1. Governing Equations

Transcription

1 1. Governng Equatons 1a. Governng Equatons for Mean Varables The governng equatons descrbe the varaton n space and tme of the zonal, merdonal and vertcal wnd components, densty, temperature, specfc humdty and other scalars transported n the PBL. These sets of equatons contan the tme and space dervatves that requre ntal and boundary condtons for ther soluton. As stated above, we don t forecast all eddy motons, nstead we pck a cut-off eddy sze and below ths we nclude only the statstcal effects of turbulence. For mesoscale and synoptc models the cut-off s on the order of 10 to 100km, whle for large eddy smulaton, the cut-off s on the order of 100m. The methodology we wll use s: Step 1. Identfy the basc governng equatons for boundary layer. Step 2. Expand dependent varables nto mean and turbulent parts. Step 3. Apply Reynolds averagng to get mean varables wthn turbulent flow. Step 4. Obtan equatons for turbulent departure from mean. A. Equaton of state The deal gas law for most ar, expressed n terms of vrtual temperature s, p = ρrt v (1) where p s pressure, ρ s the densty of most ar, T v s the vrtual temperature n K and R s the gas constant for dry ar [R = 287Jkg 1 K 1 ]. After Reynolds averagng (1) becomes: p R = ρt v + ρ T v ρt v (2) The last approxmaton s due to the fact that ρ T v << ρt v. Ths s the equaton of state for mean varables. 1

2 Notce that f we subtract the equaton for mean varables from the orgnal equaton expressed as the sum of the mean and turbulent parts of every varable, and neglect the small covarance terms we obtan the lnearzed perturbaton deal gas law. p p = T v + ρ T v ρ Wthn the PBL, we can neglect the pressure perturbaton over average pressure, so that: T v = θ v = ρ (4) T v θ v ρ Ths mples that warmer (colder) than average ar s less dense (more dense) than average. Ths also allows us to substtute temperature fluctuatons n place of densty fluctuatons. B. Conservaton of mass The conservaton of mass equaton s wrtten as: ρ + (ρu j) = ρ + U j ρ + ρ U j (3) = dρ dt + ρ U j = 0 (5) Wthn the boundary layer, dρ dt /ρ << U j, so we can assume ncompressblty: After Reynolds averagng, the equaton becomes: U j = 0 (6) U j = 0 (7) Ths s the conservaton of mass for mean varables, also called ncompressble contnuty equaton. The conservaton equaton for turbulent varables s obtaned by subtractng (7) from (6): u j = 0 (8) C. Conservaton of mosture 2

3 The conservaton of mosture s wrtten as: q + U q 2 q j = ν q x 2 j + S q ρ where q s the specfc humdty (mass of water per unt mass of most ar), ν q s the molecular dffusvty for water vapor n the ar, S q s mosture source term as, for example, a phase change and we assume t only affects the mean varables. We then Reynolds average and express the turbulent advecton term n flux form u q j = u j q usng (8) to obtan the conservaton equaton for mean total mosture: q I q 2 q + U j = ν q x }{{ j x } 2 j II III + S q ρ IV u j q V (9) (10) where Term I represents the storage of mean mosture, Term II the advecton of mean mosture by mean wnd, Term III mean molecular dfusson of water vapor (generally neglected because t s much smaller than the other terms), Term IV mean net body source term for addtonal mosture processes, and Term V represents the dvergence of turbulent total mosture flux. By subtractng (10) from (9) (whch ncludes both mean and turbulent components for each varable) and omttng the source term (S q ) for perturbatons we obtan the prognostc equaton for the perturbaton part (q ). q + U q j + u q j + u q 2 q j = ν q D. Conservaton of scalar quantty + u j q (11) Conservaton of a scalar quantty s expressed n a very smlar form as the conservaton of mosture. C + U C 2 C j = ν c + S c ρ (12) where C s the concentraton of a tracer (mass of scalar per unt mass of most ar), ν c s the molecular dffusvty for that scalar n the ar, S c s net source term for the mean varables. We perform Reynolds averagng on (12) and express the 3

