Flow over a hill covered with a plant canopy

Transcription

1 C:\My Douments +\My Siene and Projets\illanopy\Word dos\canhill v7.do Stepen eler Page 8/4/ Flow over a ill overed wit a plant anopy y J. J. FINNIGAN # and S. E. ELCHER * # CSIRO Atmosperi Resear, F C Pye Laboratory, lak Mountain, ACT 6, AUSTRALIA * Department of Meteorology, University of Reading, Reading, RG6 6, UK SUMMARY We develop an analytial model for atmosperi boundary layer flow over a ill tat is overed wit a vegetation anopy. Te slope of te ill is assumed to be small enoug tat te flow above te anopy an be treated witin te linear framework of Hunt et al (988). Perturbations to te flow witin te anopy are driven by te pressure gradient assoiated wit te flow over te ill. In te upper anopy tis pressure gradient is balaned by downwards turbulent transport of momentum and te anopy drag. Te flow tere an be alulated from linearized dynamis, wi sow tat te maximum streamwise winds are were te perturbation pressure is at a minimum, i.e. near te rest of te ill. Deep witin te anopy te pressure gradient assoiated wit te flow over te ill is balaned by te anopy drag, ere te nonlinear anopy drag. Tis nonlinear balane sows ow te streamwise winds are largest were te perturbation pressure gradient is largest, i.e. on te upwind slope of te ill. In te lee of te ill tis nonlinear solution sows ow te pressure gradient deelerates te wind deep witin te anopy and leads to separation wit a region of reversed flow wen te anopy is suffiiently deep. Coupling between te out-of-pase flows witin and above te anopy means tat te maximum veloity is furter upwind of te ill rest tan in flow over a roug ill, wile te extra turbulent mixing aused by te anopy signifiantly redues te magnitude of te veloity speed-up over te ill. Finally, we find tat tere is no formal limit proess were te solutions wit a anopy yield te well-known solutions for flow over a roug ill. Tis finding alls into question te very use of a rougness lengt in aelerating or deelerating turbulent boundary layers. Key Words: Atmosperi oundary Layer, Flow over Hills, Canopy flow, Turbulene (Reeived?? and in revised form??). Introdution We investigate te flow of a neutrally-stratified atmosperi boundary layer over a ill overed wit a deep plant anopy. Te work as four motivations. Firstly our understanding of atmosperi boundary layer flow over plant anopies is mainly restrited to flow over omogeneous anopies on level terrain, so tere is a fundamental need to understand more general, eterogeneous, anopy-flows (see Finnigan ). Seondly, tere is a large investment worldwide in ontinuous measurements of te

2 surfae-atmospere exange of energy and greenouse gases (aldoi et al, ), wi as produed an urgent need to interpret measurements made over forests in omplex topograpy. Tirdly, te even-larger worldwide investment in wind energy is seeing wind turbines sited in regions of mixed topograpy and forests and so tere is a need to develop quantitative understanding of te output of su turbines (Ayotte, et al ). Fourtly, parameterization of orograpi rougness as been sown to improve te skill of numerial weater predition models (see eler and Hunt 998). Reent omputations by rown et al () and Allen and rown () suggest tat a ill overed wit a plant anopy exerts a substantially stronger drag on te atmosperi flow tan a smooter ill, but te meanism tat gives tis inrease is not urrently known.. Ruk and Adams (99) measured flow over a two-dimensional ridge overed wit a tall anopy in a wind-tunnel. Tey observed signifiant redutions in te magnitude of veloity speed-up at te rest of te ill ompared to te values measured over smooter ills. Interpretation of teir results is ompliated, owever, by te ange in rougness tat appears to our lose to te foot of teir ill and by te fat tat tey limited teir measurements to te flow above te anopy. Finnigan and runet (995) made a more ompreensive set of wind-tunnel measurements, bot witin and above te anopy. Tey observed striking anges to te inflexion point in te mean veloity profile tat is arateristi of flow at te top of a anopy. On te windward slope, about alfway up te ill, te inflexion point disappeared and te turbulent sear stress penetrated mu deeper into te anopy tan in te unperturbed flow. At te rest of te ill in ontrast, te sear at te inflexion point was substantially stronger and te turbulent sear stress ad an assoiated large peak at te anopy top and ten dropped to zero in te upper tird of te anopy. Finnigan and runet advaned a qualitative explanation for tese features in terms of differenes in te adjustment of te flow witin and above te anopy to te pressure gradient assoiated wit flow over te ill. One aim of te present work is to develop a quantitative model for tese proesses. Wilson et al (998) were able to reprodue tese main arateristis of te flow using a numerial model wit a k λ eddy-visosity turbulene losure. Te use of eddy visosities witin anopy flows requires justifiation beause tey are known to be poor desriptors of turbulent transport in tere. In tis paper we sow ow tey an desribe te mean-flow perturbations aused by te ill, but not te unperturbed bakground flow. Reently, rown et al () and Allen and rown () ave onduted large-eddy simulations of flow over ills. Issues related to resolving te flow in te dynamially important inner region led tem to represent te rougness elements on te surfae of te ill using a anopy. In teir simulations te large-sale turbulene above te anopy was resolved and turbulene in te flow witin te anopy was represented wit an eddyvisosity model. Te primary aim of tis work was to evaluate te suess of large-eddy simulation of boundary-layer flow over ills. Neverteless, it an be seen from teir results tat te anopy redues te speed-up of te flow over te ill. At te same time it inreases te asymmetry of te flow about te rest of te ill, and tene also inreases bot te drag on te ill and te tendeny for te flow to separate. Te dynamial proesses tat yield tese differenes are urrently obsure.

