LES in the Surface Layer: Surface Fluxes, Scaling, and SGS ...

VOL. 55, NO. 10 15 MAY 1998J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 1998 American Meteorological Society 1733

LES in the Surface Layer: Surface Fluxes, Scaling, and SGS Modeling

J. C. WYNGAARD,* L. J. PELTIER,1 AND S. KHANNA

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

(Manuscript received 9 January 1996, in final form 27 August 1997)

ABSTRACT

The surface fluxes in the fine-mesh numerical codes used in small-scale meteorology are typically diagnosedfrom resolvable-scale variables through surface-exchange coefficients. This is appropriate if the aspect ratio(length/height) of the grid volume adjacent to the surface is very large, as in mesoscale models. The aspectratio can approach unity in large-eddy simulation (LES) codes for the planetary boundary layer, however. Inthat limit the surface-exchange coefficients are random variables, and it is shown through analysis of surface-layer measurements and LES results that their fluctuation levels can be large.

As an alternative to surface-exchange coefficients, the authors derive conservation equations for the surfacescalar and momentum fluxes in LES. Scaling relations for resolvable-scale variables in the surface layer aredeveloped and used to simplify these equations. It is shown that, as the grid aspect ratio decreases toward unity,local time change, horizontal advection, production due to horizontal velocity convergence, and random noiseterms cause the local surface-exchange coefficients to fluctuate. A simple closure of the equations is adopted,which has little effect on surface-layer structure calculated through LES with a Smagorinsky-based subgrid-scale (SGS) model. Through analysis of very high-resolution LES fields, the authors find the SGS model to bea poor representation of surface-layer physics and conclude that the surface-flux conservation equations needto be coupled with a greatly improved SGS model in the surface layer.

1. Introduction

Large-eddy simulation, or LES, is now very widelyused in small-scale meteorology. Applications rangefrom severe storms (Klemp 1987) to small-mesoscalephenomena (Cotton et al. 1993) and boundary layermeteorology (Moeng 1984; Mason and Thomson 1987;Nieuwstadt et al. 1993; Schumann 1993).

The strength of LES lies in its explicit calculation ofthe energy-containing eddies of a turbulent flow. Theunresolvable or subgrid-scale (SGS) eddies, which areformally removed by spatial filtering of the governingequations (Leonard 1974), manifest themselves throughSGS terms in the filtered or resolvable-scale equations.If the scale of the filter is small compared to the scaleof the energy-containing eddies, the resolvable-scale ed-dies do contain most of the turbulent kinetic energy andfluxes. Furthermore, they are fairly insensitive to the

* Also Department of Mechanical Engineering, The PennsylvaniaState University, University Park, Pennsylvania.

1 Current affiliation: Applied Research Laboratory, The Pennsyl-vania State University, University Park, Pennsylvania.

Corresponding author address: Dr. John C. Wyngaard, Departmentof Meteorology, 503 Walker Building, The Pennsylvania State Uni-versity, University Park, PA 16802-5013.E-mail: [email protected]

details of the SGS model used to close the filtered equa-tions. Thus, well-resolved LES has an appealing ro-bustness.

Because the horizontal length scale of the verticalvelocity fluctuations scales with distance from the sur-face, LES inevitably has inadequate spatial resolutionnear the surface. As a result, the SGS model takes onmuch more importance there than in other regions ofthe flow. Furthermore, the specification of resolvable-scale surface fluxes comes into question since the widelyused surface-exchange coefficients are not obviouslyvalid locally. Thus, the fidelity of LES within the surfacelayer is problematic.

In this paper we study three closely related aspectsof LES in the surface layer. First we examine the be-havior of the surface-exchange coefficients when thegrid aspect ratio (horizontal dimension/vertical dimen-sion) is of order 1, as in LES. As an alternative tosurface-exchange coefficients, we develop conservationequations for the resolvable-scale surface fluxes. Sim-plifying these surface-flux equations requires our seconddevelopment, scaling relations for LES variables in thesurface layer. These determine how turbulent variancesare apportioned between the resolvable and SGS com-ponents. Finally, in order to interpret the impact of theresolvable-scale surface-flux equations on LES fields weevaluate some aspects of the performance of the widelyused eddy-diffusivity SGS model in the surface layer.

Unauthenticated | Downloaded 01/07/22 09:16 PM UTC

1734 VOLUME 55J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. A schematic of the ‘‘coupling eddies’’ in the grid volume adjacent to the surface. Theycouple the resolvable-scale flow at height z1 to the SGS surface flux. For a unity aspect ratio grid(left) there is only one such coupling eddy; for a large aspect ratio (right) there are many.

2. Surface-exchange coefficients forresolvable-scale fluxes

a. Background

The LES codes used in boundary layer meteorology(e.g., Nieuwstadt et al. 1993) involve the surface fluxesof momentum, temperature, water vapor, and trace con-stituents. Since these codes use local spatial averagingof the governing equations rather than ensemble aver-aging, these are resolvable-scale surface fluxes that con-tain not only the ensemble mean but also the fluctuationsabout this mean on scales up to the filter cutoff. Mostsuch codes take this resolvable-scale surface flux as pro-portional to the product of the horizontal wind speedand the difference in the transported quantity betweenthe surface and the first grid point. The proportionalityfactor is the surface-exchange coefficient.

Surface-exchange coefficients for ensemble-averagefluxes under horizontally homogeneous conditions areknown from several decades of surface-layer research.In coarse-resolution (e.g., mesoscale) models, the hor-izontal scale D of the local spatial averaging and of thenumerical grid is typically much larger than the heightz1 of the first grid level. The grid aspect ratio D/z1 islarge, meaning that there are many ‘‘coupling eddies’’in the first grid volume, as illustrated in Fig. 1. In thiscase these mean exchange coefficients, as we will callthem, are applicable. In LES, however, the grid aspectratio is typically in the range from 1 to 3; there are notmany coupling eddies and the surface-exchange coef-ficient has a random component. Thus, for (x1, x2, t),rF 0

the resolvable-scale surface flux of a scalar constituent,for example, the appropriate surface-exchange coeffi-cient is not the mean value CH but a random variable,CH. (A superscript r, for resolvable scale, indicates thatthe variable has been spatially filtered; a tilde denotesthat it has both mean and fluctuating parts.) Thus, theexpression for surface flux is more properly written as

rF (x , x , t)0 1 2

r r r˜5 C (x , x , z , t)[u (x , x , z , t)Dc (x , x , z , t)] .H 1 2 1 1 1 2 1 1 2 1

(1)

Here and Dcr are the streamwise components of re-ru1

solvable-scale wind at the first grid level and the re-solvable-scale c difference between the first grid pointand the surface, respectively. We call CH the local sur-face-exchange coefficient; we expect that CH → CH asz1/D → 0.

b. Observed fluctuations in local surface-exchangecoefficients

Consider two LES grids of horizontal spacing D, inone the first grid point being close to the surface, at zs

K D, and in the other at a greater height z1 ø D. Inview of these grid aspect ratios, we take the scalar-fluxsurface-exchange coefficients as CH(zs) for the first gridand CH(z1) for the second. We can write from (1), in-dicating only the z dependence,

r r r rF 5 C (z )[u (z )Dc (z )]0 H s 1 s s

r r r˜5 C (z )[u (z )Dc (z )] . (2)H 1 1 1 1

Solving for the local surface-exchange coefficient atheight z1 yields

r r r[u (z )Dc (z )]1 s sC (z ) 5 C (z ) . (3)H 1 H s r r r[u (z )Dc (z )]1 1 1

We decompose the resolvable-scale fields into the sumof the ensemble mean and a fluctuating part, using uppercase symbols for the former:

r r r ru 5 U 1 u , c 5 C 1 c ,i i i

r rDc 5 DC 1 Dc . (4)

Using the decomposition (4) and expanding (3) yields

DC(z )U (z )s 1 sC (z ) 5 C (z )H 1 H s DC(z )U (z )1 1 1

rr rDc (z ) u (z )s 1 s1 1 1 11 2 1 2[ ]DC(z ) U (z )s 1 s

3 . (5)r

r rDc (z ) u (z )1 1 11 1 1 11 2 1 2[ ]DC(z ) U (z )1 1 1


15 MAY 1998 1735W Y N G A A R D E T A L .

We evaluated (5) using LES data for the convectiveboundary layer and observations from the STORM-FEST experiment. The LES domain was 5 km 3 5 kmin the horizontal and 2 km deep. It used a nested meshin the surface layer so the resolution there was equiv-alent to that of a 2563 simulation (Khanna and Brasseur1996). The inversion height zi was about 1000 m. Inone run the geostrophic wind was 5.0 m s21 and thesurface temperature flux was 0.14 m s21 K, giving 2zi/L 5 63. The other run had a geostrophic wind of 1.0m s21 and a surface temperature flux of 0.24 m s21 K,giving 2zi/L 5 730. We took zs 5 3.9 m, the lowestgrid point, and evaluated the local surface-exchange co-efficients at z1 5 12 and 27 m, using temperature as thescalar.

The STORMFEST data, taken by the NCAR Atmo-spheric Surface Transfer and Exchange Research facil-ity, included time series of fluctuating temperature andthe horizontal components of velocity from in situ sen-sors at z 5 1 m, 4 m, and 10 m. The mean wind speedat 10 m was 10.7 m s21, the surface temperature fluxwas 0.23 m s21 K, the Monin–Obukhov length L was263 m, and 2zi/L ø 10. We took zs 5 1.0 m, the lowest

measurement point, and evaluated the local surface-exchange coefficients for temperature at z1 5 4 m and10 m.

The randomness factor in (5) involves wave-cutofffiltering at wavenumber magnitude kc in the horizontalwavenumber plane. We did this straightforwardly withthe LES data, using several values of kc 5 p/D. Thisgave zs/D # 0.13. For the STORMFEST data we ap-proximated the averaging by wave-cutoff filtering of thetime series at frequency vc, using Taylor’s hypothesisin the form kc 5 vc/U. Here zs/D # 0.10.

Figure 2 shows the fluctuations in the local surface-exchange coefficient CH and its counterpart for mo-mentum, CD, evaluated from measured time series andLES data, using (5) and its counterpart for CD. Thefluctuation level increases as grid aspect ratio decreases,as we anticipated through the coupling-eddies notiondepicted in Fig. 1, and it increases as 2zi/L increases.It can be quite large.

