Nonlocal boundary layer: The pure buoyancy-driven

and the buoyancy-shear-driven cases

N. M. Colonna,1 E. Ferrero,1 and U. Rizza2

Received 30 June 2008; revised 27 October 2008; accepted 5 November 2008; published 4 March 2009.

[1] A Reynolds-averaged Navier-Stokes (RANS) model including nonlocality wasdeveloped and tested. The aim of this paper was to investigate different boundary layerconditions with the RANS model. In particular we focused our attention on boundarylayers where convection plays a major role: the shear-free buoyancy-driven boundarylayer and the buoyancy-shear-driven one. We presented a new model which solvesdynamical equations for all the turbulence moments up to the third order. Themodel assumes the fourth-order moments to be Gaussian-distributed according to thequasi-normal approximation. A new approach to avoid the unphysical growth ofthe third-order moments, connected to the quasi-normal hypothesis, is suggested. Thenew physical ingredient of this method is in essence the parameterization of therelaxation and dissipation time scales related to the return to isotropy terms in thetriple-pressure correlations and to the dissipation rate of the potential temperaturevariance, respectively. This new parameterization takes into account the dependenceof the third-order moments on the integral length scale of the dissipation rate, aquantity related to the eddy size variation across the boundary layer. The time scalesare reduced as the integral length increases, and these reductions provide theproper damping of the turbulent vertical transport. In order to test our model,we carried out large eddy simulations (LESs) of boundary layers with shearand buoyancy. We present the results of the comparison with LESs, aircraftmeasurements, and DNS, which show that the model performs rather well in all thestability conditions and that the proposed damping gives reliable third-order moments thatsatisfy the realizability constraints.

Citation: Colonna, N. M., E. Ferrero, and U. Rizza (2009), Nonlocal boundary layer: The pure buoyancy-driven and the buoyancy-

shear-driven cases, J. Geophys. Res., 114, D05102, doi:10.1029/2008JD010682.

1. Introduction

[2] It is generally recognized that nonlocal transport ofturbulence is critically important in the PBL’s mixingprocesses, especially in those fluxes where the buoyancyplays a major role. Among the several procedures wellknown and strongly validated in the literature, the Reynolds-AveragedNavier-Stokes (RANS)models, firstly proposed byReynolds, represent a useful tool to describe many turbulentphenomena.[3] RANS models should be divided in local and nonlocal

models. To the first category belong those models thatinclude up to the second-order moments (SOMs) equationsand adopt a parameterization for the third-order moments(TOMs) [Mellor and Yamada, 1982;Byggstoyl and Kollmann,1986; Durbin, 1993; Cheng and Canuto, 1994; Canuto et

al., 1999; Speziale et al., 1996; Jones and Musange, 1988;Trini-Castelli et al., 2001]. The usual parameterizationsemployed in local models are not appropriate when appliedto convective fluxes since they do not properly account forthe complex mixing processes that affect the turbulentquantities [Moeng and Wyngaard, 1989; Holstag andBoville, 1993]. In fact, in a local model it is usually assumeda one to one correspondence between turbulent fluxes at agiven height and other parameters of the flow, namely thelocal gradients of the mean fields, at the same height. Suchan approximation is justified only when the turbulentmixing length is much less than the length scale of hetero-geneity of the mean flow. However, observations and LargeEddy Simulation (LES) studies have demonstrated thatturbulence in the convective boundary layer is associatedwith the nonlocal integral properties of the boundary layer[Moeng and Sullivan, 1994; Holstag and Moeng, 1991;Schmidt et al., 2006]. This means that mixing is present alsoin those regions of the boundary layer where there is nolocal production of turbulence, thanks to nonlocal effects ofthe vertical transport. These effects are essentially due to thepresence in the convective boundary layer of large-scalesemiorganized structures, namely buoyancy-driven cells,

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, D05102, doi:10.1029/2008JD010682, 2009

1Dipartimento di Scienze e Tecnologie Avanzate, Universita delPiemonte Orientale, Alessandria, Italy.

2Consiglio Nazionale delle Ricerche, Istituto di Scienze dell’ Atmosferae del Clima, Lecce, Italy.

Copyright 2009 by the American Geophysical Union.0148-0227/09/2008JD010682

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that embrace the entire boundary layer. A similar effect ispresent also in the sheared flows due to large turbulentstructures like rolls or streaks [Moeng and Sullivan, 1994].Recent studies [Ferrero and Racca, 2004; Ferrero, 2005]have demonstrated that nonlocality plays an important rolein these kind of flows even though the structures entailedare smaller than those present in convective boundarylayers. As a matter of fact, a model including the TOMsis able to better determine the boundary layer height in theneutral case [Ferrero and Racca, 2004].[4] Therefore, to properly represent nonlocality it requires

that the model would be able to encompass all the effectsfrom the bottom of the boundary layer to the height wherethe fluxes are computed. Several improvements of localmodels have been made by including nonlocal adjustingterms [Deardorff, 1972;Holstag andMoeng, 1991;Wyngaardand Weil, 1991; Canuto et al., 2005] but to overcome theshortcomings of local models, one should employ a nonlocalmodel by including the dynamical equations for the TOMs[Canuto, 1992; Cheng and Canuto, 1994; Zilitinkevich et al.,1999; Canuto et al., 2001; Cheng et al., 2005; Ferrero andRacca, 2004;Ferrero, 2005;Gryanik et al., 2005; Ferrero andColonna, 2006].[5] In this paper we focus our attention on the turbulence

closure problem for convective sheared and unshearedboundary layers. We present simulations performed with athird-order closure model whose TOMs equations wererecently proposed and tested by us [Ferrero and Colonna,2006]. This model is able to account for both the turbulentproduction processes, namely shear and buoyancy, at thesame time thus allowing to simulate more complex flowscorresponding, for example, to the actual atmosphericboundary layer.[6] The first flow investigated is a buoyancy-driven

shear-free boundary layer (hereafter case B) while thesecond one is a buoyancy-driven sheared flow (hereaftercase SB). Most of the boundary layer flows studied inliterature are, for sake of simplicity, pure convective andpure shear cases which represent very particular conditions.On the contrary, the case SB represents a realistic combi-nation of shear and convection such as the one thatcharacterizes the actual boundary layer, thus representinga case of particular interest.[7] In section 2 the model equations system is presented.

