Acta Geophysica vol. 57, o. 4, pp. 98- DOI:.478/s6-9-7-4 Structure of Numerically Simulated Katabatic ad Aabatic Flows alog Steep Slopes Evgei FEDOROVICH ad Ala SHAPIRO School of Meteorology, Uiversity of Oklahoma, Norma, USA e-mails: fedorovich@ou.edu (correspodig author), ashapiro@ou.edu Abstract Direct umerical simulatio (DNS) is applied to ivestigate properties of katabatic ad aabatic flows alog thermally perturbed (i terms of surface buoyacy flux) slopig surfaces i the absece of rotatio. Numerical experimets are coducted for homogeeous surface forcigs over ifiite plaar slopes. The simulated flows are the turbulet aalogs of the Pradtl (94) oe-dimesioal lamiar slope flow. The simulated flows achieve quasi-steady periodic regimes at large times, with turbulet fluctuatios beig modified by persistet low-frequecy oscillatory motios with frequecy equal to the product of the ambiet buoyacy frequecy ad the sie of the slope agle. These oscillatory wave-type motios result from iteractios betwee turbulece ad ambiet stable stratificatio despite the temporal costacy of the surface buoyat forcig. The structure of the mea-flow fields ad turbulece statistics i simulated slope flows is aalyzed. A itegral dyamic similarity costrait for steady slope/wall flows forced by surface buoyacy flux is derived ad quatitatively verified agaist the DNS data. Key words: katabatic flow, aabatic flow, umerical simulatio, boudary layer, turbulece.. INTRODUCTION Slope wids (flows) are typical for vast areas of the Earth, ad ofte play a importat role i the weather ad climate of these areas. From the stadpoit of basic fluid dyamics, the slope wids are buoyatly drive boudary- 9 Istitute of Geophysics, Polish Academy of Scieces
98 E. FEDOROVICH ad A. SHAPIRO layer-type flows alog heated or cooled slopig surfaces i a stratified fluid. Commoly, meteorologists distiguish betwee aabatic wids, which are drive by surface heatig, ad katabatic wids, which result from surface coolig. From the viewpoit of formal descriptio ad modelig, slope flows are challegig physical pheomea because they coflate three characteristic aspects of geophysical fluid dyamics: buoyat forcig, stratificatio, ad turbulece. Although much progress has bee made i the coceptual uderstadig ad umerical simulatio of the slope wids, there are still may ope questios regardig the structure ad properties of these flows. Of particular iterest for practical applicatios are the mea-flow ad turbulece structure of slope wids as fuctios of the surface thermal forcig ad slope agle. Kowledge of these structural features of slope flows could be useful for desigig parameterizatios of slope-wid pheomea i atmospheric models. I areas where basis are largely sheltered from syoptic effects, katabatic ad aabatic flows are the buildig blocks of local weather. Eve i cases where syoptic forcig is importat, proouced katabatic ad aabatic flow sigals may still be apparet. I regios where heavily idustrialized populatio ceters exted across variable topography (e.g., Los Ageles ad Phoeix i the USA), these local flows exert major cotrols over eergy usage, visibility, fog formatio, ad air pollutat dispersio (Lu ad Turco 994, Ferado et al., Hut et al. 3, Lee et al. 3, ad Brazel et al. 5). I agricultural regios, these local wids sigificatly affect microclimates. They also eed to be take ito accout i aerial-sprayig ad fire-fightig operatios. O the larger scale, persistet katabatic wids cover vast areas of the Earth (e.g., Greelad, Atarctica), ad play a importat role i the weather ad climate of these areas (Parish ad Waight 987, Gallée ad Schayes 994, Oerlemas 998, Refrew 4). A early milestoe i the coceptual uderstadig of katabatic/aabatic flows was the developmet of the Pradtl (94) oe-dimesioal model for the lamiar atural-covectio flow of a viscous stably-stratified fluid alog a uiformly cooled or heated slopig plaar surface. Flow i the model has a boudary-layer character (low-level jet topped by weak reversed flow). The model solutio, which is exact withi the Boussiesq framework, satisfies coditios for both mechaical ad thermodyamical equilibrium. The alog-slope advectio of evirometal (mea) temperature balaces thermal diffusio, ad the alog-slope compoet of buoyacy balaces diffusio of alog-slope mometum. All other terms i the equatios of motio ad thermodyamic eergy are idetically zero. Observatios suggest that, with appropriately tued mixig parameters, this simple model provides a good descriptio of the vertical structure of slope flows at ight ad a reasoable approximatio of slope flows durig the day (e.g., Defat 949, Ty-
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 983 so 968, Papadopoulos et al. 997, ad Oerlemas 998). With a suitable chage of variables, the Pradtl model also describes the alog-slope flow ad perturbatio saliity field i a oceaic mixig layer at a slopig sidewall (Phillips 97, Wusch 97, ad Peacock et al. 4). I this oceaic cotext, the flow is geerated soleoidally by isopycals that are forced to approach the slopig boudary at a right agle (zero ormal flux coditio). The model also describes the free covective flow of a stratified fluid alog heated vertical plates (Gill 966, Elder 965, ad Shapiro ad Fedorovich 4a,b), ad the familiar Ekma (95) flow of a homogeeous viscous rotatig fluid i the presece of a imposed wid stress or pressure gradiet force (Batchelor 967). The equivalece of the classical Pradtl ad Ekma models is a remarkable maifestatio of the geeral aalogy betwee stratified ad rotatig flows (Verois 97). Advaces i computer techology have made possible umerical mesoscale modelig (Parish 984, Parish ad Waight 987, Nappo ad Rao 987, Gallée ad Schayes 994, Atkiso 98, Bromwich et al., Klei et al., Heiema ad Klei, Rampaelli et al. 4, Refrew 4) of slope wids ad eve umerical large eddy simulatio (LES) of particular slope flow cases (see, e.g. Schuma 99 ad Skylligstad 3). Of particular iterest for our research are results of those umerical studies that aalyzed simulated slope flow dyamics i the cotext of coceptual models like the exteded Pradtl model or the hydraulic model of Ball (956). Nappo ad Rao (987), who applied a e-l turbulece closure to umerically study the spatial ad temporal evolutio of katabatic flows alog fiitelegth slopes, foud that ambiet stratificatio profoudly affects the structure of the flow ad, i the case of sufficietly log slope, ultimately results i a oe-dimesioal Pradtl-like flow regime. Heiema ad Klei (), Rampaelli et al. (4), ad Refrew (4) used three differet state-ofthe-art umerical model systems to reproduce idealized slope/valley wids with a remarkable degree of detail. The umerical results of Heiema ad Klei () for Greelad katabatic wids show that flow patters above vast portios of this islad may be reasoably cosidered as twodimesioal, ad allow iterpretatio i terms of Ball s theory. Rampaelli et al. (4) looked at drivig mechaisms of umerically reproduced upslope wids ad foud that basic features of observed flows i idealized twodimesioal settigs are similar, as least qualitatively, to predictios of the Pradtl model. Refrew s (4) umerical study of Atarctic katabatic wids supported by aalyses of field measuremets also revealed a quasitwo-dimesioal structure of the large-scale katabatic flow ad poited to the triggerig of iteral gravity waves which propagate eergy away from the regio of strogly deceleratig flow. The theoretical aalysis i Shapiro ad Fedorovich (7) also idicated that iteral gravity waves would de-