As discussed by Wyngaard and Peltier (1996), we cangain insight into the fluctuations of the local surface-exchange coefficient through a simple model of Eq. (5).If Dcr/DC and /U1 are small parameters, we can lin-ru1

earize (5) to find

r r r rDC(z )U (z ) Dc (z ) u (z ) Dc (z ) u (z )s 1 s s 1 s 1 1 1C (z ) 5 C (z ) 3 1 1 1 2 2 . (6)H 1 H s 1 2DC(z )U (z ) DC(z ) U (z ) DC(z ) U (z )1 1 1 s 1 s 1 1 1

Taking the ensemble mean gives

DC(z )U (z )s 1 sC (z ) 5 C (z ) 5 C (z ), (7)H 1 H s H 1DC(z )U (z )1 1 1

which indicates that for small fluctuation levels the meanof the local surface-exchange coefficient is the tradi-tional surface-exchange coefficient. For sufficientlylarge fluctuation levels the two can differ. In the con-vective surface layer the fluctuation level of Dcr/DC istypically much less than that of /U1, so that we canru1

write (6) as

r ru (z ) u (z )1 s 1 1C (z ) ø C (z ) 1 1 2 , (8)H 1 H 1 1 2U (z ) U (z )1 s 1 1

which implies that

22 r r˜(C (z ) 2 C (z )) u (z ) u (z )H 1 H 1 1 s 1 1ø 2 . (9)2 1 2C (z ) U (z ) U (z )H 1 1 s 1 1

As D/z1 becomes large, the separation z1 2 zs becomessmall compared to the horizontal scales of the fluctu-ations in /U1. Thus, as D/z1 increases, we expect (zs)r ru u1 1

and (z1) to become increasingly well correlated andru1

their rms difference to decrease; from (9) the fluctuationlevel in the local surface-exchange coefficient should

then approach zero. This is consistent with the resultsin Fig. 2.

This suggests that the surface-exchange coefficientfluctuations are due to the decreasing correlation be-tween resolvable-scale fields very near the surface andat z1 as kcz1 → 1. Further analysis of these LES andSTORMFEST data by Wyngaard and Peltier (1996) sup-ports this notion.

An alternative to treating the resolvable-scale surfacefluxes through surface-exchange coefficients is predict-ing them directly through their conservation equations.We discuss this next, focusing on the scalar flux. Wetreat the momentum flux in appendix C.

3. The subgrid-scale flux near the surface

The Boussinesq equations for velocity ui and a con-servative scalar c in the surface layer are

]u gi ˜1 u u 5 2p 1 ud 1 nu , u 5 0, (10)i , j j , i 3i i , j j i,i]t T0

]c1 (cu ) 5 gc , (11)j , j , j j]t

where p is kinematic pressure, g is the molecular dif-



fusivity of c, and u is potential temperature (Lumleyand Panofsky 1964; Stull 1988). We assume Corioliseffects are negligible. We spatially low-pass filter eachfield over horizontal planes (Leonard 1974), designatingthe part that passes through the filter as resolvable andthe remainder as subgrid scale, denoted with super-scripts r and s, respectively. This yields the decompo-sition

ui 5 1 ,r su ui i c 5 cr 1 cs,

5u 1r s˜ ˜u u , p 5 pr 1 ps. (12)

We assume that in horizontal wavenumber space thefilter has a sharp cutoff at wavenumber magnitude kc

5 p/D f , where D f is the filter width in physical space.In principle D f need not be the same as Dg, the horizontalgrid spacing. As pointed out by Mason and Callen(1986), Df defines the spatial scales contained in theresolvable fields, whereas Dg determines the accuracyof the numerical solutions of the filtered equations. Onecould obtain very high accuracy by taking Dg K Df ,for example. Computer limitations require that Df andDg be essentially the same value, however, so we willnot distinguish them further here, taking Df 5 Dg 5 D.With our sharp cutoff filter the r and s fields have nowavenumber components in common; this implies thattheir covariance vanishes since nonoverlapping Fouriermodes are uncorrelated (Lumley and Panofsky 1964).

Equation (11) holds for any advected scalar that isconservative, that is, has no sources or sinks. Filteringit yields the evolution equation for its resolvable-scalepart:

r]cr r r r s s r s s r1 (c u ) 1 (c u 1 c u 1 c u ) 5 gc . (13)j , j j j j , j , j j]t

We write this as

r]cr r r r˜1 (c u ) 1 F 5 0,j , j j,j]t

r r s s r s s rF 5 (c u 1 c u 1 c u ) 2 gc . (14)j j j j , j

Naming is a challenge since it involves four terms,rF j

multiple filtering operations, and both turbulent and mo-lecular components. The name ‘‘resolvable-scale flux’’does not distinguish it from (cr )r. Its molecular termruj

is important only in a very thin layer adjacent to thesurface, and each of its three remaining terms has asubgrid-scale contribution. Thus, we will call therF j

subgrid-scale (SGS) flux.The terms in (14), being resolvable scale, are bounded

as z → 0 (we will use x3 and z interchangeably to denoteheight above the surface). In particular, is boundedrF 3,3

so that for sufficiently small z, (z) ø , the resolv-r r˜ ˜F F3 0

able-scale surface flux. Thus, we expect a ‘‘constant-SGS-flux’’ layer near the surface within which is arF 3

surrogate for the resolvable-scale surface flux. In section4 we will confirm this through scaling estimates.

We decompose each resolvable-scale field into the

sum of the ensemble mean and a fluctuating part, usingupper case symbols for the former:

r r r r r r˜u 5 U 1 u , c 5 C 1 c , u 5 Q 1 u ,i i i

r r r r˜p 5 P 1 p , F 5 F 1 F . (15)j j j

Since the ensemble means of the subgrid-scale field andof the fluctuating part of the resolvable-scale field van-ish, we can write

5 , cs 5 cs, 5 us, ps 5 ps. (16)s s s˜u u ui i

At heights z sufficiently large that the molecular con-tribution to the SGS flux is negligible the vertical com-ponent of the flux is, from (14)–(16),

r r s s r s s rF 5 [(C 1 c )u 1 c (U 1 u ) 1 c u ]3 3 3 3 3

r s s r s s r5 (c u 1 c u 1 c u ) , (17)3 3 3

assuming a wave-cutoff filter.We can determine the relative magnitudes of the three

components of the SGS flux by using the Peltier et al.(1996) model of the two-dimensional spectrum E(k) inthe surface layer, where k is horizontal wavenumbermagnitude ( 1 )1/2. This spectrum integrates to the2 2k k1 2

variance

`

E(k) dk 5 variance. (18)E0

They used a single form for both horizontal velocityand a conservative scalar,

2 2c l s k1E(k) 5 , (19)2 4/3[c 1 (kl) ]2

with c1 and c2 adjustable constants and l and s the lengthand intensity scales. For vertical velocity they multipliedthis form by a transfer function that models the effectsof continuity and the u3 5 0 surface condition by at-tenuating E(k) at the smallest wavenumbers.

The spectrum (19) has a peak at k ; 1/l and a k25/3

inertial range asymptote at k k l21. Peltier et al. chosethe constants c1 and c2 by requiring the variance andthe inertial-range spectral level to agree with observa-tions. They chose the length and intensity scales l ands as z and u* under neutral conditions. In free convectionthey chose the scales as the PBL depth zi and the con-vective velocity scale w* 5 (gQ0zi/T0)1/3, rather thanthe local-free-convection scales z and uf 5(gQ0z/T0)1/3 (Wyngaard et al. 1971); here Q0 is the en-semble-mean surface temperature flux. As justificationthey cited the evidence that horizontal velocity fluctu-ations in the unstable surface layer are not Monin–Obu-khov (M–O) similar, but instead scale with the mixed-layer scales zi and w* (Kaimal 1978; Wyngaard 1988).

Peltier et al. then combined the neutral and free con-vection forms to give an interpolation formula for theentire stability range between them. The spectrum E(k)cannot be directly measured with conventional instru-



FIG. 2. Behavior of the local surface-exchange coefficients as calculated from high-res-olution LES and STORMFEST data taken by the NCAR ASTER facility. Circles: STORM-FEST at 4 m (open) and 10 m (solid); 2zi/L ø 10. Squares: LES at 12 m (open) and 27m (solid); 2zi/L ø 63. Triangles: LES at 12 m (open) and 27 m (solid); 2zi/L ø 800. Plussymbols: LES at 16 m using surface-flux equations.

ments, but Peltier et al. showed that the correspondingone-dimensional spectra agree well with existing mea-surements.

The modeled two-dimensional spectra are shown in

Fig. 3 in area-preserving coordinates. For all but near-neutral conditions the horizontal spectrum Eh, whichintegrates to ( 1 )/2, peaks at wavenumbers of the2 2u u1 2

order of kh 5 1/zi. The vertical velocity spectrum Ey



peaks at larger wavenumbers, those of the order of kz

5 1/z. A third significant wavenumber is kc 5 p/D, thecutoff wavenumber of the LES grid mesh. In Moeng’s963 LES code, for example, D is typically 5000 m/64. 80 m.

Let us now examine the contributions to the SGS fluxas shown in (17) for heights z K D and inversion depthszi k D. Since z K D K zi, there is a large separationin these three wavenumbers: kh K kc K kz. Let us write

each factor in the first cross term, (cr )r, as a Fourier–su3

Stieltjes integral, noting the admissible region in the k1,k2 plane:

r ik9 xj jc (x ) 5 e dT(k9, |k9| # k ),i E j j c

s ik0 xj ju (x ) 5 e dW(k0, |k0| . k ). (20)3 i E j j c

It follows that

r s r i(k91k0)xj j j(c u ) 5 e dT(k9, |k9| # k ) dW(k0, |k0| . k ). (21)3 E E j j c j j c

|k91k0 |,kj j c

Since | | # kc, the restriction | 1 | # kc impliesk9 k9 k0j j j

that | | , 2kc. Thus, only u3 modes of wavenumberk0jmagnitude less than 2kc can contribute to (cr )r. Figuresu3

3 shows that, as we approach the surface, 2kc falls belowthe range of wavenumbers contributing significantly to

u3 so that (cr )r → 0 as z/D → 0. The second crosssu3

term in (17), (cs )r, is also eliminated because → 0r ru u3 3

as z/D → 0.We conclude that for z/D K 1 the SGS flux is due

entirely to the third term in (17),

r s s r i(k91k0)x˜ j j jF ø (c u ) ø e dT(k9, |k9| . k ) dW(k0, |k0| . k ), (22)3 3 E E j j c j j c

|k91k0 |,kj j c

so that the SGS flux is carried by cs and modes whosesu3

wavenumbers are beyond the cutoff but whose differ-ence wavenumber falls within the cutoff.

Since at heights z much smaller than the scale of thespatial filter (i.e., z/D K 1) the SGS flux (z) ørF 3

(cs )r is a good surrogate for , we can form a con-s r˜u F3 0

servation equation for :rF 0

r rr s s r s s˜]F ](c u ) ]u ]c0 3 3s sø 5 c 1 u , z/D K 1.31 2 1 2]t ]t ]t ]t(23)

The and cs equations are derived by high-pass filteringsui

(10) and (11). Carrying out the operations in (23) andsimplifying as discussed in appendix A yields the resolv-able-scale surface flux conservation equation, or budget:

]rF ø0]t

r r s s r2((u u ) c )3 , j j resolvable shear, cross flux interaction (RSCFI)

r r s s r2 ((c u ) u ), j j 3 resolvable gradient, cross stress interaction (RGCSI)r s s r r2 (u (u c ) )3 , j j resolvable-scale shear production (RSSP)r r r˜2 (u F )j 3 , j large-scale advection (LSA)

r˜2 U Fj 0, j mean advection (MA)s s s r2 (u u c )j 3 , j small-scale advection (SSA)r s s r r2 (c (u u ) ), j j 3 resolvable-scale gradient production (RSGP)

s s r2 C (u u ),3 3 3 mean gradient production (MGP)



gs s r1 (u c )

T0

buoyant production (BP)

s s r2 (p c ),3 pressure destruction (PD). (24)

FIG. 3. The two-dimensional, surface-layer spectra of a scalar(dashed line), vertical velocity (bold line), and horizontal velocity(fine line), as modeled by Peltier et al. (1996). The vertical linesindicate the kz value below which lies 50% of the variance. Top:neutral; center: sightly unstable; bottom: free convection.