The problem of turbulent damping of the TOMs is dis-cussed in section 3 and a new approach to solve the problemis suggested. large eddy simulations and data for compar-ison are presented in section 4. Finally, in section 5, theTOM model settings are discussed and the results arepresented. For sake of the completeness the details relatedto the equations have been listed in Appendix A, while inAppendix B a preliminary test of the model, where theresults are compared with a Direct Numerical Simulation(DNS) data set, is shown.

2. Model

[8] We consider a horizontally homogeneous boundarylayer. The model consists of a system of 34 dynamicalequations derived from the Navier-Stokes equations follow-ing the procedure described in the work of Canuto [1992].

The model numerically solves the equations for mean fields,SOMs and TOMs. A closure scheme is required for thefourth-order moments (FOMs) appearing in the TOMsequations. In this work we have adopted the quasi-normalapproximation though it exhibits some well known short-comings (divergences of the TOMs in neutral and convec-tive boundary layers and spurious oscillations in the stableregions of the flow) [Canuto et al., 2005; Cheng et al.,2005; Moeng and Randall, 1984]. For this reason, a newapproach, based on physical grounds, is employed to avoidthe anomalous growth of the TOMs due to the use of thequasi-normal approximation. The model does not includerotation and viscosity effects since their contributions arenegligible in the atmospheric boundary layer. Exception ismade for the mean wind equations in which rotation effectsdue to the Coriolis force are included.

2.1. Mean Fields and SOM

[9] According to the horizontal homogeneity hypothesisthe ensemble averaged equations of motion with the hydro-static approximation can be written as follows.

@U

@t¼ �

@uw

@z� f Vg � V

� �

; ð1Þ

@V

@t¼ �

@vw

@zþ f Ug � U

� �

: ð2Þ

where the upper cases indicate the mean values whilethe lower cases stand for their fluctuations. U and V are thecomponents of the mean wind velocity, Ug and Vg are thegeostrophic wind velocity components and f is the Coriolisparameter ( f = 10�4 s�1). The dynamical equation governingthe mean potential temperature, Q, is

@Q

@t¼ �

@qw

@zð3Þ

where qw is the potential temperature flux.[10] The first moments entail second-order correlations,

such as uw, vw and qw. The dynamical equations for theSOMs are given by the following expressions:

@uiuj

@t¼ � ujuk

@Ui

@xkþ uiuk

@Uj

@xk

� �

�@uiujuk

@xk�Pij þ liujq

þ ljuiq� �ij; ð4Þ

@uiq

@t¼ �uiuj

@Q

@xj� quj

@Ui

@xj�@quiuj

@xj�P

qi þ liq

2; ð5Þ

@q2

@t¼ �2uiq

@Q

@xi�@uiq

2

@xi� 2�q: ð6Þ

where li = gia, being gi = (0, 0, g) the gravitationalacceleration and a the thermal expansion coefficient, �ij and�q are the Reynolds stress and the potential temperaturevariance dissipation rates respectively. The first can be

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related to the turbulent kinetic energy dissipation rate, �, bythe following expression:

�ij � 2n@ui

@xk

@uj

@xk¼

2

3dij� ð7Þ

while the second is given by

�q ¼ c@q

@xj

� �2

ð8Þ

[11] Concerning the dissipation term �q Zeman and Lumley[1976] proposed a dynamical equation, but here we simplyadopt the parameterization suggested by Andre et al.[1982]:

�q ¼ 2c2t�1q2 ð9Þ

where c2 is a coefficient depending on the integral lengthscale, as proposed by Ferrero and Colonna [2006] anddiscussed in detail in section 3.[12] Pij and Pi

q, representing the pressure correlationterms, are discussed in detail in Appendix A. For thedissipation rate of the turbulent kinetic energy, �, we haveadopted the following dynamic equation in the standardform, as suggested by Lumley and Khajeh-Nouri [1974].

@

@t�þ

@�uj

@xj¼ a1t

�1 Pq þ Psð Þ � a2�t�1 ð10Þ

where t =q2

e, q2 being uiui = 2E (E is the turbulent kinetic

energy), is the characteristic turbulence life time.[13] In the LHS of equation (10) the second term represents

the divergence of the turbulent flux of � (see Appendix Afor discussion) while the terms Pq and Ps, in the RHS ofequation (10), represent the buoyancy and the shear pro-duction, respectively. These terms are defined as the trace ofthe following tensors.