984 E. FEDOROVICH ad A. SHAPIRO velop i deceleratig katabatic flows. All op. cit. umerical studies reported difficulties i fidig a appropriate parameterizatio for the ear-surface portio of the flow, which is either subject to strog buoyacy dampig of turbulece i the case of katabatic flow or to absolute static istability iducig covective motios above a heated surface i the case of aabatic (upslope) wid. The above-metioed iheret shortcomig of the mesoscale modelig approach may be overcome, at least partially, withi the LES framework. The LES has the potetial to cover the gap betwee the mesoscale features of slope wids ad the resolved, i a LES sese, turbulet motios that directly impact the dyamics of these flows. By resolvig most of the eergycarryig motios, LES accouts for slope-flow turbulece effects i a much more accurate ad cosistet maer tha do mesoscale models. However, this realism comes at a high computatioal cost. Nevertheless, reported LES studies of aabatic (Schuma 99) ad katabatic (Skylligstad 3) wids show that for small-scale (of the order of to km i the horizotal) slope flows, the LES method appears to be practicable. I the study of Schuma (99), LES was used to ivestigate the turbulet steady-state aabatic boudary layer alog a uiformly heated, ifiitely log iclied (i the limitig case vertical) plate immersed i a stably stratified fluid. This setup permitted adoptio of a quasi-homogeeity flow costrait i the alog-plate directio, which made it much more tractable i the LES sese ad permitted compariso with the classical Pradtl model. I fact, Schuma (99) foud that for small- ad moderate-agle slope flows, fudametal predictios of the Pradtl theory are supported fairly well by the umerical results, apart from structural peculiarities of the flow. Amog iterestig fidigs of Schuma s (99) study was the oscillatory behavior of mometum ad temperature solutios, which was quite persistet ad especially oticeable i the flow cases with large plate icliatio agles. The frequecy of these oscillatios was N siα (N beig the buoyacy frequecy of the ambiet fluid ad α the plate icliatio agle). Seekig to attai steady-state solutios for the simulated flow, Schuma (99) forced dampig of these oscillatios by applyig a specially desiged solutio relaxatio algorithm. He did ot, however, preset a argumet for a steadystate solutio beig the oly possible termial state of the flow. Adaptig a LES algorithm from oceaic studies to simulate spatially evolvig atmospheric katabatic flow, Skylligstad (3) obtaied isights ito the structure of mea fields ad turbulece characteristics i a threedimesioal flow settig without Coriolis effects or ambiet stratificatio. I these simulatios the slopes were relatively short ad the flow was developig dow slope from a state of rest. Accordigly, the quasi-homogeeous
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 985 flow (Pradtl-like) regime i the alog-slope directio was ot achieved i the simulatios. The LES results suggested the importace of turbulece i cotrollig the stregth ad depth of katabatic flows, ad revealed the shallowess of these flows whe ambiet wids were light. However, the subgrid closure used i Skylligstad s (3) study may ot be optimal for the reproductio of stably stratified flow turbulece, so the reported results regardig the turbulet structure should be viewed with cautio. I the preset study, the structural features of aabatic ad katabatic flows are ivestigated usig a updated versio of the direct umerical simulatio (DNS) code previously employed to study buoyatly drive flows alog vertical plates ad iclied surfaces with costat ad time-depedet surface thermal perturbatios (Shapiro ad Fedorovich 4b, 5, 6, 8, Fedorovich ad Shapiro 9). The DNS experimets explore prototypical physical effects i idealized dowscaled atmospheric slope flows with the goal of improvig the coceptual uderstadig of the turbulece cotributio to dyamics ad thermodyamics of katabatic ad aabatic wids. Particular emphasis is placed o studyig the depedece of the mea profiles ad turbulece statistics o the sig ad magitude of surface buoyacy forcig (i terms of surface buoyacy flux) ad the slope steepess. Due to computatioal limitatios ad loger times eeded to obtai steady statistics i the flows over shallow slopes (see Sectio 4), DNS rus are coducted oly with slopes 3º ad steeper. The layout of the paper is as follows. The basic equatios of slope flow dyamics i a doubly-ifiite slope settig ad correspodig boudary coditios are cosidered i Sectio. I Sectio 3, the Pradtl (94) model solutios for slope flows are reviewed for the origial slope-flow sceario with prescribed surface buoyacy, as well as for a flow resultig from a costat surface buoyacy flux (eergy productio rate). The umerical experimets ad the subsequet simulatio results are described i Sectio 4. Sectio 5 summarizes fidigs of the study.. EQUATIONS OF SLOPE FLOW DYNAMICS Mometum balace equatios for a small-scale (very large Rossby umber) flow i the Boussiesq approximatio are the followig: u u u u π u u u + u + v + w = + βθ si α + ν + +, t x y z x x y z v v v v v v v u v w π + + + = + ν + + t x y z y x y z, () ()
986 E. FEDOROVICH ad A. SHAPIRO w w w w π w w w + u + v + w = + βθ cos α + ν + +, t x y z z x y z with the heat balace give by θ + u θ + v θ + w θ = γ( usiα + wcos α) + ν θ θ θ h + +, t x y z x y z ad mass coservatio represeted by the cotiuity equatio for a icompressible fluid, u v w + + =. (5) x y z I the above equatios, u, v, w are velocity compoets i the right-had slope-followig Cartesia coordiate system (Fig. ) with x, y ad z beig the upslope, cross-slope, ad slope-ormal coordiates, respectively, π = [p p e (z')] / ρ r is the ormalized pressure perturbatio (p e (z') is the evirometal pressure, z' is the true vertical coordiate, ρ r = cost is the referece desity value), θ = Θ Θ e (z') is the potetial temperature perturbatio, γ = dθ e / dz' = cost is the gradiet of evirometal potetial temperature, β = g / Θ r is the buoyacy parameter (Θ r = cost is the referece potetial temperature value, g is the gravitatioal acceleratio), α is the slope agle, v is the kiematic viscosity, ad v h is the thermal diffusivity. The heat balace equatio (4) may be rewritte i terms of the buoyacy b = βθ as b b b b b b b + u + v + w = N ( usiα + wcos α) + ν h + +, t x y z x y z (3) (4) (6) where N / = ( βγ ) is the Brut Väisälä (or buoyacy) frequecy. z z ' g x α Fig.. Slope-followig coordiate system.