4 turbulent advecton term n flux form u j c = u j c equaton for mean tracer C: C I C 2 C + U j = ν c x }{{ j x } 2 j II III + S c ρ IV to obtan the conservaton u j c V (13) where each of the terms s analogous to (10). By subtractng (13) from (12), we obtan the prognostc equaton for the perturbaton part (c ). E. Conservaton of heat c + U c j + u C j + u C 2 c j = ν c + u j c (14) Conservaton of enthalpy per unt mass (C p θ) s derved from the frst law of thermodynamcs. The equaton ncludes contrbutons from both sensble and latent heat as water vapor has the potental to release or absorb latent heat durng phase change. θ + U θ 2 θ j = ν θ 1 ρc p Q j L pe ρc p (15) where ν θ s the thermal dffusvty, L p s the latent heat assocated wth the phase change of E. Q j s the component of net radaton n the j th drecton, and C p s the specfc heat of most ar, related to the specfc heat of dry ar C pd = 1005Jkg 1 K 1 by C p = C pd ( q). We perform Reynolds averagng and, wth the ad of the contnuty equaton, express the turbulent advecton term n flux form u j θ = u j θ prognostc equaton for the mean potental temperature (θ). θ I + U j θ II = ν θ 2 θ III 1 Q j ρc p x }{{ j } IV L ve ρc p V u j θ V I to obtan the (16) where Term I represents storage of heat, Term II advecton of heat by mean wnd, Term III mean molecular conducton of heat (generally neglected because t s much smaller than the other terms), Term IV mean net body source assocated 4

5 wth radaton dvergence, Term V mean net body source assocated wth latent heat release, and Term VI represents the dvergence of turbulent heat flux. By subtractng (16) from the orgnal equaton (15) we can obtan the prognostc equaton for the perturbaton part (θ ). θ + U θ j + u θ j + u θ 2 θ j = ν c + u j θ 1 ρc p Q j (17) Fgure 1: From Stull F. Conservaton of momentum Newton s second law of moton can be wrtten as: 5

6 Fgure 2: From Stull U I + U j U II = gδ }{{ 3 + f } c ɛ j3 U j 1 p ρ x III IV V + ν 2 U V I (18) where Term I represents storage of momentum (nerta), Term II represents advecton, Term III the vertcal effect of gravty, Term IV s the Corols effect where f c = 2ω snφ (wth φ beng lattude and ω beng the angular velocty of the earth), Term V pressure gradent forces and Term VI represents vscous stress (generally neglected because t s much smaller than the other terms n the equaton of mean moton, but not n the perturbaton equatons). Densty varatons are essentally what drve buoyancy n the vertcal drecton so that warmer (than average) ar rses and cooler ar snks. Wth ths fact and the ncompressblty assumpton (6), we can neglect densty varatons except n the 6

7 Fgure 3: From Stull buoyancy term assocated wth gravty. Ths s called Boussnesq approxmaton. To show ths clearly we can multply the the vertcal component of the momentum equaton by (ρ/ρ) and separate varables nto mean and perturbaton components. Applyng Boussnesq approxmaton, we can neglect all terms ρ /ρ, except n the gravty term [so (1 + ρ /ρ) 1 n all but gravty terms]. W + w + (U j + u j) (W + w ) = ) (1 + ρ g 1 (p + p ) + ν 2 (W + w ) ρ ρ z (19) We can use (4) to express the equaton n terms of vrtual potental temperature fluctuatons, and generalze for the three components: 7