3 Present knowledge of te role of a anopy in te flow over a ill, and te flow witin a anopy on a ill, is tus unertain. Wat is needed is a dynamial framework witin wi to understand te flow and to identify te ontrolling pysial parameters. Su a framework will elp organize and interpret measurements or omputations. We address tis need ere by developing an analytial model of flow above and witin a anopy on a ill. Te slope of te ill is supposed to be suffiiently low tat te perturbations to te bakground flow above te anopy an be analyzed wit linearized equations. We an ten build upon te well-developed framework of Hunt et al (988), eler (99) and eler et al (993) for flow over roug ills. Te flow of air above and witin a anopy on flat terrain is desribed in setion. In setion 3 te problem of te anges to tis bakground flow aused by a ill is stated matematially and in setion 4 te linear model of te perturbations to tis bakground flow by a low ill is reviewed briefly. In setion 5 te perturbed flow witin te anopy is analyzed. Te oupling between te flow above and witin te anopy is alulated in setion 6. Results are presented in setion 7 and onlusions in setion 8.. Flow witin and above a omogeneous anopy Consider neutrally-stratified atmosperi boundary layer flow witin and above a uniform omogeneous anopy of eigt driven by a synopti pressure gradient tat yields a wind above te anopy wit frition veloity u *. Above te anopy, z >, te boundary layer is modelled as a neutrally-stratified surfae layer, so tat te turbulent sear stress, τ = u, is approximately onstant wit eigt, ρ and te mean veloity profile,u, is logaritmi, namely, U * ( z) u z+ d ln, () = κ z were d is te displaement eigt assoiated wit te anopy, z is te rougness lengt of te anopy, κ =.4 is von Karman s onstant and te origin of vertial oordinate, z >, is taken at te top of te vegetation. Te subsript is used trougout to denote te basi or bakground flow. Te kinemati sear stress,, is represented by a mixing lengt model, wit mixing lengtl : τ du = lm dz m. () Above te anopy te mixing lengt inreases wit eigt asl m = κ ( z+. Witin te anopy, < z <, te turbulent stress gradient balanes te anopy drag, wi omes predominantly from pressure fores on te anopy elements and so is parameterized ere as CaU (see also runet et al., 994). C is te drag oeffiient of individual anopy elements and a is te leaf area per unit volume of spae. Te anopy drag is written ere τ d) 3

4 asu L, were L = Ca is an adjustment lengt sale of te anopy (eler et al, ). Hene te momentum balane in te anopy is dτ dz U = L. (3) Te turbulent stress witin te anopy is modelled using a mixing lengt (), but now wit a onstant mixing lengt, namelyl m = l. A disussion of te appropriateness of tis turbulene losure witin te anopy is given in Appendix. Trougout te following analysis we fous on deep anopies, L, were all te momentum is absorbed by te foliage. Te momentum balane (3), wit te mixing lengt model (), ten sows (Inoue 963) tat te mean wind profile is exponential β z l U z = U e, (4) were U = U ( ) mixing lengt, l, is given by is te mean wind speed at te top of te anopy and te anopy l β L. (5) 3 = Here β = u U quantifies te mass flux troug te anopy. For losed uniform natural anopies, typial values are β.3; C.5 (Raupa et al, 996: runet et al, 994). For a anopy wit a =.4 m, tese relations yield L = m and l.5m. Te mean wind speed and turbulent sear stress are ontinuous at te top of te anopy z =, and if te mixing lengt is also assumed to be ontinuous, ten u * d l β κ U = ln ; d = l κ ; z = e. (6) κ z κ Hene, te synopti pressure gradient fixes u *, β is an empirial onstant and te anopy elements fix. Te model for te basi flow ten determinesl, U = u β, d and. L * z 3. Matematial model of flow over ill overed wit a anopy We now develop a model for te anges to te basi flow aused by a low ill. Te geometry of te ill and anopy and te upwind veloity profile is illustrated in Figure. z = H f x L, wit eigt H and alf lengt L, Te ill sape is desribed by s defined to be te distane from te rest to te point were z = H. s 4

5 Te perturbations to te flow are analyzed naturally in a displaed oordinate system( X, Z ) wose lines follow te surfae at low levels and ten relax bak to Cartesian oordinates ( x, z) by (, )= ( φ(, ), ψ (, )) at iger altitudes. Tis approa ensures tat te mean flow follows te topograpy near te surfae and relaxes to te basi undisturbed flow far above te surfae (eler, 99; eler et al, 993). A suitable oie for te lines of onstant Z is te streamlines of invisid, irrotational flow over te ill fored by a uniform basi wind of unit magnitude. Te streamwise oordinate X is ten defined so tat te oordinate system is ortogonal. Hene te displaed oordinate system is given X Z x z x z, were φ ( x, z) is te veloity potential and ψ ( x, z ) te stream funtion of te potential flow over te ill. For te anonial geometry of a sinusoidal ill, zs = H oskx, te displaed oordinates to leading order in te slope are kz X = x+ H sin kx e, kz Z = z+ H os kx e. (7) were k = π L. Te equations of motion are transformed into tis oordinate frame using standard metods (e.g. Finnigan, 983), wi in te most general ase as several onsequenes. Firstly, te veloity vetor and stress tensor are now referred to an ortogonal vetor 3 basis tat onsists of te tangent vetors to te oordinate lines, e.g. u= ue + we, 3 were te tangent vetors are e = x X, e = x Z and x = ( x, z) is te position vetor in te Cartesian oordinates. Seondly, extra terms appear in te equations of motion to ompensate for te fat tat te orientation of te vetor basis anges as te streamlines urve over te ill. Tirdly, te partial differentials tat appear in te equations are replaed by diretional derivatives along te oordinate lines. For te ills of low slope, H L <<, tat are of interest ere te extra terms in te equations and te errors involved in treating te dependent variables as onventional pysial quantities wile at te same time treating te diretional derivatives as partial derivatives are all O H L or smaller. Te ill produes perturbations to te basi flow wi are denoted ere by, u = U ( Z) + u( X, Z ), w= w( X, Z ), = + p P p X, Z, τ = τ + τ ( X, Z, (8) ) for te streamwise veloity, vertial veloity, kinemati pressure and kinemati turbulent sear stress. Te streamwise veloity is written as te undisturbed profile (expressed in te displaed oordinates) plus a perturbation u. Te atual flow streamlines depart from te streamlines of invisid flow so tere is a vertial veloity perturbation, w. In 5

