OCTOBER 08 T O N G A N D D I N G 369 Mnn Obukhv Smlarty cal-free-cnvectn Scalng n the Atmspherc Bundary ayer Usng Matched Asympttc Expansns CHENNING TONG AND MENGJIE DING Department f Mechancal Engneerng Clemsn Unversty Clemsn Suth Carlna (Manuscrpt receved January 08 n fnal frm 8 June 08) ABSTRACT The Mnn Obukhv smlarty thery (MOST) s the fundatn fr understng the atmspherc surface layer. It hyptheses that nndmensnal surface-layer statstcs are functns f / nly where are the dstance frm the grund the Obukhv length respectvely. In partcular t predcts that n the cnvectve surface layer lcal free cnvectn (FC) ccurs at heghts / / where s the nversn heght. Hwever as a hypthess MOST s based n phenmenlgy. In ths wrk we derve MOST the FC scalng frm the equatns fr the velcty ptental temperature varances usng the methd f matched asympttc expansns. Our analyss shws that the dmnance f the buyancy shear prductn n the uter (/ ) nner (/ & ) layers respectvely results n a nnunfrmly vald slutn a sngular perturbatn prblem that s the thckness f the nner layer. The nner slutns are fund t be functns f / nly prvdng a prf f MOST fr the vertcal velcty ptental temperature varances. Matchng between the nner uter slutns results n the FC scalng. We then btan the crrectns t the FC scalng near the edges f the FC regn (/ ; / ; ). The nndmensnal ceffcents n the expansns are determned usng measurements. The resultng cmpste expansns prvde unfed expressns fr the varance prfles n the cnvectve atmspherc surface layer shw very gd agreement wth the data. Ths wrk prvdes strng analytcal supprt fr MOST.. Intrductn The Mnn Obukhv smlarty thery (MOST; Mnn Obukhv 954; Obukhv 946) s the fundatn fr ur understng f the atmspherc surface layer. It hyptheses that the surface-layer ( )dynamcss gverned by the knematc surface stress (the square f the frctn velcty) u the surface temperature flux Q the buyancy parameter b the heght frm the surface where s the bundary layer (nversn) heght. Any nndmensnal statstcs therefre s a functn f / where 5u 3 /(kbq) s the Obukhv length k s the vn Kármán cnstant. The thery predcts that fr / the shear prductn f the turbulent knetc energy dmnates fr /. the buyancy prductn dmnates. In partcular t predcts that fr the specal case f / (but/ ) the nfluence f mean shear becmes neglgble resultng n the s-called lcal-free-cnvectn (FC) scalng (Tennekes 970; Wyngaard et al. 97). In ths layer the vertcal velcty Crrespndng authr: Prf. Chennng Tng ctng@clemsn. edu ptental temperature varances d nt depend n u have the frms w ; (bq) /3 u ; Q 4/3 (b) /3 respectvely. Hwever when the cndtns / / are nt satsfed (e.g. / ; r/ ; ) we expect departure frm the FC scalng (Wyngaard et al. 97). Feld measurements supprt the predctn fr the vertcal velcty ptental temperature varances (sectn 3e). Whle MOST has been successfully used n predctng the surface-layer scalng t s nevertheless a hypthess based largely n the phenmenlgy f the surface layer dmensnal arguments. Measurements can prvde supprt t MOST but t cannt pstvely prve t. In the present study we derve MOST the FC scalng frm frst prncples usng the equatns fr the velcty ptental temperature varances the methd f matched asympttc expansns (Bender Orsag 978; Van Dyke 975; Custex Mauss 007). We als derve frm the expansns the crrectns t accunt fr the departure frm the FC scalng fr / ; r/ ;. The cmpste expansns nclude the nfluence f bth / / are based n the physcs f the surface layer. The frmer s an apprxmatn f the Mnn Obukhv DOI: 0.75/JAS-D-8-006. Ó 08 Amercan Meterlgcal Scety. Fr nfrmatn regardng reuse f ths cntent general cpyrght nfrmatn cnsult the AMS Cpyrght Plcy (www.ametsc.rg/pubsreusecenses).