Let us relate Eq. (24) to the more familiar budget ofthe Reynolds flux cu3 , where the overbar denotes theensemble mean. Consider a horizontally homogeneousflow and let kc → 0. In this limit each filtered field hasonly a k 5 0 component, the horizontal mean value,which because of the homogeneity is equal to the en-semble mean. Thus, → 0, cr → 0, and our decom-rui

position (15) and (16) reduces to

ui 5 Ui 1 ,sui c 5 C 1 cs,

5u Q 1 us, p 5 P 1 ps. (25)

It follows thats sas k → 0, u → u , c → c,c i i

s su → u, p → p. (26)

In this limit the first through fifth and the seventh termson the right side of the SGS flux budget (24) vanish.Thus, as kc → 0 the SGS flux budget becomes, underhorizontally homogeneous conditions,

] gu c 5 2(u u c) 2 C u u 1 uc 2 p c . (27)3 3 3 ,3 ,3 3 3 ,3]t T0

This is the budget of the vertical component of the Reyn-olds scalar flux (Wyngaard et al. 1971).

We need to scale the terms in the resolvable-scalesurface flux conservation equation (24) so that we candrop the smallest ones. Monin–Obukhov similarity (Pa-nofsky and Dutton 1984) provides expressions for thevariances of velocity and scalar fluctuations in the sur-face layer, but here we need to know also how the par-titioning of these variances between resolvable and sub-grid-scale contributions,

f 2 5 ( f r)2 1 ( f s)2 , (28)

depends on distance from the surface and the cutoffwavenumber of the filter. We develop these scaling re-lations next.

4. Scaling LES fields in the surface layer

Consider a zero-mean random variable f 5 f r 1 f s

(a fluctuating velocity component or scalar) in the sur-face layer. With wave-cutoff filtering we have

` k `c

2f 5 E(k) dk 5 E(k) dk 1 E(k) dkE E E0 0 kc

r 2 s 25 ( f ) 1 ( f ) . (29)

For horizontal (subscript h) gradients we have

2 kcr] f2; k E(k) dk, h 5 1, 2. (30)E1 2]xh 0

We will use (29) to determine how the variances ofuh, c, and u3 for z/D ; kcz K 1 are partitioned intotheir resolvable and SGS parts. We will use (30) todetermine how the variances of , , and dependr r ru u ch,h 3,h ,h

on z and D for z/D K 1.

a. Horizontal velocity and scalars

For horizontal velocity and scalars the Peltier et al.(1996) model for E(k) given in Eq. (19) yields

k lc2c s kl1r 2( f ) 5 d(kl),E4/3 4/3c 22 0 (kl)1 11 2c2

2 k lcr 2 3] f c s (kl)15 d(kl). (31)E2 4/3 4/31 2]x l c 2h 2 0 (kl)1 11 2c2

Under neutral conditions Peltier et al. chose l 5 z,



which makes (kl)2/c2 a small parameter in (31), and wefind

22 2 r 2 4c s (k l) ] f c s (k l)1 c 1 cr 2( f ) ø , ø . (32)4/3 2 4/31 22c ]x 4l c2 h 2

For horizontal velocity and scalars these imply

neutral:

2z z zr r su ; u , c ; C , u ; u 1 2 ,h h* * * 1 2[ ]D D D

2zsc ; C 1 2 ;* 1 2[ ]D

u z C zr r* *u ; , c ; . (33)h,h , h1 2 1 2D D D D

In free convection Peltier et al. chose l 5 zi. Forhorizontal velocity (kcl)2/c2 is then a large parameter.Equation (31) yields in that limit

22 2/3 r 2 4/33c s c ] f 3c s k1 2 1 ccr 2( f ) ø 1 2 , ø ,1/3 2/3 2/31 2 1 22c (k l) ]x 4l2 c h

(34)

which implies

free convection:

2/3D

ru ; w 1 2 ,h * 1 2[ ]zi

1/3 21/3 21/3D z u zfs ru ; w ; u , u ; .h f h,h*1 2 1 2 1 2z D D Di

(35)

For a scalar in free convection Peltier et al. foundthat c2 ; (zi/z)2, making (kl)2/c2 a small parameter in(31) so that (32) applies. They found that c1 ;(zi/z)2/3, so that

free convection:

2z zr sc ; C , c ; C 1 2 ,f f 1 2[ ]D D

C zfrc ; . (36),h 1 2D D

b. Vertical velocity

Peltier et al. used a slightly different model for thetwo-dimensional spectrum Ey of vertical velocity u3 inthe surface layer:

Ey (k) 5 T(k)E(k). (37)

T(k) is a ‘‘continuity transfer function,’’

2(kz)T(k) 5 , (38)

720.62 1 (kz)

8

that models the influence of the surface at kz K 1 andensures that Ey behaves as required by local isotropy atkz k 1.

Under neutral conditions l 5 z so that kcz K 1. Inthis limit we can simplify (37) to

Ey (k) ; 1.6(kz)2E(k) (39)

so that the variance expressions arek zc2 31.6c s (kz)1r 2( f ) ø d(kz),E4/3 2 4/3c (1 1 (kz) /c )2 20

2 k zcr 2 5] f 1.6c s (kz)1ø d(kz). (40)E2 4/3 2 4/31 2]x z c (1 1 (kz) /c )h 2 20

These imply

neutral:

2 4z zr su ; u , u ; u 1 2 ,3 3*1 2 * 1 2[ ]D D

2u zr *u ; . (41)3,h 1 2D D

Under free convection l 5 zi and the expressions arek zc i2 21.6c s (kz) (kz )1 ir 2( f ) ø d(kz ),E i4/3 2 4/3c (1 1 (kz ) /c )2 i 20

2 k zc ir 2 2 3] f 1.6c s (kz) (kz )1 iø d(kz ). (42)E i2 4/3 2 4/31 2]x z c (1 1 (kz ) /c )h i 2 i 20

Here, kc zi is a large parameter and we have

free convection:

2/3 4/3z zr su ; u , u ; u 1 2 ;3 f 3 f1 2 1 2[ ]D D

2/3u zfru ; . (43)3,h 1 2D D

c. Vertical gradients

The vertical inhomogeneity of the surface layermakes rms vertical gradients more difficult to estimate.For resolvable-scale variables this inhomogeneity cantake two forms: a z-dependence of amplitude and a z-dependence of vertical spatial scale. We separate thesetwo inhomogeneities by writing a zero-mean, resolva-ble-scale variable f r as

f r(xi, t) 5 A(x3)s(xi, t). (44)

Here A is a nonrandom scaling function that contains



the vertical inhomogeneity in amplitude. The functions is random, with zero mean and unit variance; it con-tains whatever vertical inhomogeneity in vertical spatialscale exists in f r. It is possible that the vertical spatialscales of vary with z, for example. Since s has unitru3

variance, it follows from (44) that ( f r)2 5 A2. In thecase of under neutral conditions, for example, (41)ru3

implies that A ; u*(z/D)2.Differentiating (44) with respect to x3, squaring, and

averaging gives

( )2 5 A2(s,3)2 1 (s2 ),3A,3A 1 (A,3)2rf ,3

5 A2(s,3)2 1 (A,3)2, (45)

the cross term vanishing because we chose s2 to beindependent of z. The gradient variances (s,h)2 and (s,3)2

are related to s2 through the Taylor microscales lh andlz (Tennekes and Lumley 1972):

5 s2 /(s,h)2 , 5 s2 /(s,3)2 .2 2l lh z (46)

With (46) we can write (45) as2lhr 2 2 2 2( f ) 5 A (s ) 1 (A ) . (47),3 ,h ,32lz

Differentiating (44) with respect to xh, squaring, aver-aging, and combining with (47) yields

2lhr 2 r 2 2( f ) 5 ( f ) 1 (A ) . (48),3 ,h ,32lz

Our results in sections 4a and 4b show that for most ofthe resolvable-scale variables the horizontal Taylormicroscale lh is D, the smallest surviving horizontalscale in the filtered fields. We will assume that lz is ofthis order as well so that (48) becomes

( )2 5 ( )2 1 (A,3)2.r rf f,3 ,h (49)

Let us illustrate with under neutral conditions.ru3

Equation (49) yields

4 22 2u z u zr 2 * *(u ) ; 1 . (50)3,3 2 21 2 1 2D D D D

The second term on the right dominates at small z/D sothat our assumption that lz ; lh ; D is not critical;we obtain the result (50) for lz ; z as well. This is thecase for almost all variables.

This procedure yields the scaling estimates

neutral:

u z C ur r r* * *u ; , c ; , u ; ;3,3 ,3 h,31 2D D D D

free convection:

21/3 21/3u z C u zf f fr r ru ; , c ; , u ; .3,3 ,3 h,31 2 1 2D D D D D(51)

From (33), (35), and (51) we see that , , andr ru u1,1 2,2

are of the same order, consistent with the resolvable-ru3,3

scale continuity equation.

5. Simplifying the resolvable-scale surface-fluxbudget

a. The ‘‘constant-SGS-flux’’ layer

We can use our surface-layer scaling relations of sec-tion 4 to quantify the variation of SGS flux with heightnear the surface. We first decompose the SGS flux intoensemble-mean and fluctuating parts,

5 F0 1 f 3.rF 3 (52)

We assume that the two flux-divergence terms in theresolvable-scale scalar conservation equation (14) areof the same order:

; ; ( )r.r r r r˜ ˜F F c u3,3 j, j , j j (53)

Our scaling estimates in section 4 then imply

2F z0rF ; (neutral),3,3 1 2D D

2/3 1/3F z z0 i (free convection). (54)1 2 1 2D D D

This says that the SGS flux divergence vanishes as z/D→ 0 because the forcing terms in the resolvable-scalescalar equation vanish in that limit.

Equation (54) implies that the rms difference betweenthe resolvable-scale surface flux and the SGS fluxrF 0

at height z, (z), scaled by the mean surface flux F0,rF 3

is of order

3r r˜ ˜ ˜F 2 F (z) zF z0 3 3,3; ; (neutral),1 2F F D0 0

5/3 1/3z zi (free convection). (55)1 2 1 2D D

Let us compare this rms flux difference to the rms fluc-tuations in SGS flux at height z K D. Estimating therms value of the fluctuating SGS flux from the usualF93surface-exchange formulation gives

F9 s3 u; , (56)F U0 1

where su is the rms fluctuation in the streamwise wind,typically in the range (0.1–0.5)U1.

We conclude that at small z/D, the rms difference inSGS flux between the surface and height z, Eq. (55), issmall compared to the rms fluctuation level in SGS fluxat height z, Eq. (56). If so, when z/D K 1 the SGS fluxat z is a reliable surrogate for the resolvable-scale sur-face flux.

