Editorial Type:
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Article Type:
Research Article

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

J. C. Wyngaard Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

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L. J. Peltier Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

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S. Khanna Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

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Print Publication:
01 May 1998
DOI:
https://doi.org/10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2
Page(s):
1733–1754
Full access

Abstract

The surface fluxes in the fine-mesh numerical codes used in small-scale meteorology are typically diagnosed from 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 aspect ratio can approach unity in large-eddy simulation (LES) codes for the planetary boundary layer, however. In that 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 surface scalar and momentum fluxes in LES. Scaling relations for resolvable-scale variables in the surface layer are developed 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 noise terms 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 be a poor representation of surface-layer physics and conclude that the surface-flux conservation equations need to be coupled with a greatly improved SGS model in the surface layer.

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

+ Current affiliation: Applied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania.

Corresponding author address: Dr. John C. Wyngaard, Department of Meteorology, 503 Walker Building, The Pennsylvania State University, University Park, PA 16802-5013.

Abstract

The surface fluxes in the fine-mesh numerical codes used in small-scale meteorology are typically diagnosed from 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 aspect ratio can approach unity in large-eddy simulation (LES) codes for the planetary boundary layer, however. In that 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 surface scalar and momentum fluxes in LES. Scaling relations for resolvable-scale variables in the surface layer are developed 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 noise terms 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 be a poor representation of surface-layer physics and conclude that the surface-flux conservation equations need to be coupled with a greatly improved SGS model in the surface layer.

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

+ Current affiliation: Applied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania.

Corresponding author address: Dr. John C. Wyngaard, Department of Meteorology, 503 Walker Building, The Pennsylvania State University, University Park, PA 16802-5013.

1. Introduction

Large-eddy simulation, or LES, is now very widely used in small-scale meteorology. Applications range from severe storms (Klemp 1987) to small-mesoscale phenomena (Cotton et al. 1993) and boundary layer meteorology (Moeng 1984; Mason and Thomson 1987; Nieuwstadt et al. 1993; Schumann 1993).

The strength of LES lies in its explicit calculation of the energy-containing eddies of a turbulent flow. The unresolvable or subgrid-scale (SGS) eddies, which are formally removed by spatial filtering of the governing equations (Leonard 1974), manifest themselves through SGS terms in the filtered or resolvable-scale equations. If the scale of the filter is small compared to the scale of the energy-containing eddies, the resolvable-scale eddies do contain most of the turbulent kinetic energy and fluxes. Furthermore, they are fairly insensitive to the details of the SGS model used to close the filtered equations. Thus, well-resolved LES has an appealing robustness.

Because the horizontal length scale of the vertical velocity fluctuations scales with distance from the surface, LES inevitably has inadequate spatial resolution near the surface. As a result, the SGS model takes on much more importance there than in other regions of the flow. Furthermore, the specification of resolvable-scale surface fluxes comes into question since the widely used surface-exchange coefficients are not obviously valid locally. Thus, the fidelity of LES within the surface layer is problematic.

In this paper we study three closely related aspects of LES in the surface layer. First we examine the behavior of the surface-exchange coefficients when the grid aspect ratio (horizontal dimension/vertical dimension) is of order 1, as in LES. As an alternative to surface-exchange coefficients, we develop conservation equations for the resolvable-scale surface fluxes. Simplifying these surface-flux equations requires our second development, scaling relations for LES variables in the surface layer. These determine how turbulent variances are apportioned between the resolvable and SGS components. Finally, in order to interpret the impact of the resolvable-scale surface-flux equations on LES fields we evaluate some aspects of the performance of the widely used eddy-diffusivity SGS model in the surface layer.

2. Surface-exchange coefficients for resolvable-scale fluxes

a. Background

The LES codes used in boundary layer meteorology (e.g., Nieuwstadt et al. 1993) involve the surface fluxes of momentum, temperature, water vapor, and trace constituents. Since these codes use local spatial averaging of the governing equations rather than ensemble averaging, these are resolvable-scale surface fluxes that contain not only the ensemble mean but also the fluctuations about this mean on scales up to the filter cutoff. Most such codes take this resolvable-scale surface flux as proportional to the product of the horizontal wind speed and the difference in the transported quantity between the surface and the first grid point. The proportionality factor is the surface-exchange coefficient.

Surface-exchange coefficients for ensemble-average fluxes under horizontally homogeneous conditions are known from several decades of surface-layer research. In coarse-resolution (e.g., mesoscale) models, the horizontal scale Δ of the local spatial averaging and of the numerical grid is typically much larger than the height z1 of the first grid level. The grid aspect ratio Δ/z1 is large, meaning that there are many “coupling eddies” in the first grid volume, as illustrated in Fig. 1. In this case these mean exchange coefficients, as we will call them, are applicable. In LES, however, the grid aspect ratio is typically in the range from 1 to 3; there are not many coupling eddies and the surface-exchange coefficient has a random component. Thus, for F̃r0(x1, x2, t), the resolvable-scale surface flux of a scalar constituent, for example, the appropriate surface-exchange coefficient is not the mean value CH but a random variable, C̃H. (A superscript r, for resolvable scale, indicates that the variable has been spatially filtered; a tilde denotes that it has both mean and fluctuating parts.) Thus, the expression for surface flux is more properly written as
i1520-0469-55-10-1733-e1
Here ũr1 and Δc̃r are the streamwise components of resolvable-scale wind at the first grid level and the resolvable-scale c̃ difference between the first grid point and the surface, respectively. We call C̃H the local surface-exchange coefficient; we expect that C̃H → CH as z1/Δ → 0.

b. Observed fluctuations in local surface-exchange coefficients

Consider two LES grids of horizontal spacing Δ, in one the first grid point being close to the surface, at zs ≪ Δ, and in the other at a greater height z1 ≈ Δ. In view of these grid aspect ratios, we take the scalar-flux surface-exchange coefficients as CH(zs) for the first grid and C̃H(z1) for the second. We can write from (1), indicating only the z dependence,
i1520-0469-55-10-1733-e2
Solving for the local surface-exchange coefficient at height z1 yields
i1520-0469-55-10-1733-e3
We decompose the resolvable-scale fields into the sum of the ensemble mean and a fluctuating part, using upper case symbols for the former:
i1520-0469-55-10-1733-e4
Using the decomposition (4) and expanding (3) yields
i1520-0469-55-10-1733-e5

We evaluated (5) using LES data for the convective boundary layer and observations from the STORMFEST experiment. The LES domain was 5 km × 5 km in the horizontal and 2 km deep. It used a nested mesh in the surface layer so the resolution there was equivalent to that of a 2563 simulation (Khanna and Brasseur 1996). The inversion height zi was about 1000 m. In one run the geostrophic wind was 5.0 m s−1 and the surface temperature flux was 0.14 m s−1 K, giving −zi/L = 63. The other run had a geostrophic wind of 1.0 m s−1 and a surface temperature flux of 0.24 m s−1 K, giving −zi/L = 730. We took zs = 3.9 m, the lowest grid point, and evaluated the local surface-exchange coefficients at z1 = 12 and 27 m, using temperature as the scalar.

The STORMFEST data, taken by the NCAR Atmospheric Surface Transfer and Exchange Research facility, included time series of fluctuating temperature and the horizontal components of velocity from in situ sensors at z = 1 m, 4 m, and 10 m. The mean wind speed at 10 m was 10.7 m s−1, the surface temperature flux was 0.23 m s−1 K, the Monin–Obukhov length L was −63 m, and −zi/L ≈ 10. We took zs = 1.0 m, the lowest measurement point, and evaluated the local surface-exchange coefficients for temperature at z1 = 4 m and10 m.