369 J O U R N A O F T H E A T M O S P H E R I C S C I E N C E S VOUME 75 functns near / ; whse functnal frms cannt be btaned analytcally frm dmensnal analyss. Instead they must be btaned emprcally frm bservatns. The nfluence f / s absent n MOST. Thus the dervatn s a majr step tward an analytcal prf f MOST. The analytcal predctn f the varance prfles can beneft numercal weather predctn mdels under cnvectve cndtns. It wll als be mprtant fr mdelng atmspherc dspersn. In the fllwng we frst examne the varance equatns fr the velcty cmpnents ptental temperature t dentfy the mathematcal structure f the prblem (a sngular perturbatn prblem). We then perfrm the methd f matched asympttc expansns t btan MOST the FC scalng as well as the crrectns t the latter fr / ; r/ ;. The analytcal results fr FC are then cmpared wth measurements are fllwed by the cnclusns.. The mathematcal structure f the prblem The equatns fr the velcty cmpnents ptental temperature varances n a hrntally hmgeneus atmspherc bundary layer are (e.g. Wyngaard Cté 97) w 5 w 3 t p w pw g T wu «() 3 u U 5uw t wu p u x «() y t 5 wy p y y «(3) u 5wu t wu «(4) u where «««3 «u are dsspatn rates fr u / y / w / u / respectvely; p s the knematc pressure. The upper- lwercase letters dente the mean fluctuatng varables respectvely. The mean wnd s algned wth the U drectn. When the shear prductn s absent (free cnvectn) Eqs. () (4) have the mxed-layer scalng; therefre the resultng nndmensnal slutn depends n the nndmensnal ndependent varable / : w u 5 w w 5 u w y u 5 yy w (Q/w ) 5 u (5) where w 5 (bq ) /3 s the mxed-layer velcty scale. The subscrpt wll be used t dente the uter varables defned n the next sectn. The mxed-layer scalng als hlds n the presence f the mean shear prductn fr snce the effects f the shear prductn (f u ) are small wthn ths range f heghts. Nte that the terms n the u y equatns acqure the mxed-layer scalng (except the shear prductn term) as a result f the pressure stran-rate crrelatn. Hwever the slutns n Eq. (5) are nt vald fr / & where the shear prductn whch has the surface-layer scalng (as a result f the parameters u ) becmes a leadng term n Eq. (); that s the presence f the shear prductn term results n a nnunfrmly vald slutn. Ths shear-prductn-dmnated layer always exstsaslngastsabvetherughnesslayerfrany small but nner mean shear. Therefre er mean shear s a sngular lmt fr the slutn; that s the structure f the slutn fr a case wth the mean shear apprachng er (but nt equal t er) s fundamentally dfferent frm that wth the mean shear equalng er. Cnsequently the system descrbed by Eqs. () (4) has the structure f a sngular perturbatn prblem whse slutn can be btaned usng the methd f matched asympttc expansns. The layers wth / / & are the s-called uter nner layers respectvely. 