra. Duraton tme T mples a hgh pass flter removng varance from frequences below 1/T, and averagng tme τmples a low pass flter removng varance from frequences larger than 1/τ. Therefore our resultng varance can be wrtten: 38 Lectures n Mcro Meteorology

240 (1.16) σ = HP( ωt ) LP( ωτ ) S( ω) dω Where the resultng varance s derved from the Hgh Pass and Low Pass fltered power spectrum for u(t) and ú(t), respectvely. The two spectra correspond to 1 and 4 n Fgure 1.6. Spectrum 4 can be consdered derved from spectrum 1 by multplcaton by f, because ú s derved from u by dfferentaton, see agan secton, and termnated n the hgh frequency end by that turbulence becomes strongly dsspated n ths end, see dscusson n secton 5. Indeed for sgnals wth the type 4 spectra, the Gaussan assumpton s rarely very good. It s only ncluded here because t s smple. Fgure 1.6. Surface Boundary Layer turbulence spectra, fs(f) showed versus the normalzed frequency, n= fz/u (wth f beng frequency n Hz) taken from secton. Type 1 corresponds to the spectrum for u(t), ) for w(t), 4) for ú(t). The Rce-system summarzed above allow one to extrapolate the gust statstcs from one averagng tme/tme wndow to another, provded one can estmate the spectrum range, needed. Indeed the system has been used wth success also to estmate gust statstcs from mesoscale models, correctng for the model grd sze nduced spectral low pass attenuaton. Extrapolaton of a wnd clmate statstc from one locaton to another. Very rarely does one have clmate records avalable at the exact locaton, where t s needed. Therefore one has to transform the statcs, determned at one ste, to smlar statstcs at the locaton, where t s needed. A very often used method s developed n the European Wnd Atlas (Troen and Petersen, 1989) and has been a central method for a quanttatve estmate of the wnd resource at a gven locaton all over the world. Bascally one uses a combnaton of the theores descrbed n sectons 5-9, about homogeneous and nhomogeneous boundary layers, and as llustrated n Fg. 1.7 and 1.8. In Fgure 1.7 s shown a meteorology mast and a wnd turbne separated not too far from ts other, and the wnd at both stes nfluenced by multple dfferent nternal boundary layers. From the models n secton 9, we can derve the clmate for the Geostrophc wnd aloft, from the clmate measured at the mast. Ths Geostrophc wnd clmate s not senstve to the small scale changng surfaces to the frst order. Therefore the same Geostrophc clmate exst over the wnd turbne, and one can now derve the wnd clmate for the turbne, agan usng the nternal boundary layer models for condtons around the wnd turbne. The methods s llustrated as well n Fgure 1.8, where the Geostrophc wnd clmate s denoted Lectures n Mcro Meteorology 39

241 Generalsed Wnd Clmate. In Fgure 1.9 one takes nto account also nearby obstacle to the both the measurng ste and the wnd turbne ste, snce such obstacles were found to exst and be relevant for the wnd energy development. The sub-models used to handle obstacles are llustrated n Fgure Fg Smlar procedures are used, when data avalable are model outputs wth a coarse resoluton, than necessary for the wnd estmaton to be undertaken see e.g. secton 11 on the use of reanalyss data. Here the coarse resoluton model wnds are converted nto the necessary fner resoluton, from a model chan wth fner and fner resoluton, each beng embedded n the coarser resoluton model n the chan, usng the fne resoluton nternal boundary layer around the wnd turbne ste from the WAsP-lke models. In practcal applcaton, one often uses a numercal flow model, ncorporatng the orography varaton n a model correspondng to the model summarzed n the appendx to secton 9. Snce the wnd s nfluenced strongly by local features (due to the pressure wnd dependency n the equatons), reflectng local terran, obstacles and roughness, for whch reason a course resoluton model (> 1 km) s not adequate for predcton of the avalable wnd power potental at a gven ste. On the other hand features further away tend to merge, the more the further away they are. Therefore the flow model n the Wnd Atlas program (WAsP) uses Bessel functon expanson of the terran maps, rather the snus and cosnes functons used n the secton 9 appendx. The Bessel functon allows for a varable grd sze, wth very small grd steps around the ponts of nterests and larger grd step further away, compare Fgure The estmated wnd clmate at the locaton of nterest s subsequently used to perform the computatons to estmate the wnd resource, the wnd loads and whatever objectve one has. The wnd atlas method has been shown to work also to transfer Gumbel derved extreme wnd data from one ste to another, n connecton wth the Extreme wnd load analyss for the Great Belt Brdge n Denmark (Abld et al, 199). For some wnd loads studes turbulence s needed as well. If they are not measured locally they must be modelled, see secton10. Also more conventonal models employng constant grd szes are used, wth the addton that one can use telescopng models wth smaller and smaller grd szed for smaller and smaller subareas. The mprovng computer technology makes ths approach more feasble, and n practce unavodable for standard numercal weather model types of all scales. Stll the WAsP approach wth the varable grd sze seems the most effcent method for obtanng the necessary small grd sze near the pont of nterest. In the prncple one could use smlar method for transfer of other clmate varables from ste to ste, but the clmate for larger wnds s n some sense easer because the boundary layer for stronger wnd s domnated by wnds wth well defned drectons and thermally neutral condtons. Ths means that one can often neglect other aspects that the wnd, the roughness and topography. A more general clmatology must nclude as well thermal stablty and the water budgets, whch can be neglected for the wnd, and whch would need ncluson of sol characterstcs as well. 40 Lectures n Mcro Meteorology