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 987 The lateral boudary coditios for progostic variables (u, v, w, b) ad ormalized pressure π are periodic (the slopig surface is supposed to be doubly-ifiite alog x ad y). The upper boudary coditios (large z) are ϕ / z =, where ϕ is ay of (u, v, w, b), ad π / z is obtaied from (3). The surface (z = ) coditios are o-slip ad impermeability (u = v = w = ), with π / z obtaied from (3), ad v h ( b / z) = B s, where B s is the surface buoyacy flux which also has a meaig of the surface eergy productio rate. 3. PRANDTL MODEL OF SLOPE FLOW Cosider the followig reduced versio of ()-(3) ad (6) for the case of a statioary lamiar flow parallel to the slope, which correspods to the slope flow model of Pradtl (94), though with v = v h : u bsiα + ν =, z b Nusiα + ν =, z with the followig boudary coditios: u() =, b() = b s or ν (d b/ d z) = B z = s, ad u ad b as z. The cotrollig parameters of the reduced problem are therefore α, v, N, ad either b s or B s. Itroducig geeric legth L, velocity V, ad buoyacy B scales, ad applyig these scales i eqs. (7) ad (8), we obtai the followig odimesioalized mometum ad buoyacy balace equatios of the Pradtl model: νv u b + =, (9) L siα B z ν B b u + = L N V z siα, with the boudary coditios trasformig ito u () =, ad u, b as z, ad either b () = b s /B (hereafter called the type I surface coditio; the coditio actually cosidered by Pradtl) or (d b / d z ) = = B ν L B (hereafter called the type II surface coditio). s 3. Type I surface coditio z / / / Assigig legth, velocity, ad buoyacy scales as L= ν N si α, V = b s N, ad B = b s, eqs. (9) ad () may be reduced to (7) (8) ()
988 E. FEDOROVICH ad A. SHAPIRO u b b + =, + =, u z z () / / / where z = zν N si α, u = ub s N, ad b = bb s, with the boudary coditios: u () =, b () =, ad u, b as z. This problem has the followig aalytical solutio (Shapiro ad Fedorovich 4a): u = si( z / ) exp( z / ), b = cos( z / ) exp( z / ), () whose dimesioal forms are idetical to the origial Pradtl model solutios. As demostrated i Shapiro ad Fedorovich (4a), the peak ormalized velocity, u max = (/ ) exp( π/4), occurs i this flow case at z max = π /4. Assumig values of the exteral parameters characteristic of atmospheric slope wids: v = m s, N = s, ad b s = m s, we come up with u max = 3. m s at z max = m for α = 6, at z max = 6 m for α = 3, ad at z max = 83 m for α =. Thus, with the prescribed costat surface buoyacy (surface coditio of type I), the velocity maximum i the Pradtl model is idepedet of the slope agle, but the elevatio of this maximum icreases with decreasig / slope agle as si α. u u 5...3.4 5.5 5 5 z z 5 5.5 b 3 4 b Fig.. Velocity u (i m s, solid lies) ad buoyacy b (i m s, dashed lies) profiles (z is i meters) from the Pradtl model with type I (b s = m s, left) ad type II (B s = m s 3, right) surface coditios for differet values of slope agle α: 9 (thi black lies), 6 (black lies), 3 (red lies), ad (blue lies) for v = m s, N = s.
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 989 Dimesioal u [m s ] ad b [m s ] profiles, calculated from () with v = m s, N = s, b s = m s, are show i Fig. (left plot). They illustrate the depedece of the Pradtl model solutio o the slope agle for the case of slope flow with costat surface buoyacy. 3. Type II surface coditio Redefiig the legth, velocity, ad buoyacy scales as L / / / = ν N si α, / 3/ / / / / V = ν N B s si α, B = ν N B s si α, respectively, reduces the dimesioless problem (9)-() with the surface coditio of type II to u b b + =, + =, u z z (3) / / / / 3/ / / 3/ / where z = zν N si α, u = uν N Bs si α, b = bν N Bs si α, with the boudary coditios u () =, (d b / d z ) =, ad u =, z b as z. This problem has the followig aalytical solutio (Shapiro ad Fedorovich 4a): u = si( z / )exp( z / ), b = cos( z / )exp( z / ). (4) I this case, as i the case of type I surface boudary coditio, the peak ormalized velocity also occurs at zmax = π /4, but ow it has a differet magitude of u max = exp( π / 4). Assumig typical atmospheric values of exteral parameters: v = m s, N = s, ad B s = m s 3, we fid u max = 4.9 m s at z max = m for α = 6, u max = 6.5 m s at z max = 6 m for α = 3, ad u max = 35 m s at z max = 83 m for α =. Thus, with the prescribed costat surface buoyacy flux, both the velocity maximum ad the elevatio at which the maximum occurs icrease with decreasig slope agle / as si α. Dimesioal u [m s ] ad b [m s ] profiles i Fig. (right plot), calculated from (4) with v = m s, N = s, ad B s = m s 3, illustrate the depedece of the Pradtl model solutio o the slope agle for the case of slope flow drive by a costat surface buoyacy flux. 4. NUMERICAL SIMULATION OF TURBULENT SLOPE FLOWS I the preset study, idealized turbulet aabatic ad katabatic flows alog double-ifiite slopes are ivestigated usig direct umerical simulatio (DNS), which implies resolvig all scales of turbulet motio dow to the viscous dissipatio scale. The umerical algorithm employed to directly solve ()-(3), (5) ad (6) with Pr = v/v h = is geerally the same as the oe
99 E. FEDOROVICH ad A. SHAPIRO used to reproduce lamiar (Shapiro ad Fedorovich 4b, 6) ad turbulet (Fedorovich ad Shapiro 9) covectio flows of a stably stratified fluid alog a heated vertical plate, which may be cosidered a case of a ultimately steep heated slopig surface, ad turbulet katabatic flows alog a homogeeously cooled slopig surface (Shapiro ad Fedorovich 8). I the curret versio of the umerical code, the time advacemet is performed with a hybrid leapfrog/adams-moulto third-order scheme (Shchepetki ad McWilliams 998). The spatial derivatives are approximated by secod-order fiite-differece expressios o a staggered grid. The Poisso equatio for pressure is solved with a fast Fourier trasform techique over the x-y plaes ad a tri-diagoal matrix iversio method i the slope-ormal directio. No-slip ad impermeability coditios are applied o the velocity field at the slope surface. Equatio (3) is used to formulate a Neuma boudary coditio for the pressure at the surface ad at the outer boudary of the domai (large z). Normal gradiets of progostic variables (velocity compoets ad buoyacy) are set to zero at the outer computatioal boudary, ad periodic boudary coditios are imposed at the x-z ad y-z boudaries of the computatioal domai. 4. Parameters of simulated flows We ivestigate idealized turbulet aabatic ad katabatic flows alog double-ifiite slopes usig DNS. The values of the surface buoyacy flux B s are take as.3 m s 3,.5 m s 3 for the katabatic flow cases ad.3 m s 3,.5 m s 3 for the aabatic flow cases. Each flow type is ivestigated with the slope agle α of 3º ad 6º. For compariso, data from the Fedorovich ad Shapiro (9) study of turbulet flow alog a vertical heated wall (α = 9 ) are also cosidered. I the wall-flow case (α = 9 ), the magitudes of all flow variables are uchaged whe the sig of the surface buoyacy flux B s chages from positive (heated wall) to egative (cooled wall). Although the sigs of the mea velocity, buoyacy, ad turbulet fluxes of mometum ad heat reverse, the variaces of the velocity compoets ad buoyacy remai ivariat with respect to the sig of the surface buoyacy forcig. Cosideratio of the averaged slope flow provides a coveiet framework for the aalysis of the DNS data. Applyig the Reyolds decompositio to the flow variables i the goverig equatios ()-(3) ad (6) ad averagig these equatios over time ad spatially over x-y plaes parallel to the slope, we obtai: ' ' si u u w b α + ν =, z z (5)