3. Matched asympttc expansns In ths sectn we use the methd f matched asympttc expansns t slve the sngular perturbatn prblem t derve MOST the FC scalng fr the vertcal velcty (the hrntal cmpnents d nt have ths scalng) ptental temperature varances as well as t btan the secnd-rder crrectns t the FC scalng. Matched asympttc expansns are a methd t slve a set f dfferental equatns havng a slutn that has dfferent scalng n dfferent parts f the slutn dman that s a nnunfrmly vald slutn. In ths study they are the mxed-layer scalng surface-layer scalng. The slutn n each part f the dman s expressed as seres expansns wth ther respectve scalng. The expansns n the dfferent parts are then asympttcally matched t btan cmpste expansns (unfrmly vald slutn). a. Outer expansns We defne the dmensnless uter varables n the sngular perturbatn prblem w u y u (/ ) p wu wu t usng the mxed-layer scales as fllws: Q w 5 w w u 5 w u y 5 w y u 5 u w 5 Q 5 w p 5 w p Q wu 5 Qwu wu 5 w w wu t 5 t. (6) w
OCTOBER 08 T O N G A N D D I N G 3693 The nndmensnal frms f Eqs. () () (4) n terms f the uter varables are w t 5 w 3 p w pw wu «3 u U 5uw t u t 5wu (7) w u p u «w 3 x (8) wu «u. (9) Fr cnvenence the equatn fr y s nt gven as t s nt explctly used n the analyss. In the uter layer the buyancy prductn term the pressure stran-rate terms are f rder ne (leadng terms) whle the nndmensnal shear prductn term uw( U/ )( /w 3 ) (u3 /)( /w 3 ) ; therefre t s a secnd-rder term can be wrtten as uw U U 5 uw w 3 (0) where s a small parameter whse rder f magntude has yet t be determned. Hwever t wll becme a leadng term when s suffcently small therefre results n a nnunfrmly vald slutn a sngular perturbatn prblem. Unlke ths shear prductn term the prductn f u n Eq. (9) s a leadng term therefre s nt the surce f sngularty as / 0. The uter expansns f the velcty cmpnents ptental temperature ther varances n terms f the pwer f can be wrtten as w ( ) 5 w ( ) w ( ) O( ) u ( ) 5 u ( ) u ( ) O( ) u ( ) 5 u ( ) u ( ) O( ) () w ( ) 5 w ( ) w ( )w ( ) O( ) u ( ) 5 u ( ) u ( )u ( ) O( ) u ( ) 5 u ( ) u ( )u ( ) O( ). () The equatns fr the leadng-rder terms w u u are btaned by substtutng Eqs. () () nt Eqs. (7) (8)(9) cllectng terms f rder ne ( 0 ). b. Inner expansns a prf f MOST As dscussed abve when s suffcently small (n the nner layer) the term cntanng the mean shear n Eq. (8) becmes a leadng term the uter slutn s n lnger vald. A new scalng s needed n the nner layer. We defne the dmensnless nner varables w n u n u n (/ ) n n p n wu n wu n as fllws: Q w 5 u w n u 5 u u n u 5 u n u 5 Q u 0 5 0 n p 5 u p n n Q wu 5 Qwu n wu 5 u u wu (3) n where 0 s the nner length scale (thckness f the nner layer) has yet t be determned. Here u s used as the velcty scale as shwn n the dervatn f the multpnt Mnn Obukhv smlarty (MMO; Tng Nguyen 05; Tng Dng 08 manuscrpt submtted t J. Flud Mech.). The nndmensnal frms f Eqs. () () (4) n terms f the nner varables are u w w n t 5 w 3 u 3 n n 0 p w u 3 n 0 pw u 3 n 0 g T Qwu «u 3 n 3n 0 (4) u w u n U t 5uw u 3 n n 0 w n u u 3 n n 0 p u u 3 x n 0 «u 3 n 0 (5) Q w u n u t 5Q Q u 0wu n n Q u wu u 0 n Q «un n u 0. (6) In the nner layer & 0 uw( U/ ) needs t be a leadng-rder term; thus t must scale as uw U ; u u 0 ; p u x ; p w ; g T Qwu (7) n leadng t gq/t ; u 3 /0. The fact that the pressure stran-rate crrelatn terms are f the same rder f magntude are leadng-rder terms n bth the u n w n equatns s used n dervng Eq. (7). Therefre the nner length scale 0 s the Obukhv length. Equatns (4) (5) (6) becme 0 w n t 5 w 3 n p w n n pw wu n «3n n (8)