242 Fgure Inhomogeneous terran wth a wnd turbne and meteorology mast for montorng clmate data and a wnd turbne. Lectures n Mcro Meteorology 41

243 Fg Summary of the Wnd Atlas methodology from Troen and Petersen (1989). The fgure shows how the observed wnd clmate s cleaned for nfluence of nearby obstacles and shelterng, roughness changes, terran changes untl a generalzed clmate that essentally apples to the Geostrophc wnd. The Geostrophc wnd s per defnton not nfluenced by the local terran features and therefore can be consdered vald also above the locaton of nterest for the nstallaton under consderaton. The generalzed clmatology can now be moved down to nto the boundary layer at the new locaton, takng nto account the new terran features, roughness and obstacles pertanng to that ste. 4 Lectures n Mcro Meteorology

244 Fg. 1.9 Example of an obstacle often encountered both n connecton and wth wnd observaton statons and wth stes of nterest. (Troen and Petersen, 1989) Fgure Terran wth grd based on the Bessel functons. Notce the dmnshng grd sze towards the pont of nterests. (Mortensen et al, 199, Troen and Petersen,1989) Lectures n Mcro Meteorology 43

245 13. Instruments, measurements and data. Many and dverse nstruments are used n mcro scale meteorology. Here we shall try to summarse mportant aspects of conductng measurements and understandng the data wthout gong nto two many detals. We wll dscuss: Instruments, platforms, expermental set-ups and data nterpretaton, but frst we specfy the sgnals, as dscussed n the other sectons. We shall not dscuss the sensor physcs and technology, but only the way each sensor and sensor system nteracts wth the atmosphere. The secton s structures as follows: Instruments and response characterstcs Sgnals and Instruments. In stu Instruments wth no movng parts and manly tme response. Thermometers, hot-flms, hygrometers, radaton nstruments In stu Instruments wth no movng parts and manly spatal response sonc anemometers/thermometers, concentraton nstruments by absorpton path In stu moton based wnd measurements: Cup-anemometers, propellers, wnd -vanes Remote sensng nstruments Acoustc, electromagnetc Screens and shelters Platforms Flow dstorton and calbraton Calbraton Expermental set-ups Statonary and horzontally homogenous condtons Statstcal consderatons and averagng Statstcal consderatons, dgtal data, alasng Instatonary and nhomogenous condtons Spatal resoluton and alasng. Sgnals and nstruments. We assume the sgnals to be gven as χ(x,t) or χ (x,t) χ (r,t). Ths means our sgnal can be any velocty component, u,or any scalar,γ, and that t s a functon of the three space coordnates and tme. χ s a turbulent sgnal but may nclude as well low frequency behavour as dscussed n secton 1. We wll use our knowledge about the sgnal structure derved n earler sectons. We wll to a large extent utlze spectral analyss n the descrpton. Followng our dscussons n secton we have: ( kx + ωt) ( kx ) χ( x,) t dz( k, ωt) e or dz( k,) t e (13.01) k ω k ωt or dz( x, ωt) e or dz( x,, t k, k ) e ω = Yeldng the followng power spectrum: k1, k 3 1 ( kx 1 1+ kx) 44 Lectures n Mcro Meteorology

246 (13.0) χ ' = S ( k, ω) dk dω or χ'() t = S (, t k ) dk s s k ω k or χ' ( x ) = S ( x, ω) dω or χ' ( x,) t = S ( x,) t dk dk s 3 s 3 1 ω k1, k Where we wll choose the useful type of Fourer analyss for the problems encountered. Characterstc sensors are presented n Fgure We shall characterse the ndvdual sensors by these response characterstcs, whch are relevant for the way each sensor nteract wth the ar and produces ts estmate of an atmospherc varables. All nstruments are assocated wth some knd of spatal averagng and temporal averagng. We start consderng nstruments wth no movng parts and where the spatal averagng although exstng s normally not of mportance for nterpretaton of the sensor sgnals. We start wth nstruments. Fgure Sketch of dfferent nstruments types used wthn mcro scale meteorology (Stull, 1999). Of such nstruments we start consderng nstruments that are manly charactersed as: In-stu nstruments wth no movng parts and manly tme response.. Of such nstruments one can menton thermometers, hot-flm wnd nstruments, hygrometers and other concentraton nstruments. As characterstc nstrument, we consder a thermometer: Its response to a chance n ar temperature can be wrtten: (13.03) heat storage electrcal heatng heat loss + radatonal heatng We neglect the radaton terms, descrbng sensors radaton balance wth the surroundng, manly beng heated by the Sun and cooled aganst the nght sky. The remanng terms are defned by:τ Lectures n Mcro Meteorology 45