6 te outer layer (see 4 below) te flow relaxes to invisid potential flow over te ill and so follows te oordinate lines and w. For ills wit low slopes H L, perturbations indued to te basi flow by te ill are small enoug, uu= O( H L ) tat tey an be alulated using linearized equations, wit produts of perturbations and nonlinear parts of te oordinate transformation negleted. In tis limit te transformed flow equations are, u U p τ U u U + w = + ( -Η( Z) ) X Z X Z L w ψ p τ U w U U = + ( H ( Z) ) b (9) X X Z X L u w + = X Z We ave also negleted linear terms involving te gradients of te normal Reynolds stresses as tey play no signifiant role in te momentum balane. Te last terms on te rigt and sides of (9) a and b represent te absorption of momentum by te foliage, so H Z is te Heaviside step funtion, defined byh Z = Z < andh ( Z) =, Z >. tat It is lear tat, wen te veloity perturbations are defined as in (8), te linearized momentum equations ave te same form as in Cartesian oordinates. u, w, p, τ as Z L (), Finally we note tat tese transformed equations differ from tose presented in eler (99) and eler et al (993). In tose papers te oordinate frame was transformed but te dependent variables were not. Instead te dependent variables remained in te original Cartesian referene frame. Here we prefer to transform also te dependent variables. Firstly, tis approa is pysially intuitive. Seondly, it is in aord wit te standard pratie in field experiments in omplex terrain to rotate measurements eiter into surfae following oordinates or into te Cartesian frame aligned wit te loal mean wind vetor: te dependent variables in (9) give a lose approximation to tis pratie. Te outer boundary onditions are tat te perturbations deay far above te ill, so tat a 6

7 At te top of te anopy, Z = te veloity and turbulent stress are ontinuous. Here it is assumed tat te mixing lengt is also ontinuous, so tat ontinuity of stress implies tat te veloity gradient is also ontinuous at te top of te anopy. Finally we note tat for te deep anopies onsidered ere it is appropriate to use a free slip boundary ondition at te ground surfae on te ill. Te reason is tat in a deep anopy >> l te stress perturbations deay to negligible levels for Z >> l and so we an treat any boundary layer tat develops atually at te surfae beneat te pressure driven lower-anopy flow perturbation as deoupled from te rest of te solution. 4. Flow above te anopy Te dynamis of te flow above a roug ill, were te rougness elements are represented wit a rougness lengt z, are now well understood. A sket of te analysis is given ere, for a full disussion see Hunt et al (988) (enefort, HLR), eler (99) and eler et al (993). As te atmosperi boundary layer flow is defleted over te ill, te onsequent vertial motion leads to a pressure perturbation, wit te minimum pressure at te ill rest. Tis pressure field aelerates te streamwise flow on te upwind slope and ten deelerates te flow in te lee of te ill. Troug te bulk of te flow above te anopy te gradients of te turbulent sear stress in (9) an be ignored beause tey are mu smaller tan te inertial terms: H τ Z L u L u ~ ~ H U u X U L U L * * << Witin tis outer region te veloity perturbation is, to leading order in u* U, invisid. In te upper part of tis region, were te sear in te bakground wind is also small, te veloity perturbations are well desribed by potential flow so tat te displaed streamlines follow te Z-lines and te additional perturbation vertial veloity in (9) approaes zero, i.e. w. Solution of te vertial momentum equation (9)b ten sows tat te magnitude of te outer-region pressure perturbation is p( X, Z) U ψ x = U R, were R is te loal radius of urvature of te Z oordinate lines (Finnigan, 983). Tese invisid proesses lead to flow perturbations tat deay to zero far above te ill but are not zero at te surfae of te ill and so do not satisfy te no-slip ondition. Of ourse te real flow must satisfy no slip and tere is a relatively tin layer near te ground, known as te inner region (Jakson & Hunt, 975), were te effet of perturbations to te turbulent sear stress beome omparable to te invisid proesses. In tis inner region sear stress perturbations progressively redue te streamwise veloity perturbation as te top of te anopy is approaed. 7

8 Te perturbation sear stress witin te inner region is parameterized by te mixing lengt model, te mixing lengt inreasing linearly wit Z as in te basi flow desribed in. On linearising about te basi state tis model yields, τ u τ = κuz *. () Z Te eigt of te inner region, i ln L ( z ) = κ i, is defined by te impliit relation, i. () Ten, sine ln ( i z ) is typially large, te inner region is a tin layer, i.e. i L<<. Tis allows tree approximations. Firstly, te vertial momentum equation (9)b ten sows OHLand equal to tat te pressure perturbation aross te inner region is onstant to [ ] te value of te outer-region pressure perturbation at Z perturbation troug te dept of te inner region is (, ) ψ. Hene te pressure p kz p k = U x (3) were te veloity sale for te pressure perturbation isu = U ( m), and m is te eigt of te middle layer, wi is defined as follows. If L is less tan about % of te boundary layer dept, ten te approa flow is logaritmi up to a eigt ~L and m is given by, = m ( z ) m / ln m = L (4) If L is greater tan te dept of te logaritmi region ten m may be taken as equal to te dept of te boundary layer (HLR, eler, 99). Wen te ill is sinusoidal wit te oordinate lines defined by (7), te pressure perturbation tat drives te flow in te inner region is, ikx { } Re p X = U H ke, (5) were te tilde denotes a solution for a sinusoidal ill. A seond impliation of te tinness of te inner region is tat te vertial veloity perturbation is smaller tan te streamwise veloity perturbation, by a fator i L<<. Tirdly, to a leading-order approximation, te bakground wind speed in te inner region an be approximated by its value at te inner region eigt, namelyu U. Wit tese approximations te i 8

9 streamwise momentum equation (9)a witin te inner region of te flow above te anopy beomes u U τ d p U i w X Z Z dx + =. (6) Hene in te inner layer, te foring by te pressure gradient indued by te ill is balaned by streamwise aeleration and te divergene of te perturbation sear stress. Te approximate solution of (6) for a ill of sinusoidal sape is ( ( ) ( ) p u ( k, Z) = Re δ ln Z d i AK i ik Z d i ), (7) U i were δ = ln d and i K is te modified essel funtion of zero t order, wi deays at large Z, and te solution aounts for te small O δ variation in te unperturbed wind profile. Tis solution ontains an integration onstant, determined below. Te orresponding solution for te sear stress is A i, wi is p Z + d τ ( kz, ) = Re u* + Ai K. (8) U( li) i Here prime denotes differentiation wit respet to Z. Te solution for ills of more general sape is onstruted by Fourier superposition, wit te amplitude, H of te sinusoidal mode wit wavenumber k replaed by te Fourier amplitude f ( k ) of te ill, wi is defined by, ikx f ( k) = zs ( x)e dk. Te solution for te streamwise veloity perturbation, for example, is ten reonstruted using ikx u( X, Z) = (, ) π u k Z e dk. For low ills x and X an be used interangeably in te Fourier transforms, wit errors of seond order in H L. For a roug ill, enforing te no slip ondition troug a logaritmi law of te wall leads to = 4. For flow over a anopy, te oeffiient A k is fixed by ensuring tat A i i 9