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 99 b b' w' Nu α + ν = si, z z (6) π ww ' ' + b cosα =, z z where primes sigify deviatios from averages, deoted by overbars. Equatio (7) itegrates to π = ww ' ' cosα bdz give that both π ad ww ' ' vaish at z =. Boudary coditios for eqs. (5) ad (6) are: u =, ν (d b/d z) = Bs at z = (surface) ad u =, b = as z (far away from the slope). The turbulet fluxes are supposed to vaish o the surface ad far above the slope. Itegratig (6) over z from to ad otig that both molecular ad turbulet fluxes of the buoyacy vaish at, we obtai a itegral form of the buoyacy balace: I I z B (7) s d, (8) N siα VL u z= where the product V I L I is the volume flux. Note that we do ot defie either the itegral velocity scale V I or the itegral legth scale L ( u / V )dz. As follows from eq. (8), katabatic ad aabatic flows represeted by the same value of Bs /( N si α ) should have the same absolute value of the mea velocity itegral. Therefore, the relatio (8) may be iterpreted as a itegral dyamic similarity costrait for slope flows forced by the surface buoyacy flux (ote that this criterio applies to both turbulet ad lamiar slope flows). Based o the above cosideratios, oe may itroduce a itegral slopeflow Reyolds umber as where the dimesioless combiatio Fp Re VL I I Bs B I, ν = νn siα siα (9) Fp B Bsν N, () hereafter called the flow forcig parameter, is egative for a katabatic flow (B s < ) ad positive for a aabatic flow (B s > ). The magitude of Fp B represets the ratio betwee the eergy productio at the surface ad the I I
99 E. FEDOROVICH ad A. SHAPIRO work agaist buoyacy ad viscous forces. From this defiitio of Re I, we expect a particular slope flow to be more turbulet with icreasig Fp B. I the performed DNS, values of Re I were withi the rage of 3 to,. These values were large eough to obtai reasoably developed turbulece while allowig use of relatively compact umerical grids ad providig sufficietly log time series of variables to track the flow developmet. The simulatios were coducted o the x y z = 56 56 N z uiformly-spaced (Δx = Δy = Δz = Δ) grids, with N z varyig from 4 to 8 depedig o the Re I of the simulated flow. The grid spacig Δ was chose to esure that the resolvability coditio Δ L m is satisfied (Pope ), 3/4 /4 with Lm = ν Bs beig a aalog of the Kolmogorov microscale. Notig that the dimesioal goverig parameters of the slope flow v, N, ad B s have, respectively, dimesios of [L T ], [T ], ad [L T 3 ], oe ca itroduce some geeric slope flow scales L (for distace), V (for velocity), ad B (for buoyacy), ad, based o the Π theorem (Laghaar 95), write ( ) ( ) ( ) L = ν N f Fp, V = ν N f Fp, B = ν N f Fp, () / / / / / 3/ L B V B B B where f L, f V, ad f B, are fuctios of the dimesioless forcig parameter Fp B = B s v N. Normalizig the averaged mometum ad buoyacy balace equatios (5) ad (6) with the scales give i () we obtai the followig scaled equatios: ( uw ' ') f u f b siα + =, f f z V V L B fl fb z B L V ( bw ' ') f b f B u siα + =, f f z f z with ormalized boudary coditios: u L () (3) = db fl ad Fp at z, dz = f = (4) B B u ad b as z, (5) where z = z/ L, u = u / V, b = b/ B, ( u' w' ) = u' w'/ V, ad ( ' ') bw = bw ' '/( VB ). Equatios ()-(5) idicate that for a give slope agle α, ay scalig relatioship for L, V ad B, which results i the same Fp B = B s v N, should
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 993 yield u, ' ' bw ' ' that are uiversal fuctios of z. A particular form of the discussed uiversal behavior for the special case of buoyatly drive flow alog a heated vertical wall, where si α = ad Re I = Fp B >, has bee umerically verified i Fedorovich ad Shapiro (9). b, ( uw ), ad ( ) 4. Spatial ad temporal evolutio of simulated slope flows The spatial (i the z directio) ad temporal evolutio of the simulated velocity (u compoet) ad buoyacy b fields i the cetral poit of the x-y plae is illustrated i Fig. 3. Results are show for katabatic flows with the same value of Fp B = B s v N = 5, but with two differet slope agles: 3 ad 6. After passig through relatively short trasitio stages, both flows become turbulet. They display radom, large-amplitude fluctuatios of velocity ad buoyacy fields i the core regios, where their behavior is characteristic of a developed turbulet flow, ad show a quasiperiodic oscillatory behavior at larger distaces from the slope. Notably, oly fluctuatios with a frequecy equal to the atural buoyacy frequecy N siα i the evirometal fluid domiate at large z. Fluctuatios with other frequecies rapidly decay away from the slope beyod the turbulet core of the flow. These domiat N siα oscillatios are apparet far beyod the thermal ad dyamic turbulet boudary layers developig alog the slope. Note that the red shadig i the color bar was chose to draw attetio to the weak retur flow (positive u) ad positively buoyat air above the primary katabatic jet. The thermal boudary layers, whose depth may be estimated from the positio of a arrow white regio separatig areas of egative (blue) ad positive (red) buoyacy values, i both flow cases are cosiderably shallower tha the correspodig dyamic boudary layers, whose depth is marked by the trasitio regio (white) betwee the dow-slope flow (blue) ad upslope retur flow (red). It ca be oted, however, that have we defied the top of the dyamic boudary to be the height of the jet, the thermal ad dyamic boudary layers will be of comparable thickess. The overall thickesses of the primary katabatic jet as well as the magitudes of the buoyacy ad velocity i the katabatic jet are larger i the 3 slope case tha i the 6 slope case (ote that values of all other cotrollig parameters of the flows are the same). This idicates that the qualitative predictio of the Pradtl model regardig the itesificatio ad vertical expasio of the slope flow with decreasig slope agle i the case of specified surface buoyacy flux (cf. with right plot of Fig. ), which applies strictly speakig oly to a lamiar slope flow, holds also for a turbulet slope flow.