247 (13.04) dt Storage = ρ C V Electrcal heatng = I R α T T dt heat loss = k Nu T T wth Nu Λ+ u s s s s. o(1 ( s o)); 0.5 a ( s a); ( ) Where subscrpt, a, ndcates ar values, and s sensor values. V s s sensor volume. Cs s the heat capacty of the sensor. Λ s a coeffcent of order 1, and u s the wnd speed, Nu s called a Nusselt number. Insertng we obtan as wth good approxmaton: (13.05) dt s τ + Ts = Ta, dt Where the tme constant, τ ρ s C s V s /k a Nu, reflects the raton the heat capacty of the sensor and ts rate of heat exchange wth the ar, as shown n (13.04). For now, we have neglected the radatonal heatng and coolng. Equaton (13.5) s seen to descrbe how the sensor temperature, whch s the measured varable always s tryng to approach the ar temperature, but wth delay measured by the tme constantτ. To see the mplcatons we use the Fourer transforms from (14.1) ωt (13.06) T () t = dz ( x, ωt) e ; T () t = dz ( x, ωt) e a Ta s Ts ω ω Insertng (14.6) nto (14.5) and neglectng the x-coordnate, we obtan: (13.07). dz ( ω) = dz ( ω)/(1 + ωτ ) Ts Gven the followng expresson for the power spectra: Ta ωt (13.08) S Ts ( ω) = Sa ( ω)/(1 + ( ωτ ) ). Whch shows how the temperature s low pass fltered by a smple low pass flter reflectng the tme constant, τ or alternatvely a cut-off frequency, ω 0 ~ 1/τ, above whch the sgnal s T a s attenuated. t s seen From ts defnton t s seen that τ s varyng slowly wth condtons, specfcally here the wnd speed. Bascally however the low pass characterstcs s fxed n frequency space (Hz or radans/sec). Recall the plot of atmospherc spectra n Fgures 6.8 and 6.9. Here t s seen that the sgnal spectra slde along the frequency axs, mowng to hgher Hzfrequences wth wnd speed and ncreasng stablty (z/l) and mowng to lower frequences for ncreasng measurng heght. Therefore, a temperature sensor wll resolve more or less of the temperature sgnal dependent on not only of the value of τ but also on atmosphere parameters lke wnd speed, measurng heght and stablty. The temperature sensor s here consdered an example of sensor wth no movng parts, wth response manly beng charactersed by a tme constant. Examples can resstance, thermocouple thermometers, dfferent nstruments for measurng ar concentratons, hot-flm and hot-wres anemometers for wnd measurements. Although we here neglect the ssue of spatal resoluton, t does of course exst also for these sensors, but t s relevant for specal studes only (Larsen and Højstrup, 198). The tme constant, τ, of the nstruments range from several mnutes for some of the classcal nstruments to less than a mcrosecond for fast respondng hot-wre anemometers. The physcal sze of the sensors ranges from less than a mcrometer for some thermometers and several tens of centmetres for others. 46 Lectures n Mcro Meteorology

248 In-stu nstruments wth no movng parts and manly spatal response Fgure 13. Prncple for a sonc anemometer/thermometer and examples of ther practcal forms. These nstruments use ether sound or electromagnetc/optcal sgnals. They have one or several sgnal paths between an emtter and a recever. They utlse ether a tme of flght measurement for wnd speed, an ar densty measure for temperature and scatterng/ absorpton measurements for concentraton of trace consttuents n the ar. The propagaton speed of sound and lght and the electroncs s mostly fast enough for the nstrumental tme constants can be neglected. Therefore focus s on the spatal averages along the transmsson paths. As we used a temperature sensor as example above, we shall here consder the so called sonc anemometer/thermometer as an example. Fgure 13. shows how sonc anemometers normally employ a number of sound paths n dfferent confguratons. A schematcs sound path s shown on the fgure as well. Here sound pulses are alternatvely send n opposte drectons, wth the transmsson tmes, t 1 and t beng measured. They are seen to be related to the speed of sound and the wnd speed as follows: d d (13.09) t1 = ; t = c cosα + Vd c cosα Vd Here V d s the wnd speed parallel to the sound path, V n s the wnd component perpendcular to the sound path, c s the speed of sound and α = sn -1 (V n /c), see Fgure13.. From (13.9) we now derve the sonc estmate of the wnd speed along the sound path and the ar temperature as follows: d V = = ( V + V ) + ( c cosα c cos α ) V (13.10) ds d d d t1 t d s = + = d1 d sv 4γdRd t1 t 4γdRd (13.11) ( cosα cosα ) T c c V V T Here subscrpt 1 and refer to the tmes, when the sound pulses are transmtted n the two drectons, compare Fgure 13.. If we assume the tme dfference between the two sound pulses s small, and also that we can neglect the angle,α, then the last approxmatons n the equatons hold. Addtonally, the relaton between the speed of sound and the sound vrtual temperature s used as follows: (13.1) c = γ RT ; T = ( q) d d sv sv Lectures n Mcro Meteorology 47

249 Where γ d and R d s the rato of specfc heat and the gas constant respectvely, q s the specfc humdty, and subscrpt d means referrng to dry ar. T sv s called the sound vrtual temperature, and s a close approxmaton of to T v, the vrtual temperature, although they are not qute the same. See defnton of T v n secton n (3.4). The sonc s seen to measure wth close approxmaton the sound vrtual temperature and the wnd speed along the sound path. Hence by havng three dfferent sound paths we can obtan all three velocty components. If the sgnals at tmes 1 and are dfferent and α cannot be neglected, correctons have to be appled (Larsen et al, 1993). In equatons (13.10 and 13.11) we have neglected that the V d or T sv are not just smple scalars but represent ar temperature and velocty components, v d or T v that change n space and tme. The tme varaton of V d or T sv derves from that we have turbulent tme and space varyng sgnals of v d or T v passng the sensor. The lmtatons from sonc tme response to the resoluton derves from the emsson rate and emsson cycle of the transducers, and on sgnal flters n the nstrument. Most modern sonc can resolve sgnal frequences up to 0-100Hz, adequate for most statonary nstruments. The spatal resoluton arses because V d or T sv n Fgure 13. derve from the averagng of the true ar varables v d and T v along the sound path, d. To analyse the characterstcs, we chose the spatal Fourer transform n (13.1), neglectng the tme varaton, and wth (x ) meanng (x 1,x, x 3 ): (13.13) χ = k ( x,) t dz( k,) t e ( kx) Where χ s any velocty component or temperature n a turbulent feld. Assume now that the sound path d s along the x d -axs. The correspondng averaged sgnal Χ(x,t) s obtaned through: xd + d/ 1 ' ' (13.14) Χ ( x ) = χ( x ) dx d d, xd d/ From (13.13 and 13.14)) we obtan the relaton between the Fourer mode and the power spectra of the two sgnals: sn( kd/ ) sn( kd/ ) d d (13.15) dz ( k ) = dz ( k ); S ( k ) = S ( k ) Χ χ Χ χ kd/ kd/ d d As seen the spectrum of the averaged sgnal s averaged along the drecton of the sound path, x d. A statonary sonc wll measure the spectra along the x 1 -drecton, beng the drecton along the mean wnd speed. Hence the measured S-spectrum of the nstrument wll be: (13.16) sn( kd/ ) d = 1 = Χ 3 χ 3 kd/ d S ( k ) S ( k ) dk dk S ( k ) dk dk Χ Therefore the modfcaton of the measured spectrum by the spatal averagng wll depend on the drecton of the sound path d relatve to the wnd drecton and also on the assumptons about the behavour of the three dmensonal spectrum. In general on defnes a pseudo-transfer functon as: SΧ ( k1 ) (13.17) Tχ ( k1 ) = S ( k ) χ Smlar expressons can be derved for cross-spectra e.t.c Lectures n Mcro Meteorology