3694 J O U R N A O F T H E A T M O S P H E R I C S C I E N C E S VOUME 75 0 u n U t 5uw w n u n p u «n n n x n n (9) n 0 u n t 5wu n n wu n n «un (0) smlar. We can therefre wrte the nner expansns fr the vertcal velcty ptental temperature varances as w n ( n ) 5 w n ( n ) 0 w n ( n )w n ( n ) O( 0 ) u n ( n ) 5 u n ( n ) 0 u n ( n )u n ( n ) O( 0 ). () where 0 5 w 5 k /3 /3 () u s a small parameter. We nte that fr the vertcal velcty temperature varances t fllw MOST that s t be Mnn Obukhv (M-O) smlar t have a slutn that scales wth the nner varables the varance equatns must als be M-O smlar. Althugh the hrntal velcty varances have mxed-layer scalng are nt M-O smlar ther rate equatns are (r have apparent M-O smlarty; Dng et al. 08 manuscrpt submtted t J. Flud Mech.; Tng Dng 08 manuscrpt submtted t J. Flud Mech.). Therefre the dynamcs f the hrntal vertcal velcty cmpnents n the surface layer are M-O The results n Eq. () are f fundamental mprtance: They shw that the nndmensnal vertcal velcty ptental temperature varances are functns f / nly thus prvdng a prf f MOST fr these varables. c. Asympttc matchng t derve the FC scalng Snce the uter nner expansns descrbe the dynamcs at the uter nner scales respectvely are vald fr / / there exsts an verlappng regn where bth cndtns are satsfed the expansns represent the same functn. Therefre f we wrte the uter expansn as a functn f the nner varable the nner expansn as a functn f the uter varable they shuld be equal. The nner expansn f the uter expansn f w s " # w 5 w w n w n w n as / 0 wth n fxed ; u /3w n ; u /3 a n (3) keepng nly ne term. The uter expansn f the nner expansn f w s w 5 u w n 0 w n w n as 0 / 0 wth fxed ; w /3 w n ; u a n. (4) Matchng Eqs. (3) (4) results n a 5 /3. Thus /3 w ; w ; u /3 5 u f. (5) Ths s the FC scalng fr the vertcal velcty varance u f s the velcty scale. The nner expansn f the uter expansn f u s " Q u 5 u w Q ; /3 u u # n u n u n as / 0 wth n fxed n ; /3 Q g u n (6)
OCTOBER 08 T O N G A N D D I N G 3695 keepng nly ne term. The uter expansn f the nner expansn f u s Q u 5 u n u Q ; w /3u 0 u n n ; u n as 0 / 0 wth fxed Q g n u. (7) Matchng Eqs. (6) (7) results n g 5/3. Thus u ; /3. (8) Ths s the FC scalng fr the temperature varance. d. Secnd-rder crrectns t the leadng-rder slutns (FC scalng) When the cndtns fr the verlappng regn (/ / ) are nt satsfed departures frm the FC scalng [Eqs. (5) (8)] are expected. Crrectns t accunt fr the departures can be made by ncludng the hgher-rder terms n the expansns n Eqs. () (). T btan the secnd-rder terms we need t frst determne the scalng f the shear prductn term uw( U/ ) the small parameter. Snce the surface layer s a cnstant flux layer (e.g. Haugen et al. 97) the turbulent flux uw s apprxmately