8 U + u + (U j + u j) (U + u ) ( ) = g 1 + θ v δ 3 +f c ɛ j3 (U j + u j) 1 (p + p ) + ν 2 (U + u ) ρ x θ v (20) Fnally, we can do Reynolds averagng and express the turbulent advecton term n flux form u u j mean varables: U I + U j U II = u j u to obtan the conservaton of momentum for = gδ }{{ 3 + f } c ɛ j3 U j 1 p ρ x III IV V + ν 2 U V I u j u } {{ } V II (21) where the frst sx terms represent the same physcal mechansms as (18), and the new Term VII represents the nfluence of Reynold s stress on mean motons. Ths last term s also descrbed as dvergence of turbulent momentum flux. If we subtract (21) from (20), we get the equaton of conservaton of momentum for perturbatons: + U j + u U j + u u j = gθ v δ 3 + f c ɛ j3 u j 1 p + ν 2 u θ v ρ x u u We wll re-vst ths equaton when dscussng turbulent knetc energy. + u j u (22) It s sometmes useful to substtute the horzontal pressure gradent terms usng the defnton of geostrophc wnd: f c U g = 1 p and f ρ y cv g = + 1 p. In ths way, ρ x we can express the zonal and merdonal components of the momentum equaton n smplfed form as: U + U U j = f c (V g V ) (u j u ) (23) V + U V j = +f c (U g U) (u j v ) (24) In these equatons we have neglected the molecular dffuson/vscosty terms because they are much smaller than the other terms n the equaton. In addton, 8

9 notce that n the momentum equaton for far-weather condtons we can usually neglect subsdence, so the prognostc equaton for mean vertcal wnd s not presented. Notably, PBL numercal models generally use ths notaton for the horzontal components of the momentum equaton. 1b. Prognostc Equatons for Turbulent Fluxes and Varances A. Prognostc equatons for turbulent fluxes Equatons for the mean varables n turbulent flow (10), (13), (16) and (21) contan dvergence terms of turbulent fluxes q u j, c u j, θ u j and u u j. These terms arse drectly because of the nonlnearty of the advecton terms n the orgnal equatons. They act as source/snk terms that change the mean varables due to the transport of momentum, mosture, heat and/or scalars by perturbatons of wnd, mosture, heat and/or scalars. We can fnd prognostc equatons for these fluxes usng the followng methodology descrbed for any varable ζ, whch can denote wnd, humdty, potental temperature or a scalar. 1. Multply perturbaton equaton for ζ by u and Reynolds average. 2. Multply the momentum perturbaton equaton (u ) by ζ 3. Add the two resultng equatons 4. Use the contnuty equaton to get the turbulent transport terms nto flux form and merge any other terms Example: mosture flux Start wth the momentum perturbaton equaton and multply by mosture perturbaton, then Reynolds average. q u +U jq u +q x u j j U +q x u j j u = g q θ v δ 3 +f c ɛ j3 q x u j j θ q v ρ p x +νq 2 u (25) Smlarly, we start wth the mosture perturbaton equaton and multply by u and Reynolds average. u q + U ju q + u x u j j q + u q x u j = ν q u j 2 q (26) 9

10 Now we add (25) and (26), assume ν = ν q and use these smplfyng expressons: q 2 u q p = 1 (p q ) p q (27) ρ x ρ x ρ x + u 2 q = 2 (q u ) ( ) ( ) u 2 q (28) The boxed terms can be neglected based on scalng arguments (Stull, 1988), and we can denote the last term n (28) as 2ɛ u q. Other terms (e.g., ɛ u u j and ɛ) used n equatons below can be smlarly defned. We then express the turbulent flux dvergence terms nto flux form, neglect the Corols term and express the smplfed expresson as: u q I u + U q j + u q x u j + q }{{ j x u U j }}{{ j x }}{{ j } II III IV g q θ v δ ( ) p ρ q 2ɛ u q x θ v } {{ } V I } {{ } V II V III + (q u u j ) = x }{{ j } V (29) Term I s the storage term, Term II s the advecton term, III, IV and VI are producton/consumpton terms, V s a turbulent transport term, VII represents redstrbuton, and Term VIII s a molecular destructon (dsspaton of turbulent mosture flux). Wthout formal dervaton, we present the momentum flux budget equaton [See Stull (1988) for a formal dervaton]. Ths equaton s derved n a smlar fashon as the turbulent mosture flux, and wll be re-vsted later when dscussng turbulence closure: 10