250 Many Soncs has one path as vertcal. Usng the sotropc formulatons n Secton, Kamal et al (1968) derved the pseudotransfer functon for the vertcal velocty for such a sonc confguraton, as presented n Fgure Fgure 13.3 Transfer functon for vertcal averagng over length, l, accordng to Kamal et al (1968). Here T 3 (k 1 l) s the functon gven n (13.17) for the vertcal velocty spectrum, whle the curves denoted Gurvch relates to the temperature spectrum. Measurements of ar concentraton of the trace consttuents of gases and partculates H O, aerosols and CO s performed across an optcal path smlar to the one descrbed for a sonc above. The physcal prncple s then sgnal absorpton or extncton by the relevant molecules. The path averagng however yelds the same knd of transfer functon as gven by Fgure 13.3, dependent of the orentaton of the sensor path.. In stu moton based wnd measurements: Ths group of nstruments ncludes many of the classcal wnd nstruments that utlse the force of the wnd for the measurement of the wnd. These nstruments nclude cup anemometers, wnd vanes, propeller anemometers, wnd socks etc. It s characterstc for these nstruments that also they have a response flter lke the thermometer, descrbed above. However, such nstruments are charactersed by a response length scale rather than by a tme constant. Typcal versons of the nstruments are shown on Fgure Fgure 13.4 characterstc versons of cup-anemometers and propeller anemometers. For llustraton, we shall summarse the response of a wnd vane, see Fgure The governng equaton for ths nstrument s: d (13.18) θ = M / I D/ I, dt Lectures n Mcro Meteorology 49

251 Where θ = υ +α s the angle from the reference drecton to the poston of the wnd vane, see Fgure M s the forcng torque, I the moment of nerta and D the dampng torque. Fgure Schematcs of the wnd vector seen from above. U s the ar velocty, drvng the moton of the vane. k s an arbtrary reference drecton. The fgure shows how the effectve velocty forcng the vane, v r, s the dfference between U and the vane velocty due to the moton of the vane, v r. The forcng torque s derved from the total force on the vane multpled by the dstance to ts centre of mass: (13.19) M = F r = 1 ρ( v) A K( α β) r, r where A s the area of the vane and K s a drag coeffcent. From Fgure 13.5 s seen that the vane velocty can be wrtten: dθ (13.0) v = r r dt If we further assume that U>>v r the effectve wnd drvng the vane can be summarsed as, see Fgure 13.5: v r dθ r (13.1) v U. β = r U U dt Insertng nto (13.19), we obtan: d θ 1 = M / I D/ I = ρu A K( α β) r / I D/ I dt (13.) 1 1 r dθ = ρu A Krα / I ρu A K r / I D/ I U dt To smplfy we assume that D/I s proportonal to the speed of rotaton of the wane D/I ~ C(U)dθ/dt. A smplfcaton s ntroduced by the defnton of a natural frequency and a natural wavelength 1 I U r λ (13.3) λ = π ; ; CU ( ) w ω = π ς = + e ρrak λ λ U w w Equaton (13.) can now be brought on the followng form: 50 Lectures n Mcro Meteorology

252 (13.4) 1 d θ ς dθ + + θ = ϑ ω dt ω dt e e Where we have used that α =θ-υ. It s seen that λ s derved from characterstcs of the vane, whle ω e s determned from the wnd speed dvded by λ w. Insertng the respectve Fourer ntegrals nto (13.4), t s transformed to the followng spectral transfer functon: (13.5) ( ω) = ( ω)/(1+ ςω / ω ( ω / ω ) ) θ ϑ e e dz dz That related the Fourer mode for the turnng of the vane to those of the wnd drecton through a transfer functon, charactersed manly by the wnd speed, the length constant, λ w, and the length of the vane arm. Assumng that the frequency s related to the Taylors hypothess (6.30): ω = Uk 1, where U as usual s the mean wnd and k 1 the wave number component along that mean wnd axs, x 1, axs. (13.6) dz ( k ) = dz ( k )/(1+ ςk λ ( k λ ) ) θ 1 ϑ 1 1 w 1 w Therefore, as seen the wnd vane has a response that s constant n wave number space, although the dampng coeffcent, ζ show some wnd dependency that can be mnmsed through desgn. The ampltude transfer functon s shown on Fgure13.6. Fgure Ampltude transfer functon for a wnd vane versus ω/ω e = k 1 λ w wth ζ as parameter. It s clearly seen why ζ s called the dampng coeffcent Remote sensng nstruments As the name ndcates, these nstruments obtan the measured varable at a dstance from the transducers. The nstruments typcally measure the radaton emtted from a target measurement pont. It s ether n the form of radaton emtted from the target, passve measurements lke nfrared surface temperature measurements, or back scatter from an electromagnetc radaton or sound waves emtted by the sensor and a subsequent samplng of the backscattered sgnal from the target(s). As opposed to the other types of nstruments above, we wll not go nto detals wth the response of the nstruments here, because the feld s too large and specalsed and s presently evolvng extremely fast. You are referred to Pena et al (013) or more specalsed lterature. However we try to summarse the characterstc types here. For measurements wthn the atmospherc boundary layer, a well establshed passve remote sensng method s the estmaton of the surface temperature by the samplng the nfra read lght Lectures n Mcro Meteorology 51