The randomness factor in (5) involves wave-cutoff filtering at wavenumber magnitude κc in the horizontal wavenumber plane. We did this straightforwardly with the LES data, using several values of κc = π/Δ. This gave zs/Δ ⩽ 0.13. For the STORMFEST data we approximated the averaging by wave-cutoff filtering of the time series at frequency ωc, using Taylor’s hypothesis in the form κc = ωc/U. Here zs/Δ ⩽ 0.10.

Figure 2 shows the fluctuations in the local surface-exchange coefficient C̃H and its counterpart for momentum, C̃D, evaluated from measured time series and LES data, using (5) and its counterpart for C̃D. The fluctuation level increases as grid aspect ratio decreases, as we anticipated through the coupling-eddies notion depicted in Fig. 1, and it increases as −zi/L increases. It can be quite large.

As discussed by Wyngaard and Peltier (1996), we can gain insight into the fluctuations of the local surface-exchange coefficient through a simple model of Eq. (5). If Δcr/ΔC and ur1/U1 are small parameters, we can linearize (5) to find
i1520-0469-55-10-1733-e6
Taking the ensemble mean gives
i1520-0469-55-10-1733-e7
which indicates that for small fluctuation levels the mean of the local surface-exchange coefficient is the traditional surface-exchange coefficient. For sufficiently large fluctuation levels the two can differ. In the convective surface layer the fluctuation level of Δcr/ΔC is typically much less than that of ur1/U1, so that we can write (6) as
i1520-0469-55-10-1733-e8
which implies that
i1520-0469-55-10-1733-e9
As Δ/z1 becomes large, the separation z1 − zs becomes small compared to the horizontal scales of the fluctuations in ur1/U1. Thus, as Δ/z1 increases, we expect ur1(zs) and ur1(z1) to become increasingly well correlated and their rms difference to decrease; from (9) the fluctuation level in the local surface-exchange coefficient should then approach zero. This is consistent with the results in Fig. 2.

This suggests that the surface-exchange coefficient fluctuations are due to the decreasing correlation between resolvable-scale fields very near the surface and at z1 as κcz1 → 1. Further analysis of these LES and STORMFEST data by Wyngaard and Peltier (1996) supports this notion.

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

3. The subgrid-scale flux near the surface

The Boussinesq equations for velocity ũi and a conservative scalar c̃ in the surface layer are
i1520-0469-55-10-1733-e10
where p̃ is kinematic pressure, γ is the molecular diffusivity of c̃, and θ is potential temperature (Lumley and Panofsky 1964; Stull 1988). We assume Coriolis effects are negligible. We spatially low-pass filter each field over horizontal planes (Leonard 1974), designating the part that passes through the filter as resolvable and the remainder as subgrid scale, denoted with superscripts r and s, respectively. This yields the decomposition
i1520-0469-55-10-1733-e12
We assume that in horizontal wavenumber space the filter has a sharp cutoff at wavenumber magnitude κc = π/Δf, where Δf is the filter width in physical space. In principle Δf need not be the same as Δg, the horizontal grid spacing. As pointed out by Mason and Callen (1986), Δf defines the spatial scales contained in the resolvable fields, whereas Δg determines the accuracy of the numerical solutions of the filtered equations. One could obtain very high accuracy by taking Δg ≪ Δf, for example. Computer limitations require that Δf and Δg be essentially the same value, however, so we will not distinguish them further here, taking Δf = Δg = Δ. With our sharp cutoff filter the r and s fields have no wavenumber components in common; this implies that their covariance vanishes since nonoverlapping Fourier modes are uncorrelated (Lumley and Panofsky 1964).
Equation (11) holds for any advected scalar that is conservative, that is, has no sources or sinks. Filtering it yields the evolution equation for its resolvable-scale part:
i1520-0469-55-10-1733-e13
We write this as
i1520-0469-55-10-1733-e14
Naming F̃rj is a challenge since it involves four terms, multiple filtering operations, and both turbulent and molecular components. The name “resolvable-scale flux” does not distinguish it from (c̃rũrj)r. Its molecular term is important only in a very thin layer adjacent to the surface, and each of its three remaining terms has a subgrid-scale contribution. Thus, we will call F̃rj the subgrid-scale (SGS) flux.

The terms in (14), being resolvable scale, are bounded as z → 0 (we will use x3 and z interchangeably to denote height above the surface). In particular, F̃r3,3 is bounded so that for sufficiently small z, F̃r3(z) ≈ F̃r0, the resolvable-scale surface flux. Thus, we expect a “constant-SGS-flux” layer near the surface within which F̃r3 is a surrogate for the resolvable-scale surface flux. In section 4 we will confirm this through scaling estimates.

We decompose each resolvable-scale field into the sum of the ensemble mean and a fluctuating part, using upper case symbols for the former:
i1520-0469-55-10-1733-e15
Since the ensemble means of the subgrid-scale field and of the fluctuating part of the resolvable-scale field vanish, we can write
ũsiusic̃scsθ̃sθsp̃sps
At heights z sufficiently large that the molecular contribution to the SGS flux is negligible the vertical component of the flux is, from (14)–(16),
i1520-0469-55-10-1733-e17
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(κ) in the surface layer, where κ is horizontal wavenumber magnitude (κ21 + κ22)1/2. This spectrum integrates to the variance
i1520-0469-55-10-1733-e18
They used a single form for both horizontal velocity and a conservative scalar,
i1520-0469-55-10-1733-e19
with c1 and c2 adjustable constants and l and s the length and intensity scales. For vertical velocity they multiplied this form by a transfer function that models the effects of continuity and the u3 = 0 surface condition by attenuating E(κ) at the smallest wavenumbers.

The spectrum (19) has a peak at κ ∼ 1/l and a κ−5/3 inertial range asymptote at κ ≫ l−1. Peltier et al. chose the constants c1 and c2 by requiring the variance and the inertial-range spectral level to agree with observations. They chose the length and intensity scales l and s as z and u∗ under neutral conditions. In free convection they chose the scales as the PBL depth zi and the convective velocity scale w∗ = (gQ0zi/T0)1/3, rather than the local-free-convection scales z and uf = (gQ0z/T0)1/3 (Wyngaard et al. 1971); here Q0 is the ensemble-mean surface temperature flux. As justification they cited the evidence that horizontal velocity fluctuations in the unstable surface layer are not Monin–Obukhov (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 convection forms to give an interpolation formula for the entire stability range between them. The spectrum E(κ) cannot be directly measured with conventional instruments, but Peltier et al. showed that the corresponding one-dimensional spectra agree well with existing measurements.

The modeled two-dimensional spectra are shown in Fig. 3 in area-preserving coordinates. For all but near-neutral conditions the horizontal spectrum Eh, which integrates to (u21 + u22)/2, peaks at wavenumbers of the order of κh = 1/zi. The vertical velocity spectrum Eυ peaks at larger wavenumbers, those of the order of κz = 1/z. A third significant wavenumber is κc = π/Δ, the cutoff wavenumber of the LES grid mesh. In Moeng’s 963 LES code, for example, Δ is typically 5000 m/64 ≃ 80 m.