10 veloity and sear stress are bot ontinuous at te top of te anopy. Hene (7) and (8) are mated to te solutions for flow witin te anopy, wi we develop in 5 below. Te ontinuity onditions are treated in Flow witin te anopy As in te inner region above te anopy, te main effet of te ill on te flow in te anopy is to apply a varying orizontal pressure gradient. Te effets of tis pressure gradient are analyzed first by onsidering te linearized momentum equations (9) a, b witin te anopy. Teir solution determines te flow perturbations in te upper portion of te anopy. Tese linearized solutions, owever, ease to be valid deeper witin te anopy, and we sall find tat a different nonlinear balane is dominant tere. 5. Flow perturbations in te upper anopy Witin te upper part of te anopy te linear perturbation to te sear stress is determined by substituting te deomposition (8) into te mixing lengt model, now 3 using te onstant mixing lengt l = β L as for te bakground flow. Tis use of a mixing lengt model for te perturbed flow witin te anopy is justified in Appendix A. Tis proedure yields, τ = βlu u Z. (9) Te basi wind profile in te anopy is given by = U ( U Z exp β Z l so tat te arateristi sale for mean veloity in te upper anopy is U and its sale of vertial variation is l. Hene vertial variations of te flow, inluding te vertial derivatives, sale on l, wereas te streamwise variations and ene te orizontal derivatives, ontinue to sale on L. Te dynamis in te upper anopy an ten be simplified onsiderably beausel << L. Firstly, te ontinuity equation (9) sows tat te vertial veloity perturbation w l β L u. Seondly, te vertial momentum equation (9)b sows tat te vertial ~ variation of te pressure perturbation indued by te ill may be ignored at first order, just as in te inner region of te flow above te anopy. Tirdly, te magnitude of te advetion term in te streamwise momentum equation (9)a an be ompared to te magnitude of te sear stress gradient: U u X U u L l ~ ~ <<, β U u Z U u l L sowing tat advetion is mu smaller tan te sear stress gradient in te upper anopy. Employing tese tree simplifiations, te linearized streamwise momentum equation in te upper anopy beomes )

11 u u U u d p β + l = Z Z L d βu X () Hene, te streamwise pressure gradient indued by te flow over te ill is balaned by a ombination of vertial momentum transport by te turbulent sear stress and a loss of momentum to te anopy drag. Te solution of () is βz l βz l L d p u = U A( X) e + ( X) e e U dx βz l. () HereU is te saling for te streamwise veloity perturbation witin te anopy given by U U HL d p dx = = O. () U U L U L Hene te linearized solution () is valid provided tat te perturbation veloity is smaller tan te veloity in te basi state, i.e. U U <<. Tis ondition ensures tat te fore due to te pressure gradient indued by te ill is mu smaller tan te drag in te anopy due to te basi state. Equation () sows tat tis ondition is satisfied provided te ill as suffiiently low slope, H L and tat te adjustment lengt of te anopy flow, is small ompared wit te lengt of te ill. L Te first and seond terms in () are solutions to te omogeneous equation and ave oeffiients (X tat are determined by boundary onditions at te top A ) and ( X ) and bottom of te anopy. Tese terms represents te response of te anopy to turbulent transfer of momentum from te flow above. Te tird term in () is te response in te anopy to te varying pressure gradient indued by te ill. Te sear stress perturbation is alulated by substituting te veloity perturbation () into (9) to give βzl βzl L d p τ = β U U( Ae e ) + U dx (3) Deep witin te anopy, as β Z l, te seond term beomes exponentially large. Tis makes no pysial sense, and so we set = to keep te perturbation to te sear stress finite deep witin te anopy.

12 For a sinusoidal ill te pressure perturbation is given by (5) and so te veloity and stress perturbations in te upper anopy an be written { ( βz l ) } βz l i u = Re U Ae + ie e kx, (4) { } β β Z l i τ = Re UU Ae i e kx, (5) were ill. U U H k L U and as before te tilde indiates a solution for a sinusoidal = 5. reakdown of te upper anopy solution Te solution obtained above for te orizontal veloity perturbation must be added to te undisturbed wind profile to obtain te total wind profile in te anopy, namely, u U u U e U Ae e L d p βzl βzl βzl = + = +. (6) U dx Hene, deep witin te anopy, were β Z l, u inreases beause of te tird term on te rigt and side, but te undisturbed wind speed dereases. At some dept, denoted ere for breakdown dept, tese two terms beome of te same order of magnitude and, below Zb, u an no longer be onsidered to be a small perturbation. Tis dept is determined by, βzb l β Z b l Ue Ue so tat Z l= ln ( U U ) (7) Z b b Tis argument sows tat te linear solutions obtained above in (5.), on te ondition tat U U, are valid only witin te upper anopy. Furtermore, wen << U U <<, ten Z l = ln U U >> and tere is formally a distint asymptoti layer, Z Z = O(), were different dynamis dominate. Tese are desribed next. b b 5.3 Flow perturbations deep witin te anopy Deep witin te anopy, wat fores balane te orizontal pressure gradient assoiated wit te flow over te ill? Te argument given above sows tat for Z >> Z te dynamial balane must be nonlinear beause te bakground wind, around wi we linearized in te upper anopy, as beome exponentially small. Hene te full nonlinear form of te anopy drag must be retained in te momentum budget. Wind speeds deep in te anopy are expeted to be even smaller tan in te upper anopy and so we expet advetion to remain small. Wat of te role of te stress gradient deep witin te anopy? b