253 emtted from a surface, typcally wth the sensor place 1- meter above the surface, but can be mounted on arcrafts or satelltes as well for larger area averages. Today surface nformaton can be obtaned from a farly large selecton of actve sensors mounted on arcrafts and satelltes. b b Fgure 13.7 Examples of remote sensng systems. The upper fgure depcts a satellte montorng the surface condtons, and the prncple behnd LIDAR measurements, usng aerosol scatterng. The lower fgures dsplay how one can obtan several velocty components by rotatng the LIDAR beam n several drecton, and fnally a measurement of the wnd profles wth several velocty components, utlsng two (or three) SODAR transmtters amng smultaneous at the same ponts. The fgure llustrates as well that the speed of sound s small enough for the wnd to bend the beam. Most remote sensng systems employs actve sensors emttng ether electromagnetc waves or sound waves, nterpretng the backscatter return sgnal, usng a gatng technque to determne the dstance. The SODARs (SOund radar) emts one or several sound waves and receve the back scattered sgnal, where the ampltude reflect the ar densty at the target volume, whle the Doppler shft of the sgnal reflects the radal velocty of the ar. A smlar nstrument, LIDAR (LIght radar) works after the same prncple, but usng Laser lght. The scatterng elements for SODARs are densty gradents of the ar, whle t s suspended aerosol for LIDARs, see fgure Therefore SODARs work best for non-neutral ar, and LIDARs are nhbted when the ar s too clean of aerosols, Both SODARs and LIDARs are often employed wth several beams to 5 Lectures n Mcro Meteorology

254 derve the wnd vector. The range of SODARs s up to about 1 klometre, whle the LIDARs presently reach up to several klometres. Some LIDARs are operated as Celometres that scatter prmarly from cloud drops and are used to estmate the cloud heghts and/or boundary layer heghts. The equaton for the Doppler frequency shft looks smlarly for SODARS and LIDARS. ω ω = k V (13.7) r 0 Where subscrpt r means reflected and 0 means emtted radaton, k s the wave number vector for the emtted radaton and V s the wnd vector. 0 A range of ground placed remote sensng systems, denoted proflers, employ radar frequences and Doppler shfts, to montor the wnd vector up to heghts of several tens of klometres. Some of these are ntegrated nto wde nternatonal networks. A sute of LIDARs and radar sensors are nstalled on arplanes and satelltes, used n much the same observatons as the ground based nstruments, but now lookng down, both on the atmosphere and on the ground. The ground lookng nstruments here add to the passve nstruments dscussed n the begnnng of ths secton. As for the sensor descrbed as n stu nstrument above, the response of the remote sensng nstruments can be descrbed by ts spatal or temporal characterstcs. The temporal characterstcs are mostly seen as a queston about samplng, and the spatal response s descrbed by the volume of the scatterng volume that s often qute complcate beng of the type of very elongated cgars. For the systems utlsng beam at dfferent drectons to compose one estmate of the wnd vector, one must consder also that the target volumes are dfferent and at dfferent samplng tmes for the dfferent wnd vector components, just as for the sonc anemometer /thermometers consdered n Fg For the surface lookng system the descrpton of the foot- prnt of the beam s essental. Screens and shelters All sensors must be protected as much as possble, but wthout nfluencng the measurements. Weather protecton s of course one aspect, but some for some sensors the protecton must be amed also on the surroundng and the sgnal qualty. Fgure 13.7 shows a radaton screen to be employed to protect temperature sensors aganst the radatonal terms n (13.3). SODAR screens also depcted n Fg.13.7 serve both to protect the senstve recevers aganst envronmental nose, and to protect the surroundngs agan the hgh sound pressure from the SODAR transmtter. For smlar reasons systems usng lght must be shelded, for recevers from exposure to the Sun and for some LIDAR transmtters to avod eye damage. For nstruments used to measure ar concentraton, usng absorpton of lght across a gven path, as n Fgure 13.1, one dstngushes between closed path and open path sensors: The closed path sensors are often completely enclosed n chambers, wthn whch one can control envronmental parameters. The ar s pumped nto the sensor wthn the enclosure, whch obvously gves rse to many problem of nterpretaton. For the open path sensors, the ar s blowng freely through the lght beams, but the wndows are protected, by varous means. Lectures n Mcro Meteorology 53