Let us now examine the contributions to the SGS flux as shown in (17) for heights z ≪ Δ and inversion depths zi ≫ Δ. Since z ≪ Δ ≪ zi, there is a large separation in these three wavenumbers: κh ≪ κc ≪ κz. Let us write each factor in the first cross term, (crus3)r, as a Fourier–Stieltjes integral, noting the admissible region in the κ1, κ2 plane:
i1520-0469-55-10-1733-e20
It follows that
i1520-0469-55-10-1733-e21
Since |κ′j| ⩽ κc, the restriction |κ′j + κ"j| ⩽ κc implies that |κ"j| < 2κc. Thus, only u3 modes of wavenumber magnitude less than 2κc can contribute to (crus3)r. Figure 3 shows that, as we approach the surface, 2κc falls below the range of wavenumbers contributing significantly to u3 so that (crus3)r → 0 as z/Δ → 0. The second cross term in (17), (csur3)r, is also eliminated because ur3 → 0 as z/Δ → 0.
We conclude that for z/Δ ≪ 1 the SGS flux is due entirely to the third term in (17),
i1520-0469-55-10-1733-e22
so that the SGS flux is carried by cs and us3 modes whose wavenumbers are beyond the cutoff but whose difference wavenumber falls within the cutoff.
Since at heights z much smaller than the scale of the spatial filter (i.e., z/Δ ≪ 1) the SGS flux F̃r3(z) ≈ (csus3)r is a good surrogate for F̃r0, we can form a conservation equation for F̃r0:
i1520-0469-55-10-1733-e23
The usi and cs equations are derived by high-pass filtering (10) and (11). Carrying out the operations in (23) and simplifying as discussed in appendix A yields the resolvable-scale surface flux conservation equation, or budget:
i1520-0469-55-10-1733-e24
Let us relate Eq. (24) to the more familiar budget of the Reynolds flux cu3, where the overbar denotes the ensemble mean. Consider a horizontally homogeneous flow and let κc → 0. In this limit each filtered field has only a κ = 0 component, the horizontal mean value, which because of the homogeneity is equal to the ensemble mean. Thus, uri → 0, cr → 0, and our decomposition (15) and (16) reduces to
i1520-0469-55-10-1733-e25
It follows that
i1520-0469-55-10-1733-e26
In this limit the first through fifth and the seventh terms on the right side of the SGS flux budget (24) vanish. Thus, as κc → 0 the SGS flux budget becomes, under horizontally homogeneous conditions,
i1520-0469-55-10-1733-e27
This is the budget of the vertical component of the Reynolds scalar flux (Wyngaard et al. 1971).
We need to scale the terms in the resolvable-scale surface flux conservation equation (24) so that we can drop the smallest ones. Monin–Obukhov similarity (Panofsky and Dutton 1984) provides expressions for the variances of velocity and scalar fluctuations in the surface layer, but here we need to know also how the partitioning of these variances between resolvable and subgrid-scale contributions,
i1520-0469-55-10-1733-e28
depends on distance from the surface and the cutoff wavenumber of the filter. We develop these scaling relations next.

4. Scaling LES fields in the surface layer

Consider a zero-mean random variable f = fr + fs (a fluctuating velocity component or scalar) in the surface layer. With wave-cutoff filtering we have
i1520-0469-55-10-1733-e29
For horizontal (subscript h) gradients we have
i1520-0469-55-10-1733-e30

We will use (29) to determine how the variances of uh, c, and u3 for z/Δ ∼ κcz ≪ 1 are partitioned into their resolvable and SGS parts. We will use (30) to determine how the variances of urh,h, ur3,h, and cr,h depend on z and Δ for z/Δ ≪ 1.

a. Horizontal velocity and scalars

For horizontal velocity and scalars the Peltier et al. (1996) model for E(κ) given in Eq. (19) yields
i1520-0469-55-10-1733-e31
Under neutral conditions Peltier et al. chose l = z, which makes (κl)2/c2 a small parameter in (31), and we find
i1520-0469-55-10-1733-e32
For horizontal velocity and scalars these imply
i1520-0469-55-10-1733-e33
In free convection Peltier et al. chose l = zi. For horizontal velocity (κcl)2/c2 is then a large parameter. Equation (31) yields in that limit
i1520-0469-55-10-1733-e34
which implies
i1520-0469-55-10-1733-e35
For a scalar in free convection Peltier et al. found that c2 ∼ (zi/z)2, making (κl)2/c2 a small parameter in (31) so that (32) applies. They found that c1 ∼ (zi/z)2/3, so that
i1520-0469-55-10-1733-e36

b. Vertical velocity

Peltier et al. used a slightly different model for the two-dimensional spectrum Eυ of vertical velocity u3 in the surface layer:
EυκTκEκ
T(κ) is a “continuity transfer function,”
i1520-0469-55-10-1733-e38
that models the influence of the surface at κz ≪ 1 and ensures that Eυ behaves as required by local isotropy at κz ≫ 1.
Under neutral conditions l = z so that κcz ≪ 1. In this limit we can simplify (37) to
Eυκκz2Eκ
so that the variance expressions are
i1520-0469-55-10-1733-e40
These imply
i1520-0469-55-10-1733-e41
Under free convection l = zi and the expressions are
i1520-0469-55-10-1733-e42
Here, κczi is a large parameter and we have
i1520-0469-55-10-1733-e43

c. Vertical gradients

The vertical inhomogeneity of the surface layer makes rms vertical gradients more difficult to estimate. For resolvable-scale variables this inhomogeneity can take two forms: a z-dependence of amplitude and a z-dependence of vertical spatial scale. We separate these two inhomogeneities by writing a zero-mean, resolvable-scale variable fr as
frxitAx3sxit
Here A is a nonrandom scaling function that contains the vertical inhomogeneity in amplitude. The function s is random, with zero mean and unit variance; it contains whatever vertical inhomogeneity in vertical spatial scale exists in fr. It is possible that the vertical spatial scales of ur3 vary with z, for example. Since s has unit variance, it follows from (44) that (fr)2 = A2. In the case of ur3 under neutral conditions, for example, (41) implies that A ∼ u∗(z/Δ)2.
Differentiating (44) with respect to x3, squaring, and averaging gives
(fr,3)2A2(s,3)2s2,3A,3AA,32A2(s,3)2A,32
the cross term vanishing because we chose s2 to be independent of z. The gradient variances (s,h)2 and (s,3)2 are related to s2 through the Taylor microscales λh and λz (Tennekes and Lumley 1972):
λ2hs2(s,h)2λ2zs2(s,3)2
With (46) we can write (45) as
i1520-0469-55-10-1733-e47
Differentiating (44) with respect to xh, squaring, averaging, and combining with (47) yields
i1520-0469-55-10-1733-e48
Our results in sections 4a and 4b show that for most of the resolvable-scale variables the horizontal Taylor microscale λh is Δ, the smallest surviving horizontal scale in the filtered fields. We will assume that λz is of this order as well so that (48) becomes
i1520-0469-55-10-1733-e49
Let us illustrate with ur3 under neutral conditions. Equation (49) yields
i1520-0469-55-10-1733-e50
The second term on the right dominates at small z/Δ so that our assumption that λz ∼ λh ∼ Δ is not critical; we obtain the result (50) for λz ∼ z as well. This is the case for almost all variables.
This procedure yields the scaling estimates
i1520-0469-55-10-1733-e51

From (33), (35), and (51) we see that ur1,1, ur2,2, and ur3,3 are of the same order, consistent with the resolvable-scale continuity equation.