994 E. FEDOROVICH ad A. SHAPIRO (a) (b) (c) (d) Fig. 3. Temporal variatios of alog-slope velocity compoet (u, plots a ad c, i m s ) ad buoyacy (b, plots b ad d, i m s ) at differet distaces from the slope i the ceter of simulatio domai for the katabatic flow cases (B s =.5 m s 3, v = 4 m s, N = s, resultig i Fp B = 5) alog slopes of differet steepess: 3 (plots a ad b) ad 6 (plots c ad d). Distace is i meters ad time is i secods. A oscillatory flow patter similar to the patters show i Fig. 3 was observed by Shapiro ad Fedorovich (6) i their study of a lamiar atural covectio flow alog a wall with a temporally periodic surface thermal forcig. I the preset case, however, the oscillatory flow motios result from iteractios betwee turbulece ad ambiet stable stratificatio uder the coditios of a temporally costat surface buoyacy forcig. Direct estimatio of the oscillatio period T from the plotted data provides T values very close to π /(N siα) for each of the two cosidered slope agles: T =.6 s for α = 3 ad T = 7.3 s for α = 6. Aalogous flow oscillatios
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 995 have bee ecoutered by Schuma (99) i large eddy simulatios of aabatic wids (see discussio i the Itroductio) ad predicted/observed i the katabatic wid studies of Moti et al. () ad Pricevac et al. (8), where the frequecy N siα has bee idetified as the frequecy of the iteral gravity waves that arrive ormal to the slope. The flow evolutio patters i Fig. 3 reveal that both simulated katabatic flows become statistically quasi-statioary with time. Such a quasistatioarity of established katabatic flow is additioally illustrated i Fig. 4, where the u-velocity compoet ad buoyacy fields averaged over x-y plaes parallel to the slope are show at icreasig distaces from the surface. The couterpart velocity ad buoyacy distributios i a aabatic flow are demostrated i Fig. 5. The preseted time series of u ad b refer to times i the simulatio well beyod the trasitio stages, that is, after flows have evolved from the lamiar to the developed turbulet state. A typical duratio of the trasitio stage i the coducted simulatios is about several secods. I the katabatic flow case (Fig. 4), the buoyacy field displays strog temporal variability i the turbulet core of the flow (at z =.7 m), where variability of the velocity field is also sigificat. Deeper ito the flow, at z =.3 m, the magitude of the buoyacy fluctuatios drops oticeably while the average buoyacy chages its sig, which idicates that the cosidered elevatio is i the relatively warm layer that caps the katabatic jet. However, the average velocity at z =.3 m remais egative, idicatig that the flow at this level is still, o average, withi the dyamic boudary layer. Such a vertical phase differece betwee the velocity ad buoyacy profiles is aother feature of the simulated turbulet flow that is qualitatively cosistet with predictios of the Pradtl model (see right plot of Fig. ). The discussed flow feature is clearly also see i the spatial ad temporal flow patters preseted i Fig. 4. The buoyacy field variatios at z =.3 m are almost periodic with frequecy N si α, while the variability of the average velocity at this level still happes o a relatively broad rage of time scales, but with the modulatio of velocity fluctuatios by the domiat mode oscillatio already evidet. At z =.6 m, the remaiig radom fie-scale fluctuatios of both velocity ad buoyacy become very small compared to the iteral gravity wave oscillatio, with the flow oscillatio at z =.9 m beig almost purely periodic ad etirely determied by the iteral gravity waves. Time series of velocity ad buoyacy i the aabatic flow (Fig. 5) show the variability features that are i may respects similar to the above discussed features of the katabatic flow. Far away from the slope, small-scale flow fluctuatios associated with turbulet motios become egligible ad the residual variability is etirely due to the iteral-wave oscillatios.