255 b Fgure13.8 Screen ad protect for dfferent nstruments, temperature radaton screen, and SORAR enclosures. Platforms Instruments must be placed somewhere to conduct the measurements. The platforms however can modfy/dstort the measurements, modfcatons that have to be understood and mnmzed. For boundary layer meteorology many platforms are n use, e.g.: Mast, cars, drop-vehcles, arplanes, shps, balloons, satelltes. Examples are shown n Fg The platform modfcatons can be organsed nto platform moton, and flow dstorton by the physcal platform structure. The nfluence of the platform moton can be dscussed usng the Euler equaton, derved n secton 3 (3.7). d (13.8) ϕ ϕ v ϕ = + + u ϕ dt t x x Where ϕ s the atmospherc varables, beng measured. The two veloctes, v and u are the velocty of the sensor platform and the flow relatve to the sensor, dϕ/dt s the tme varaton observed by the sensor. If the sensor wnd speed s zero, for example for a statonary mast, we recover the Taylor hypothess, frst tme ntroduced n (.0), as follows: d (13.9) ϕ ϕ u ϕ = + u ϕ, dt t x x where we have assumed as well that the Euleran tme dervatve s much smaller than the advecton term on the rght hand sde. For ar planes the sensor velocty wll often domnate the wnd speed, and the ar craft Taylor hypothess wll nvolve the ar plane speed only, or the vector sum of the two veloctes. For cars and shps, one wll often have to consder (13.8) ncludng both wnd and sensor velocty. An nstrumented balloon, lke a radosonde, t wll bascally follow the horzontal flow, whle s wll addtonally undergo a vertcal moton drven by the dfferental buoyancy of the balloon. Fnally, we menton that the sensor velocty often s varable, not only as varyng advecton, but also rotaton, twstng and oscllaton etc. As an example we see the shp characterstcs of a shps moton, relatve to the crusng speed. 54 Lectures n Mcro Meteorology

256 Fgure Illustraton of the moton of a shp. Wth a sensor placed on such a shp, ths moton can be corrected ether by placng the sensor packet on a so called stable platform, or by measurng the shps moton for later correcton n the data analyss. Ths knd of moton wll obvously also characterse floatng buoys (AutoFlux, 1998). Flow dstorton and calbraton All sensors and sensor platforms to a larger or smaller extent dstort the flow by ther physcal presence n the flow. For practcal reasons, one mostly dstngushes between flow dstorton due to the sensors themselves and due to the platforms. In passng we note that for remote sensng nstruments, measurements wll be nfluenced very lttle by the flow dstorton, precsely because the target volume s remote. Platform moton however must be accounted for. Examples of flow dstortons for dfferent platforms and sensor confguratons are llustrated n Fgure 13.10, llustratng as well dfferent methods appled to establsh the flow dstorton: Fgure A llustrates the nfluence of a mast on cup-anemometer measurement of wnd speed. The fgure shows the rato between wnd measurements from two cup-anemometers, placed on dametrcally opposte to each other on each sde of the mast, shown versus wnd drecton. Fgure B shows the result of boom nfluence on a cup anemometer, showng the relatve varaton of the measured wnd speed versus the angle between the wnd and the boom drecton, based on wnd tunnel measurements. Fgure C shows results of wnd tunnels calbraton of a sonc anemometer. The fgure shows the rato between the tunnel wnd speed and the horzontal wnd speed measured by the sonc, the tlt-angle, and the drecton deflecton, as functon of the azmuth angel, relatve to the sonc array (Mortensen et al, 1987). Fgure D shows results of modelled flow dstorton around a shp, based on numercal modellng. Lectures n Mcro Meteorology 55

257 -- A B C D Fgure Flow dstortons. A: Relatve dfference between the two cup- anemometers opposte each other, relatvely to a meteorologcal measurement mast, as functon of wnd drecton, full scale measurements. B: Wnd tunnels study of the boom nfluence on cupanemometer readngs as functon of the angle between the wnd and the boom drecton. C: Wnd tunnels study (Mortensen et al, 1987) of the measured sonc wnd, relatve to the free stream tunnel velocty for an omn-drectonal sonc anemometer, as s e.g. shown n 13.. D: Model study of wnd flow around a shp (AutoFlux, 1998). Generally, two dfferent ways of handlng flow dstorton effects are used. One method conssts of comprehensve wnd-tunnel calbratons that deally allow to compute a ``true'' velocty vector for each measured value of the same vector. Ths method s llustrated Fgure A,B.C. The other method nvolves estmatng the effects on mean values and turbulence statstcs, usng more or less sophstcated models for the physcs of the flow dstorton. These models 56 Lectures n Mcro Meteorology