5. Simplifying the resolvable-scale surface-flux budget

a. The “constant-SGS-flux” layer

We can use our surface-layer scaling relations of section 4 to quantify the variation of SGS flux with height near the surface. We first decompose the SGS flux into ensemble-mean and fluctuating parts,
F̃r3F0f3
We assume that the two flux-divergence terms in the resolvable-scale scalar conservation equation (14) are of the same order:
F̃r3,3F̃rj,jc̃r,jũrjr
Our scaling estimates in section 4 then imply
i1520-0469-55-10-1733-e54
This says that the SGS flux divergence vanishes as z/Δ → 0 because the forcing terms in the resolvable-scale scalar equation vanish in that limit.
Equation (54) implies that the rms difference between the resolvable-scale surface flux F̃r0 and the SGS flux at height z, F̃r3(z), scaled by the mean surface flux F0, is of order
i1520-0469-55-10-1733-e55
Let us compare this rms flux difference to the rms fluctuations in SGS flux at height z ≪ Δ. Estimating the rms value of the fluctuating SGS flux F′3 from the usual surface-exchange formulation gives
i1520-0469-55-10-1733-e56
where σu is the rms fluctuation in the streamwise wind, typically in the range (0.1–0.5)U1.

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

It remains to evaluate the importance of the cross contributions to the SGS scalar flux, Eq. (17). In appendix B we show their rms values are of the order of F0 (z/Δ)3. Thus, at height z ≪ Δ they are not larger than the rms difference in SGS flux between the surface and height z. Since we judged the latter to be negligible, we will continue to neglect the cross contributions to SGS 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 direction to be that of the mean wind so that Ui = (U1, 0, 0). The scaling results are summarized in Table 1. We cannot directly scale pressure destruction, but from experience with the Reynolds flux budget close to the surface (Wyngaard et al. 1971) we expect that it is a leading term.

To scale the time-change term, not included in Table 1, we assume that the local rate of change of the resolvable-scale surface flux is determined by resolvable-scale processes. Thus, we take the time-change term to be of the order of mean advection at neutral and large-scale horizontal advection in free convection, giving
i1520-0469-55-10-1733-e57

c. A hierarchy of resolvable-scale surface-flux budgets

According to the scaling results of Table 1 the budget of surface scalar flux takes different forms depending on the value of z1/Δ.

1) z1/Δ ≪ 1

When z/Δ ≪ 1 we need retain only the O(1) terms in the resolvable-scale surface-flux conservation equation (24), and it reduces to
i1520-0469-55-10-1733-e58
The κ = 0 mode of Eq. (58), its expected value, is the quasi-steady form of Eq. (27), the locally homogeneous Reynolds budget of scalar flux. The terms in Eqs. (58) and (27) represent, in order, the rates of vertical transport of turbulent flux by small-scale turbulence, production through the interaction of vertical velocity fluctuations with the mean vertical gradient of the quantity being transported, production through buoyancy, and destruction by pressure gradient interactions.
In the locally homogeneous, quasi-steady surface layer, there exists well-established flux-gradient and surface-exchange relations (Businger et al. 1971):
i1520-0469-55-10-1733-e59
Here z0c is the “roughness length” for the scalar. In (59) we have shown explicitly only the dependence on z. The dependence on stability is represented well through the Monin–Obukhov similarity hypothesis (Panofsky and Dutton 1984).

The turbulence active in the processes represented in (58) is subgrid scale; it has a dominant length scale z and timescales Ts ∼ z/u∗ and z/uf in the neutral and free-convection limits, respectively. The final filtering of each term can be thought of roughly as averaging over horizontal spatial scales Δ ≫ z. The time-change term is negligible because its timescale is much greater than Ts. Similarly, the horizontal inhomogeneity of filtered variables is weak because it occurs on a spatial scale much larger than z. Thus, we argue that for z/Δ ≪ 1 the resolvable-scale surface flux budget (58) represents a quasi-steady, locally homogeneous state of“local grid-volume equilibrium.”

The analogy between the Reynolds flux budget (27) and the z/Δ → 0 limit of the resolvable-scale surface-flux budget (58), the well-behaved surface-exchange relation (59) for the Reynolds flux, and the “coupling eddies” notion of Fig. 1 make it plausible that in this limit the resolvable-scale surface flux displays a surface-exchange relation like (59), but with resolvable-scale variables replacing the ensemble averages:
i1520-0469-55-10-1733-e60
Here CH is the mean surface-exchange coefficient that depends on stability; here the stability is interpreted locally, however. For that purpose we define the local stability parameter for the grid volume, z1/L̃, with L̃ the local Monin–Obukhov length,
i1520-0469-55-10-1733-e61
and ũ∗ and Q̃0 the local SGS friction velocity and surface temperature flux. Based on Eq. (58) we would expect ΔC rather than Δc̃r to appear in (60), but section 4 indicates that this difference is negligible for z/Δ ≪ 1 and (60) is consistent with typical practice in LES.

2) z1/Δ small but not negligible

Table 1 shows that in free convection the horizontal part of large-scale advection (LSA) is significant at z1 such that
i1520-0469-55-10-1733-e62
For Δ = 10 m and zi = 103 m this occurs at z1 = 1 m. The same arguments hold for the time-change term, Eq. (24). We conclude that in application to fine-mesh LES with z1 on the order of a few meters, the time-change and horizontal LSA terms in the surface-flux budget (24) can be significant.
If we combine the LSA and MA terms by writing U1 + urh = ũrh, as we previously did with the RSGP and MGP terms, Eq. (24) becomes
i1520-0469-55-10-1733-e63
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 horizontal advection and time change of resolvable-scale surface flux in addition to the production, small-scale advection, and pressure-destruction processes.
Let us model the local time-change and horizontal advection effects through a hypothesis advanced by Bradshaw (1969) and discussed by Wyngaard (1982). We assume that the SGS turbulence remains in local grid volume equilibrium in the sense that
i1520-0469-55-10-1733-e64
where (c̃r,3)eq is the local-grid-volume-equilibrium value of the resolvable-scale scalar gradient in the vertical. Because z1/Δ is small, there are many coupling eddies in the grid volume and to first approximation we can take (c̃r,3)eq to be nonrandom and to have a similarity form identical to that of the traditional M–O function for temperature, ϕh:
i1520-0469-55-10-1733-e65
Using (64) in (63) yields
i1520-0469-55-10-1733-e66
In order to solve Eq. (66) the local value of the SGS vertical velocity variance (us3us3)r is needed; it could be obtained through a local similarity model
us3us3rũ2∗fz1L̃
where f is the M–O function for vertical velocity variance. With such a model for (us3us3)r, Eq. (66) indicates how the scalar gradient differs from the local-grid-volume-equilibrium value if there is local time change or horizontal advection of resolvable-scale surface flux.
If we approximate (66) by
i1520-0469-55-10-1733-e68
and introduce the mean and local surface-exchange coefficients,
i1520-0469-55-10-1733-e69
combining (68) and (69) then gives
i1520-0469-55-10-1733-e70
This determines the departure of the local surface-exchange coefficient C̃H from its mean value CH due to the effects of local time change and horizontal advection of SGS flux.