It remains to evaluate the importance of the crosscontributions to the SGS scalar flux, Eq. (17). In ap-



TABLE 1. Scaling of terms in the budget of surface scalar flux.

r r s s r r r r r r((u u ) c ) ; u u dc ; u u cRSCFI: 3, j j 3, j j s 3, j j

r r s s r r r s r r r((c u ) u ) ; c u du ; c u uRGCSI: , j j 3 , j j 3 j j 3

r s s r r r s s(u (u c ) ) ; u u cRSSP: 3, j j 3, j j

r ru F uh 0 h,hr r r r r˜ ˜(u F ) ; u F ; , j 5 1, 2;LSA: j 3 , j j 3, j U1

r r r r r; u (c u 1 c u ), j 5 33 ,h h ,3 3

r r˜U F ; u FMA: j 0, j h,h 0

s s s r s s s(u u c ) ; (u u c )SSA: j 3 ,j j 3 , j

r s s r r r s s(c (u u ) ) ; c u uRSGP: , j j 3 , j j 3

s s r s sC (u u ) ; C u uMGP: ,3 3 3 ,3 3 3

g gs s r s s(u c ) ; u cBP:

T T0 0

s s r s s(p c ) ; p cPD: ,3 ,3

Mean surface flux F 5 u C 5 u C0 f f* *

Term

u*Neutral units of F01 2z

j 5 1, 2 j 5 3

Free convectionuf

units of F01 2z

j 5 1, 2 j 5 3

RSCFI5z1 2D

5z1 2D

7/3 1/3z zi1 2 1 2D D

7/3z1 2D

RGCSI5z1 2D

5z1 2D

7/3 1/3z zi1 2 1 2D D

7/3z1 2D

RSSP3z1 2D

2z1 2D

4/3z1 2D

2/3z1 2D

LSA3z u*1 2 1 2D U1

5z1 2D

2/3 1/3z zi1 2 1 2D D

7/3 1/3z zi1 2 1 2D D

MA2z1 2D

0 0 0

SSAz1 2D

1z1 2D

1

RSGP2z1 2D

z1 2D

2z1 2D

z1 2D

MGP 1 1

BP 0 1

PD 1 1

pendix B we show their rms values are of the order ofF0 (z/D)3. Thus, at height z K D they are not largerthan the rms difference in SGS flux between the surfaceand height z. Since we judged the latter to be negligible,we will continue to neglect the cross contributions toSGS flux as well.

b. Scaling the flux budget

We can now scale some of the terms in the resolvable-scale surface flux budget (24). We choose the x1 direc-tion to be that of the mean wind so that Ui 5 (U1, 0,0). The scaling results are summarized in Table 1. Wecannot directly scale pressure destruction, but from ex-perience with the Reynolds flux budget close to thesurface (Wyngaard et al. 1971) we expect that it is aleading term.

To scale the time-change term, not included in Table1, we assume that the local rate of change of the re-solvable-scale surface flux is determined by resolvable-scale processes. Thus, we take the time-change term tobe of the order of mean advection at neutral and large-scale horizontal advection in free convection, giving

2r˜]F u z0 *; F (neutral),0 1 2]t z D

2/3 1/3u z zf i; F (free convection). (57)0 1 2 1 2z D D

c. A hierarchy of resolvable-scale surface-fluxbudgets

According to the scaling results of Table 1 the budgetof surface scalar flux takes different forms dependingon the value of z1/D.

1) z1/D K 1

When z/D K 1 we need retain only the O(1) termsin the resolvable-scale surface-flux conservation equa-tion (24), and it reduces to

gs s s r s s r s s r s s r0 ø 2(u u c ) 2 C (u u ) 1 (u c ) 2 (p c ) .3 3 ,3 ,3 3 3 ,3T0

(58)

The k 5 0 mode of Eq. (58), its expected value, is thequasi-steady form of Eq. (27), the locally homogeneousReynolds budget of scalar flux. The terms in Eqs. (58)and (27) represent, in order, the rates of vertical transportof turbulent flux by small-scale turbulence, productionthrough the interaction of vertical velocity fluctuationswith the mean vertical gradient of the quantity beingtransported, production through buoyancy, and destruc-tion by pressure gradient interactions.

In the locally homogeneous, quasi-steady surface lay-er, there exists well-established flux-gradient and sur-face-exchange relations (Businger et al. 1971):

wc 5 2KC ; F 5 C (z)U (z)[C(z 2 C(z)],3 0 H 1 0c

5 C (z)U (z)DC(z). (59)H 1

Here z0c is the ‘‘roughness length’’ for the scalar. In (59)we have shown explicitly only the dependence on z. Thedependence on stability is represented well through theMonin–Obukhov similarity hypothesis (Panofsky andDutton 1984).

The turbulence active in the processes represented in



(58) is subgrid scale; it has a dominant length scale zand timescales Ts ; z/u* and z/uf in the neutral andfree-convection limits, respectively. The final filteringof each term can be thought of roughly as averagingover horizontal spatial scales D k z. The time-changeterm is negligible because its timescale is much greaterthan Ts. Similarly, the horizontal inhomogeneity of fil-tered variables is weak because it occurs on a spatialscale much larger than z. Thus, we argue that for z/DK 1 the resolvable-scale surface flux budget (58) rep-resents a quasi-steady, locally homogeneous state of‘‘local grid-volume equilibrium.’’

The analogy between the Reynolds flux budget (27)and the z/D → 0 limit of the resolvable-scale surface-flux budget (58), the well-behaved surface-exchange re-lation (59) for the Reynolds flux, and the ‘‘couplingeddies’’ notion of Fig. 1 make it plausible that in thislimit the resolvable-scale surface flux displays a surface-exchange relation like (59), but with resolvable-scalevariables replacing the ensemble averages:

rF (x , x , t)0 1 2

r r r˜5 C (z/L)[u (x , x , z, t)Dc (x , x , z, t)] ,H 1 1 2 1 2

z/D K 1. (60)

Here CH is the mean surface-exchange coefficient thatdepends on stability; here the stability is interpretedlocally, however. For that purpose we define the localstability parameter for the grid volume, z1/L, with L thelocal Monin–Obukhov length,

3u T0˜ *L 5 2 , (61)˜kgQ0

and u* and Q0 the local SGS friction velocity and sur-face temperature flux. Based on Eq. (58) we would ex-pect DC rather than Dcr to appear in (60), but section4 indicates that this difference is negligible for z/D K1 and (60) is consistent with typical practice in LES.

2) z1/D SMALL BUT NOT NEGLIGIBLE

Table 1 shows that in free convection the horizontalpart of large-scale advection (LSA) is significant at z1

such that

2/3 1/3 3/2z z D1 i ; 1, or z ; . (62)1 1/21 2 1 2D D zi

For D 5 10 m and zi 5 103 m this occurs at z1 5 1 m.The same arguments hold for the time-change term, Eq.(24). We conclude that in application to fine-mesh LESwith z1 on the order of a few meters, the time-changeand horizontal LSA terms in the surface-flux budget (24)can be significant.

If we combine the LSA and MA terms by writing U1

1 5 , as we previously did with the RSGP andr ru uh h

MGP terms, Eq. (24) becomes

] gr r r r s s s r r s s r s s r˜ ˜F 1 (u F ) 1 (u u c ) 1 c (u u ) 2 (u c )0 i 0 , i 3 3 ,3 ,3 3 3]t T0

s s rø 2(p c ) ,,3 (63)

where the index i is summed over 1 and 2. Equation(63) represents a departure from the local-grid-volume-equilibrium state of Eq. (58) in that it includes horizontaladvection and time change of resolvable-scale surfaceflux in addition to the production, small-scale advection,and pressure-destruction processes.

Let us model the local time-change and horizontaladvection effects through a hypothesis advanced byBradshaw (1969) and discussed by Wyngaard (1982).We assume that the SGS turbulence remains in localgrid volume equilibrium in the sense that

gs s s r r s s r s s r s s r(u u c ) 1 (c ) (u u ) 2 (u c ) 5 2(p c ) ,3 3 ,3 ,3 eq 3 3 ,3T0

(64)

where ( )eq is the local-grid-volume-equilibrium valuerc,3

of the resolvable-scale scalar gradient in the vertical.Because z1/D is small, there are many coupling eddiesin the grid volume and to first approximation we cantake ( )eq to be nonrandom and to have a similarityrc,3

form identical to that of the traditional M–O functionfor temperature, f h:

rF kzu ]Q0r ˜ *(c ) 5 2 f (z /L), f 5 2 . (65),3 eq h 1 hku z Q ]z1 0*

Using (64) in (63) yields

]r r r r s s r r r˜ ˜F 1 (u F ) . (u u ) ((c ) 2 c ). (66)0 h 0 ,h 3 3 ,3 eq ,3]t

In order to solve Eq. (66) the local value of the SGSvertical velocity variance ( )r is needed; it could bes su u3 3

obtained through a local similarity model

( )r 5 f (z1/L),s s 2u u u3 3 * (67)

where f is the M–O function for vertical velocity vari-ance. With such a model for ( )r, Eq. (66) indicatess su u3 3

how the scalar gradient differs from the local-grid-vol-ume-equilibrium value if there is local time change orhorizontal advection of resolvable-scale surface flux.

If we approximate (66) bys s r] (u u )3 3r r r r r r˜ ˜F 1 (u F ) ø (Dc 2 Dc ) (68)0 h 0 ,h eq]t z1

and introduce the mean and local surface-exchange co-efficients,

r r˜ ˜F F0 0r r(Dc ) 5 , (Dc ) 5 , (69)eq r r˜C u C uH 1 H 1

combining (68) and (69) then gives

s s r r˜] (u u ) F 1 13 3 0r r r r˜ ˜F 1 (u F ) 1 2 5 0. (70)0 h 0 ,h r ˜1 21 2]t z u C C1 1 H H



This determines the departure of the local surface-ex-change coefficient CH from its mean value CH due tothe effects of local time change and horizontal advectionof SGS flux.

3) z1/D # 1

In order to evaluate other candidate terms in the re-solvable-scale surface flux budget, let us write (24) sym-bolically as

nr r˜ ˜]F F0 05 T 2 , (71)O i]t ti51

where the final term, a parameterization of the rate ofpressure destruction, is patterned after its behavior inthe Reynolds flux budget (Wyngaard et al. 1971), andTi represents the remaining terms. We estimate the rmscontribution to from each component of Ti as di

r r˜ ˜F F0 0

; tTi. As z1/D → 1 these contributions increase since,according to Table 1, the Ti increase. In fact, however,(24) describes the budget at z1, not at the surface, andwe showed in (55) that as z1/D increases the rms dif-ference between the fluxes at the surface and at heightz1 also increases. This suggests a criterion for retaininga term in (24): that its rms contribution to the flux belarge compared to the rms difference between the re-solvable-scale surface flux and the SGS flux at heightz1:

di k 2 (z1).r r r˜ ˜ ˜F F F0 0 3 (72)

This ensures that the term does reflect a contributionprimarily to the resolvable-scale surface flux.