258 are subsequently reversed to yeld ``true'' mean values and turbulence statstcs from the values measured. The smplest ``model'' s here just to rotate the coordnate system along the measured mean flow (Dyer, 1981,198; Edson et al, 1981; Wyngaard, 198). More refned descrptons have been developed by Wyngaard (1981) for stuatons where the dstortng body s much smaller than the scale of turbulence, as s mostly the case for over-land measurements n the surface layer. Generally, the flow dstorton s less mportant for scalar fluxes than for velocty (Wyngaard, 1988), due to the absence of pressure effects for dstorton of scalars. Calbraton Bascally all sensors should be calbrated. Some are usng scentfc/techncal features that are emprcal or sem-emprcal, these obvously must be calbrated before use, meanng ther measurements must be compared to other well establshed sensors for measurng the same parameters. Of the sensors consdered above, hot-wres and cold wres, cup-anemometers and propellers all employ such sem-emprcal measurng prncples, and therefore should be calbrated. Sonc anemometers, LIDARS and SODARS all use the Doppler shft for wnd measurements and dffracton/absorpton of the sound or electromagnetc beam that nvolve fundamental physcal prncples, wth no emprcal coeffcents, for whch reason they should need no calbraton. However, also these nstruments nvolve uncertantes, n the measurement of the frequency change and for charactersaton of the effects of the measurement volumes and transmsson propertes wthn the sensors that necesstates calbraton. Expermental set-ups The plans for measurng actvtes n the atmospherc boundary layer can be organzed n many ways. Below we shall try to organze the relevant ponts as: 1) What s the objectve of the measurements, and whch confguraton of nstruments wll meet the objectves. ) Temporal and spatal scales that should be resolved, as well as the postonng and samplng rates of the dfferent sensors. 3) Statstcal demands to the data. 4) Instrumentaton technology, wth spatal and temporal resoluton, calbraton ssues, and flow dstorton, as we have touched upon above. The last three ponts can all be consdered wthn ths secton of the notes. The frst two of these have often been summarsed by the followng questons: How hgh s hgh enough?, How fast s fast enough? How close s close enough?, and how long s long enough (Krstensen et al, 1997, Lenschow et al, 1994, Larsen, 1993, Wyngaard,1973). The frst two questons focus the demands to spatal and temporal resoluton. The thrd queston consders use of several sensors, measurng dfferent parameters, whle the last queston concerns the need for relable statstcs. To consder these ponts n slghtly more detals, we wll below start wth a measurng campagn for statonary and horzontally homogenous condtons. Lectures n Mcro Meteorology 57

259 Statonary and horzontally homogenous condtons For a smple horzontally homogeneous statonary boundary layer average changes take place only n the vertcal dmenson, compare the Fg 6.1 for the Kansas experment. Here, one can use several measurng heghts. Accordng to the logarthmc profles, the levels for the mean values (e.g. wnd, temperature, humdty etc.) should be closer at low levels than at hgher levels. One wll often try to measure turbulence structure and fluxes drectly at dfferent ponts along the mast, as shown on the fgure 6.1 Theoretcally, we expect the turbulence characterstcs to be related to dfferental gradents of the mean values (e.g. Monn-Obuchov smlarty), whle the measurements yeld dfference gradents between the dfferent measurement levels only. In the surface layer t follows from secton 6 that any profles of mean values for a farly large stablty and heght nterval around neutral, can be wrtten as: (13.30) X( z) = a+ b ln( z) + c z, where the a, b and c-coeffcents contan all profle parameters other than z. If measurements of mean values are conducted n the two heghts, z 1 and z, and the profle s assumed to follow (13.9) n a heght nterval, the measured dfference gradent wll correspond to the theoretcal dfferental gradent at the level, z = (z -z 1 )/ ln(z /z 1 ), as can be seen by comparng the two expressons. If flux-profle relatons are measured, ths heght s therefore the best heght for flux nstrumentaton, as well as for any other turbulence measurements that have to be related to the vertcal gradents of the mean values. Equaton (13.30) s an example of how the expermenter must strve to match the deal world of the equatons wth real nstruments and measurement confguratons. A smlar queston arses for the determnaton of the coordnate system, specfcally the true vertcal drecton, whch s mportant as the man drecton for the vertcal flux. The queston has two aspects, one s phlosophcal: what should be consdered true vertcal n the real world, lkely wth some slopng terran? Here consensus has developed that true vertcal s the drecton perpendcular to the average stream lnes, meanng that the true <w> =0. Next comes the practcal determnaton of such drecton. It obvously necesstates the use of a three dmensonal velocty sensor, lke a 3D sonc anemometer. Hence, one can use the method of rotatng the measurement coordnate system, startng wth the mean values of the ntal data, <u s >, <v s > and <w s >, nto a coordnate system where all the mean speed s along the u-drecton (<u>+u, v,w ). The tlt ϕ = tan -1 (<w s >/(<u s > +<v s > ) 1/ ), Addtonally one has to develop a true lateral velocty and longtudnal wnd by a horzontal rotaton θ = tan -1 (<v s >/<u s >) of the ntal data.the angle ϕ can be nterpreted a the average true vertcal durng a partcular experment, as seen from the sensor coordnate system, although t can contan both terran slopes, and flow dstorton around the sensor head and a smple msalgnment of the sensor (Nelsen and Larsen, 00). Next we turn to the temporal and spatal scale n the fluctuatons. As mentoned n sectons, 6 and 8, they are convenently descrbed n terms of frequency or wave number spectra, where the relaton between wave numbers and frequency s descrbed through Taylor's hypothess, see (6.30), and e.g. Fg. 6.8 and 6.9. The nstrumental response, n both frequency and wave number space, s descrbed above n ths secton, for dfferent types of nstruments. Usng the dfferent spectral response functons derved, one can test f the nstruments have suffcent spectral resoluton to resolve turbulence sgnals one wants to resolve. For example one may fnd that the nstruments have tme scales 58 Lectures n Mcro Meteorology