13 Te stress gradient in te undisturbed flow dτ dz balanes te anopy drag, U L at all depts and ene needs to be onsidered deep in te anopy. Conversely, te solution for te perturbation stress (3) tends to a onstant deep in te anopy, and ene te perturbation stress gradient tends to zero. Tese arguments suggest tat deep witin te anopy te streamwise perturbation pressure gradient is balaned by te portion of te anopy drag not supported by te undisturbed stress, i.e. uu U d p L L dx =. (8) Te modulus sign arises beause te anopy drag is always direted against te loal flow diretion. Tis is an algebrai equation and its solution is, ( dx) / sgn u= U Ld p dx U Ld p, (9) were sgn( x ) = if x > and sgn( x ) = if x <. Towards te upper part of te anopy, were te term in te modulus signs is positive, tis solution an be approximated tus, Ld p dx βz l Ld p dx βz l βz l e U U u = U e U e = U U (3) were we ave used (4) for te undisturbed veloity profile. Tis is preisely te same as te solution (6) wen β Z l. Hene te solutions mat in a formal matematial way, wi gives us onfidene tat no important pysis as been missed in making te approximations. Te ondition for te nonlinearity to ave deayed to small values in te upper anopy is tat te tird term in te binomial expansion in (3) sould be small ompared wit te seond. Te retained term is four times bigger tan te negleted term if d p dx L H k UL = <, (3) U U wi provides a quantitative limit on tis ondition of te analysis. Suffiiently deep into te anopy U Z may beome so small tat U L d p dx beomes zero and ten negative on te lee side of te ill. Te solution obtained ere ten implies reversed flow and te appearane of a separation region. Te meanism of separation in te anopy is as follows. eause advetion is small, flow in te positive X diretion in te lee of te ill is maintained by te turbulent transfer of momentum from te faster moving flow in te upper anopy and opposed by anopy drag. Te 3

14 perturbation pressure gradient passes undiminised troug te anopy but te bakground flow deep in te anopy is strongly diminised by anopy drag. Similarly, downward transport of momentum by te perturbation sear stress gradient is strongly damped. As a result, if te anopy is suffiiently deep, we always enounter a region were te adverse pressure gradient in te lee an exeed te proesses ating to maintain te flow in te positive X diretion. For a sinusoidal range of ills and valleys, wen maximum positive value of d p dx = U ( H ) = π d p dx = U H k kx te sin k ours at kx alf way down te lee slope. Zero wind speed and ene te onset of separation ten ours at a dept into te anopy given by, Z s l U Zs = ln ( H ) k L (3) β U Hene, if te anopy dept,, exeeds ten we expet a region of reversed flow deep witin te anopy. Te flow witin te anopy as separated, wereas te flow above te anopy remains attaed. Even on ills of arbitrarily low slope, H L, te flow witin te anopy will separate if Z s. Te onset of separation ours wen = Zs. For anopies tat are deeper tan Z s Z s tere are separation and re-attament points at kx ( β l) U exp = sin U H k L. (33) As te anopy beomes very deep so tat ex p( β l) beomes small, or te slope of te ill beomes large, ten te separation and re-attament points tend respetively to X = and X = π, i.e. te rest and troug of te ill. Te separation region ten spans te entire region of adverse pressure gradient. 5.5 A uniformly valid solution for te anopy flow Te detailed analysis as sown tat te flow in te anopy separates naturally into two distint layers. However for pratial purposes it is useful to ave a single, uniformly valid expression for te wind profile trougout te anopy. Te solution (6) for u in te upper anopy is omposed of two parts. Firstly, a part tat deays exponentially into te anopy and is driven by te downward turbulent transfer of momentum from te faster flow above te anopy. And seondly a response to te streamwise pressure gradient assoiated wit te flow over te ill. Te solution deep into te anopy is a nonlinear response to te pressure gradient assoiated wit te flow over te ill. Hene a uniformly valid solution for te total streamwise veloity (basi plus perturbation) is 4

15 onstruted by adding te first part of te upper anopy solution () to te nonlinear solution in te deep anopy (9), to yield, / Z / l = + = sgn + (34) u U u U L d p dx U L d p dx AU e β Te vertial veloity and stream funtion tat arise from tis solution are alulated in Appendix. Te alulation of te vertial veloity naturally brings te dept of te anopy, into te solution and leads to a ondition on for te model developed ere to be valid. Tis ondition an be obtained by alulating w at te rest of a sinusoidal ill, wi is Z dz ll d p dx βl βzl w( X = ) = L d P dx = ( e e ). (35) U βu Te signifiane is tat te vertial veloity inreases exponentially towards te top of anopy at te rest of te ill. Tis large vertial veloity is te result of onvergene, u X <, trougout te dept of te anopy at te rest. Hene as te dept inreases so must te vertial veloity to remove te mass. Now in te upper anopy advetion, inluding vertial advetion, was assumed small ompared to te pressure gradient indued by te ill. Hene, te analysis of te upper anopy requires wdu dz << d p dx kl e β << (36) l A Te oeffiient in (34) remains to be determined by mating to te solution for te flow above te anopy. Tis will be done in 6 below. 6. Coupling at te top of te anopy Te unknown oeffiients remaining in te solutions are now determined. Te solutions for te veloity and stress in te flow above te anopy (7) and (8) are equated to te expressions for te veloity and stress in te anopy (4) and (5) at te top of te anopy Z =. Tis proedure yields te oupling oeffiients: A ( k) A k i = = d i { δ ( ln i ) } δ{ } K iskl + d K is kl d ( δ ) SkL K SK { δ ( ln i ) } is kl S d i, (37) d. (38) K SK δ i 5