ndependent f heght frm the surface scales as u. We cnsder the shear stress uw budget equatn (e.g. Wyngaard et al. 97) uw w U t g uw uu T w p x u p 5 0. (9) Nndmensnalng uw by u t by /u f w by results n u f u f u uw w U t u f u u f u! g uw uu T w p x u p 5 0. (30) The shear prductn term must be a leadng term [O()] n Eq. (9). Thus resultng n w U u f u ; u U u f u f ; (3) U ; u /3 (3) whch s the same as that btaned usng dmensnal analyss (Carl et al. 973). Therefre the apprprate uter scale fr U/ s (u / )( /) /3. We can then defne the uter varable ( U/ ) as U 5 U u /3 4/3 5 5 4/3. (33) The shear prductn term n Eq. (8) can then be wrtten as uw U 5 u uw u /3 U 5 u3 /3uw U. (34) Its nndmensnal frm s uw U w 3 5 u3 w 3 5 /3uw U 4/3uw U Therefre the small parameter n Eq. (0) s. (35) 5 4/3. (36) Substtutng the uter expansns n Eq. () nt Eqs. (7) (8) (9) cllectng the terms f rder we btan the secnd-rder equatns fr the uter varables w w 5 3 w w p w pw t u u U 5uw t wu «3 (37) p u x w u u u u «(38)
3696 J O U R N A O F T H E A T M O S P H E R I C S C I E N C E S VOUME 75 u u 5w t u w u w u w u u «u. (39) Snce the frst term n the rght-h sde f Eq. (38) s nw a leadng term we have U uw ; p u ; p w ; w w x / ; r w w (40) where r s the crrelatn between w / w «3 ««u are the dsspatn rates fr w w u u u u respectvely. Here we assume that the crrelatn ceffcent between w w s als r.the fact that the pressure stran-rate crrelatn terms are f the same rder f magntude are leadng terms n Eqs. (37) (38) s used n dervng Eq. (40). In the last step the estmate w w ; r w w / s used. Therefre frm Eqs. (33) (5) gvng 4/3 / ; r w w (4) r w / ;. (4) Thus the secnd-rder crrectn term s w w ; r w / / w ; /3. Frm Eqs. (39) (4) w u ; w u u (43) u 5 u ( ) u ( )u ( ) O( ) /3 5 A u B u k /3 4/3 O( ) (46) respectvely. Rewrtng the last tw equatns n dmensnal frm drppng the O( ) terms we have w 5 w w 5 Ak/3 w /3 /3 Bk /3 w 4/3 5 Au /3 Bu /3 (47) Q Q u 5 u w 5 A u w /3 Q B u k /3 4/3 w Q 5 A u k /3 /3 Q Bu (48) u u respectvely. Each f these expressns cntans tw nndmensnal ceffcents whch wll be determned usng measurements (sectn 3e). Smlarly substtutng the nner expansns n Eq. () nt Eqs. (8) (0) cllectng the rder 0 terms we have the equatns fr the secnd-rder nner varables w n t 53 w n w n p w pw n n n r u u / ; 5/3 (44) where r u s the crrelatn ceffcent between u u. Therefre the secnd-rder term u u ; r u u / u / ;. Thus w u wth the secndrder crrectns fr / ; are w 5 w ( ) w ( )w ( ) O( ) /3 5 Ak /3 Bk /3 4/3 /3 O( ) (45) wu n «3n (49) u n t 5w u w n n n u n n n w u u n n n w n u n «n un (50) n where «3n «un are the dsspatn rates fr w n w n u n u n. Nw the term n the left-h sde f Eq. (49) s a leadng term thus w n t ; w n w n ; r n w n w n. (5) n n /