996 E. FEDOROVICH ad A. SHAPIRO u (m s - ) - - b (m s - ) -3 - -4 4 6 8 t (s) - 4 6 8 t (s) Fig. 4. Temporal evolutio of the (x-y) plae-averaged velocity (left plot) ad buoyacy (right plot) fields i the katabatic flow with B s =.3 m s 3, v = 4 m s, ad N = s (Fp B = 3) alog a 3 -slope at four differet distaces from the slope:.7 m (black),.3 m (blue),.6 m (red), ad.9 m (mageta)..5.5 u (m s - ) b (m s - ).5 - -.5 4 6 8 t (s) 4 6 8 t (s) Fig. 5. Temporal evolutio of the (x-y) plae-averaged velocity (left plot) ad buoyacy (right plot) fields i the aabatic flow with B s =.5 m s 3, v = 4 m s, ad N = s (Fp B = 5) alog a 6 -slope at four differet distaces from the slope:. m (black),.4 m (blue),.75 m (red), ad. m (mageta). Also, as i the katabatic flow case, the depth of the thermal boudary layer is cosiderably smaller tha that of the dyamic boudary layer. The latter feature is revealed by compariso of relative buoyacy ad velocity chages over the iterval of heights betwee z =. m ad z =.4 m. Note that the
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 997 average velocity remais positive ad approximately costat betwee these two layers, whilst the average buoyacy chages from a slightly positive value to a relatively large egative value which correspods to the egatively buoyat flow above the thermal boudary layer. A compariso of the velocity time series i Fig. 5 ad Fig 4 also reveals that mometum i the turbulet regio of the aabatic flow is distributed vertically more uiformly tha i the core of the katabatic flow. Ispectio of the evolutio of velocity ad buoyacy oscillatios at large distaces from the slope (z =.75 m ad z =. m) idicates that there is a delay of the wavy motio developmet i the remote flow regios. This delay is icreasig with distace from the wall: flow oscillatios at z =. m develop oticeably later tha at z =.75 m. 4.3 Mea-flow ad turbulece structure Figure 6 shows the mea flow profiles of mometum ad buoyacy i katabatic ad aabatic flows with the same absolute value of the flow forcig parameter Fp B alog slopes of three differet agles. The mea profiles were obtaied by averagig the simulated flow fields spatially over x-y plaes ad temporally over at least 7 oscillatio periods beyod the trasitio stage. The plots i Fig. 6 illustrate the slope-agle depedece of the mea structure of the turbulet flows. The structural features of these flows ca be compared to those of the Pradtl slope flow (Sectio 3.). The velocity ad buoyacy profiles i Fig. 6 show cosiderable sesitivity to the slope agle for both katabatic ad aabatic flow cases. The shape of the katabatic-flow velocity profile for both slope agles less tha α = 9 is very differet from the shape of the velocity profile i the aabatic flow. Differeces i shape betwee the buoyacy profiles for both flow cases are less proouced which is partially due to the fact that i both flow cases the buoyacy sharply drops (icreases) i the very close viciity of the wall. Buoyacy ad velocity profiles i the simulated katabatic flows very closely resemble their couterparts observed i the field study of katabatic flow by Moti et al. (; Fig. 9a). I the katabatic flow, stable evirometal stratificatio i combiatio with egative surface buoyacy forcig lead to a effective suppressio of vertical turbulet exchage i the flow regio i the immediate viciity of the slope. Associated with this shallow layer of strog egative buoyacy is a arrow mea-velocity jet with peak velocity foud at very low levels. The magitude of this jet icreases with decreasig slope agle, thus revealig a qualitative similarity to the same feature of the velocity profile i the Pradtl model (right plot of Fig. ). However, due to the stroger impedace of the slope-ormal motios by the egative buoyacy i the case of more shallow slope (see eq. 3), the jet maximum i the umerically simulated katabatic flow does ot shift away
998 E. FEDOROVICH ad A. SHAPIRO from the slope with decreasig α as oticeably as i the Pradtl model (right plot of Fig. ). Aother structural feature of the turbulet katabatic flow that is similar to the Pradtl-model flow is the icrease of the surface buoyacy magitude with decreasig slope agle. A sharpess of the ear-surface buoyacy icrease i the turbulet katabatic flow is apparetly ehaced by the suppressio of vertical exchage of heat (buoyacy) by hydrostatic stability effects. Because of the aforemetioed features of the mometum ad buoyacy trasport i the ear-surface regio of the katabatic flow, the resultig jet is much more asymmetric i the simulated turbulet katabatic flow tha i the Pradtl flow. Structurally, the simulated aabatic flows differ from their katabatic couterparts primarily with respect to the shape, magitude ad vertical extesio of the jet. I the ear-surface regio of the aabatic flow, the positive surface buoyacy flux acts as a turbulece productio mechaism. I combiatio with shear geeratio of turbulece, it opposes the turbulece destructio effect of the stable stratificatio. The iteractio betwee these three turbulece geeratio/destructio mechaisms results i a vertical mixig of mometum i the aabatic jet that becomes more efficiet with a decreasig slope agle. This happes because the surface buoyacy productio apparetly offsets the suppressig effect of stratificatio o the turbulet exchage i the slope-ormal directio more effectively as slope agle decreases ad leads to the ehacemet of vertical mixig. The effect of this ehaced mixig is also see i the reduced values of the surface buoyacy profiles for the aabatic flow case ad a ear idepedece of the surface buoyacy o the agle. These turbulece effects make turbulet aabatic flow differ more, i a mea qualitative sese, from its Pradtl model aalog tha the couterpart katabatic flow. Notably, despite all these structural differeces, the overall vertical extesio of the mea flow disturbace i terms of velocity appears to be practically the same for katabatic ad aabatic flows alog slopes of the same agle. Furthermore, direct evaluatio of the itegrals of mea velocity profiles show i Fig. 6 cofirms validity of the earlier obtaied itegral dyamic similarity costrait (8) for the simulated slope flow cases. I particular, the velocity profiles i Fig. 6 that correspod to the same α should itegrate to the same value as they ideed do. I additio, the velocity itegrals for the same values of B s ad N but for differet slope agles relate as ratios of the sies of slope agles. As revealed by the buoyacy variace profiles i Fig. 7, the buoyacy fluctuatios i both katabatic ad aabatic flow cases attai their maximum magitude extremely close to the wall, withi the regio where maximum gradiets are observed i the mea buoyacy profiles (Fig. 6). The drop of bb ' '
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 999 u (m s - ) -7-6 -5-4 -3 - - u (m s - ) 3 4 5 6 7.8.8.6.6 z (m).4 z (m).4.. -7-6 -5-4 -3 - - b (m s - ) 3 4 5 6 7 b (m s - ) Fig. 6. Mea alog-slope velocity (u, solid lies) ad buoyacy (b, dashed lies) profiles i the katabatic (left) ad aabatic (right) flows with B s =.5 m s 3, v = 4 m s, ad N = s ( Fp B = 5) for three differet slope agles: 3 (blue lies), 6 (red lies), ad 9 (black lies). u'u' (m s - )..4.6.8 u'u' (m s - )..4.6.8.8.8.6.6 z (m).4 z (m).4.. 3 4 b'b' (m s -4 ) 3 4 b'b' (m s -4 ) Fig. 7. Slope agle depedece of the alog-slope velocity (solid lies) ad buoyacy (dashed lies) variaces i the katabatic (left) ad aabatic (right) flows preseted i Fig. 6.