260 or spatal averagng that wll dampen the hgh frequency end of the co-spectra and spectra resultng n an underestmaton of the fluxes and varances. Insertng the expected wnd speeds, measurng heghts, and stabltes, one can go through the detals. Smplstcally, one can see that ths problem become the more serous the closer to the ground the measurements are conducted, because generally the spectra are unversal functon of the normalzed frequency, n= fz/u= z/λ, where, f s the frequency (Hz) of the turbulence fluctuatons, and λ s the correspondng wave length. As seen from the defnton, f one wshes to resolve a gven n, then one needs to resolve hgher and hgher f (Hz), the hgher s the wnd sped and the lower the measurng heght. Correspondngly, one needs a better and better spatal resoluton the lower s the measurng heght, z to resolve smaller and smaller λ. The more detaled calculaton, mentoned above wll also show that ncreasng stablty wll lead to ncreasng demands to the sensor resoluton, because the spectra sldes to hgher frequency and smaller wave lengths for ncreasng stablty. The above dscusson reflects the consderatons about some of questons above: How hgh s hgh enough?, for the nstrumental response to be adequate. Alternatvely, for a selected heght, whch nstrumental resoluton s necessary? In Fgure 13.11, the nstruments shown are bascally sze or length scale lmted. The cups, propellers, wnd vanes are exactly of the type that are charactersed by a length scale, whle the remanng ones, soncs (and many concentraton sensors) and hot/cold wres have frequency lmtatons as well see (13.8), but ther length scale (meanng here sensor sze) are generally more mportant for some measurement of atmospherc turbulence. The next of these questons: How close s close enough?, becomes relevant when one combnes sgnals from dfferent sensors that necessarly must be dsplaced from each other, wth unavodable loss of correlaton, as s often the case, when one estmates scalar fluxes from cross-correlaton between sgnals from dfferent sensors. Consder such a flux measurng confguraton n Fg Here, we see a sonc anemometer/thermometer wth two open path CO and H O sensors, based on nfra-red absorpton. A thrd, closed path, gas sensor s ndcated by ts nlet tube fxed to the sonc. As argued above the response lmtatons of all these sensors are lmted by the spatal averagng along ther dfferent paths, and ther ndvdual response s determned above, n the secton dscussng the sonc response. For the closed path nstrument one must addtonally consder the delay and addtonal dampng of the fluctuatons when the ar flows through the ppe see e.g. Ebrom et al (007). The fluxes of the dfferent scalars, all denoted C, are determned by correlaton between the w sgnal determned by the sonc and the concentraton sgnals from the varous gas sensors, and here one must addtonally nclude the dsplacement between the dfferent sensors. Note that f the temperature sgnal s derved from the sonc, ths dsplacement s zero for the turbulent heat flux. Lectures n Mcro Meteorology 59

261 Fgure Comparson between the characterstc length scales of the turbulence fluctuatons, wth the correspondng length scale,λ nterval for characterstc nstruments. For example s seen that f we want to resolve n= 10, we need λ < 1m for z= 10m and λ < 5 cm for z= 1m. We have neglected the frequency response of the nstruments, because the nstruments are all lmted by ther length scale, rather by ther frequency response for atmospherc measurements. The nstruments can bascally resolve turbulence λ.> nstrument,λ or normalsed frequences= z/ λ >z/λ, the dagonal lnes correspond to gven normalsed frequences n=z/ λ. λ k s the Kolmogorov wave number λ k = πη, wth η the Kolmogorov scale, see e.g. Fgure 4..Hence λ k consttutes the small scale lmt for the nertal sub-range. The n resoluton necessary for a gven measurng heght and a gven parameter can be found from the spectral plots n Fgures From Krstensen et al (1997) one fnds, for horzontal dsplacement, D, a correcton of the type: z 4/3 (13.31) wc ' ' m = wc ' ' exp[ βδ (, )( D/ z) ], L Where β s a coeffcent between and 0.5, dependng on the angle δ between the dsplacement drecton and the wnd drecton, and on stablty, z/l. It s seen that the correcton depends on the dsplacement relatve to the measurng heght, ndcatng that the equaton pertans to surface layer condtons. For vertcal separaton condton are more dffcult, and the correctons are least wth the anemometer over the scalar sensor. The enclosed sensor measurng pont n Fgure 13.1 s obvously close to the measurng pont of the sonc. However, one must account for the ar transport tme through the tube between the sensor pont and the enclosed nstrument below. Ths can be done from the pumpng strength of the pump controllng the tube flow, and also account for possble mxng durng the transport n the tube, see more n (e.g. Ibrom et al, 007), 60 Lectures n Mcro Meteorology

262 Fgure Example of an expermental set-up for measurement of fluxes of heat,: Central s a sonc anemometer (Ka, DA600) and two open path systems for measurements of CO and water vapor from nfra red absorpton, LI7500, and OP, fnally a close path system for water vapor and CO s ndcated by the tube wth nlet strnged to the sonc. The tube leads down to the enclosed system below the pcture (Hrano and Sagusa, 007). Before leavng Fgure 13.1, we notce n the passng that the nstrument confguraton shown here certanly needs to be evaluated from a flow dstorton pont of vew. Statstcal consderatons and averagng The dscusson about how long s long enough?, nvolves the confdence of the data and ts relaton to the choce of averagng tme (or length f spatal data are avalable). Here, we shall start by followng the dscusson n Wyngaard (1973), concernng the expected statstcal varance around the estmated mean value, resultng from our measurements. Neglectng the spatal coordnates, we now consder statstcs of the fluxes and varables n the turbulent surface layer only consderng tme varables, followng (Wyngaard, 1973, Larsen, 1993, Lenschow et al, 1994). Gven statonarty, the varance around an expected value of an averagng of a correlated tme seres, χ(t), s derved n secton, see (.33), as follows: (13.3) χ τ χ Ix δ = (1 ) ρ ( τ) τ, T T T T d xt x 0 where, ρ(τ) s the correlaton functon, T s the averagng tme, and x, the ntegral scale for seres χ(t), see the dscusson n secton. T x can often smplest be derved from the spectra. Denotng the relatve accuracy, a = δ χ, we get an averagng tme, T xt a, needed to obtan a gven accuracy as: Lectures n Mcro Meteorology 61