Pqij ¼ liquj þ ljqui; ð11Þ

Pij ¼ � uiuk@Uj

@xkþ ujuk

@Ui

@xk

� �

: ð12Þ

2.2. Third-Order Moments

[14] The dynamic equations for the third-order correla-tions can be derived from the SOMs equations as in thework of Canuto [1992]. The TOMs imply the fourth-order

correlations, uiujukul, uiujukq, uiujq2 and uiq

3 which need tobe modeled. We have adopted the approximation of

Hanjalic and Launder [1972, 1976], analyzed in detailby Zeman and Lumley [1979]. The FOMs are taken to beGaussian-distributed, according to the quasi-normal approx-imation and therefore each fourth-order term can beexpressed as a products of SOMs.

abcd ¼ ab � cd þ ac � bd þ ad � bc ð13Þ

[15] With these and the horizontal homogeneity assump-tions, the equations for the TOMs result to

@

@tþ t�1

3

� �

uiujuk ¼ � uiuju3@Uk

@x3þ ukuiu3

@Uj

@x3þ ujuku3

@Ui

@x3

� �

� uiu3@ujuk

@x3þ uku3

@uiuj

@x3þ uju3

@ukui

@x3

� �

þ 1� c11ð Þ liqujuk þ ljqukui þ lkquiuj� �

�2

3tdijq2uk þ dkiq2uj þ djkq2ui

� �

; ð14Þ

@

@tþ t�1

3

� �

uiujq ¼ � uiuju3@Q

@x3� uiu3q

@Uj

@x3þ uju3q

@Ui

@x3

� �

� uiu3@quj

@x3þ uju3

@qui

@x3þ qu3

@uiuj

@x3

� �

þ 1� c11ð Þ liq2uj þ ljq

2ui

� �

þc*tdijq2q;

ð15Þ

@

@tþ t�1

3 þ 2t�1q

� �

uiq2 ¼ � 2quiu3

@Q

@x3� u3q

2 @Ui

@x3� 2u3q

@qui

@x3

þ 1� c11ð Þliq3 � uiu3

@

@x3q2; ð16Þ

@

@tþc10

c8t�13

� �

q3 ¼ �3q2u3@Q

@x3� 3u3q

@q2

@x3: ð17Þ

where c10 is a constant of the model (see Table 1) while tq =t/(2c2) is a time scale related to the turbulent damping[Ferrero and Colonna, 2006] discussed in the next Section.The dynamical equations for the TOMs have been obtainedby modeling the pressure triple correlation terms followingAndre et al. [1982] (see Appendix A for details).

3. Turbulence Damping: A New Parameterizationof the Relaxation and Dissipation Time Scales

[16] As pointed out in previous works, those of Cheng etal. [2005], Canuto et al. [2007], and Moeng and Randall

Table 1. Constants of the Model

Case c4 c5 c6 c7 c10 c11 c* a1 a2 cm ae a0 a1 a2

B 1.9 0.4 3.25 0.7 8 0.1 0 1.44 3.8 0.02 0.77 . . . . . . . . .

SB 1.64 0.7 3.2 0.5 8 0.4 0.2 1.44 3.8 0.01 0.77 0.4 0.31 0.22

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[1984], the use of the quasi-normal approximation for thefourth-order correlations can cause an unexpected andunphysical growth of the TOMs. This singular behavior isdue to the fact that the quasi-normal approximation does notprovide sufficient damping for the third-order correlationswhich, especially in convective boundary layer, may diverge.To avoid this from happening, several approaches have beensuggested as the clipping approximation [Andre et al.,1976], the Eddy Damped quasi-normal Markovian model[Orszag, 1977; Lesieur, 1990] and ad hoc procedures likethat proposed by Canuto et al. [2001] for the unstableboundary layer. Thanks to these procedures the eddy sizesare chopped down and the transport is properly weakened.In spite of its shortcomings the quasi-normal approximationis still largely employed [Vedula et al., 2005]. Cheng et al.[2005] have suggested a new, nonlocal formulation for theFOMs based on a top-down approach, a procedure that,unlike the bottom-up one, starts from the higher-ordermoments to obtain information about the lower moments.Gryanik et al. [2005] assumed that a measure of deviationof statistics from the Gaussian behavior is provided by theskewness of vertical velocity and potential temperature.[17] As stated in the previous section, in this work we

have adopted the quasi-normal approximation with a newmethod providing the correct turbulence damping. Thismethod has already been presented by Ferrero andColonna [2006]. It should be underlined that while inthe present paper we have performed simulations of theplanetary boundary layer in which the dynamic equationsare numerically solved for mean fields, SOMs and TOMs,in a previous paper [Ferrero and Colonna, 2006], wepresented a series of validations of the TOMs. They werecarried out by numerically solving the dynamical equa-tions of the TOMs only, using as input of the model LESprofiles of the mean components and the SOMs. Wetested this new approach over a wide range of atmosphericflows [Ferrero and Colonna, 2006]. On the contrary, themodel presented here solves all the mean and turbulentquantities and LES profiles are used only for sake ofcomparison.[18] The LHS of equations (14)–(17) show terms in

which the relaxation time scale, t3, and the dissipation timescale, tq, appear. Both these parameters depend on thecharacteristic turbulent life time, t, and on the two constantsc8 and c2:

t3 ¼t

2c8; ð18Þ

tq ¼t

2c2ð19Þ

[19] The terms involving these two time scales followfrom the return-to-isotropy terms of the triple-pressurecorrelations and from the dissipation rate of the potentialtemperature variance respectively. They control the turbu-lent damping of the TOMs [Andre et al., 1982; Lesieur,1990]. A proper choice of t3 and tq may therefore avoid theunphysical growth of the TOMs.[20] The basic idea of the new approach proposed in

this work is to consi and c2 varying across the

whole planetary boundary layer depending on a physicalparameter. This parameter is represented by the integrallength scale for the dissipation rate [Ferrero and Colonna,2006]:

L� ¼q2

32

�ð20Þ

This length scale, which varies with the height, representsthe size of the eddies in the boundary layer. Since the size ofthe eddies is strictly related to the turbulent transport thevariation across the boundary layer of L� implies that theturbulence damping should be different at different heights.Therefore c8 and c2 are taken to depend on the height and onL� as follow [Ferrero and Colonna, 2006]:

c8 ¼ cA8 1þL�

cA8Zi

� �

; ð21Þ

c2 ¼ cC2 1þL�

cC2 Zi

� �

: ð22Þ

where Zi is the boundary layer height while c8A = 8 and

c2C = 2.5 are the standard values suggested for this twoconstants by Andre et al. [1982] and Canuto [1992]respectively. As can be seen from equations (21) and (22)the new expressions reduce to these values if L� � Zi.Equations (21) and (22) properly modulate the value of c8and c2 at different levels without differing excessively fromthe literature values of the two constant. As a matter offact, the linear dependence on L� assures a slight modulationof the constants across the boundary layer. This first-ordercorrection should properly damp the turbulent transport atdifferent heights thus avoiding overestimation and under-estimation of TOMs vertical profiles. It is worth noticing thatZi is a constant scale parameter whose value is not calculatedrun time by the model but merely derived as a bulkcharacteristic of the specific case considered. Although,in an actual boundary layer, Zi can vary, the simulationperformed by the RANS model refers to a stationarycondition and therefore we can assume that the boundarylayer height is constant, although it could have been easilydetermined dynamically. Furthermore, the important para-meter is the ratio between L� and Zi which variation couldbe due to one of the two considered quantities or to both ofthem. If the model would be applied to a non stationary casethe possible Zi variations should be taken into account.[21] An important point related to the use of the quasi-

normal approximation is represented by the problem of themodel realizability. Gryanik et al. [2005] demonstratedthat any model using quasi-normal approximation can berealizable only if the skewness S satisfies the relation �2 S2 2. We verified that our RANS model, with the newdamping method, is consistent with these quasi-normalapproximation constraints. The vertical velocity skewness,Sw, satisfies the realizability condition, while for the poten-tial temperature skewness, Sq, the condition is not verified(according to Gryanik et al. [2005]) only near the surfaceand near the capping inversion.

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[22] As said before, the effectiveness of the methodproposed here was demonstrated by Ferrero and Colonna[2006] with the direct comparison of tests performed withconstant values of c8 and c2 versus a test with valuesvariable following equations (21)–(22). However, in orderto further verify the model, full boundary layer simulations,under various conditions, are indispensable.

4. Data for Comparison

[23] Since previous studies have highlighted that turbu-lent quantities obtained with LESs may show trends incontrast with experimental data [Moeng and Rotunno,1990], it becomes necessary to compare the results of ourRANS model not only with LES data but also with aircraftmeasurements. Furthermore, the Direct Numerical Simula-tion (DNS) results of a channel flow [Moser et al., 1999],were used for preliminary comparison of the RANS modelperformance (see Appendix B). In order to obtain all theturbulent quantities for the comparison of the model resultswe have appositely ran two LESs (hereafter CFR). The LESdata considered include both the resolved and the subgrid-scale (SGS) statistics, except for the TOMs. In fact, thesubgrid-scale (SGS) model is able to predict only the turbu-lence kinetic energy and hence it cannot contribute to theTOMs. Further, the high resolution of the LES reduces theSGS contribution to the predicted flow. For sake of compar-ison we have also used, when available, the LES data ofMoeng and Sullivan [1994] (hereafter referred to as MS).[24] Our LES model is based on the incompressible

Boussinesq form of the Navier-Stokes equations and con-siders a horizontally homogeneous boundary layer. Theboundary layer variables (mean wind components, pressure,potential temperature etc..) are spatially filtered to defineresolved component and SGS component. More details ofthe LES model used in the present work can be found in theworks of Moeng [1984], Moeng and Sullivan [1994], andRizza et al. [2006].[25] In this work, in order to obtain a complete data set

for the comparison with the output of our third-order model,we reproduced two types of atmospheric boundary layers: abuoyancy-driven boundary layer with weak shear effects(case B) and a flow with both shear and convection (caseSB) respectively. As a matter of fact, the turbulent kineticenergy is produced by two distinct mechanisms, namelyshear and buoyancy. The turbulent boundary layer can bedriven either by both the two (SB), which is the mostfrequent and typical atmospheric situation, or by one ofthem as the buoyancy (B). Even if this last is for theatmosphere an idealized case, it can occur when the solarradiation is intense and the wind weak (low wind).[26] The dimension of the domain was 5 5 km2 in the

horizontal directions and 2 km in the vertical direction.Simulation started fro tropic conditions. Parameters

like the extension of domain, grid size, geostrophic windsand surface heat flux are listed in Table 2. In order toobtain a quasi-stationary boundary layer we advanced intime the LES code for more than ten large eddy turnovertimes, t* = Zi/w* (Tables 3 and 4).[27] As experimental data we have used the aircraft

profiles taken during the ARTIST (Arctic Radiation andTurbulence Interaction Study) campaign [Hartmann et al.,1999]. These measurements reproduce the polar convectiveboundary layer during a cold outbreak over the ocean atmoderate wind (8–12 ms�1). The stability parameter �Zi/L(L the Monin Obukhov length) ranged from 10 to 20, andthe corresponding surface parameter u*/w* ranged from 0.27to 0.34. This values indicate that convective instabilityplayed the dominant role in organizing the turbulence[Etling and Brown, 1993; Moeng and Sullivan, 1994;Lupkes and Schlunzen, 1996]. The meteorological condi-tions of the field experiment correspond to a shear-buoy-ancy case even though the boundary conditions cannot beexactly the same as the LESs and the RANS simulation.However the experimental data are considered for compar-ison using non dimensional coordinates. Moreover, due tothe light wind, the measured data can be used as abenchmark also for the purely buoyancy case, which is anidealized case.