16 S U U is te sear aross te inner region, = ( i ) ( Here = ( i ) = i K K ikd and K K ikd ), were primes denote differentiation wit respet to te argument of te funtion. Tese general results an be understood in a number of ways. Firstly, onsider te limit of a roug ill were momentum absorption would be parameterized onventionally using a onstant rougness lengt. A roug ill is usually onsidered to ave a small displaement eigt, d, weneδ K, d K and δ i i d A ( k) A k i + SkL i ln. Te expressions (37) and (38) ten beome, iskl is kl 4S + S + ( + S ) is k, (39) L, (4) wene veloity and stress in te upper anopy beome, aku iskl + is kl βzl βzl ikx u Re e U( i) = + iskl e e, + S (4) aku iskl β + is kl Zl ikx τ = Re u* e is kl e. U( i) + S (4) eler et al () sow tat te lengt sale araterises te fet required for flow witin a anopy to adjust, ene kl measures te lengt sale for te variation witin te anopy ompared to te lengt sale of te varying pressure gradient, wi drives te anopy flow away from loal equailibrium. Hene te roug flow limit orresponds to small kl. However, some are is needed to reover te limit of a roug ill, wi as kl, beause in te limits just set up te sear aross te inner region, S, ten also beomes large. Te distinguised limit is for kl and, but wit SkL remaining finite. ut ten L S i 4 A k iskl. (43) Tis result is suprising: te limit of a roug ill does not yield te result obtained by HLR, namely A k, wen a onstant surfae rougness is assumed! Pysially tis i 4 signals tat te anopy does not tend to a passive onstant rougness lengt in te aelerating and deelerating flow over a ill. In tis way te present analysis asts into doubt te very onept of using a rougness lengt in non-uniform turbulent boundary 6

17 layer flows. Te residual imaginary term in (43) moves te position of maximum speedup upwind as a response of te flow above te anopy to te flow witin te anopy. In addition to tis formal objetion to te onstant rougness lengt, te limit proess to (43) onverges extremely slowly. For any pratial values of d<< i, te required d limiting values ofδ K, K are not aieved. Instead, tese variables ave i imaginary parts of te same order of magnitude as teir real parts. So in te plots, sown below, te oeffiients are omputed from (37) and (38). Te forms of te oupling onstants elp larify te role of te upper anopy layer. It ats as te region of ommuniation between te flow witin te anopy, wi is largely in pase wit te perturbation pressure gradient assoiated wit te flow over te ill, and te flow above te anopy, wi is largely in pase wit te pressure perturbation. Te dimensionless anopy parameter kl measures ow rapidly te anopy flow omes into equilibrium wit te pressure foring. For very small kl te anopy flow losely follows te flow in te inner region above te anopy, wi itself is not mu affeted by te anopy. For larger kl te anopy flow more strongly anges te flow in te inner region above. 7. Results Te solutions above te anopy desribed in 4, sow tat te veloity perturbations are driven by te pressure perturbation p U, so tat we define a saling veloity, SC H L ~ H L U = U. Results are presented in tree ways. Firstly, te general arater of te solution for uusc is sown by fixing te slope of te ill, H L, and te anopy density L L, and ten omparing wit te solution obtained for a roug ill wit te same value of. Seondly, te variation of te veloity profiles wit anopy density, z measured by L L, is investigated. Finally, te streamline pattern is sown for different anopy depts,. 7. Comparison of te anopy and roug ill ase Consider a sinusoidal ill wit L= m, H L=., L = m, β =.3 and u =.. Wit tese oies, l =.54m, z =.35m and d =.34m, and te inner layer dept is li m. Tese parameters yield U U. and ll.54, wi satisfies te onditions of te analysis. Finally te dept of te anopy is limited by te ondition (36), wi yields < 6m. Plots are sown tat do not satisfy it in order to illustrate te signifiant features of te flow. It would be interesting to develop numerial solutions to establis weter te flow anges substantially from te analytial form wen tis limit on is exeeded. 7

18 Figure sows vertial profiles of te normalized veloity perturbations uu SC alulated from te solutions for a ill wit te anopy and te solutions for a roug ill at a series of X positions starting at te upstream troug, X = L. Reall tat solutions for te roug ill extend down to only Z = d + z. Firstly, for te roug ill tere is a distint peak in uu SC around Z = i 3, wereas wen a anopy is present te peak is less pronouned, lies around Z = i, and its magnitude is redued by about %. Seondly, te veloity perturbation deep into te anopy is zero at te peak and troug of te ill, and reaes maximum values around X =± L, but wit pronouned asymmetry upwind and downwind of te rests, due to te nonlinear solution. Finally, tere are sarp gradients in te veloity profiles in te upper anopy in te lee of te ill rest. Tese gradients also arise from te nonlinear solution in te lower anopy (9), wi for tis oie of parameters is substantial relative to te upper anopy solution. Figure 3 sows te same omparison in te form of streamwise variation of uu SC at a series of Z values. To obtain te maximum information it is useful to view tese plots in onjuntion wit te vertial profiles of Figure. Te first four panels of Figure 3 span te range d > Z > L, wi is in te lower part of te anopy, were U is small. Here te perturbation veloity losely follows ( sin kx sgn sin kx ), were k = π L (9). Tere are te peaks in uu SC at X =± L and steep streamwise gradients at X =, ± L. In te upper anopy, sown in te fourt panel (were Z = d ) and te fift panel (were Z = l ) of Figure 3, U beomes larger and te profiles beome / asymmetri about X =. In tis upper part of te anopy te linear term AU eβ Z l, wi is assoiated wit oupling wit te inner region flow, also moves te peak in downwind of te ill rest. Rigt at te top of te anopy, Z =, te variation of uu s uu s is nearly linear and follows a sinusoidal variation wit X. Tis sinusoidal variation persists troug te inner region. Te veloity perturbation uus is largest furter upstream of te rest tan te roug ill solution in te lower part of te inner region. y te top of te inner region, Z i = 9., te flow perturbations are determined largely by invisid dynamis and so te solutions obtained wit te anopy are almost idential to te roug ill solution. Figure 4 sows vertial profiles of dimensionless total veloity U u U + s. In tese panels it is lear tat above te anopy te differene between te anopy and no-anopy total veloities is small but signifiant in te lower part of te inner layer but tat witin te anopy, te pressure driven, non-linear part of te anopy perturbation dominates. In partiular, we an observe te rapid onset of reversed flow in te anopy between X = and X = L as we would expet from te streamwise profiles sown in Figure 3. Quantative omparison of tis model wit te measurements of Finnigan and runet (995) (F95) is not appropriate as teir wind tunnel model ill was steep and flow in te lee was ompletely separated bot witin and above te anopy. A qualitative 8