OCTOBER 08 T O N G A N D D I N G 3697 In the last step the estmate w n w n ; r n w n w n/ s used where r n s the crrelatn between w n w n/ we assume that the crrelatn ceffcent between w n w n s als r n. Therefre frm Eq. (5) resultng n /3 n / ; r n w n w n (5) n r n w / n ; n. (53) Thus the secnd-rder crrectn term s w n w n ; r n w / / n w n ; 4/3 n. Frm Eqs. (50) (5) u n t ; w n u n u n n (54) r un u n / ; /3 n (55) where r un s the crrelatn between u n u n.therefre the secnd-rder term u n u n ; r un u / / n u n ; 0 n.thusw n u n wth the secnd-rder crrectns are w n 5 w n ( ) n 0 w n ( n )w n ( n ) O( 0 ) 5 A /3 Ck /3 /3 4/3 O( 0 ) (56) u n 5 u n ( ) n 0 u n ( n )u n ( n ) O( 0 ) 5 A u k /3 /3 Cu k /3 /3 0 O( 0 ) (57) respectvely. Rewrtng the last tw equatns n dmensnal frm drppng the O( 0 ) terms we have /3 w 5 u w n 5 Au Ck /3 u /3 4/3 /3 4/3 5 Aw k/3 Cw (58) Q Q u 5 u n u 5 A u k/3 u Q C u k /3 /3 0 u /3 5 A u Q w /3 C u Q w 0 (59) respectvely. The nndmensnal ceffcents n the abvementned equatns wll be determned belw. Summng the tw asympttc expansns subtractng the cmmn parts we btan the cmpste (unfrm) expansns " /3 w 5 w Ak /3 Bk /3 4/3 /3 # 4/3 C 5 u A /3 B /3 Ck /3 /3 4/3 (60) /3 Q u 5 "A w u B u k /3 4/3 # 0 C u Q 5 A u u k /3 /3 Bu Cu k /3 /3 0 (6) whch are vald n bth the nner uter layers. e. Cmparsn wth measurements The nndmensnal ceffcents A B C A u B u C u are nw determned usng measurements frm the Kansas (Wyngaard et al. 97) Mnnesta (Kamal et al. 976; Ium Caughey 976) Atmspherc Radatn Measurement (ARM; Mather Vyles 03; Berg et al. 07) Ashchurch (Caughey Palmer 979) feld prgrams. By fttng the FC scalng term f w t
3698 J O U R N A O F T H E A T M O S P H E R I C S C I E N C E S VOUME 75 FIG.. Cmparsn f the cmpste expansn fr the vertcal velcty varance wth the Kansas (968) data n terms f the nner (surface layer) varables. The FC lmt (sld lne) FC the secnd-rder crrectn (dashed lne) are marked. the Kansas data (Fg. )fr/. we fnd A 3:(the sld lne n Fg. ). Fttng Eq. (47) fr / dwn t 0. (dashed lne n Fg. ) we btan B 0:. Fttng the FC scalng term f u t the Kansas data (Fg. )fr/. gves A u :8 (the sld lne n Fg. ). Fttng Eq. (48) t the Kansas data fr / dwn t 0.03 (dashed lne n Fg. ) we btan B u 0:0038. Here the secnd-rder crrectns accunt fr departure frm the FC scalng caused by the mean shear prductn fr / ;. Applyng the Mnnesta Ashchurch ARM data (Fgs. 3 4) t Eqs. (58) (59) we fnd C :35 C u :. Usng these values fr the ceffcents the cmpste expansns shw very gd agreement wth the feld data. Wth these A B values the expansn fr w als fts reasnably well the Nrthern Hemsphere Clmate Prcesses Surface Experment (NOPEX) data (Jhanssn et al. 00; nt shwn) whch have larger scatters. Fr the uncertanty levels fr the NOPEX data (6% fr u 0% fr ) t s clear frm Fgs. 3 that self-crrelatn effects d nt alter the trends f the data n any sgnfcant way. Furthermre judgng frm the scatter f the data pnts the uncertantes n the Kansas data are lwer. As a result any effects f self-crrelatn wuld be even smaller. Therefre the bserved FC scalng departure frm t fr / ; areduetthe surface-layer physcs nt self-crrelatn effects. FIG.. Cmparsn f the cmpste expansn fr the temperature varance wth the Kansas (968) data n terms f the nner (surface layer) varables. ne styles are the same as n Fg..