5. Numerical Solutions and Results

[28] As said before, in order to test the new ensembleaverage model we have investigated the development of anunsheared (case B) and a sheared (case SB) buoyancy-driven boundary layer, both heated from below at a constantand uniform rate switched on at the initial time. Thephysical parameters and the initial and boundary conditionswere chosen to simulate the conditions of our large eddysimulations. Equations (1)–(2), (3), (4)–(6) and (14)–(17)were numerically solved using a standard finite differencesnumerical scheme, with forward derivatives in space andtime. The horizontal homogeneity hypothesis allows all thespatial derivatives but the vertical to be set to zero. Furtherall the terms involving the mean vertical velocity areneglected. We have adopted a staggered vertical grid: theSOMs and the dissipation, �, were evaluated on the outergrid, while the mean fields and the TOMs were computedon the inner grid. Therefore boundary conditions for theSOMs and the dissipation rate are required only. They wereset as a function of the surface layer parameters and of thestability (see Table 5): Q* is the surface layer heat flux, w* =[gaQ*Zi]

1/3 the convective velocity scale, u* the frictionvelocity and q* = Q*/w* the potential temperature scale. Thevalues assigned to these parameters are listed in Tables 3and 4 respectively. As discussed in detail in next subsection,for the case B only five boundary conditions are reportedsince in this simulation it is not needed to solve all the

Table 2. External LES (CFR) Parameters

Case N (LxLyLz) (km) Ug (ms�1) Q*(ms�1 K)

B 256 (5, 5, 2) 10 0.24SB 256 (5, 5, 2) 15 0.03

Table 3. Numerical External Parameters of the Simulations

Case dt (s) dz (m) Zi (m) Q*(ms�1 K) Ug (ms�1) Vg (ms�1)

B 0.04 26 1060 0.19 . . . . . .

SB 0.04 13 1006 0.028 15 0

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SOMs equations. It can be noted that, in case SB, thecoefficient of u2 is about twice that of v2 as suggested byDobrinski et al. [2004]. The coefficients for the momentumfluxes are chosen so that the surface value of vw is one tenthof uw according to experimental and simulation data[Moeng and Sullivan, 1994; Hartmann et al., 1999]. Atthe upper boundary all turbulent quantities are taken to bezero.[29] As said before, the simulations results have been

compared with the LES (CFR) (the profile of mean compo-nents, SOMs and TOMs were available) and, for theavailable quantities, with the LES (MS) by Moeng andSullivan [1994]. All the simulations were carried out forabout 8 large eddy turnover time (see Table 4 for numericalvalues), in order to match the potential temperature profileof the LES.[30] The model resolution was 26 m for the B simulation

and 13 m in the SB simulation consequently, the number ofgrid points in the BL was about 40 and 77 respectively (seeTable 3), allowing to well resolving the boundary layer.

5.1. Unsheared Buoyancy-Driven Boundary Layer:Case B

[31] Firstly we simulated a buoyancy-driven boundarylayer without shear. To simulate this case of pure convec-tion, we took U = 0, V = 0; this simplification implies thatthe model does not solve the full system of 34 dynamicequations but only some of these. In particular, besides theequations for the mean potential temperature, there are fourSOMs to be considered, the equations for the dissipationrate and five TOMs.

Q; u2;w2; qw; q2; �; u2w;w3; qw2; q2w; q3: ð23Þ

where q2 = w2 + 2 u2 according to Cheng et al. [2005]. Thevalues of the model parameters and constants are listed inTables 1, 3 and 4.[32] In Figures 1 and 2, the results of a simulation

performed with a second-order model based on the closurescheme for the TOMs proposed by Daly and Harlow [1970]are included for sake of comparison (hereafter DH model).[33] The model initial condition corresponds to a poten-

tial temperature profile increasing with height which repre-sents a stable condition of the flow. At the top of theboundary layer, a strong thermal inversion characterizesthe profile. Then the heating of the boundary layer from

below at a constant rate, Q* (see Table 3), generates a regionof instability near the surface in which the profile ofpotential temperature decreases with height. Above thislayer the buoyancy forces provide and maintain a well-mixed layer in which the potential temperature is quiteuniform. Figure 1 shows the final profile of potentialtemperature; as can be seen, the agreement of the modeloutput with the LES is good, the superadiabatic conditionnear the surface is reached and a strong mixing layer isobtained; in the upper boundary layer, at the inversion layer,the model output accurately reproduces the LES data.[34] In Figures 2 and 3, the SOMs and TOMs profiles are

compared with the two LESs: CFR (solid line) and, whenavailable, MS (thin dotted line), with the aircraft measure-ments (crosses) and with the second-order DH model (dots).Each turbulent quantity is properly normalized with appro-priate combinations of the convective velocity (w*) and thepotential temperature (q*) scales (see Table 4 for theirnumerical values).[35] The q2 profile (Figure 2a) shows efficient mixing in

the middle part of the boundary layer and the modelreproduces well this feature although underestimates theLES (CFR) profile. The agreement with the data LES (MS)is satisfactory.[36] The heat flux vertical profile, reported in Figure 2b,

decreases linearly with height and becomes negative at theinversion layer, which is associated to the entrainment ofthe overlying air. The agreement with both the LESs and theaircraft data is good. From Figures 1 and 2, it can beargued that the TOM model is generally superior in repro-ducing the LES data than the lower order closure (DHmodel).[37] The TOMs are reported in Figure 3. As said in the

previous sections, the triple correlations are essential for acorrect description of the nonlocal properties of the flow.The results obtained with the TOM model generally agreesatisfactorily with both the LES and the aircraft data.Concerning u2w (Figure 3a), the profile predicted by themodel is more similar to the aircraft data, especially inthe lower part of the boundary layer, than to the LES data.The profile rapidly decreases in the middle of the boundarylayer underestimating both the LES profiles. This may bedue to the fact that the third-order model simulated a shear-free flow, while the LES, although very weakly, accountsfor shear effects. In Figure 3b the w3 profile is shown. Thegeneral trend of the LESs is well reproduced by the modeleven if the maximum of the curve occurs at higher levelthan in the LES data.[38] In Figures 3d, 3e, and 3f, the TOMs involving the

potential temperature field are shown. The q2w profilepredicted by the TOM model differs from the LES (CFR)near the surface, as also found in the works of Andre et al.[1982] and Cheng and Canuto [1994], but agrees with the

aircraft data. Concerning qw2 and q3, it can be noted that the

Table 4. Numerical Internal Parameters of the Simulations

Case u* (ms�1) w* (ms�1) t* (s) q* (K)