3) z1/Δ ⩽ 1

In order to evaluate other candidate terms in the resolvable-scale surface flux budget, let us write (24) symbolically as
i1520-0469-55-10-1733-e71
where the final term, a parameterization of the rate of pressure destruction, is patterned after its behavior in the Reynolds flux budget (Wyngaard et al. 1971), and Ti represents the remaining terms. We estimate the rms contribution to F̃r0 from each component of Ti as δiF̃r0 ∼ τTi. As z1/Δ → 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, and we showed in (55) that as z1/Δ increases the rms difference between the fluxes at the surface and at height z1 also increases. This suggests a criterion for retaining a term in (24): that its rms contribution to the flux be large compared to the rms difference between the resolvable-scale surface flux and the SGS flux at height z1:
δiF̃r0F̃r0F̃r3z1
This ensures that the term does reflect a contribution primarily to the resolvable-scale surface flux.
Using our estimate (55) for F̃r0 − F̃r3(z1) and taking τ ∼ z1/u∗ at neutral and τ ∼ z1/uf in free convection then yields
i1520-0469-55-10-1733-e73
Let us interpret “≫” in (73) as meaning between one and two orders less in z1/Δ. We therefore also include the vertical part of RSSP. The surface flux budget then is
i1520-0469-55-10-1733-e74
Using the local-grid-volume-equilibrium hypothesis (64) for the SGS turbulence then gives an extension of (70):
i1520-0469-55-10-1733-e75
Equation (75) further corrects the equilibrium surface-exchange expression for the production of resolvable-scale surface flux through the interaction of the SGS flux and the resolvable-scale velocity gradient ur3,3.

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

d. Resolution in z

Our scaling analysis of section 4 treats the effects of filtering the fields in the horizontal plane. Such filtering is done explicitly in Moeng’s (1984) LES code, for example. Our analysis does not treat the effects of filtering or finite-difference approximations in z. Thus, our surface-layer scaling results are directly applicable only to LES with high vertical resolution.

The vertical resolution of LES also figures in the choice of a resolvable-scale surface flux model. The tradeoff is clear: if z1/Δ ≪ 1, which requires the first grid point to be very close to the surface, a standard surface-exchange model for flux, Eq. (60), can be used. If z1/Δ ≈ 1, the first grid point can be much higher but then more physics enters the resolvable-scale surface flux conservation equation, as indicated in Eq. (75). Thus, when z1/Δ ≪ 1, the complicated connection between the resolvable-scale surface flux and the resolvable-scale flow at z ∼ Δ must be made entirely through the SGS model; when z1/Δ ≈ 1, some elements of that connection can be made through the resolvable-scale surface flux model.

6. Use of resolvable-scale surface-flux equations in LES

a. Implementation and results

We implemented the z1/Δ ⩽ 1 form of the conservation equations for resolvable-scale surface temperature flux Q̃r0 and resolvable-scale surface stress τ̃r0k in Moeng’s LES code (Moeng 1984). Using (69) for the surface-exchange coefficients, Eq. (75) can be rewritten for Q̃r0 as
i1520-0469-55-10-1733-e76
We used (65) for the local-grid-volume-equilibrium value of the resolvable-scale temperature gradient, where
i1520-0469-55-10-1733-e77
(Businger et al. 1971), L̃ is given by (61), and ũ∗ is given by
ũ2∗(τ̃r01)2 + (τ̃r02)2
The local SGS vertical velocity variance is modeled as Eq. (67), where the nondimensional function is taken as
i1520-0469-55-10-1733-e79
(Panofsky et al. 1977).
Our modeled conservation equation for the subgrid–subgrid part of the SGS stress, (uskus3)r = S̃r0k, reduces to (appendix C)
i1520-0469-55-10-1733-e80
where k = 1, 2. The local-grid-volume-equilibrium value of the resolvable-scale velocity gradient is given by
i1520-0469-55-10-1733-e81
where ϕm is taken as
i1520-0469-55-10-1733-e82
(Businger et al. 1971).
In section 2 of appendix C we show that the cross stress C̃r0k = (urkus3 + uskur3)r has zero mean but an rms value of order urkur3. We model it as
C̃r0kurku′3u′kur3
where a prime denotes the rms value of the resolvable-scale field. The model (83) has zero mean, has an rms value of the required order, and depends on both the horizontal and vertical components of the resolved velocity.
Equations (76) and (80) were solved using a second-order, explicit Adams–Bashforth time marching scheme in which the temperature flux, for example, at time step n + 1 is given by
i1520-0469-55-10-1733-e84
where Δt is the time step, and RQ (n) is the right-hand side of (76) at time step n.

Since z1/Δ is a critical parameter in our analysis, we carried out three sets of simulations: 643, 1282 × 64, and 1922 × 64, with z1/Δ of 0.13, 0.27, and 0.4, respectively, for a moderately convective boundary layer (−zi/L ≈ 10). Figure 2 shows the fluctuation levels in the surface-exchange coefficients obtained by using conservation equations for resolvable-scale surface fluxes in the simulations. These levels agree well with the findings of Wyngaard and Peltier (1996). The fluctuation levels in S̃r0k and C̃r0k are roughly of the same order.

Figures 4–6 show the nondimensional mean shear ϕm, mean potential temperature gradient ϕh, and vertical velocity variance, respectively, from 643 and 1282 × 64 simulations using the conservation equations (curves A and B) and the standard surface-exchange coefficients (curves 1 and 2). The profiles observed in past field experiments are also plotted in these figures.

Consistent with the experience of Mason and Thomson (1992) and Sullivan et al. (1994) ϕm is overpredicted, with respect to the observations, by the Smagorinsky-based SGS model (used in Moeng’s code) with surface-exchange coefficients. Moreover, the profile kink moves closer to the surface as the horizontal resolution is increased.

In the 643 simulation with surface-exchange coefficients, ϕh is somewhat lower than the measured value at the first grid point; above that it is slightly overpredicted. Increasing the horizontal resolution to 1282, while keeping the vertical resolution the same, limits the overprediction of ϕh to the second grid point. The values at other grid points agree well with the field observations.

The nondimensional vertical-velocity variance in both the 643 and the 1282 × 64 simulations with surface-exchange coefficients are slightly lower than the field measurements reported by Panofsky et al. (1977). The change in resolution does not alter the profile significantly.

The use of flux-conservation equations instead of the surface-exchange coefficients yields slight improvements in all three profiles, most noticeably in that of vertical-velocity variance. However, the improvement is only marginal and not wholly satisfactory. We believe that the mild influence of improved lower boundary conditions on the mean surface-layer structure is due to deficiencies in the SGS model. The objective of surface-flux budgets is to capture more reliably the local structure of the resolvable-scale surface fluxes, but, as we show next, the Smagorinsky-based SGS model represents the surface-layer physics poorly and therefore presumably responds to enhanced structure in surface fluxes improperly as well.

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

The Smagorinsky-type SGS model has been standard in LES since the early work of Lilly (1967) and Deardorff (1970). Its popularity stems from its simplicity and, most importantly, the insensitivity of resolved-scale fields to the SGS model in well-resolved turbulent flows. Near the surface, however, the vertical motions are always inadequately resolved and as a result the SGS closure becomes crucial there.

We analyzed the accuracy of Smagorinsky closure in the surface layer using highly resolved LES with nested meshes. Such tests have been done previously using direct numerical simulation (DNS) of isotropic turbulence (Clark et al. 1979; Bardina et al. 1983) and the neutral turbulent boundary layer (Piomelli et al. 1991). The unstable atmospheric surface layer, however, has certain distinct features not present in homogeneous turbulence and neutral boundary layers. Specifically, the horizontal motions in the surface layer of an unstable atmospheric boundary layer (ABL) are strongly influenced by the zi-scaled mixed-layer eddies, causing a strong anisotropy of length and intensity scales between the horizontal and vertical motions. There is also a substantial horizontal mean temperature flux in the surface layer that is absent in the neutral boundary layer. Thus, past studies using DNS fields are not directly relevant for our analysis.