11 u u k u + U u k j = u U k x u j u k j x u j j B. Prognostc equatons for varance terms U u j u u k + g ( ) [δ k3 u θ θ v + δ 3 u k θ v] + p u + u k v ρ x k x 2ɛ u u k (30) Prognostc equatons for varances are also useful for solvng the equatons for mean flow. In addton, the varance equatons provde a physcal descrpton for the evoluton of the boundary layer as wll be seen later when turbulent knetc energy s dscussed. The procedure for obtanng the equatons for the varance terms s smlar, although not dentcal, to that for the fluxes. 1. Begn wth the prognostc equaton for the perturbaton of the varable ζ. 2. Multply by 2ζ and convert the terms 2ζ ζ nto (ζ ) 2 3. Reynolds average 4. The terms that look lke u ζ 2 j can be turned nto flux form by usng the contnuty equaton multpled by ζ 2 (whch s equal to zero). Example: momentum varance Applyng the frst two steps to (22), we obtan: u ζ 2 j + ζ 2 u j = ζ 2 u j (31) u 2 u 2 + U j + 2u u U j + u u 2 j = u 2δ θ v 3 g + 2f c ɛ j3 u u j 2 u p + 2νu 2 u u + 2u j u θ v ρ x x 2 (32) j We can smplfy the dsspaton term usng the followng expresson: 11

12 2νu 2 u = ν 2 u 2 2ν ( ) u 2 (33) x We can neglect the boxed term based on scalng arguments (Stull, 1988) and express the last term n (33) as 2ɛ, transform the pressure perturbaton term n flux (u p ) form 2 ρ x usng the fact that the turbulence contnuty equaton s zero, and neglect Corols for velocty varances: u 2 u 2 + U j = 2δ 3 u θ v θ v g 2u U u j + u j u 2 ( ) 2 (u p ) 2ɛ (34) ρ x We can do the same for the prognostc equatons for mosture varance q 2, potental temperature varance θ 2, and varance of a scalar quantty c 2. The smplfed equatons are lsted below wthout formal dervaton: q 2 + U q 2 j = 2q x u j j θ 2 + U θ 2 j = 2θ x u j j c 2 + U c 2 j = 2c x u j j q (u j q 2 ) 2ɛ q (35) θ (u j θ 2 ) 2ɛ θ 2 θ ρc Q j (36) p c (u j c 2 ) 2ɛ c (37) where the defntoon of ɛ q, ɛ θ, and ɛ c s smlar to that of ɛ n (34). These three equatons are presented n smlar form where the frst term s the local storage, the second term s the advecton term, the thrd s a producton term assocated wth turbulent motons wthn a mean gradent, term four s the turbulent transport term, term fve denotes the molecular dsspaton, and the potental temperature varance has an addtonal term assocated wth radaton destructon. 1c. Turbulent Knetc Energy The prognostc equaton for turbulent knetc energy (TKE) s one of the most useful equatons for turbulence studes. The TKE per unt mass s defned as: T KE/m = e = 1 2 (u 2 + v 2 + w 2 ) = 1 2 u 2 (38) 12