B 0.35 1.79 592 0.1SB 0.67 0.98 1026 0.028

Table 5. Boundary Conditions

Case u2 v2 w2 uv uw vw qw qu qv q2 e

B 13.8u*2

. . . 0.24w*2

. . . . . . . . . Q*

. . . . . . 2.1q*2 5.1w

*3/Zi

SB 4.9u*2 2.85u

*2 0.06w

*2 0.08u

*2 �0.4w

*2 �0.04w

*2 Q

*�1.7Q

*�0.1Q

*4.5q

*2 21.3w

*3/Zi

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results compare well with both the LES (CFR) andthe experimental measurements. The model output for q3

reproduces well the aircraft data near the surface while theLES (CFR) shows a different behavior. However, it is notunusual that the TOM model (and measurements) featurescannot be reproduced by the LES near the surface [Moengand Rotunno, 1990] since the LES is not completely reliablein the surface layer because of the subgrid limitations.

5.2. Sheared Buoyancy-Driven Boundary Layer:Case SB

[39] The second kind of boundary layer we have inves-tigated represents a more realistic flow characterized by amixture of shear and convection. A flow with similarcharacteristics was considered for model comparison inthe work of Gryanik and Hartmann [2002] but it wasanalyzed according to a mass-flux approach. In a previouspaper [Ferrero and Colonna, 2006], we tested our TOMs inthis kind of flow, as above described. Numerical parametersof the present simulation can be found in Tables 3 and 4while the model constants are listed in Table 1. The model

results were compared with our LES (CFR) and LES (MS,case SB1) [Moeng and Sullivan, 1994] and the aircraftmeasurements utilized for the previous simulation althoughthe flow they represent is characterized by weak sheareffects. The initial condition corresponds, as in the previoussimulation, to a stable profile for the mean potentialtemperature with a capping inversion at the top of theboundary layer. Furthermore, for this simulation the meanvelocity components were also initialized with values equalto the geostrophic wind (see Table 3 for numerical values).As in the case B, a simulation using the DH second-ordermodel was also carried out. The results are shown inFigures 4 and 5 together with the TOM model results.[40] Figure 4 displays the mean potential temperature and

the mean velocity components final profiles. The agreementwith the LES (CFR) is satisfactory across the whole domainand all the features are well reproduced by the modelalthough small discrepancies appear in the layer very closeto the surface, probably due to the boundary conditions. Itcan be noted that the three profiles of mean quantities showa pronounced mixing layer comparable to that of a pureconvective flow. This indicates that even a relatively smallbuoyancy force is strong enough to create a considerablemixing process in the boundary layer.[41] The SOMs are shown in Figure 5. The agreement

of q2 (Figure 5a) with both the LESs can be consideredsatisfactory. Concerning the heat flux vertical profile(Figure 5b), the third-order model output shows a differentslope respect to the LES (CFR) which may depend on thesurface heat flux used in the simulation, which is slightlydifferent than that of the LES simulation. In Figures 5c and 5d,the momentum fluxes uw and vw are reported. The typicalnegative decreasing trend expected for these fluxes is wellreproduced by the model and the agreement with the LES(CFR) data is good. With respect to the aircraft data, the uwprofile yields larger (negative) values, while vw fits theexperimental data.[42] Mean values and all the SOMs are better predicted

by the TOM model than by the second-order DH model,except for the case of qw where the latter reproduces betterthe aircraft data but it shows an overestimation of the LES(CFR) data.

Figure 1. Case B: vertical profile of mean potentialtemperature. Circles represent the third-order model output;solid line, our LES (CFR); and dotted line, the model outputof Daly and Harlow [1970] (DH).

Figure 2. Case B: SOMs normalized vertical profiles of (a) q2 and (b) qw. Circles represent the third-order model output; solid line, LES (CFR); thin dotted line, LES (MS); crosses, aircraft measurements;and dotted lin DH model output.

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[43] This is a very important point, in fact, the resultsdemonstrate that including TOMs in the RANS modelallows not only to reproduce these moments but alsoimproves the simulated SOMs respect to the lower orderclosure.[44] The TOMs obtained in the case SB are reported in

Figure 6. With respect to the LES, the profiles of u2w, v2wand w3 (Figures 6a, 6b, and 6c) simulated using the TOMmodel show an underestimation in the lower and in the midpart of the boundary layer and overestimation in the upper

part. Concerning w3 it can be noted that the model flattensthe characteristic trend of this turbulent quantity. Thecomparison with the aircraft data is not particularly signif-icant for this profile bei aircraft measurements mainly

affected by the buoyancy than by the shear, which insteadinfluences the model result.[45] The overestimation and underestimation found out

for the TOMs previously shown, do not occur for themoments involving the potential temperature field, as canbe observed in (Figures 6d, 6e, and 6f). The vertical flux of

the potential temperature variance (q2w) shows a satisfac-tory agreement with the LES and the experimental data. Inthe surface layer, as also in simulation B, the model differsfrom the LES reproducing better the experimental data.Concerning qw2, the model agrees with the LES while itunderestimates the aircraft measurements in which the

buoyant effect prevails. Finally, the q3 profile matches wellboth the LES and the aircraft data. Also for this profile, in

Figure 3. Case B: TOMs normalized vertical profiles of (a) u2w, (b) w3, (c) q2w, (d) q2w, (e) qw2, and

(f) q3. Circles represent the third-order model output; solid line, LES (CFR); dotted line, LES (MS);crosses, aircraft measurements.