Using our estimate (55) for 2 (z1) and takingr r˜ ˜F F0 3

t ; z1/u* at neutral and t ; z1/uf in free convectionthen yields

3u z1*T units of F k (neutral),i 01 2 1 2z D1

5/3u zf 1T units of F k (free convection). (73)i 01 2 1 2z D1

Let us interpret ‘‘k’’ in (73) as meaning between oneand two orders less in z1/D. We therefore also include thevertical part of RSSP. The surface flux budget then is

]r r r r r r r s s s r˜ ˜ ˜F 1 (u F ) 1 (u F ) 1 (u u c )0 h 0 ,h 3,3 0 3 3 ,3]t

gr s s r r s s r s s r1 [c (u u ) ] 2 (u c ) ø 2(p c ) . (74),3 3 3 ,3T0

Using the local-grid-volume-equilibrium hypothesis(64) for the SGS turbulence then gives an extension of(70):

]r r r r r r r˜ ˜ ˜F 1 (u F ) 1 (u F )0 h 0 ,h 3,3 0]t

s s r r˜(u u ) F 1 13 3 01 2 5 0. (75)r ˜1 21 2z u C C1 1 H H

Equation (75) further corrects the equilibrium surface-exchange expression for the production of resolvable-scale surface flux through the interaction of the SGSflux and the resolvable-scale velocity gradient .ru3,3

According to our scaling analysis in appendix B, thefractional contribution of the cross terms to the SGSflux is one to two orders smaller in z/D than the SGSflux fluctuations produced by the terms retained in (74).Thus, there seems no need to include the cross contri-butions to SGS flux.

d. Resolution in z

Our scaling analysis of section 4 treats the effects offiltering the fields in the horizontal plane. Such filteringis done explicitly in Moeng’s (1984) LES code, for ex-ample. Our analysis does not treat the effects of filteringor finite-difference approximations in z. Thus, our sur-face-layer scaling results are directly applicable only toLES with high vertical resolution.

The vertical resolution of LES also figures in thechoice of a resolvable-scale surface flux model. Thetradeoff is clear: if z1/D K 1, which requires the firstgrid point to be very close to the surface, a standardsurface-exchange model for flux, Eq. (60), can be used.If z1/D ø 1, the first grid point can be much higher butthen more physics enters the resolvable-scale surfaceflux conservation equation, as indicated in Eq. (75).Thus, when z1/D K 1, the complicated connection be-tween the resolvable-scale surface flux and the resolv-able-scale flow at z ; D must be made entirely throughthe SGS model; when z1/D ø 1, some elements of thatconnection can be made through the resolvable-scalesurface flux model.

6. Use of resolvable-scale surface-flux equations inLES

a. Implementation and results

We implemented the z1/D # 1 form of the conser-vation equations for resolvable-scale surface tempera-ture flux and resolvable-scale surface stress inr rQ t0 0k

Moeng’s LES code (Moeng 1984). Using (69) for thesurface-exchange coefficients, Eq. (75) can be rewrittenfor asrQ0

]r r r r r r r˜ ˜ ˜Q ø 2(u Q ) 2 (u Q )0 h 0 ,h 3,3 0]t

s s r r r r˜ ˜2 [(u u ) (u 2 (u ) )] . (76)3 3 ,3 ,3 eq

We used (65) for the local-grid-volume-equilibrium val-ue of the resolvable-scale temperature gradient, where



FIG. 4. Nondimensional mean shear in a moderately convectiveboundary layer (2zi/L ø 10). Curves 1 and 2 are from 643 and 1282

3 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 643 and 1282 3 64 simulations, respectively,using surface-flux equations; solid line is the empirical function ofBusinger et al. (1971)

21/2z˜f (z/L) 5 0.74 1 2 9 (77)h ˜1 2L

(Businger et al. 1971), L is given by (61), and u* isgiven by

5 ( )2 1 ( )2.2 r ru t tÏ 01 02* (78)

The local SGS vertical velocity variance is modeled asEq. (67), where the nondimensional function is taken as

2/3z˜f (z/L) 5 1.6 1 2.9 2 (79)˜1 2L

(Panofsky et al. 1977).Our modeled conservation equation for the subgrid–

subgrid part of the SGS stress, ( )r 5 , reducess s r˜u u Sk 3 0k

to (appendix C)

]r r r r r r r˜ ˜ ˜S ø 2(u S ) 2 (u S )0k h 0k ,h 3,3 0k]t

s s r r r r2 [(u u ) (u 2 (u ) )] , (80)j 3 k, j k, j eq

where k 5 1, 2. The local-grid-volume-equilibrium val-ue of the resolvable-scale velocity gradient is given by

t u0kr ˜*(u ) 5 2 f (z /L)d , (81)k, j eq m 1 j32u kz1*

where fm is taken as

21/4z˜f (z /L) 5 1 2 15 (82)m 1 ˜1 2L

(Businger et al. 1971).In section 2 of appendix C we show that the cross

stress 5 ( 1 )r has zero mean but an rmsr r s s rC u u u u0k k 3 k 3

value of order . We model it asr ru uk 3

5 1 ,r r rC u u9 u9u0k k 3 k 3 (83)

where a prime denotes the rms value of the resolvable-scale field. The model (83) has zero mean, has an rmsvalue of the required order, and depends on both thehorizontal and vertical components of the resolved ve-locity.

Equations (76) and (80) were solved using a second-order, explicit Adams–Bashforth time marching schemein which the temperature flux, for example, at time stepn 1 1 is given by

3r r˜ ˜Q (n 1 1) 5 Q (n) 1 Dt R (n)0 0 Q2

12 Dt R (n 2 1), (84)Q2

where Dt is the time step, and RQ (n) is the right-handside of (76) at time step n.

Since z1/D is a critical parameter in our analysis, wecarried out three sets of simulations: 643, 1282 3 64,and 1922 3 64, with z1/D of 0.13, 0.27, and 0.4, re-

spectively, for a moderately convective boundary layer(2zi/L ø 10). Figure 2 shows the fluctuation levels inthe surface-exchange coefficients obtained by usingconservation equations for resolvable-scale surface flux-es in the simulations. These levels agree well with thefindings of Wyngaard and Peltier (1996). The fluctuationlevels in and are roughly of the same order.r r˜ ˜S C0k 0k

Figures 4–6 show the nondimensional mean shearf m, mean potential temperature gradient f h, and ver-tical velocity variance, respectively, from 643 and 1282

3 64 simulations using the conservation equations(curves A and B) and the standard surface-exchangecoefficients (curves 1 and 2). The profiles observed inpast field experiments are also plotted in these figures.

Consistent with the experience of Mason and Thom-son (1992) and Sullivan et al. (1994) f m is overpre-dicted, with respect to the observations, by the Sma-gorinsky-based SGS model (used in Moeng’s code) withsurface-exchange coefficients. Moreover, the profilekink moves closer to the surface as the horizontal res-olution is increased.

In the 643 simulation with surface-exchange coeffi-cients, f h is somewhat lower than the measured valueat the first grid point; above that it is slightly overpre-dicted. Increasing the horizontal resolution to 1282,while keeping the vertical resolution the same, limits



FIG. 5. Nondimensional mean temperature gradient in a moderatelyconvective boundary layer (2zi/L ø 10). Curves 1 and 2 are from643 and 1282 3 64 simulations, respectively, using surface-exchangecoefficients; curves A and B are from 643 and 1282 3 64 simulations,respectively, using surface-flux equations; solid line is the empiricalfunction of Businger et al. (1971).

the overprediction of f h to the second grid point. Thevalues at other grid points agree well with the fieldobservations.

The nondimensional vertical-velocity variance inboth the 643 and the 1282 3 64 simulations with surface-exchange coefficients are slightly lower than the fieldmeasurements reported by Panofsky et al. (1977). Thechange in resolution does not alter the profile signifi-cantly.

The use of flux-conservation equations instead of thesurface-exchange coefficients yields slight improve-ments in all three profiles, most noticeably in that ofvertical-velocity variance. However, the improvement isonly marginal and not wholly satisfactory. We believethat the mild influence of improved lower boundaryconditions on the mean surface-layer structure is due todeficiencies in the SGS model. The objective of surface-flux budgets is to capture more reliably the local struc-ture of the resolvable-scale surface fluxes, but, as weshow next, the Smagorinsky-based SGS model repre-sents the surface-layer physics poorly and therefore pre-sumably responds to enhanced structure in surface flux-es improperly as well.

b. Analysis of the SGS model from highly resolvedsurface-layer fields

The Smagorinsky-type SGS model has been standardin LES since the early work of Lilly (1967) and Dear-dorff (1970). Its popularity stems from its simplicityand, most importantly, the insensitivity of resolved-scalefields to the SGS model in well-resolved turbulent flows.Near the surface, however, the vertical motions are al-ways inadequately resolved and as a result the SGSclosure becomes crucial there.

We analyzed the accuracy of Smagorinsky closure inthe surface layer using highly resolved LES with nestedmeshes. Such tests have been done previously usingdirect numerical simulation (DNS) of isotropic turbu-lence (Clark et al. 1979; Bardina et al. 1983) and theneutral turbulent boundary layer (Piomelli et al. 1991).The unstable atmospheric surface layer, however, hascertain distinct features not present in homogeneous tur-bulence and neutral boundary layers. Specifically, thehorizontal motions in the surface layer of an unstableatmospheric boundary layer (ABL) are strongly influ-enced by the zi-scaled mixed-layer eddies, causing astrong anisotropy of length and intensity scales betweenthe horizontal and vertical motions. There is also a sub-stantial horizontal mean temperature flux in the surfacelayer that is absent in the neutral boundary layer. Thus,past studies using DNS fields are not directly relevantfor our analysis.

We generated highly resolved surface-layer fields us-ing a three-level nested-mesh LES discussed in Khannaand Brasseur (1996). The full boundary layer was sim-ulated at 2zi/L ø 10 using a 1283 mesh covering adomain of 5zi 3 5zi 3 2zi. The next level of refinement

in the surface layer was attained by an effective 2563

mesh covering a domain of 5zi 3 5zi 3 0.125zi, andthe final level was attained by an effective 5123 meshcovering 5zi 3 5zi 3 0.06125zi of the boundary layer.The upper boundary conditions for the embedded mesh-es were obtained from the next-level coarser domainswith a one-way communication between the domains.At a height where the effective 5123 simulation resolvedapproximately 90% of the vertical fluxes and variances(the sixth grid level, at z/zi ø 0.02) the resolved vari-ables were treated as fully resolved fields. They weredecomposed into a resolvable part and a subgrid partby two-dimensional horizontal wave-cutoff filteringwith a cutoff wavenumber corresponding to a 962 hor-izontal mesh.

Table 2 compares the calculated SGS stresses andfluxes and their divergences with the values predictedby the SGS model used by Moeng (1984). The mag-nitudes of the cross-correlation coefficients of the mod-eled and actual SGS flux components are very low (lessthan 0.20). Those for SGS flux divergences, which ap-pear in the dynamical equations, are particularly low(less than 0.05 for the horizontal components of thedivergences). The SGS model also fails in capturing themean diagonal SGS stress components, presumably dueto its inability to reproduce the anisotropic distribution



FIG. 6. Nondimensional vertical-velocity variance in a moderatelyconvective boundary layer (2zi/L ø 10). Curves 1 and 2 are from643 and 1282 3 64 simulations, respectively, using surface-exchangecoefficients; curves A and B are from 643 and 1282 3 64 simulations,respectively, using surface-flux equations; solid line is the empiricalfunction of Businger et al. (1971).