OCTOBER 08 T O N G A N D D I N G 3699 FIG. 3. Cmparsn f the cmpste expansn fr the vertcal velcty varance wth the Mnnesta (crcles) Ashchurch (trangles) ARM (astersks) data n terms f the uter (mxed layer) varables. ne styles are the same as n Fg.. The cmpste expansn fr w s vald up t / : well beynd the surface layer. Here the secnd-rder crrectns accunt fr the nfluence f the nversn heght (atmspherc bundary layer depth) whch s reflected n the tme dervatve terms n Eq. (8). Unlke n the surface layer where / the turbulence s n lnger n equlbrum wth the external cndtns (nfluences) when / ;. The cmpste expansn fr u s vald up t / 0:6 because at the nversn layer there s a prductn surce wth dfferent scalng whch s beynd ur surface-layer analyss. Frm the pnt f vew f sngular perturbatn prblems there s a secnd nner layer near 5 whch needs t be cnsdered n the matched asympttc expansns f we want t predct the behavr there. Therefre the current crrectn cannt capture the trend f u near / 5. Fr w the nversn damps the fluctuatns whch des nt necessarly result n a secnd nner layer. The abvementned cmparsns shw that by addng nly the secnd-rder crrectns the functnal frms f the cmpste expansns already shw very FIG. 4. Cmparsn f the cmpste expansn fr the temperature varance wth the Mnnesta data n terms f the uter (mxed layer) varables. ne styles are the same as n Fg.. gd agreement wth the data demnstratng the effcacy f the methd f matched asympttc expansns fr analyng the surface layer. Prevusly vertcal prfles f turbulence statstcs have been emprcal expressns btaned by curve fttng feld data [e.g. Caughey Palmer (979) fr vertcal velcty prfles]. Furthermre the emprcal curves fr / / & are separate curves. The cmpste expansns btaned n the present study prvde unfed expressns fr the vertcal velcty ptental temperature varances frm / 0: t/ ;. Equally mprtant each part f the expansns has a clear physcal nterpretatn (rgn). 4. Cnclusns dscussns In the reprted study we used the methd f matched asympttc expansns t derve analytcally Mnn Obukhv smlarty thery fr the vertcal velcty ptental temperature varances the lcal-freecnvectn scalng whch prevusly have been a hypthess based n phenmenlgy. We fcused n the vertcal velcty ptental temperature varances.
3700 J O U R N A O F T H E A T M O S P H E R I C S C I E N C E S VOUME 75 The equatns fr the hrntal velcty vertcal velcty ptental temperature varances are used t derve MOST the FC scalng. The dmnance f buyancy shear prductn terms n the uter nner layers whch have dfferent scalng prpertes results n a nnunfrmly vald slutn a sngular perturbatn prblem whch s slved usng the methd f matched asympttc expansns. We btaned as the thckness f the nner layer. The nner expansns were fund t depend n / nly prvdng a prf f MOST fr the vertcal ptental temperature varances. The FC scalng was btaned by matchng the leadngrder nner uter expansns. Crrectns fr the departure frm the scalng fr / ; / ; whch cannt be btaned analytcally usng dmensnal analyss are als derved by ncludng the secnd-rder expansns. The cmpste expansns btaned shw very gd agreement wth the Kansas Mnnesta Ashchurch Atmspherc Radatn Measurement (ARM) feld data acheved wth nly leadng- secnd-rder expansns demnstratng that matched asympttc expansns prvde an effectve methd fr analyng understng the atmspherc bundary layer. In dervng the nner equatns [Eqs. (8) (0)] we have used the surface-layer scalng f the terms n these equatns whch s supprted by bservatnal evdence (e.g. Kamal et al. 976; Wyngaard