E. FEDOROVICH ad A. SHAPIRO beyod the maximum is also rather sharp which meas that sigificat fluctuatios of the buoyacy are restricted to a comparatively thi ear-wall layer. The variace magitudes are larger i the katabatic flows compared to the variaces i aabatic flows with the same Fp B value. The buoyacy variace decay with distace from the wall displays a clear slope-agle depedece. This decay is weaker i the aabatic flows, which apparetly is a result of the icreasig cotributio to the variace from buoyacy-related turbulet mixig with smaller slope agles. I the katabatic flow, the icrease of variace caused by larger ear-surface mea buoyacy gradiet at smaller slope agles is partially offset by stroger dampig of turbulece by stratificatio as the slope agle decreases, so the resultig decay of bb ' ' with z is oly slightly weaker i the case of a relatively shallow slope (α = 3 ) tha i the case of a steeper slope (α = 6 ) ad the vertical wall case (α = 9 ). Alog-slope velocity fluctuatios of otable magitudes are distributed over layers that are typically a few times thicker tha the layers which cotai most of the buoyacy variace i both the katabatic ad aabatic flow cases (see profiles of velocity variace uu ' ' i Fig. 7). I both flow cases, a tedecy to develop arrow secodary maxima of uu ' ' i the close viciity of the slope is observed. These secodary maxima, which are ot foud i the flow alog a vertical wall (α = 9 ), become more proouced with decreasig slope agle ad are more promiet i katabatic flow (i the aabatic flow with a 6 slope, a ear-surface bed i the uu ' ' profile rather tha a maximum is observed). After uu ' ' reaches its global maximum for ay idividual slope case, it decays away from the slope i a more gradual maer tha the bb ' ' variace. As may be cocluded from the profiles of slope-ormal fluxes of mometum, uw, ' ' ad buoyacy, bw, ' ' i Fig. 8, zero crossigs i the mea profiles of b ad u are closely co-located with the miima ad maxima of the fluxes uw ' ' ad bw. ' ' This co-locatio is i agreemet with iviscid forms of (5) ad (6) for ay z except for locatios very close to the wall, where molecular effects are importat. Typically, molecular fluxes i the simulated flows become egligible at distaces from the surface that are sigificatly smaller tha the elevatios of the mea velocity maxima (jet elevatios). Narrow ear-surface bads of positive (egative) mometum flux i katabatic (aabatic) flow correspod to the flow regios where egative (positive) mometum is trasported to the wall (that is, i the z directio). As may be additioally deduced from relatig profiles of velocity ad buoyacy i Fig. 6 to profiles of mometum ad turbulet fluxes i Fig. 8, the ordiates
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES w'u' (m s - ) -. -.5 -. -.5 w'u' (m s - ).5..5..8.8.6.6 z (m).4 z (m).4.. -.5 -.4 -.3 -. -. w'b' (m s -3 )...3.4.5 w'b' (m s -3 ) Fig. 8. Slope agle depedece of the kiematic turbulet mometum (solid lies) ad buoyacy (dashed lies) fluxes i the katabatic (left) ad aabatic (right) flows preseted i Fig. 6. of zero fluxes are closely co-located with the ordiates of zero gradiets of the correspodig mea profiles. These flow features imply that turbulet fluxes are i apparet ati-correlatio with the correspodig mea gradiets for both simulated flow types. Therefore, turbulet trasport i these katabatic ad aabatic flows allows a descriptio i terms of uiquely defied positive proportioality coefficiets (eddy viscosities ad diffusivities). We coducted a prelimiary evaluatio of the slope-ormal of the turbulet Pradtl umber (which is the ratio of the eddy viscosity to the eddy diffusivity) for oe of simulated katabatic flow cases ad foud that, except for the regio of very strog static stability close to the surface, the Pradtl umber values do ot sigificatly depart from uity over the mai portio of the flow ad are withi the rages observed by Moti et al. (). Iterestigly, the magitude of uw ' ' for the katabatic flow case shows very weak depedece o the slope agle i sharp cotrast to the mometum flux behavior i the aabatic flow case where it is characterized by relatively strog decay of the flux magitude ad upward shift of the flux maximum with decreasig slope agle. O the other had, the shapes of the bw ' ' profiles i katabatic ad aabatic flows show some degree of similarity with respect to the depedece o the slope agle. I both cases, the buoyacy flux magitude icreases as the slope agle decreases.
E. FEDOROVICH ad A. SHAPIRO Flux profiles show i Fig. 8 idicate that there is o regio i the flow domai with costacy (eve approximate) of ay flux with distace from the wall. Arguably, this may be a specific feature of the slope flows i the presece of ambiet stratificatio. I more covetioal boudary-layer type flows, the existece of height itervals with slowly chagig (i the first approximatio, costat) mometum ad buoyacy fluxes is used as a foudatio for similarity aalyses ad scaligs (Teekes ad Lumley 97). Such a costat-flux formalism does ot apply, at least i a straightforward maer, to the slope flows cosidered i our study. O a related ote, we have ot foud ay evidece of scale separatio i the simulated flows that would allow the flow to be subdivided ito regios where ay of the goverig parameters (B s, v, N, or α) could be dropped from cosideratio. Eve at relatively large distaces from the slope, the molecular viscosity/diffusivity i combiatio with the surface buoyacy flux ifluece the local flow structure through the ear-wall peak velocity value that is directly determied by their combied effect. O the other had, the ambiet stratificatio (i terms of N) is started to be felt i the immediate viciity of the wall, so it is impossible to isolate a flow regio where the depedece o N may be eglected. These coclusios are cosistet with the flow parameter cosideratios preseted i Sectio 4.. The developmet of velocity fluctuatios ormal to the wall ( w ') is apparetly hampered by the presece of the wall. This explais the relatively slow growth with z ad smaller values of ww ' ' i Fig. 9 as compared to uu ' ' i Fig. 7 ad vv ' ' i Fig. 9. Curiously, profiles of the latter variace cosistetly display secodary maxima very close to the wall. This feature is remiiscet of the previously discussed maxima i the uu ' ' distributios (Fig. 7). O the whole, the vv ' ' profiles are geerally rather similar to the uu ' ' profiles, especially i the case of aabatic flow. However, ulike the uu ' ' variace, secodary maxima of vv ' ' are foud also i the flows alog a vertical wall. It is our guess that these oted features of the velocity compoet variaces may be associated with the ifluece of iteral gravity waves (discussed i Sectio 4.) whose effect o the spatial distributio of velocity fluctuatios i the ear-surface flow regio apparetly depeds o the slope agle. To quatitatively explai the secodary maxima i uu ' ' ad vv, ' ' the estimates of secod-order turbulece momet budgets would be eeded, but those are ot available at this poit. We ow aalyze the ifluece of the magitude of the surface forcig (the surface buoyacy flux B s ) o the mea velocity ad buoyacy profiles (Fig. ) ad turbulece structure parameters (Figs. ad ).