TABLE 2. A comparison of predicted and actual subgrid-scale tem-perature flux and stress components from an effective 5123 simula-tion, using nested meshes, of the atmospheric boundary layer (2zi/L ø 8). The comparisons are made at z/zi ø 0.04 and z/D ø 0.75;(q 5 Fi,i; ai 5 ij,j).t

Actualmean

Actualrms

Predicted/actual(mean)

Predicted/actual(rms)

Cross-correlationcoefficient

SGS fluxF1

F2

F3

F1,1

F2,2

20.14—0.16——

0.620.450.470.430.42

0.01—0.94——

0.170.330.500.220.33

0.190.250.260.210.27

F3,3

q20.1320.13

0.620.80

1.041.04

0.920.81

0.050.16

13t23t12t11t22t

20.18—

20.010.10

20.11

0.680.500.690.790.69

0.41—0.020.02

20.05

0.210.250.120.110.16

0.180.110.140.140.06

33t11,1t12,2t13,3t

a1

0.01——0.160.16

0.590.570.670.951.21

20.89——0.410.41

0.190.120.110.420.37

0.280.140.180.040.08

21,1t22,2t23,3t

a2

31,1t

—————

0.520.630.741.020.47

—————

0.090.150.450.380.17

0.010.100.000.020.09

32,2t33,3t

a3

—0.000.00

0.471.081.36

—5.885.88

0.170.190.12

0.070.260.26

of SGS energy, and in capturing the mean horizontalSGS temperature flux, due to the failure of the eddy-viscosity model of that flux (Wyngaard et al. 1971).These deficiencies in the SGS model, we hypothesize,can mask the effects of improved resolution of the struc-ture of resolvable-scale surface fluxes.

c. Results with improved subgrid-scale model

Mason and Thomson (1992) found that adding sto-chastic fluctuations to the SGS stress divergence im-proved the nondimensional mean shear in a neutralboundary layer calculated through LES. These fluctu-ations simulate the local transfer of energy from unre-solved to resolved scale, or ‘‘backscatter,’’ in the Sma-gorinsky SGS closure. We implemented this modifica-tion to the SGS model used in Moeng’s code along withthe surface-flux conservation equations.

Moeng’s code uses the SGS model proposed by Dear-dorff (1980) in which the subgrid-scale energy, e 5( )r/2, is calculated explicitly and used as the velocitys su ui i

scale for the SGS eddy diffusivity (K):

r r˜ ˜t 5 22KS , (85)i j i j

K 5 c Ïe l, (86)k

r˜ ˜K ]urQ 52 , (87)i Pr ]xi

where is the resolvable strain rate tensor, ck is arS ij

constant, l is the length scale of unresolved eddies, andPr is the turbulent Prandtl number. Following the de-velopment of Mason and Thomson, we added a white-noise random acceleration, ai, and a random temperaturesource, q, to the resolvable-scale equations:

r r]u ]ti i j5 · · · 2 1 a , (88)i]t ]xj

r r˜˜]u ]Qj5 · · · 2 1 q, (89)]t ]xj

to account for stochastic backscatter. To satisfy conti-nuity ai was taken to be the curl of another randomvector: ai 5 e ijkbk,j. As indicated by Mason and Thom-son, this random forcing adds a production rate aiaiDt tothe resolved-scale kinetic energy equation and 2q2Dtto the equation for resolved-scale temperature variance,where Dt is the time step. Mason and Thomson showthat when the filter scale is of the order of the scale of



FIG. 7. Effect of SGS model and lower boundary conditions onthe nondimensional mean shear. Curves 1 and 2 are from the originalSGS model (without backscatter) and curves A and B from the mod-ified SGS model (with backscatter). In both sets, the first curve isbased on surface-exchange coefficients and the second on the surfaceflux equations.

the unresolved eddies, the mean backscatter of resolved-scale kinetic energy due to stochastic fluctuations inunresolved eddies is cbe, where e is the ensemble-av-erage dissipation rate and cb 5 1.4; the mean backscatterof resolved-scale temperature variance is 2cbux, wherex is the destruction rate of one-half temperature varianceand cbu 5 0.45. The random components bi and q arescaled such that

a a Dt 5 c e, (90)i i b

2q Dt 5 c x. (91)bu

We increased the constant ck in (86) to (1 1 cb)ck toaccount for the additional production of resolved-scalekinetic energy. The turbulent Prandtl number in (87)was increased by a factor of (1 1 cb)/(1 1 cbu) to ac-count for the additional production of resolved-scaletemperature variance. The length scale l in (86) wastaken to be the minimum of D, the grid scale, and kz/ck, where k is the von Karman constant and z is thedistance from the surface.

Figure 7 shows the effect of the SGS model and lowerboundary conditions on the f m profile in the 1282 364 simulation of the moderately convective boundarylayer. The original SGS model and drag coefficientsoverestimate f m at the first grid point (z/zi ø 0.03) by

50%. Use of surface-flux conservation equations withthe original SGS model makes a marginal improvement.The modified SGS model (with stochastic backscatter)with the drag coefficients gives a smooth profile for f m,although it is not entirely in agreement with the ob-served profile. The simulated value increases with z/Lbefore falling off, while the observed profile decreasesmonotonically. Our results are not in complete agree-ment with those of Mason and Thomson, presumablydue to three factors: first, we are simulating a moderatelyconvective boundary layer (with both buoyancy andshear effects) while their work concerned neutral bound-ary layers; second, we use a 1282 3 64 mesh, whereasMason and Thomson used a much finer vertical meshcompared to the horizontal mesh; finally, some of theparameters used in our implementation of the stochastic-backscatter model are different from those of Masonand Thomson. Nevertheless, we infer from these resultsthat the subgrid-scale model does have a significant in-fluence on the mean structure of the surface layer. Useof surface-flux conservation equations along with thestochastic backscatter model makes a marginal improve-ment. We conclude that the improved lower boundaryconditions need a compatible subgrid-scale model tomake better predictions of atmospheric surface layers.

7. Summary and conclusions

We argued physically and showed through analysis ofobservations and LES data that the local surface-exchangecoefficient relating the resolvable-scale surface flux to re-solvable-scale properties of the overlying flow is a randomvariable. Its fluctuation level increases with z/D, where zis height above the surface and D is the spatial scale ofthe filter that separates fields into resolvable-scale and sub-grid-scale parts. As z/D → 0, the surface-exchange co-efficient approaches its traditional definition.

An alternative to using a local surface-exchange co-efficient to diagnose the resolvable-scale surface flux ispredicting that flux through its conservation equation.We showed that near the surface this flux is dominatedby the s–s component and we derived the equation forthat component. We used the surface-layer spectral mod-el of Peltier et al. (1996) to develop scaling expressionsfor the partitioning of the variances of surface-layerfields between the resolvable and subgrid-scale com-ponents. We used these scaling expressions to simplifythe flux conservation equations in three limits.

For high aspect ratio grids, z1/D K 1, the surface-flux conservation equation is in a state of local gridvolume equilibrium and the mean surface-exchange co-efficient can be used locally, as is generally done inLES. For smaller aspect ratio grids it has horizontaladvection and time-change terms, consistent with theappearance of fluctuations in the local surface-exchangecoefficient in this regime. As the grid aspect ratio ap-proaches unity, the flux conservation equation gains aproduction term proportional to the convergence of hor-



izontal velocity near the surface. The observed fluctu-ations in the local surface-exchange coefficient in thisregime are quite large.

Based on Bradshaw’s (1969) suggestion that hori-zontal inhomogeneity of the surface layer causes thelocal mean gradient of a transported quantity to deviatefrom its equilibrium value, we proposed a simple closurefor the flux conservation equations and implementedthem in the Moeng (1984) LES code.

Use of the surface flux conservation equations yieldsslight improvements in the nondimensional mean shear,mean potential temperature gradient, and vertical-ve-locity variance profiles. The commonly observed kinkin the mean shear and temperature gradient profiles pre-dicted by Smagorinsky-based SGS models at z ø D isreduced. This improvement, however, is not substantial.

We showed through highly resolved LES that theSmagorinsky-based SGS models perform poorly in theatmospheric surface layer; a better SGS model is need-ed. We are currently extending the present analysis toderive conservation equations for SGS stresses and flux-

es in the atmospheric surface layer, which, combinedwith dynamic lower boundary conditions, will hopefullymake significant improvements in the LES predictionsof atmospheric surface-layer structure.

Acknowledgments. We are grateful to T. Horst and G.Maclean of NCAR SSSF for kindly providing ASTERdata from the STORMFEST experiment; to Andrew R.Brown of U.K. Meteorological Office for helpful dis-cussions on stochastic backscatter; and to J. Brasseur,P. Mourad, and P. Sullivan for making helpful sugges-tions on the manuscript. This work was supported byArmy Research Office Grant DAAL03-92-G-0117 andby Office of Naval Research Grant N00014-92-J-1688.

APPENDIX A

Simplifying the Resolvable-Scale Surface ScalarFlux Budget

Deriving the budget of resolvable-scale surface scalarflux as indicated in (23) yields

]r r r s s rF ø 2((u u ) c ) resolvable shear, cross flux interaction (RSCFI)0 3, j j]t

r r s s r2 ((c u ) u ) resolvable gradient, cross stress interaction (RGCSI), j j 3

r s s s r2 ((u u ) c ) resolvable-scale shear production (RSSP)3, j j

r s s s r s s s r2 ((u u ) c 1 (u c ) u ) large-scale advection (LSA)j 3, j j , j 3

r˜2 U F mean advection (MA)j 0, j

s s s s s s s s r2 ((u u ) c 1 (u c ) u ) small-scale advection (SSA)j 3, j j , j 3

r s s s r2 ((c u ) u ) resolvable-scale gradient production (RSGP), j j 3

s s r2 C (u u ) mean-gradient production (MGP),3 3 3

gs s r1 (u c ) buoyant production (BP)

T0

s s r2 (p c ) pressure destruction (PD),3

s s r s s r1 g(c u ) 1 n(u c ) molecular terms (M)., jj 3 3, jj (A1)

We can simplify a number of the terms in (A1). Werewrite the resolvable-scale shear production term(RSSP) as

RSSP 5 2[( )scs]r 5 2[( )cs]r 1 [( )rcs]r.r s r s r su u u u u u3,j j 3,j j 3,j j

(A2)

The term on the far right has the form (arbs)r. Thewavenumber components of ar have magnitudes be-tween 0 and kc. The wavenumber components of bs havemagnitudes greater than kc. Let us write this symboli-cally as

(arbs)r 5 {[0, kc](kc, `]}r 5 [0, kc]. (A3)

In our notation, brackets mean that the set’s end point isincluded in the range and parentheses mean that the set’send point is not included in the range. The wavenumbercomponents of arbs are the vector sum of all those of ar

and bs. Therefore wavenumbers of bs larger than 2kc can-not contribute to (arbs)r, so we can write (A3) as

(arbs)r 5 {[0, kc](kc, 2kc]}r. (A4)

The variable b is the temperature field, whose dominant



FIG. A1. The spectra of vertical temperature flux (long dashes),horizontal temperature flux (short dashes), and vertical velocity (sol-id) from high-resolution LES at z 5 35 m (fine) and 66 m (bold) for2zi/L . 65.

wavenumbers have magnitudes of the order of 1/z (Pel-tier et al. 1996). Thus, as kcz → 0, there are no modesof b in the range (kc, 2kc] so that (arbs)r vanishes, andwe can write

(arbs)r K (abs)r as kcz → 0. (A5)

We will refer to the development that leads to (A5) asthe scale-separation argument.