13 Smlarly, we can defned e = u 2 /2. TKE ndcates whether the BL wll become more turbulent or f turbulence wll decay. Turbulence producton s due to buoyant thermals and mechancal eddes whle turbulence suppresson s drven by statcally stable lapse rate, and dsspated nto heat by molecular vscosty. To obtan a prognostc equaton for TKE, we can multply (34) by 0.5: e I + U j e II u = δ θ v 3 g θ }{{ v } III u U u j } {{ } IV u j e V 1 ρ (u p ) x } {{ } V I ɛ V II (39) where Term I represents local storage of TKE whch ncreases from early mornng wth a peak n the early afternoon. Over oceans, the storage term can be neglected (due to the small durnal cycle), whch means that the ntensty of turbulence doesn t change sgnfcantly wth tme. Term II represents advecton of TKE by mean wnd. Many tmes t s neglected assumng horzontal homogenety, however, ths mght not be vald for heterogeneous terran. Term III s a buoyant producton or consumpton term actng only n the vertcal drecton. It can be a producton term f the flux of vrtual potental temperature w θ v s postve (assocated to rsng thermals over land durng day) or a loss term when ths flux s negatve (as over land at nght, when statc stablty tends to suppress TKE). Ths term s very mportant for days of free convecton and can be used to normalze the TKE equaton. Term IV s a mechancal producton/loss term assocated wth mean wnd shear. Ths term s prmarly contrbuted by the vertcal gradent of horzontal wnd. Because mean horzontal wnd s zero at surface, t always ncreases wth heght near surface. Ths also mples that rsng parcels (wth w > 0) would have u < 0. Therefore term IV s generally postve and represents the mechancal producton of TKE. Term IV s largest near the surface, because of the large wnd shear, however, large wnd shear can exst at the top of the mxed layer wth geostrophc wnds above. Ths term s largest on wndy days, durng synoptc cyclones. Term V s the turbulent transport of TKE. It s a flux dvergence term - f ntegrated throughout the BL, t s zero- so t only redstrbutes turbulence and s not a producton/loss term. The maxmum vertcal transport occurs at z/z = 0.3 (wth z beng the PBL heght). Term VI descrbes how TKE s redstrbuted by pressure perturbatons, often assocated to oscllatons (buoyancy or gravty waves). Pressure perturbatons are usually small and s dffcult to measure, so ths term s usually calculated as a resdual. In very unstable condtons, ths term 13

14 can be a sgnfcant source of TKE near the top of the mxed layer (Hogstrom, 1990). Term VII s the vscous dsspaton of TKE representng the converson of TKE nto heat. Molecular destructon s greatest for the smallest eddy szes, so ntense small-scale turbulence nduces larger dsspaton. Ths term s largest near the surface, and then becomes constant, rapdly decreasng to zero above the PBL. Fgure 5 presents the terms of the TKE equaton as a functon of heght for steady-state daytme condtons. If we assume horzontal homogenety and neglect subsdence, the TKE equaton becomes: e = w θ v g (w u U θ v z + w v V z ) w e 1 (w p ) ɛ (40) z ρ z Shear producton and buoyant producton terms are large and postve for large eddy szes. However, there s a cascade of energy away from the large eddes towards the small eddes. At small eddy szes, the producton s close to zero and the dsspaton s very large. Ths can be thought of as an nertal process where large eddes bump nto smaller ones and transfer ther nerta, and the mddle porton of the spectrum s the nertal subrange. It can be shown from (21) that the prognostc equaton for the mean knetc energy contans the same mechancal producton/loss term as n the TKE equaton but they have opposte sgns. Therefore, the energy that s mechancally produced as turbulence s lost from the mean flow. 1d. Scalng and Stablty Crtera As we have shown, turbulence s generated by buoyant convecton (thermals) and by mechancal processes (wnd shear). Free convecton s the lmt when buoyant processes domnate, whle durng forced convecton, mechancal processes domnate. Here some useful scalng varables are defned for each of these two cases. They are used extensvely n scalng analyss and for presentaton of the data. A. Forced convecton scalng We defne a velocty scale called the frcton velocty u : u 2 = τ / ρ = [(u w ) 2 s + (v w ) 2 s] 1/2 (41) where τ refers to the total Reynolds stress, and subscrpt s refers to surface. 14