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the surface layer, the model reproduces the trend of theaircraft measurements better than the LES.[46] The two time scales defined by the equations (21)

and (22), more accurately, their ratios to t, are, by physicalarguments, reduced as the integral length increases. Theresults demonstrate that these reductions effectively dampthe growth of the TOMs, as desired. Since the weakest partof the Mellor-Yamada type of second-order closure (SOC)model is the constant time scale ratios, there is a lot of roomof improvement for the standard SOC models. As a matterof fact, recently Canuto et al. [2008] parameterized the timescale ratio tpq/t (where tpq is the time scale of the return toisotropy term of the pressure-potential temperature correla-tion) and this single new ingredient made their earlier model[Cheng et al., 2002] match data up to Richardson numberRi = 100 thus no Ri(critical) exists.

6. Conclusions

[47] In this paper we have presented a Reynolds Stressmodel in which turbulence is treated nonlocally and theresults of two complete simulations of turbulent boundarylayers buoyancy and shear-buoyancy-driven respectively.[48] The model numerically solves a system of dynamical

equations up to the TOMs. To close the system the quasi-normal approximation is adopted for the FOMs. The knownshortcoming of the quas al approximation, namely the

insufficient damping of the third-order correlations in pres-ence of convection, is overcome by adopting a newapproach to evaluate the relaxation and dissipation timescales, which are related to the TOMs turbulence damping.The new parameterizations are now functions of the integrallength scale [Ferrero and Colonna, 2006] and hence depen-dent on height rather than being constant. These dependen-cies provide a proper modulation of turbulent transport. Themodel is able to account for both shear and convection at thesame time thus allowing to simulate complex flows such as,for example, the actual atmospheric boundary layer.[49] Nonlocal treatment of turbulent quantities is an

essential requirement for a successful description of thoseflows involving convection. The intrinsic nature of theseflows is in fact nonlocal being characterized by largeenergetic eddies which transport energy through the mixinglayer. The nonlocality of turbulence is also important in thecase of pure shear flows as shown by Ferrero and Racca[2004] and Ferrero [2005]. Many authors [Canuto, 1992;Zilitinkevich et al., 1999; Gryanik and Hartmann, 2002]also stressed that the triple correlations should be correctlytreated to provide realistic Reynolds stress.[50] The results are compared with a LES performed by

us (CFR), the LES of Moeng and Sullivan [1992] (MS) andaircraft measurements [Hartmann et al., 1999] and prelim-inarily with a DNS [Moser et al., 1999]. Many of theturbulent quantities profiles presented in this work, in fact,show differences between LES and aircraft measurementsand the third-order model often reproduces the experimentaldata rather than the LES, especially near the surface layer.Use of LES data and experimental measurements is partic-ularly significant to test the validity of the model, as pointedout in previous work [Moeng and Rotunno, 1990]. Further-more the comparison with two different LES data set haveshown that there may be differences also between the largeeddy simulations themselves, especially near the surfacelayer and at the bottom of the boundary layer, near theinversion layer.[51] Most of the profiles presented in the paper are in a

satisfactory agreement with both LES and experimentaldata, thus demonstrating that our model is able to performsimulations of boundary layers where the turbulence isproduced by mechanical and thermal processes separatelyor by a mixture of them as it occurs in the usual atmosphericflows. The success in correctly reproducing the TOMsconfirms that the damping proposed in this work is correct,further it satisfies the realizability constraints. The reductionof scale ratios may be also useful for developing newsecond-order models.[52] Finally, the comparison with a second-order model

[Daly and Harlow, 1970] demonstrates that to correctlyreproduce the turbulence features in the atmospheric bound-ary layer, the nonlocal transport must be included, and thisis accomplished here by accounting for the TOMs.

Appendix A: Modeled Terms in the SOMS andTOMS Equations

[53] In equation (4) the term including the pressurevelocity correlation gradient (� @

@xk

23puk) has been omitted

[see Cheng et al., 2005], its contribution being negligible.The pressure correlation terms Pij and Pi

q can be modeled

Figure 4. Case SB: vertical profiles of (a) mean potentialtemperature and (b) mean horizontal wind velocity compo-nents. Circles represent the third-order model output; solidline, LES (CFR); and dotted line, the DH model output.

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(neglecting rotation) considering three distinct contributionsaccounting for the three sources from which pressurefluctuations arise:

Pij ¼ P1ð Þij þP

2ð Þij þP

3ð Þij ; ðA1Þ

Pqi ¼ P

qi1 þP

qi2 þP

qi3; ðA2Þ

the turbulence self-interaction represented by Rotta ‘‘returnto isotropy’’ term (Pij

(1) and Pi1q ), the contribution due to the

turbulence-buoyancy interaction (Pij(2) and Pi2 q) and those

of the turbulence-mean flow interaction (Pij(3) and Pi3

q ),respectively. The explicit form of the pressure-velocity-gradient Pij is prescribed following Canuto [1992].