We can now rewrite (A2) as

RSSP 5 2[ ( cs)]r 5 2[ ( cs)r]r 2 [ ( cs)s]r.r s r s r su u u u u u3,j j 3,j j 3,j j

(A6)

In order to proceed we need to know the shape of thespectrum of cs. We calculated it for j 5 1 and j 5 3suj

from high-resolution (2563) LES data. The results,shown in Fig. A1, indicate that the spectrum of cs issuj

like that of u3 in that it peaks at k ; 1/z. Thus, we canuse the scale-separation argument to drop the secondterm on the right side of (A6). This yields

RSSP → 2[ ( cs)r]r as kcz → 0.r su u3,j j (A7)

We can rewrite the large-scale advection terms asr s s s r s s s rLSA 5 2[(u u ) c 1 (u c ) u ]j 3, j j , j 3

r s s r s s r5 2[(u u )c 1 (u c )u ]j 3, j j , j 3

r s r s r s r s r1 [(u u ) c 1 (u c ) u ] . (A8)j 3, j j , j 3

We can drop the second pair of terms through the scale-separation argument. Rewriting the first pair gives

LSA 5 2[ ( cs),j]rr su uj 3

5 2[ ( cs) ]r 1 [ ( cs) ]r.r s r r s su u u uj 3 ,j j 3 ,j (A9)

The scale-separation argument allows us to drop thesecond term in (A9), yielding finally

LSA 5 2[ ( cs) ]r 5 2[ ]r.r s r r r˜u u u Fj 3 ,j j 3,j (A10)

The small-scale advection term isr s s s s s s s rSSA 5 2[(u u ) c 1 (u c ) u ]j 3, j j , j 3

s s s s s s r5 2[(u u )c 1 (u c )u ]j 3, j j , j 3

s s r s s s r s r1 [(u u ) c 1 (u c ) u ] . (A11)j 3, j j , j 3

We drop the second pair of terms through the scale-separation argument and obtain

SSA 5 2[ cs] .s s ru uj 3 ,j (A12)

The resolvable-scale gradient production term is

RSGP 5 2[( )s ]rr s sc u u,j j 3

5 2[( ) ]r 1 [( )r ]r.r s s r s sc u u c u u,j j 3 ,j j 3 (A13)

Dropping the second term through the scale-separationargument yields

RSGP 5 2[ ]rr s sc u u,j j 3

5 2[ ( )r]r 2 [ ( )s]r.r s s r s sc u u c u u,j j 3 ,j j 3 (A14)

Dropping the second term through the scale-separationargument then finally yields

RSGP 5 2[ ( )r]r.r s sc u u,j j 3 (A15)

The molecular terms can be rewrittens s r s s rg(c u ) 1 n(u c ), jj 3 3, jj

s s r s s r s s r5 (n 1 g)(c u ) 2 n(c u ) 2 g(c u )3 , jj , j 3 , j 3, j , j

s s r2 (n 1 g)(c u ) . (A16), j 3, j

We can drop these terms on the basis of arguments usedby Wyngaard (1982) for the analogous terms in the Reyn-olds flux budget. The first term, molecular diffusion, andthe second are negligible because of the large Reynoldsand Peclet numbers above the molecular sublayer. Thethird term, molecular destruction, is assumed negligibleon the grounds of local isotropy (Wyngaard et al. 1971).

Finally, we drop a mean-shear production term2U3,j( cs)r from (A1) because it vanishes under horizon-suj

tally homogeneous conditions. The simplified budget reads

]rF ø0]t

r r s s r2((u u ) c )3, j j resolvable shear, cross flux interaction (RSCFI)

r r s s r2 ((c u ) u ), j j 3 resolvable gradient, cross stress interaction (RGCSI)r s s r r2 (u (u c ) )3, j j resolvable-scale shear production (RSSP)



r r r˜2 (u F )j 3 , j large-scale advection (LSA)r˜2 U Fj 0, j mean advection (MA)

s s s r2 (u u c )j 3 , j small-scale advection (SSA)r s s r r2 (c (u u ) ), j j 3 resolvable-scale gradient production (RSGP)

s s r2 C (u u ),3 3 3 mean-gradient production (MGP)g

s s r1 (u c )T0


s s r2 (p c ),3 pressure destruction (PD). (A17)

APPENDIX B

The Cross Components of SGS Flux

We showed in Eq. (17) that with wave-cutoff filteringthe SGS flux can be written

5 (cr 1 cs 1 cs )r.r s r sF u u u3 3 3 3 (B1)

We designate (cr )r and (cs )r as the cross components.s ru u3 3

By the arguments in section 3 of the paper, the con-tributing wavenumber ranges of and cs here are kc

su3

, k # 2kc. We can therefore estimate the magnitudeof the cross fluxes as

(cr )r ; crd , cs ; dcs ,s s r ru u u u3 3 3 3 (B2)

where d means the rms value of when its wave-s su u3 3

numbers are restricted to the range kc , k # 2kc. Since,in general, we can write

2kc

s 2(d f ) 5 E(k) dkEkc

2k kc c

5 E(k) dk 2 E(k) dk, (B3)E E0 0

it follows that

(df s)2 5 [ f r(2kc)]2 2 [ f r(kc)]2. (B4)

Using (33), (36), (41), and (43) in (B4) then yields

d ; , dcs ; cr.s ru u3 3 (B5)

Our scaling estimates for the cross fluxes are then

(cr )r ; cr , (cs )r ; crs r r ru u u u3 3 3 3 (B6)

so that each is of the order of the resolvable-scale flux.It follows from our scaling estimates that

cross flux

3z; F , neutral and free convection. (B7)01 2D

Thus, we can write for the entire stability range

3cross flux z; . (B8)1 2s–s flux D

This confirms that the cross flux is negligible sufficientlyclose to the surface.

APPENDIX C

Extension to Resolvable-Scale Surface Stress

a. The resolvable-scale surface stress budget

Filtering the momentum Eq. (10) yields the evolutionequation for its resolvable-scale part:

r]u gi r r r r r r˜ ˜1 (u u ) 1 R 5 2p 1 u d , (C1)i , j j i j , j , i 3i]t T0

where the SGS stress tensor isrRij

5 ( 1 1 )r 2 n( 1 ).r r s s r s s r rR u u u u u u u uij i j i j i j i,j j,i (C2)

We again neglect the molecular contribution and definethe deviatoric SGS stress tensor:

rRkkr r˜t 5 R 2 d . (C3)i j i j i j3

We write the shear stress components and nearr rt t13 23

the surface as . Through our scaling arguments, theyrt 0k

are good surrogates for the resolvable-scale surfaceshear stress. From (C2) they are the sum of ‘‘cross’’ and‘‘s–s’’ parts:

r r s s r r s s r r r˜ ˜t 5 (u u 1 u u ) 1 (u u ) 5 C 1 S ,0k k 3 k 3 k 3 0k 0k

k 5 1, 2. (C4)

Deriving the budget of as indicated in (23) yieldsrS 0k

]rS ø0k]t

r r s s r2((u u ) u )3, j j k resolvable shear, cross flux interaction (RSCFI)

r r s s r2 ((u u ) u )k, j j 3 resolvable gradient, cross stress interaction (RGCSI)



r s s s r2 ((u u ) u )3, j j k resolvable-scale shear production (RSSP)r s s s r s s s r2 ((u u ) u 1 (u u ) u )j 3, j k j k, j 3 large-scale advection (LSA)

r˜2 U Sj 0k, j mean advection (MA)s s s s s s s s r2 ((u u ) u 1 (u u ) u )j 3, j k j k, j 3 small-scale advection (SSA)r s s s r2 ((u u ) u )k, j j 3 resolvable-scale gradient production (RSGP)

s s r2 U (u u )k,3 3 3 mean-gradient production (MGP)g

s s r1 (u u )kT0


s s r s s r2 (p u ) 2 (p u ),3 k ,k 3 pressure destruction (PD). (C5)

TABLE C1. Scaling of terms in the budget of surface stress.

r r s s r r r s((u u ) u ) ; u u duRSCFI: 3, j j k 3, j j h

r r s s r r r s((u u ) u ) ; u u duRGCSI: k, j j 3 h, j j 3

r s s s r r s s((u u ) u ) ; u u uRSSP: 3, j j k 3, j j h

r s s s r s s s r r s s r((u u ) u 1 (u u ) u ) ; u (u u )LSA: j 3, j k j k, j 3 j 3 k , j

r r r˜U S ; u SMA: j k3, j h,h k3

s s s s s s s s r s s s((u u ) u 1 (u u ) u ) ; (u u u )SSA: j 3, j k j k, j 3 j 3 h , j

r s s s r r s s((u u ) u ) ; u u uRSGP: k, j j 3 k, j j 3

s s r s sU (u u ) ; U u uMGP: k, 3 3 3 1,3 3 3

g gs s r s s(u u ) ; u uBP: k kT T0 0

s s r s s r s s(p u ) 1 (p u ) ; p uPD: ,3 k , k 3 ,3 k

Term

u*2Neutral units of u1 2* z

j 5 1, 2 j 5 3

Free convectionuf2units of uf1 2z

j 5 1, 2 j 5 3

RSCFI5z1 2D

5z1 2D

1/3z zi1 21 2D D

z1 2D

RGCSI5z1 2D

5z1 2D

1/3z zi1 21 2D D

2z1 2D

RSSP3z1 2D

2z1 2D

z1 2D

2/3z1 2D

LSA3z u*1 2 1 2D U1

5z1 2D

2/3 1/3z zi1 2 1 2D D

1/3z zi1 21 2D D

MA2z1 2D

0 0 0

SSAz1 2D

1z1 2D

1

RSGP2z1 2D

z1 2D

2/3z1 2D

2/3z1 2D

MGP 1 0

BP 0 1

PD 1 1

The scaling procedure discussed in the text yields theestimates, shown in Table C1, of the magnitudes of theterms in (C5).

b. The constant-SGS-stress layer

To verify the feasibility of solving the surface-stressconservation equation away from the surface, we eval-uate the ratio of the difference between the surface stressand the stress at height z to the fluctuation in stress atheight z. This ratio is approximately given by

z ]t h3 . (C6)t ]zh3

We estimate the vertical stress gradient from the re-solvable-scale horizontal momentum equation by as-suming the inertial term to be of the leading order. Usingour scaling estimates in section 4, we find

3z ]t zh3 ø : neutral,1 2t ]z Dh3

2/3 1/3 214/3z D zø 1 21 2 1 2 1 2[ ]D z Di

: free convection. (C7)

We conclude that for a sufficiently small z/D there existsa constant-SGS-stress layer such that the SGS stress atheight z is a reliable surrogate for the surface stress.