We generated highly resolved surface-layer fields using a three-level nested-mesh LES discussed in Khanna and Brasseur (1996). The full boundary layer was simulated at −zi/L ≈ 10 using a 1283 mesh covering a domain of 5zi × 5zi × 2zi. The next level of refinement in the surface layer was attained by an effective 2563 mesh covering a domain of 5zi × 5zi × 0.125zi, and the final level was attained by an effective 5123 mesh covering 5zi × 5zi × 0.06125zi of the boundary layer. The upper boundary conditions for the embedded meshes were obtained from the next-level coarser domains with a one-way communication between the domains. At a height where the effective 5123 simulation resolved approximately 90% of the vertical fluxes and variances (the sixth grid level, at z/zi ≈ 0.02) the resolved variables were treated as fully resolved fields. They were decomposed into a resolvable part and a subgrid part by two-dimensional horizontal wave-cutoff filtering with a cutoff wavenumber corresponding to a 962 horizontal mesh.

Table 2 compares the calculated SGS stresses and fluxes and their divergences with the values predicted by the SGS model used by Moeng (1984). The magnitudes of the cross-correlation coefficients of the modeled and actual SGS flux components are very low (less than 0.20). Those for SGS flux divergences, which appear in the dynamical equations, are particularly low (less than 0.05 for the horizontal components of the divergences). The SGS model also fails in capturing the mean diagonal SGS stress components, presumably due to its inability to reproduce the anisotropic distribution of SGS energy, and in capturing the mean horizontal SGS 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 structure of resolvable-scale surface fluxes.

c. Results with improved subgrid-scale model

Mason and Thomson (1992) found that adding stochastic fluctuations to the SGS stress divergence improved the nondimensional mean shear in a neutral boundary layer calculated through LES. These fluctuations simulate the local transfer of energy from unresolved to resolved scale, or “backscatter,” in the Smagorinsky SGS closure. We implemented this modification to the SGS model used in Moeng’s code along with the surface-flux conservation equations.

Moeng’s code uses the SGS model proposed by Deardorff (1980) in which the subgrid-scale energy, e = (usiusi)r/2, is calculated explicitly and used as the velocity scale for the SGS eddy diffusivity (K̃):
i1520-0469-55-10-1733-e85
where S̃rij is the resolvable strain rate tensor, ck is a constant, l is the length scale of unresolved eddies, and Pr is the turbulent Prandtl number. Following the development of Mason and Thomson, we added a white-noise random acceleration, ai, and a random temperature source, q, to the resolvable-scale equations:
i1520-0469-55-10-1733-e88
to account for stochastic backscatter. To satisfy continuity ai was taken to be the curl of another random vector: ai = ϵijkbk,j. As indicated by Mason and Thomson, this random forcing adds a production rate aiaiΔt to the resolved-scale kinetic energy equation and 2q2Δtto the equation for resolved-scale temperature variance, where Δt is the time step. Mason and Thomson show that when the filter scale is of the order of the scale of the unresolved eddies, the mean backscatter of resolved-scale kinetic energy due to stochastic fluctuations in unresolved eddies is cbϵ, where ϵ is the ensemble-average dissipation rate and cb = 1.4; the mean backscatter of resolved-scale temperature variance is 2cbθχ, where χ is the destruction rate of one-half temperature variance and cbθ = 0.45. The random components bi and q are scaled such that
i1520-0469-55-10-1733-e90
We increased the constant ck in (86) to (1 + cb)ck to account for the additional production of resolved-scale kinetic energy. The turbulent Prandtl number in (87) was increased by a factor of (1 + cb)/(1 + cbθ) to account for the additional production of resolved-scale temperature variance. The length scale l in (86) was taken to be the minimum of Δ, the grid scale, and kz/ck, where k is the von Kármán constant and z is the distance from the surface.

Figure 7 shows the effect of the SGS model and lower boundary conditions on the ϕm profile in the 1282 × 64 simulation of the moderately convective boundary layer. The original SGS model and drag coefficients overestimate ϕm at the first grid point (z/zi ≈ 0.03) by 50%. Use of surface-flux conservation equations with the original SGS model makes a marginal improvement. The modified SGS model (with stochastic backscatter) with the drag coefficients gives a smooth profile for ϕm, although it is not entirely in agreement with the observed profile. The simulated value increases with z/L before falling off, while the observed profile decreases monotonically. Our results are not in complete agreement with those of Mason and Thomson, presumably due to three factors: first, we are simulating a moderately convective boundary layer (with both buoyancy and shear effects) while their work concerned neutral boundary layers; second, we use a 1282 × 64 mesh, whereas Mason and Thomson used a much finer vertical mesh compared to the horizontal mesh; finally, some of the parameters used in our implementation of the stochastic-backscatter model are different from those of Mason and Thomson. Nevertheless, we infer from these results that the subgrid-scale model does have a significant influence on the mean structure of the surface layer. Use of surface-flux conservation equations along with the stochastic backscatter model makes a marginal improvement. We conclude that the improved lower boundary conditions need a compatible subgrid-scale model to make better predictions of atmospheric surface layers.

7. Summary and conclusions

We argued physically and showed through analysis of observations and LES data that the local surface-exchange coefficient relating the resolvable-scale surface flux to resolvable-scale properties of the overlying flow is a random variable. Its fluctuation level increases with z/Δ, where z is height above the surface and Δ is the spatial scale of the filter that separates fields into resolvable-scale and subgrid-scale parts. As z/Δ → 0, the surface-exchange coefficient approaches its traditional definition.

An alternative to using a local surface-exchange coefficient to diagnose the resolvable-scale surface flux is predicting that flux through its conservation equation. We showed that near the surface this flux is dominated by the s–s component and we derived the equation for that component. We used the surface-layer spectral model of Peltier et al. (1996) to develop scaling expressions for the partitioning of the variances of surface-layer fields between the resolvable and subgrid-scale components. We used these scaling expressions to simplify the flux conservation equations in three limits.

For high aspect ratio grids, z1/Δ ≪ 1, the surface-flux conservation equation is in a state of local grid volume equilibrium and the mean surface-exchange coefficient can be used locally, as is generally done in LES. For smaller aspect ratio grids it has horizontal advection and time-change terms, consistent with the appearance of fluctuations in the local surface-exchange coefficient in this regime. As the grid aspect ratio approaches unity, the flux conservation equation gains a production term proportional to the convergence of horizontal velocity near the surface. The observed fluctuations in the local surface-exchange coefficient in this regime are quite large.

Based on Bradshaw's (1969) suggestion that horizontal inhomogeneity of the surface layer causes the local mean gradient of a transported quantity to deviate from its equilibrium value, we proposed a simple closure for the flux conservation equations and implemented them in the Moeng (1984) LES code.

Use of the surface flux conservation equations yields slight improvements in the nondimensional mean shear, mean potential temperature gradient, and vertical-velocity variance profiles. The commonly observed kink in the mean shear and temperature gradient profiles predicted by Smagorinsky-based SGS models at z ≈ Δ is reduced. This improvement, however, is not substantial.

We showed through highly resolved LES that the Smagorinsky-based SGS models perform poorly in the atmospheric surface layer; a better SGS model is needed. We are currently extending the present analysis to derive conservation equations for SGS stresses and fluxes in the atmospheric surface layer, which, combined with dynamic lower boundary conditions, will hopefully make significant improvements in the LES predictions of atmospheric surface-layer structure.

Acknowledgments

We are grateful to T. Horst and G. Maclean of NCAR SSSF for kindly providing ASTER data from the STORMFEST experiment; to Andrew R. Brown of U.K. Meteorological Office for helpful discussions on stochastic backscatter; and to J. Brasseur, P. Mourad, and P. Sullivan for making helpful suggestions on the manuscript. This work was supported by Army Research Office Grant DAAL03-92-G-0117 and by Office of Naval Research Grant N00014-92-J-1688.