19 omparison of te flow upwind of te ill rest and results from te urrent model, owever, sows lose agreement. Te measured profiles of F95 displayed maximum positive veloity perturbations deep in te anopy lose to X = L wile aeleration above te anopy was weak, te disrepany being suffiient to remove te arateristi anopy-top infletion point in te mean veloity profile at tis X position. Over te ill top, in ontrast, te F95 deep anopy perturbation was near zero and aeleration above te anopy was maximal leading to very strong sear (and a large peak in sear stress). In Figure 4 we see te same ontrast between te profiles of total veloity at tese two X positions. Figure 5 sows te effet of varying te anopy density aross te range L L=.,.,.5. As in te earlier figures te plot range is i > Z > L. As anopy density inreases, firstly, te uniform eigt-independent flow in te lower anopy is aieved at progressively sallower depts. Seondly, te overall magnitude of te anopy flow is redued relative to te flow above te anopy. And tirdly, te flow in te inner region obtained wit te anopy model beome loser to te roug-ill solution. Tese findings support te interpretation tat L Lontrols te extent to wi te dynamis witin te anopy flow affet te flow in te inner region above. Sine our anopy parameterization sould be valid for any aerodynamially roug surfae overed wit bluff objets, were all te momentum is absorbed as form drag on te rougness elements, in Figure 6 we ave plotted a omparison of te anopy and noanopy ase for a very dense anopy were Ca = m L = m, wene ;.5 L L=.5, z =.8m and d i ~.. Te oter ill parameters are unanged so we are observing flow over a roug ill witout a deep anopy. It is obvious tat, wile te two solutions are lose over mu of te X range, signifiant differenes remain at te bottom of te inner layer near rest and troug. Tis, as we ave already stated, alls into te question te onept of using a onstant surfae rougness to parameterize momentum absorption on a ill. 7. Flow Streamlines and Separation. Figure 7 sows te streamlines omputed from te formula for te streamfuntion given in Appendix. Te anopy parameters are te same as for Figure and = L ~ m. Figure 7a sows te streamlines of te total flow. Tere is a reirulation assoiated wit flow separation witin te anopy flow in te lee of ill, wi indues a strong asymmetry in te flow. Figure 7b sows te streamlines of te perturbation flow to empasize two furter points. Firstly, te onvergene around te rest of te ill, disussed in 5.5, an be seen learly (it is not so evident in figure 7a wen te mean flow is added). Seondly, onentrating on te perturbation streamlines empasizes te large veloity gradients tat develop at te edges of te integration domains, and 3, defined in Appendix. If te eddy visosity model were used to represent turbulent momentum transport in tis region of te flow, ten tese large veloity gradients would render te turbulent stress gradients large witin tese narrow layers. Te effet would be to smoot sligtly te veloity fields tere, witout anging te overall flow struture. 9

20 Tis ontention is supported by numerial omputations in Katul et al, (). Hene tese orretions are not analyzed in detail ere. Finally, in Figure 8 we plot te streamlines of total and perturbation veloity for te same ill and anopy density parameters as in Figure 7 but now for a anopy wose eigt, is Z L = m so tat separation is just avoided. Here we sligtly less tan te value of.6 s see tat wile te perturbation flow reverses in te lee, te mean veloity is strong enoug to overome te negative perturbation. Neverteless, te asymmetry between te upwind and lee slope flow patterns remains marked. 8 Conlusions We ave onstruted an analyti model of atmosperi boundary-layer flow over a low ill overed by a tall anopy. Te undisturbed veloity profile U z is defined by mating a onventional logaritmi profile above te anopy to an exponential profile witin te anopy. Tis veloity profile flows over a ill tat as a suffiiently low slope, H L<<, tat flow perturbations indued by te ill above te anopy an be analyzed wit te linear analytial framework developed for flow over roug ills (HLR; eler, 99; eler et al, 993). Te model for te flow witin te anopy is based on: (a) a linearisation ondition, U U << ; (b) negleting orizontal advetion in te upper anopy momentum balane, ll<< ; () negleting vertial advetion in te upper anopy, wi provides a restrition on te anopy eigt, l kl eβ <<. Perturbations to te flow witin te anopy are driven by te streamwise pressure gradient tat is set up as te flow above te anopy goes up and over te ill. To a leading approximation, tis pressure gradient remains onstant rigt down troug te anopy. Te flow in te upper portion of te anopy an ten be analyzed wit linear equations and te pressure gradient aused by te ill is balane by vertial transport of momentum, by te turbulent sear stress and by te aerodynami drag on anopy elements. Deep witin te anopy, Z < Zb solutions to te linearized, upper-anopy momentum balane ease to be valid beause te perturbations beome of te same order as te exponentially deaying bakground wind. elow Z b a nonlinear dynamial balane is struk between te pressure gradient assoiated wit te flow over te ill and te nonlinear form of te anopy drag. A osine saped ill, z s = Hos kx leads to a pressure gradient tat varies to leading order in anti-pase wit elevation, p os kx. Te veloity perturbation above te anopy also varies wit te elevation u os kx. Deep witin te anopy, owever, u sin kx sgn(sin k X) so tat te veloity perturbation attains its largest positive value upwind of te rest at X = L, is zero at te ill rest and as its largest negative value alfway down te lee slope at X = L. Hene tere is a pase sift of π between u deep into te anopy and u above te anopy. Turbulent transfer of momentum