In a similar analysis, Mason and Thomson (1992)assumed that as the surface is approached the verticalstress gradient is balanced by the horizontal pressuregradient, which they estimated to be roughly indepen-dent of height. Their estimate of vertical stress gradient,therefore, differs from ours. The conclusion regardingthe constant-SGS-stress layer, however, still holds albeitfor a smaller z/D. The discrepancy can only be resolvedthrough direct numerical simulations.

c. The cross stresses

As shown in appendix B, the cross stresses are oforder



r s r r s r r(u u ) ; u du ; u u ,k 3 h 3 h 3

s r r s r r r(u u ) ; du u ; u u (neutral),k 3 h 3 h 3

s r; u u (free convection). (C8)h 3

Our scaling estimates in conjunction with (C8) showthat the dominant contributor to the cross stress is( )r ; ; . Thus, we can writer s r s r ru u u du u uk 3 h 3 h 3

3z2cross stress ; u (neutral),*1 2D

1/3 1/3z zi2u (free convection). (C9)f 1 2 1 2D D

We estimate the magnitude of the s–s component of theSGS stress as

s s r s s 2s–s stress 5 (u u ) ; u u ; u (neutral),k 3 h 3 *

21/3z2u (free convection), (C10)f 1 2D

From (C9) and (C10) we have

3cross stress z; (neutral),1 2s–s stress D

1/3 2/3z zi (free convection). (C11)1 2 1 2D D

Equation (C11) confirms that the cross stress is negli-gible sufficiently close to the surface.

Equation (C11) also indicates that under free-con-vection conditions the cross stress can be of the orderof the s–s stress when

1/3 2/3 1/2z z z Di ; 1, or ; . (C12)1 2 1 2 1 2D D D zi

For typical values of D/zi (1/10 to 1/100, say) (C12)indicates that in free convection the cross stress is asimportant as the s–s stress when z/D ; 0.1 to 0.3. Forlarger z/D values, the cross term can be the dominantcontributor to SGS stress. This is unlike the situationfor scalar flux, for which we showed (appendix B) thatthe cross components are quite small.

d. The surface stress budget for z1/D K 1

When z/D K 1 only the O(1) terms in (C5) are sig-nificant so the streamwise stress budget reduces to

]r s s s s s s s s rS ø 0 ø 2[(u u ) u 1 (u u ) u ]01 3 3,3 1 3 1,3 3]t

gs s r s s r2 U (u u ) 1 (u u )1,3 3 3 1T0

s s r s s r2 (p u ) 2 (p u ) . (C13),3 1 ,1 3

The k 5 0 component of this equation is the Reynoldsbudget of streamwise stress (Wyngaard et al. 1971).

The lateral stress budget for z/D K 1 becomes

]r s s s s s s s s rS ø 0 ø 2[(u u ) u 1 (u u ) u ]02 3 3,3 2 3 2,3 3]t

gs s r s s r s s r1 (u u ) 2 (p u ) 2 (p u ) . (C14)2 ,3 2 ,2 3T0

The mean (the k 5 0 component) of each term in thisequation vanishes, due to the symmetry about the x1

axis. Thus, (C14) predicts zero mean lateral stress, asexpected; however, fluctuations in the lateral stress aregenerated by small-scale advection, buoyancy, and thepressure term.

We argued in section 5 of the paper that when z1/DK 1 the turbulence in the first grid element is in localgrid volume equilibrium. A closure for the resolvable-scalesurface stress budgets (C13) and (C14) in this case is

5 CD(z1/L)[sr(z1) (z1)]r.r rS u0k k

(k 5 1, 2; z1/D K 1). (C15)

Here CD is the mean surface-exchange coefficient formomentum and sr is the resolvable-scale horizontalwind speed.

e. The surface stress budget for z1/D , 1

Table C1 shows that in free convection there is notthe clear separation in the order of the terms that existedfor scalar flux. The next-order terms in (C5) are of order(z/D)2/3, but after that come a number of zero-mean,‘‘noise terms’’ (e.g., RSCFI, RGCSI) of order z/D.

Despite this lack of separation, the two budgets aresimilar in that the next-order terms are large-scale ad-vection, resolvable-scale gradient production, and thevertical part of resolvable-scale shear production. Weargued in the paper that the time-change term is of theorder of advection, which we will take to include thatby the mean velocity.

Rewriting the resolvable-scale gradient productionterm as we did for scalar flux and using the scale-sep-aration argument (appendix A) gives

RSGP ø 2( ( )r)r.r s su u uk,j j 3 (C16)

The scale-separation argument is inapplicable to the re-solvable-scale shear production term due to the con-suk

tained within it, so the term is

RSSP 5 2( ( )r)r 2 ( ( )s)r 1 (( )r )r.r s s r s s r s su u u u u u u u u3,j j k 3,j j k 3,j j k

(C17)

Only the first term in (C17) can be calculated, so wewill drop the second and third, which are zero-mean‘‘noise’’ terms, and write for its vertical part

RSSP ø 2( ( )r)r . 2( )r.r s s r r˜u u u u S3,3 3 k 3,3 0k (C18)

The budget of then becomesrS 0k



]r r r r r r r s s s s s s s s r˜ ˜ ˜S 1 (u S ) 1 (u S ) 1 [(u u ) u 1 (u u ) u ]0k h 0k ,h 3,3 0k 3 3,3 k 3 k,3 3]t

gr s s r r s s r s s r s s r1 (u (u u ) ) 2 (u u ) ø 2(p u ) 2 (p u ) .k, j j 3 k ,3 k ,k 3T0

(C19)

This is the counterpart to Eq. (74) for the surface scalarflux.

We showed in section 2 of this appendix that the crosscontribution is relatively more important for stress thanit is for scalar flux. Thus, when z1/D is not very smallso that a surface stress conservation equation rather thanthe usual surface-exchange expression is appropriate, itappears that the cross stress could also be significant.We showed that the dominant cross stress is ( )r,r su uk 3

which in principle is inaccessible since it involves asubgrid-scale quantity. It has zero mean and we showedthat its rms magnitude is of order , which is cal-r ru uk 3

culated in LES. We discuss the modeling of this termin section 6a.

REFERENCES

Bardina, J., J. H. Ferziger, and W. C. Reynolds, 1983: Improvedturbulence models based on large-eddy simulation of homoge-neous, incompressible, turbulent flows. Tech. Rep. TF-19, Stan-ford University, Stanford, CA, 174 pp.

Bradshaw, P., 1969: Comments on ‘‘On the relation between the shearstress and the velocity profile after change in surface roughness.’’J. Atmos. Sci., 26, 1353–1354.

Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971:Flux-profile relationships in the atmospheric surface layer. J.Atmos. Sci., 28, 181–189.

Clark, R. A., J. H. Ferziger, and W. C. Reynolds, 1979: Evaluationof subgrid-scale models using an accurately simulated turbulentflow. J. Fluid Mech., 91, 1–16.

Cotton, W. R., R. L. Walko, K. R. Costigan, P. J. Flatau, and R. A.Pielke, 1993: Using the Regional Atmospheric Modeling Systemin the large-eddy-simulation mode: From inhomogeneous sur-faces to cirrus clouds. Large-Eddy Simulation of Complex En-gineering and Geophysical Flows, B. Galperin and S. Orszag,Eds., Cambridge University Press, 369–398.

Deardorff, J. W., 1970: A numerical study of three-dimensional tur-bulent channel flow at large Reynolds numbers. J. Fluid Mech.,41, 453–480., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495–527.

Kaimal, J. C., 1978: Horizontal velocity spectra in an unstable surfacelayer. J. Atmos. Sci., 35, 18–24.

Khanna, S., and J. G. Brasseur, 1996: Analysis of Monin–Obukhovsimilarity from large-eddy simulation. J. Fluid Mech., 345, 251–286.

Klemp, J. B., 1987: Dynamics of tornadic thunderstorms. Annu. Rev.Fluid Mech., 19, 369–402.

Leonard, A., 1974: Energy cascade in large eddy simulation of tur-bulent flows. Advances in Geophysics, Vol. 18A, AcademicPress, 237–248.

Lilly, D. K., 1967: The representation of small-scale turbulence innumerical simulation experiments. Proc. Tenth IBM ScientificComputing Symp. on Environmental Sciences, IBM Thomas J.Watson Research Center, Yorktown Heights, NY, 195–210.

Lumley, J. L., and H. A. Panofsky, 1964: The Structure of Atmo-spheric Turbulence. Interscience, 239 pp.

Mason, P. J., and N. S. Callen, 1986: On the magnitude of the SGSeddy coefficient in LES of turbulent channel flows. J. FluidMech., 162, 439–462., and D. J. Thomson, 1987: Large-eddy simulation of the neutral-static-stability planetary boundary layer. Quart. J. Roy. Meteor.Soc., 113, 413–443., and , 1992: Stochastic backscatter in large-eddy simula-tions of boundary layers. J. Fluid Mech., 242, 51–78.

Moeng, C.-H., 1984: A large-eddy-simulation model for the study ofplanetary boundary layer turbulence. J. Atmos. Sci., 41, 2052–2062.

Nieuwstadt, F. T. M., P. J. Mason, C.-H. Moeng, and U. Schumann,1993: Large-eddy simulation of the convective boundary layer:A comparison of four computer codes. Turbulent Shear Flows,F. Durst, R. Friedrich, B. E. Launder, F. W. Schmidt, U. Schu-mann, and J. H. Whitelaw, Eds., Vol. 8, Springer-Verlag, 343–368.

Panofsky, H. A., and J. A. Dutton, 1984: Atmospheric Turbulence.Wiley, 397 pp., H. Tennekes, D. H. Lenschow, and J. C. Wyngaard, 1977: Thecharacteristics of turbulent velocity components in the surfacelayer under convective conditions. Bound.-Layer Meteor., 11,355–361.

Peltier, L. J., J. C. Wyngaard, S. Khanna, and J. G. Brasseur, 1996:Spectra in the unstable surface layer. J. Atmos. Sci., 53, 49–61.

Piomelli, U., W. H. Cabot, P. Moin, and S. Lee, 1991: Subgrid-scalebackscatter in turbulent and transitional flows. Phys. Fluids A,3, 1766–1771.

Schumann, U., 1993: Large-eddy simulation of turbulent convectionover flat and wavy surfaces. Large-Eddy Simulation of ComplexEngineering and Geophysical Flows, B. Galperin and S. Orszag,Eds., Cambridge University Press, 399–421.

Stull, R., 1988: An Introduction to Boundary Layer Meteorology.Kluwer, 402 pp.

Sullivan, P., J. C. McWilliams, and C.-H. Moeng, 1994: A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows. Bound.-Layer Meteor., 71, 247–276.

Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence.The MIT Press, 300 pp.

Wyngaard, J. C., 1982: Planetary boundary layer modeling. Atmo-spheric Turbulence and Air Pollution Modelling, F. T. M.Nieuwstadt and H. van Dop, Eds., Reidel, 69–106., 1988: Structure of the PBL. Lectures on Air-Pollution Mod-eling, A. Venkatram and J. C. Wyngaard, Eds., Amer. Meteor.Soc., 9–61., and L. J. Peltier, 1996: Experimental micrometeorology in anera of turbulence simulation. Bound.-Layer Meteor., 78, 71–86., O. R. Cote, and Y. Izumi, 1971: Local free convection, simi-larity, and the budgets of shear stress and heat flux. J. Atmos.Sci., 28, 1171–1182.


Date post:	08-Jan-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

LES in the Surface Layer: Surface Fluxes, Scaling, and SGS ...

Documents