P1ð Þij ¼ 2c4t

�1bij; ðA3Þ

P2ð Þij ¼ c5 liquj þ ljqui �

2

3dijlkquk

� �

; ðA4Þ

�P3ð Þij ¼ 2a1 Sikbkj þ Skjbik �

2

3dijSklblk

� �

þ 2a2

� Rikbkj þ Rjkbik�

þ2a0q2Sij: ðA5Þ

with

bij ¼ uiuj �1

3q2dij; ðA6Þ

2Sij ¼@Ui

@xjþ@Uj

@xi; ðA7Þ

2Rij ¼@Ui

@xj�@Uj

@xi: ðA8Þ

[54] For the pressure gradient-potential temperature cor-relation Pi

q we have [Canuto, 1992]

Pqi1 ¼ 2c6t

�1uiq; ðA9Þ

Pqi2 ¼ c7liq

2; ðA10Þ

Pqi3 ¼ �

4

5am

@Uk

@xjdikquj �

1

4dkjqui þ dijquk� �

� �

: ðA11Þ

In equations (A3) and (A9) the turbulence life time ismodified as in the work of Cheng et al. [2005].

Figure 5. Case SB: SOMs normalized vertical profiles of (a) q2, (b) qw, (c) uw, and (d) vw. Circlesrepresent the third-order model output; solid line, LES (CFR); thin dotted line, LES (MS); crosses,aircraft measurements; and dotted line, the DH model output.

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[55] In the LHS of equation (10) the second term repre-sents the derivative of the turbulent flux of �, here evaluatedas

�uj ¼ �a�nt@�

@xjðA12Þ

where a� is a constant (Table 1) and vt is the eddy viscosity,defined according to the Prandtl-Kolmogorov hypothesis as

nt ¼ cmE2

�ðA13Þ

where cm is an empirical constant. The down gradientapproximation for the � flux may be a limitation for thenonlocal transport, although the dissipation is due to thesmaller eddies and hence it can be considered a localphenomenon.[56] In the above equations ci (i = 4, 5, 6, 7), c*, am, a1,

a2, a0 are constants of the model whose values are listed inTable 1.[57] As a matter of fact, the pressure models used in this

paper have zero trace and therefore are pressure-strainmodels. The pressure transport is therefore being neglected(it has a nonzero trace). Pressure transport can be parame-terized together with the TOMs since they are the samemagnitude and act in similar ways physically. The pressure

Figure 6. Case SB: TOMs normalized vertical profiles of (a) u2w, (b) v2w, (c) w3, (d) q2w, (e) qw2,

and (f) q3. Circles represent the third-order model output; solid line, LES (CFR); dotted line, LES (MS);crosses, aircraft measurements.

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triple correlation terms present in the dynamical equationsof the TOMs are modeled following Andre et al. [1982]:

Pijk ¼ t�13 uiujuk þ c11 uiujqlk þ perm:

� �

; ðA14Þ

Pqij ¼ t�1

3 uiujqþ c11 ljuiq2 þ liujq

2 �2

3dijlkukq

2

� �

� c*t�1dijq2q; ðA15Þ

Pqqi ¼ t�1

3 uiq2 þ c11liq

3: ðA16Þ

where t3 = t/(2c8) and c11, c* are constants whose valuesare reported in Table 1. c8 is a coefficient, related to theturbulent damping, dep g on the integral length scale

discussed in section 3 [Ferrero and Colonna, 2006]. It canbe noted that the last term in the parenthesis, in the secondof the above equations, can be set to zero in order to ensurethe realizability conditions satisfaction, as suggested byCheng et al. [2005].

Appendix B: Preliminary Test With DNS Data

[58] A preliminary test was performed by comparing theresults of our model with a DNS data set [Moser et al.,1999]. This refers to a fully developed turbulent channelflow characterized by a Reynolds number Re = 590 (Ret =u*d/v). Because this DNS refers to a pure neutral flow, weused our model in its version for shear-driven boundarylayer [Ferrero and Racca, 2004] and the boundary con-

Figure B1. Normalized vertical profiles of (a) mean horizontal wind velocity component U, (b) u2, (c) v2,(d) w2, and (e) uw. Crosses represent the third-order model output; dotted line represents the DNS data.

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ditions were set as a function of the friction velocity. Thewall-normal profiles of mean velocity and some SOMs areshown in Figure B1. It can be observed that the agree-ment between RANS model and DNS can be consideredsatisfactory except for a thin layer near the wall. This is notsurprising because the spatial resolutions of the two modelare very different. However the aim of our model is tocorrectly reproduce the main turbulent features in the wholeboundary layer and this objective is achieved as is shown inthis paper. Moreover, it should be stressed that the Reynoldsnumber of the DNS is lower than that of a typical atmo-spheric flow (even if it is enough high to guarantee a fullydeveloped turbulence), which can be obtained only usinglarge eddy simulations. The lower Reynolds number canexplain the small discrepancy found between DNS andRANS results, in fact, as shown in the Moser et al. [1999]paper, where the results of simulations at different Reynoldsnumber are presented, different Reynolds numbers yieldslightly different results of the simulated flow.

[59] Acknowledgments. We would like to thank J. Hartmann forkindly providing us with the aircraft data and V. Gryanik and V. M. Canutofor useful discussions.

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�����������������������

N. M. Colonna and E. Ferrero, Dipartimento di Scienze e TecnologieAvanzate, Universita del Piemonte Orientale, via Bellini 25, Alessandria I-15100, Italy. (colonna@unipmn.it; enrico.ferrero@mfn.unipmn.it)U. Rizza, Consiglio Nazionale delle Ricerche, Istituto di Scienze dell’

Atmosfera e del Clima, Str. Prov. Lecce-Monteroni Km 1,200, LecceI-73100, Italy. (u.rizza@isac.cnr.it)

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