21 witin te upper anopy region and te inner region above te anopy smootes tis pase sift. Te result is tat te peak in speed-up in te inner region is furter forward of te ill rest and diffused to iger levels ompared wit flow over a roug ill. Te pysial reason for tis is te extra turbulent mixing tat is produed in te anopy model beause te mixing lengt l attains a onstant value at te anopy top instead of ontinuing to derease to Z = d + z as in te HLR roug wall ase. If te anopy is deeper tan a ritial value of Z s ten tere is a region of reversed flow on te lee slope. Te reversed flow is driven by te adverse pressure gradient in te lee of te ill, wi is balaned deep witin te anopy by te nonlinear drag. Tis explains te observation in rown et al () tat numerial simulations of flow over ills overed wit a anopy seem to separate more readily tat ills wit roug surfaes. We migt expet separation above te anopy to appear wen Zs. Tis possibility and te relation to previous studies of separation from roug ills (e.g. Wood 995) deserve furter study. Te asymmetry of te non-linear solution is likely to ave two important onsequenes. Firstly, te inreased displaement of te streamlines from te surfae on te lee side of te ill will inrease te magnitude of te pressure perturbation in pase wit te ill slope. Tis omponent of te pressure perturbation yields te form drag of roug ills (eler et al 993, eler 999). We speulate tat tis is te reason for te inreased drag in simulations tat resolve a anopy (e.g. rown et al ). Seondly, we suggest tat te asymmetry in te flow, and in partiular stagnation on te lee slope, promotes strong orizontal gradients of salar quantities like CO respired from te soil. Calulations by Katul et al () sow tat te resulting advetive fluxes an easily be as large as vertial eddy fluxes above forests on even moderate topograpy. Tis finding as important impliations for te interpretation of CO fluxes measured at towers in te FLUXNET Program (aldoi et al, ), wi usually ignore tese advetive fluxes as tey annot be measured from a single tower. Te estimates of surfae-atmospere exange of arbon dioxide need to be orreted for te advetive fluxes if tey are not to be in serious error. Finally, we migt expet in te limit of a dense anopy L L te solutions tend to te solutions for flow over a roug ill derived by HLR. Firstly, te onvergene to te limit is exeedingly slow, and for reasonable values of te parameters te anopy ontinues to play a dynamial role in te flow above te anopy. Seondly, and more fundamentally, te solution does not attain te roug-ill solution in te matematial limit L L. Instead a ontribution to te flow from te anopy dynamis remains in te flow above. Tis signals tat te anopy does not tend to a passive onstant rougness lengt in te aelerating and deelerating flow over a ill and asts into doubt te very onept of using a rougness lengt in non-uniform turbulent boundary layer flows. Aknowledgements

22 JJF gratefully aknowledges te support of te Department of Meteorology, University of Reading, te Met Offie and te NERC under grant GR3/545. SE gratefully aknowledges te support of Division of Environmental Meanis, CSIRO, and te NERC under grants GR3/545 and GST//3. Appendix A: Te Turbulene Closure Tere is onsiderable evidene tat turbulent transfer in plant anopies as a non-loal arater tat is not aptured by eddy diffusivity parameterizations of te turbulent flux. Tis evidene inludes ounter-gradient transport of salars in regions of te anopy (Denmead and radley, 985) and seondary maxima or jets in mean wind speed profiles in anopies wit open trunk spaes (Wilson and Saw, 977). Su penomena would require te diffusivity to be loally negative. udgets of seond moment quantities like sear stress, uw, turbulent kineti energy, uu i i and salar fluxes all sow te importane of transport terms in exporting turbulene moments from te region of strongest prodution at te top of te anopy to te lower anopy spae (runet et al, 994; Finnigan, ). All tese penomena are manifestations of te fat tat te size of te dominant eddies in te anopy and rougness sublayer (te region of te surfae layer between te ground and about two anopy eigts) are typially of te same order as te anopy eigt, and so omparable to te sale of anges in gradients of mean quantities in tis layer (Finnigan and Saw, ). In tese irumstanes we do not expet to be able to desribe eddy fluxes in terms of eddy diffusivities and mean gradients (Corsin, 974; Finnigan 985). However, we an onstrut a simple non-loal losure, suitable for rougness sublayer flow as follows. A. Seond order losure Te single-point seond moment equations for general anopy flow are given for salars by Finnigan (984) and for veloity moments by runet et al (994). We are interested ere in te searing stress so we onentrate on te equation for uw, Du w U w u = ww uww + pu + p + p Dt z z x z (A) A series of simplifiations ave been made in writing (A). First we ave ignored te terms involving visosity, wi are small in te ovariane equations. Seond, for simpliity, we ave used an overbar to represent bot te time average operator and te volume averaging over a tin orizontal slab tat is impliit in anopy flow equations written in ontinuum form. Tird we ave ignored te dispersive ontributions to te tird moment terms tat result from te volume averaged orrelations between steady departures from te volume averaged mean. Finally we ave assumed tat prodution of stress by diret interation wit te waving foliage is negligible. Fuller disussion of all tese simplifiations an be found in runet et al (994), Ayotte et al (999) and Finnigan () and referenes terein.

23 We ave also written te rigt and side of te equation in one-dimensional form making te impliit assumption tat terms involving vertial gradients of mean quantities will dominate tose ontaining orizontal gradients. See te saling arguments in 5. Te tree terms on te RHS of (A) are usually alled: sear prodution, turbulent and pressure transport and pressure-strain interation, respetively. Pressure-strain interation is te main sink term for te ovariane, uw. A standard set of parameterizations for te tird moment expressions in terms of te first and seond moments ave been proposed by Launder (99). Tese expressions represent te tird moments tat appear in te transport term, uww + pu by an effetive diffusivity multiplied by te gradient of uw wile te pressure-strain terms are split into rapid and return to isotropy parts Launder (99). Ayotte et al (999) sowed tat tese parameterizations worked satisfatorily in plant anopies witout altering te values of Launder s oeffiients so long as te expression for kineti energy dissipation tat appears in te parameterizations is adjusted to reflet anopy dynamis Finnigan (). Te parameterized equation an be written, uw uw U A( z) A z uw K z z z z were, A z + =, = S T A z = A z K z = l ( ) S lq (A) l and q are eigt dependent lengt and veloity sales, respetively and,,, are O onstants. Te values of tese onstants are given in Ayotte et al S T () (999). Te first two terms on te LHS of (A) result from te parameterization of te transport term so tat, wen tis is negligible, te sear stress may be represented by an eddy visosity K(z). Te mixing layer analogy (Raupa et al, 996) maintains tat te prodution and arater of turbulene in te rougness sublayer is very similar to tat in a plane mixing layer. A key result is tat te dominant eddies in anopy flow are araterized by single lengt and veloity sales, wi are invariant troug te rougness sublayer. If we oose te square root of te turbulent kineti energy at te top of te anopy as te veloity sale q, 3