et al. 97). The surface-layer scalng f these terms can als be btaned frm the surface-layer smlarty f multpnt statstcs (Tng Nguyen 05) whch has als been derved mathematcally usng the methd f matched asympttc expansns (Tng Dng 08). Therefre the derved scalng n the present study s a cnsequence f MMO the dervatn s mathematcally rgrus. The present wrk s als part f a cmprehensve analytcal dervatn f MMO MOST. The present study uses the balance equatns fr the velcty temperature varances t derve MOST the FC scalng fr these varables thereby prvdng strng analytcal supprt t Mnn Obukhv smlarty thery. The expansns g beynd the prevus bservatn-based emprcal frmulas fr turbulence statstcs t prvde physcs-based analytcally derved expressns wth clear physcal rgns nterpretatns. These expressns the understng f the asscated physcs are als ptentally mprtant fr a range f applcatns. The vertcal velcty varance s ften used n eddy vscsty dffusvty mdels. Fr example n numercal weather predctn mdels usng clumn parameteratn fr the bundary layer the analytcal expressn fr the vertcal velcty prfle n cnvectve bundary layers s mprtant fr mprvng the predcted temperature prfle under cnvectve cndtns. The derved varance prfles can als beneft predctn f atmspherc dspersn wave prpagatn. Acknwledgments. Ths wrk was supprted by the Natnal Scence Fundatn thrugh Grants AGS- 335995 AGS-5690. REFERENCES Bender C. M. S. A. Orsag 978: Advanced Mathematcal Methds fr Scentsts Engneers: Asympttc Methds Perturbatn Thery. Internatnal Seres n Pure Appled Mathematcs McGraw-Hll 593 pp. Berg. K. R. K. Newsm D. D. Turner 07: Year-lng vertcal velcty statstcs derved frm Dppler ldar data fr the cntnental cnvectve bundary layer. J. Appl. Meter. Clmatl. 56 44 454 https://d.rg/0.75/ JAMC-D-6-0359.. Carl M. D. T. C. Tarbell H. A. Panfsky 973: Prfles f wnd temperature frm twers ver hmgeneus terran. J. Atms. Sc. 30 788 794 https://d.rg/0.75/50-0469 (973)0300788:POWATF..0.CO;. Caughey S. J. S. G. Palmer 979: Sme aspects f turbulence structure thrugh the depth f the cnvectve bundary layer. Quart. J. Ry. Meter. Sc. 05 8 87 https://d.rg/ 0.00/qj.4970544606. Custex J. J. Mauss 007: Asympttc Analyss Bundary ayers. Scentfc Cmputatn Sprnger-Verlag 434 pp. https://d.rg/0.007/978-3-540-46489-. Haugen D. A. J. C. Kamal E. F. Bradley 97: An expermental study f Reynlds stress heat flux n the atmspherc surface layer. Quart. J. Ry. Meter. Sc. 97 68 80 https://d.rg/0.00/qj.497097404. Ium Y. J. S. Caughey 976: Mnnesta 973 Atmspherc Bundary ayer Experment data reprt. Ar Frce Cambrdge Research abratres Tech. Rep. AFCR-TR-0038 Envrnmental Research Papers 547 8 pp. Jhanssn C. A.-S. Smedman U. Högström J. G. Brasseur S. Khanna 00: Crtcal test f the valdty f Mnn Obukhv smlarty durng cnvectve cndtns. J. Atms. Sc. 58 549 566 https://d.rg/0.75/50-0469(00) 058549:CTOTVO..0.CO;. Kamal J. C. J. C. Wyngaard D. A. Haugen O. R. Cté Y. Ium S. J. Caughey C. J. Readngs 976: Turbulence structure n the cnvectve bundary layer. J. Atms. Sc. 33 5 69 https://d.rg/0.75/50-0469(976)0335: TSITCB..0.CO;. Mather J. H. J. W. Vyles 03: The ARM Clmate Research Faclty: A revew f structure capabltes. Bull. Amer. Meter. Sc. 94 377 39 https://d.rg/0.75/bams-d- -008.. Mnn A. S. A. M. Obukhv 954: Basc laws f turbulent mxng n the grund layer f the atmsphere. Tr. Gef. Inst. Akad. Nauk SSSR 4 63 87. Obukhv A. M. 946: Turbulence n the atmsphere wth nhmgeneus temperature. Tr. Inst. Ter. Gef. Akad. Nauk SSSR 95 5.
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