15 Fgure 4: From Garrat We can ntroduce the surface layer temperature scale θ = (w θ ) s u (42) 15

16 Fgure 5: From Garrat and the surface layer humdty scale B. Free convecton scalng q = (w q ) s u (43) 16

17 To defne the free convecton velocty scale, we frst defne the buoyance flux as: (g/θ v )w θ v. To generate the velocty scale we multply by the typcal length scale whch s z (the top of the mxed layer). Ths velocty scale s on the same order of magntude as observed vertcal velocty fluctuatons. w = [ ] 1/3 gz (w θ v) s (44) θ v The characterstc temperature s on the order of how much warmer thermals are than ther envronment. θ ML = (w θ ) s w (45) The characterstc humdty scale s on the order of how much moster thermals are than ther envronment. C. Flux Rchardson number q ML = (w q ) s w (46) The rato of the buoyant producton term and the mechancal producton term n the TKE equaton s called the flux Rchardson number (R f ). Ths number characterzes the thermal stablty of the flow. R f = g (w θ θ v v) (u u j ) (47) U The denomnator conssts of 9 terms. We assume horzontal homogenety and neglect subsdence: g (w θ θ R f = v v) (u w ) U + z (v w ) V z Because the denomnator s usually negatve. (48) R f > 0 for statcally stable flows R f < 0 for statcally unstable flows R f = 0 for statcally neutral flows 17

18 At the crtcal value of R f = +1, the mechancal producton rate balances the buoyant consumpton. Because dsspaton s always negatve n the TKE equaton, even before the buoyant consumpton balances the mechancal producton, turbulence would cease to exst. Therefore, the crtcal R f separatng turbulent and lamnar flows s generally less than 1 (and s about 0.25). D. Gradent Rchardson number The value of the turbulent correlatons could be expressed as beng proportonal to the lapse rate, and the turbulent momentum flux can be proportonal to the wnd gradent: w θ v θv, z w u U and z w v V. Ths s the bass z of K-theory that we wll dscuss later. When substtutng nto (48), we get the gradent Rchardson number: R = ( U z g θ v θ v z ) 2 + ( V z ) 2 (49) where t s assumed that turbulent exchange coeffcents for temperature and wnd are the same. The advantage of R over R f s that the former can be computed from temperature and wnd gradents drectly. E. Bulk Rchardson number When measurng wnd shear and temperature gradents, the gradents are usually approxmated by measurements at dscrete heghts: R B = ( U z g θ v θ v z ) 2 + ( V z ) 2 = g θ v z θ v (( U) 2 + ( V ) 2 (50) Ths s the form most frequently used. The values of the crtcal Rchardson number separatng turbulent and lamnar flows don t apply to these fnte dfferences across thck layers. The thnner the layer, the closer the value to the theory. F. Monn-Obukhov length The Monn-Obukhov Length s a very mportant parameter n the surface layer n whch turbulent fluxes and wnd drecton don t vary much wth heght. Takng 18

19 Table 1: Stablty Crtera Free Convecton Unstable Neutral Stable Lamnar (u w ) U / z 0 R < 0, ζ < 0 R = ζ = 0 R > 0, ζ > 0 R > R c U / z = (u /kz)φ m wth k beng the Von Karman constant (0.4) and φ m beng the stablty functon to be dscussed later, R f can be rewrtten as: R f = ζ/φ m (51) where ζ = z/l s the dmensonless heght and the Monn-Obukhov length L s L = θ vu 3 kg(w θ v) s (52) At a heght of z = φ m L, R f = 1 and the buoyancy and shear producton terms are equal. Consequently, we can nterpret the Monn-Obukhov Length as the heght at whch the buoyancy producton s equal to the mechancal producton under neutral condton (wth φ m = 1). ζ s a prmary parameter of the Monn- Obukhov smlarty of the atmospherc surface layer. We summarze the stablty condton n Table 1. Notce that when the gradent Rchardson number reaches a crtcal value of R c.2.25 the transton from turbulent to lamnar flow occurs. Theoretcally the transton from turbulent to lamnar flow s better represented by R f (rather than R ). Furthermore R B s computed n modelng and data analyss to represent R. In the atmosphere, even when turbulence ceases, turbulent ntermttency stll exsts and nternal gravty waves can stll generate mxng. For these reasons, even when R > R c, some turbulent mxng s stll consdered n numercal modelng. Ths s stll an area of actve research (Fernando and Wel, 2010). 19