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APPENDIX A

Simplifying the Resolvable-Scale Surface Scalar Flux Budget

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

i1520-0469-55-10-1733-ea1
We can simplify a number of the terms in (A1). We rewrite the resolvable-scale shear production term (RSSP) as
ur3,jusjscsrur3,jusjcsrur3,jusjrcsr
The term on the far right has the form (arbs)r. The wavenumber components of ar have magnitudes between 0 and κc. The wavenumber components of bs have magnitudes greater than κc. Let us write this symbolically as
arbsrκcκcrκc
In our notation, brackets mean that the set’s end point is included in the range and parentheses mean that the set's end point is not included in the range. The wavenumber components of arbs are the vector sum of all those of ar and bs. Therefore wavenumbers of bs larger than 2κc cannot contribute to (arbs)r, so we can write (A3) as
arbsrκcκcκcr
The variable b is the temperature field, whose dominant wavenumbers have magnitudes of the order of 1/z (Peltier et al. 1996). Thus, as κcz → 0, there are no modes of b in the range (κc, 2κc] so that (arbs)r vanishes, and we can write
arbsrabsrκcz
We will refer to the development that leads to (A5) as the scale-separation argument.
We can now rewrite (A2) as
ur3,jusjcsrur3,jusjcsrrur3,jusjcssr
In order to proceed we need to know the shape of the spectrum of usjcs. We calculated it for j = 1 and j = 3 from high-resolution (2563) LES data. The results, shown in Fig. A1, indicate that the spectrum of usjcs is like that of u3 in that it peaks at κ ∼ 1/z. Thus, we can use the scale-separation argument to drop the second term on the right side of (A6). This yields
ur3,jusjcsrrκcz
We can rewrite the large-scale advection terms as
i1520-0469-55-10-1733-ea8
We can drop the second pair of terms through the scale-separation argument. Rewriting the first pair gives
urjus3cs,jrurjus3csr,jrurjus3css,jr
The scale-separation argument allows us to drop the second term in (A9), yielding finally
urjus3csr,jrurjF̃r3,jr
The small-scale advection term is
i1520-0469-55-10-1733-ea11
We drop the second pair of terms through the scale-separation argument and obtain
usjus3csr,j
The resolvable-scale gradient production term is
cr,jusjsus3rcr,jusjus3rcr,jusjrus3r
Dropping the second term through the scale-separation argument yields
cr,jusjus3rcr,jusjus3rrcr,jusjus3sr
Dropping the second term through the scale-separation argument then finally yields
cr,jusjus3rr
The molecular terms can be rewritten
i1520-0469-55-10-1733-ea16
We can drop these terms on the basis of arguments used by Wyngaard (1982) for the analogous terms in the Reynolds flux budget. The first term, molecular diffusion, and the second are negligible because of the large Reynolds and Peclet numbers above the molecular sublayer. The third term, molecular destruction, is assumed negligible on the grounds of local isotropy (Wyngaard et al. 1971).
Finally, we drop a mean-shear production term −U3,j(usjcs)r from (A1) because it vanishes under horizontally homogeneous conditions. The simplified budget reads
i1520-0469-55-10-1733-ea17

APPENDIX B

The Cross Components of SGS Flux

We showed in Eq. (17) that with wave-cutoff filtering the SGS flux can be written
F̃r3crus3csur3csus3r
We designate (crus3)r and (csur3)r as the cross components. By the arguments in section 3 of the paper, the contributing wavenumber ranges of us3 and cs here are κc < κ ⩽ 2κc. We can therefore estimate the magnitude of the cross fluxes as
crus3rcrδus3csur3δcsur3
where δus3 means the rms value of us3 when its wavenumbers are restricted to the range κc < κ ⩽ 2κc. Since, in general, we can write
i1520-0469-55-10-1733-eb3
it follows that
δfs2frκc2frκc2
Using (33), (36), (41), and (43) in (B4) then yields
δus3ur3δcscr
Our scaling estimates for the cross fluxes are then
crus3rcrur3csur3rcrur3
so that each is of the order of the resolvable-scale flux. It follows from our scaling estimates that
i1520-0469-55-10-1733-eb7
Thus, we can write for the entire stability range
i1520-0469-55-10-1733-eb8
This confirms that the cross flux is negligible sufficiently close to the surface.

APPENDIX C

Extension to Resolvable-Scale Surface Stress

The resolvable-scale surface stress budget

Filtering the momentum Eq. (10) yields the evolution equation for its resolvable-scale part:
i1520-0469-55-10-1733-ec1
where the SGS stress tensor R̃rij is
R̃rijũriũsjũsiũrjũsiũsjrνũri,jũrj,i
We again neglect the molecular contribution and define the deviatoric SGS stress tensor:
i1520-0469-55-10-1733-ec3
We write the shear stress components τ̃r13 and τ̃r23 near the surface as τ̃r0k. Through our scaling arguments, they are good surrogates for the resolvable-scale surface shear stress. From (C2) they are the sum of “cross” and“s–s” parts:
i1520-0469-55-10-1733-ec4
Deriving the budget of S̃r0k as indicated in (23) yields
i1520-0469-55-10-1733-ec5

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

The constant-SGS-stress layer

To verify the feasibility of solving the surface-stress conservation equation away from the surface, we evaluate the ratio of the difference between the surface stress and the stress at height z to the fluctuation in stress at height z. This ratio is approximately given by
i1520-0469-55-10-1733-ec6
We estimate the vertical stress gradient from the resolvable-scale horizontal momentum equation by assuming the inertial term to be of the leading order. Using our scaling estimates in section 4, we find
i1520-0469-55-10-1733-ec7
We conclude that for a sufficiently small z/Δ there exists a constant-SGS-stress layer such that the SGS stress at height 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 vertical stress gradient is balanced by the horizontal pressure gradient, which they estimated to be roughly independent of height. Their estimate of vertical stress gradient, therefore, differs from ours. The conclusion regarding the constant-SGS-stress layer, however, still holds albeit for a smaller z/Δ. The discrepancy can only be resolved through direct numerical simulations.

The cross stresses

As shown in appendix B, the cross stresses are of order
i1520-0469-55-10-1733-ec8
Our scaling estimates in conjunction with (C8) show that the dominant contributor to the cross stress is (urkus3)r ∼ urhδus3 ∼ urhur3. Thus, we can write
i1520-0469-55-10-1733-ec9
We estimate the magnitude of the s–s component of the SGS stress as
i1520-0469-55-10-1733-ec10
From (C9) and (C10) we have
i1520-0469-55-10-1733-ec11
Equation (C11) confirms that the cross stress is negligible sufficiently close to the surface.
Equation (C11) also indicates that under free-convection conditions the cross stress can be of the order of the s–s stress when
i1520-0469-55-10-1733-ec12
For typical values of Δ/zi (1/10 to 1/100, say) (C12) indicates that in free convection the cross stress is as important as the s–s stress when z/Δ ∼ 0.1 to 0.3. For larger z/Δ values, the cross term can be the dominant contributor to SGS stress. This is unlike the situation for scalar flux, for which we showed (appendix B) that the cross components are quite small.