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 3 v'v' (m s - )...3 v'v' (m s - )...3.8.8.6.6 z (m).4 z (m).4...5..5..5 w'w' (m s - ).5..5..5 w'w' (m s - ) Fig. 9. Slope agle depedece of the variace of the v (solid lies) ad w (dashed lies) velocity compoets i the katabatic (left) ad aabatic (right) flows preseted i Fig. 6. u (m s - ) -8-6 -4 - u (m s - ) 4 6 8.8.8.6.6 z (m).4 z (m).4.. -5-4 -3 - - b (m s - ) 3 4 5 b (m s - ) Fig.. Mea u (solid lies) ad b (dashed lies) profiles i the katabatic (left) ad aabatic (right) flows alog 3 slope with v = 4 m s, N = s, ad two differet surface forcig magitudes: B s =.3 m s 3 ( Fp B = 3, blue lies) ad B s =.5 m s 3 ( Fp B = 5, red lies).
4 E. FEDOROVICH ad A. SHAPIRO u'u' (m s - )..4.6.8 u'u' (m s - )..4.6.8.8.8.6.6 z (m).4 z (m).4.. 3 4 b'b' (m s -4 ) 3 4 b'b' (m s -4 ) Fig.. Surface forcig depedece of the alog-slope velocity (solid lies) ad buoyacy (dashed lies) variaces i the katabatic (left) ad aabatic (right) flows preseted i Fig.. w'u' (m s - ) -. -.5 -. -.5 w'u' (m s - ).5..5..8.8.6.6 z (m).4 z (m).4.. -.5 -.4 -.3 -. -. w'b' (m s -3 )...3.4.5 w'b' (m s -3 ) Fig.. Surface forcig depedece of the kiematic turbulet fluxes of mometum (solid lies) ad buoyacy (dashed lies) i the katabatic (left) ad aabatic (right) flows preseted i Fig..
KATABATIC AND ANABATIC FLOWS ALONG STEEP SLOPES 5 The differeces betwee the buoyacy profiles correspodig to differet B s are largest i the close viciity of the slope. With B s =.5 m s 3, the magitude of the surface buoyacy value, b s, i the katabatic flow case is.38 times larger tha with B s =.3 m s 3. The correspodig icrease i the aabatic flow case is.56. This differece poits to distictively differet ad o-liear scalig relatioships betwee B s ad b s i katabatic ad aabatic flows. The differece i depth betwee the thermal ad dyamic boudary layers i the katabatic (aabatic) flow, already oted above, is clearly see from comparig the rate of buoyacy growth (decay) with distace from the wall to that of velocity after it reached its miimum i the viciity of the wall. Although there are obvious differeces i detail betwee the mea velocity ad buoyacy profiles i the turbulet katabatic flow (left plot of Fig. ) ad the correspodig lamiar (Pradtl-model) flow profiles i Fig., the overall mea flow structure is qualitatively the same: cold (relative to eviromet) fluid desceds alog the slope, whilst warm fluid rises upslope at some distace from the slope. The aabatic flow (right plot of Fig. ) displays a more ati-symmetric structure, with the reverse flow (which is directed dow-slope i this case) beig shallower ad the upslope curret (shaped as a broad jet) beig deeper tha i the katabatic flow. However, accordig to the previously established itegral dyamic similarity costrait (eq. ), the itegrals of the velocity profiles i Fig. scale with B s for both katabatic ad aabatic flow cases show. A strikig feature of the mea flow profiles show i Fig. is a very weak depedece (i practical terms idepedece) of their vertical scales o the magitude of surface forcig B s. Ideed, vertical chages of velocity ad buoyacy profiles i flows with differet B s are closely coordiated for each flow type. This implies that the vertical flow legth scale for either flow type with fixed values of v, N, ad α is practically idepedet of B s. A aalogous behavior is displayed by the profiles of turbulece statistics show i Figs. ad. The positios of miima/maxima ad zerocrossover heights of profiles correspodig to differet B s values practically coicide for a give slope flow type, showig a miimal depedece o the B s magitude. To further ivestigate this weak depth-scale depedece of slope flows o the surface eergy productio rate, additioal umerical experimets with B s values varyig i broader rages will be eeded. 5. SUMMARY AND CONCLUSIONS I this study, buoyatly drive slope flows alog doubly-ifiite cooled/heated iclied surfaces immersed i a stably-stratified fluid have bee reexamied withi the coceptual framework of the Pradtl (94)
6 E. FEDOROVICH ad A. SHAPIRO slope-flow model, ad umerically ivestigated by meas of DNS. The umerically simulated flows, drive by a spatially-uiform surface buoyacy flux B s (eergy productio rate), were moderately turbulet, with characteristic itegral Reyolds umbers i the rage from 3 to,. These DNS complemet previous umerical large eddy simulatios of katabatic/aabatic flows reported i the literature. Solutios of the Pradtl model with prescribed surface forcig i the form of costat buoyacy flux were obtaied. (The origial Pradtl solutios correspod to slope-flow cases with prescribed costat surface buoyacy.) It was foud that the surface buoyacy value as well as the elevatio ad itesity of the velocity maximum icrease with decreasig slope / agle as si α. A correspodig scalig has bee proposed for the Pradtl model flow that icorporates depedece of the buoyacy ad velocity profiles o the slope agle. Numerical experimets show that followig the trasitio from a lamiar to a turbulet regime, the simulated katabatic ad aabatic flows eter quasistatioary oscillatory phases. The frequecy of the esuig perpetual oscillatios (associated with iteral gravity waves) is give by the product of the evirometal Brut Väisälä frequecy ad the sie of the slope agle, N siα, which is i a direct agreemet with kow hydraulic model predictios ad field observatios of slope flows. I both simulated flow types (katabatic flow with B s < ad aabatic flow with B s > ), the turbulet fluctuatios gradually fade with distace from the wall, while periodic oscillatios persist withi outer lamiar flow regios before fadig out. Notably, these oscillatory flow regimes are foud uder the coditios of temporally costat surface buoyacy forcig. They are characterized by strog iteractios betwee turbulet ad wavy motios that apparetly result i peculiar structural flow features. For istace, the slope-parallel velocity compoet variaces cosistetly display secodary maxima very close to the wall, at distaces comparable to those of the mea velocity maxima/miima. A explaatio of this ad other turbulece structure features would require estimates of the secod-order turbulece momet budgets i the simulated flow ad aalyses of the turbulece spectra. The mea (averaged over time ad wall-parallel plaes) structure of the simulated flows is depedet o a dimesioless combiatio of goverig parameters give by Fp B B s v N, which may be iterpreted as a flow forcig parameter. The sig of Fp B is determied by the slope flow type. It is egative for katabatic flow (B s < ) ad positive for aabatic flow (B s > ). The magitude of Fp B represets the ratio betwee the eergy productio at the surface ad the work agaist buoyacy ad viscous forces. The itegral Reyolds umber Re I of the flow is related to the magitude of Fp B as Re I