The surface stress budget for z1/Δ ≪ 1

When z/Δ ≪ 1 only the O(1) terms in (C5) are significant so the streamwise stress budget reduces to
i1520-0469-55-10-1733-ec13
The κ = 0 component of this equation is the Reynolds budget of streamwise stress (Wyngaard et al. 1971).
The lateral stress budget for z/Δ ≪ 1 becomes
i1520-0469-55-10-1733-ec14
The mean (the κ = 0 component) of each term in this equation vanishes, due to the symmetry about the x1 axis. Thus, (C14) predicts zero mean lateral stress, as expected; however, fluctuations in the lateral stress are generated by small-scale advection, buoyancy, and the pressure term.
We argued in section 5 of the paper that when z1/Δ≪ 1 the turbulence in the first grid element is in local grid volume equilibrium. A closure for the resolvable-scale surface stress budgets (C13) and (C14) in this case is
S̃r0kCDz1L̃s̃rz1ũrkz1rkz1
Here CD is the mean surface-exchange coefficient for momentum and s̃r is the resolvable-scale horizontal wind speed.

The surface stress budget for z1/Δ < 1

Table C1 shows that in free convection there is not the clear separation in the order of the terms that existed for scalar flux. The next-order terms in (C5) are of order (z/Δ)2/3, but after that come a number of zero-mean,“noise terms” (e.g., RSCFI, RGCSI) of order z/Δ.

Despite this lack of separation, the two budgets are similar in that the next-order terms are large-scale advection, resolvable-scale gradient production, and the vertical part of resolvable-scale shear production. We argued in the paper that the time-change term is of the order of advection, which we will take to include that by the mean velocity.

Rewriting the resolvable-scale gradient production term as we did for scalar flux and using the scale-separation argument (appendix A) gives
urk,jusjus3rr
The scale-separation argument is inapplicable to the resolvable-scale shear production term due to the usk contained within it, so the term is
ur3,jusjuskrrur3,jusjusksrur3,jusjruskr
Only the first term in (C17) can be calculated, so we will drop the second and third, which are zero-mean“noise” terms, and write for its vertical part
ur3,3us3uskrrur3,3S̃r0kr
The budget of S̃r0k then becomes
i1520-0469-55-10-1733-ec19
This is the counterpart to Eq. (74) for the surface scalar flux.

We showed in section 2 of this appendix that the cross contribution is relatively more important for stress than it is for scalar flux. Thus, when z1/Δ is not very small so that a surface stress conservation equation rather than the usual surface-exchange expression is appropriate, it appears that the cross stress could also be significant. We showed that the dominant cross stress is (urkus3)r, which in principle is inaccessible since it involves a subgrid-scale quantity. It has zero mean and we showed that its rms magnitude is of order urkur3, which is calculated in LES. We discuss the modeling of this term in section 6a.

Fig. 1.
Fig. 1.

A schematic of the “coupling eddies” in the grid volume adjacent to the surface. They couple 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.

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 2.
Fig. 2.

Behavior of the local surface-exchange coefficients as calculated from high-resolution LES and STORMFEST data taken by the NCAR ASTER facility. Circles: STORMFEST at 4 m (open) and 10 m (solid); −zi/L ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 800. Plus symbols: LES at 16 m using surface-flux equations.

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 3.
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 lines indicate the κz value below which lies 50% of the variance. Top: neutral; center: sightly unstable; bottom: free convection.

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 4.
Fig. 4.

Nondimensional mean shear in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 5.
Fig. 5.

Nondimensional mean temperature gradient in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 6.
Fig. 6.

Nondimensional vertical-velocity variance in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Fig. 7.
Fig. 7.

Effect of SGS model and lower boundary conditions on the nondimensional mean shear. Curves 1 and 2 are from the original SGS model (without backscatter) and curves A and B from the modified SGS model (with backscatter). In both sets, the first curve is based on surface-exchange coefficients and the second on the surface flux equations.

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

i1520-0469-55-10-1733-fa1

Fig. A1. The spectra of vertical temperature flux (long dashes), horizontal temperature flux (short dashes), and vertical velocity (solid) from high-resolution LES at z = 35 m (fine) and 66 m (bold) for −zi/L ≃ 65.

Citation: Journal of the Atmospheric Sciences 55, 10; 10.1175/1520-0469(1998)055<1733:LITSLS>2.0.CO;2

Table 1.

Scaling of terms in the budget of surface scalar flux.

Table 1.
Table 2.

A comparison of predicted and actual subgrid-scale temperature flux and stress components from an effective 5123 simulation, using nested meshes, of the atmospheric boundary layer (−zi/L ≈ 8). The comparisons are made at z/zi ≈ 0.04 and z/Δ ≈ 0.75; (q̃ = F̃i,i; ãi = τ̃ij,j).

Table 2.

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

i1520-0469-55-10-1733-t301
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  • Fig. 1.
    View in gallery
    Fig. 1.

    A schematic of the “coupling eddies” in the grid volume adjacent to the surface. They couple 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.

  • Fig. 2.
    View in gallery
    Fig. 2.

    Behavior of the local surface-exchange coefficients as calculated from high-resolution LES and STORMFEST data taken by the NCAR ASTER facility. Circles: STORMFEST at 4 m (open) and 10 m (solid); −zi/L ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 800. Plus symbols: LES at 16 m using surface-flux equations.

  • Fig. 3.
    View in gallery
    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 lines indicate the κz value below which lies 50% of the variance. Top: neutral; center: sightly unstable; bottom: free convection.

  • Fig. 4.
    View in gallery
    Fig. 4.

    Nondimensional mean shear in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)

  • Fig. 5.
    View in gallery
    Fig. 5.

    Nondimensional mean temperature gradient in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

  • Fig. 6.
    View in gallery
    Fig. 6.

    Nondimensional vertical-velocity variance in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

  • Fig. 7.
    View in gallery
    Fig. 7.

    Effect of SGS model and lower boundary conditions on the nondimensional mean shear. Curves 1 and 2 are from the original SGS model (without backscatter) and curves A and B from the modified SGS model (with backscatter). In both sets, the first curve is based on surface-exchange coefficients and the second on the surface flux equations.

  • View in gallery

    Fig. A1. The spectra of vertical temperature flux (long dashes), horizontal temperature flux (short dashes), and vertical velocity (solid) from high-resolution LES at z = 35 m (fine) and 66 m (bold) for −zi/L ≃ 65.

Fig. 1.

A schematic of the “coupling eddies” in the grid volume adjacent to the surface. They couple 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.
Fig. 2.

Behavior of the local surface-exchange coefficients as calculated from high-resolution LES and STORMFEST data taken by the NCAR ASTER facility. Circles: STORMFEST at 4 m (open) and 10 m (solid); −zi/L ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −zi/L ≈ 800. Plus symbols: LES at 16 m using surface-flux equations.
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 lines indicate the κz value below which lies 50% of the variance. Top: neutral; center: sightly unstable; bottom: free convection.
Fig. 4.

Nondimensional mean shear in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)
Fig. 5.

Nondimensional mean temperature gradient in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).
Fig. 6.

Nondimensional vertical-velocity variance in a moderately convective boundary layer (−zi/L ≈ 10). Curves 1 and 2 are from 643 and 1282 × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 643 and 1282 × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).
Fig. 7.

Effect of SGS model and lower boundary conditions on the nondimensional mean shear. Curves 1 and 2 are from the original SGS model (without backscatter) and curves A and B from the modified SGS model (with backscatter). In both sets, the first curve is based on surface-exchange coefficients and the second on the surface flux equations.

Fig. A1. The spectra of vertical temperature flux (long dashes), horizontal temperature flux (short dashes), and vertical velocity (solid) from high-resolution LES at z = 35 m (fine) and 66 m (bold) for −zi/L ≃ 65.
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