## 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.

*z*

_{1}of the first grid level. The

*grid aspect ratio*Δ/

*z*

_{1}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

*F̃*

^{r}

_{0}

*x*

_{1},

*x*

_{2},

*t*), the resolvable-scale surface flux of a scalar constituent, for example, the appropriate surface-exchange coefficient is not the mean value

*C*

_{H}but a random variable,

*C̃*

_{H}. (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

*ũ*

^{r}

_{1}

*c̃*

^{r}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

*C̃*

_{H}the

*local*surface-exchange coefficient; we expect that

*C̃*

_{H}→

*C*

_{H}as

*z*

_{1}/Δ → 0.

### b. Observed fluctuations in local surface-exchange coefficients

*z*

_{s}≪ Δ, and in the other at a greater height

*z*

_{1}≈ Δ. In view of these grid aspect ratios, we take the scalar-flux surface-exchange coefficients as

*C*

_{H}(

*z*

_{s}) for the first grid and

*C̃*

_{H}(

*z*

_{1}) for the second. We can write from (1), indicating only the

*z*dependence,

*z*

_{1}yields

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 256^{3} simulation (Khanna and Brasseur 1996). The inversion height *z*_{i} 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 −*z*_{i}/*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 −*z*_{i}/*L* = 730. We took *z*_{s} = 3.9 m, the lowest grid point, and evaluated the local surface-exchange coefficients at *z*_{1} = 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 −*z*_{i}/*L* ≈ 10. We took *z*_{s} = 1.0 m, the lowest measurement point, and evaluated the local surface-exchange coefficients for temperature at *z*_{1} = 4 m and10 m.

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

Figure 2 shows the fluctuations in the local surface-exchange coefficient *C̃*_{H} and its counterpart for momentum, *C̃*_{D}, evaluated from measured time series and LES data, using (5) and its counterpart for *C̃*_{D}. 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 −*z*_{i}/*L* increases. It can be quite large.

*c*

^{r}/Δ

*C*and

*u*

^{r}

_{1}

*U*

_{1}are small parameters, we can linearize (5) to find

*c*

^{r}/Δ

*C*is typically much less than that of

*u*

^{r}

_{1}

*U*

_{1}, so that we can write (6) as

*z*

_{1}becomes large, the separation

*z*

_{1}−

*z*

_{s}becomes small compared to the horizontal scales of the fluctuations in

*u*

^{r}

_{1}

*U*

_{1}. Thus, as Δ/

*z*

_{1}increases, we expect

*u*

^{r}

_{1}

*z*

_{s}) and

*u*

^{r}

_{1}

*z*

_{1}) 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 *z*_{1} as *κ*_{c}*z*_{1} → 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

*ũ*

_{i}and a conservative scalar

*c̃*in the surface layer are

*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

*κ*

_{c}=

*π*/Δ

^{f}, where Δ

^{f}is the filter width in physical space. In principle Δ

^{f}need not be the same as Δ

^{g}, the horizontal grid spacing. As pointed out by Mason and Callen (1986), Δ

^{f}defines the spatial scales contained in the resolvable fields, whereas Δ

^{g}determines the accuracy of the numerical solutions of the filtered equations. One could obtain very high accuracy by taking Δ

^{g}≪ Δ

^{f}, for example. Computer limitations require that Δ

^{f}and Δ

^{g}be essentially the same value, however, so we will not distinguish them further here, taking Δ

^{f}= Δ

^{g}= 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).

*conservative,*that is, has no sources or sinks. Filtering it yields the evolution equation for its resolvable-scale part:

*F̃*

^{r}

_{j}

*c̃*

^{r}

*ũ*

^{r}

_{j}

^{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

*F̃*

^{r}

_{j}

The terms in (14), being resolvable scale, are bounded as *z* → 0 (we will use *x*_{3} and *z* interchangeably to denote height above the surface). In particular, *F̃*^{r}_{3,3}*z,* *F̃*^{r}_{3}*z*) ≈ *F̃*^{r}_{0}*F̃*^{r}_{3}

*ũ*

^{s}

_{i}

*u*

^{s}

_{i}

*c̃*

^{s}

*c*

^{s}

*θ̃*

^{s}

*θ*

^{s}

*p̃*

^{s}

*p*

^{s}

*z*sufficiently large that the molecular contribution to the SGS flux is negligible the vertical component of the flux is, from (14)–(16),

*E*(

*κ*) in the surface layer, where

*κ*is horizontal wavenumber magnitude (

*κ*

^{2}

_{1}

*κ*

^{2}

_{2}

^{1/2}. This spectrum integrates to the variance

*c*

_{1}and

*c*

_{2}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

*u*

_{3}= 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 *c*_{1} and *c*_{2} 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 *z*_{i} and the convective velocity scale *w*∗ = (*gQ*_{0}*z*_{i}/*T*_{0})^{1/3}, rather than the local-free-convection scales *z* and *u*_{f} = (*gQ*_{0}*z*/*T*_{0})^{1/3} (Wyngaard et al. 1971); here *Q*_{0} 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 *z*_{i} 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 *E*_{h}, which integrates to (*u*^{2}_{1}*u*^{2}_{2}*κ*_{h} = 1/*z*_{i}. The vertical velocity spectrum *E*_{υ} peaks at larger wavenumbers, those of the order of *κ*_{z} = 1/*z.* A third significant wavenumber is *κ*_{c} = *π*/Δ, the cutoff wavenumber of the LES grid mesh. In Moeng’s 96^{3} LES code, for example, Δ is typically 5000 m/64 ≃ 80 m.

*z*≪ Δ and inversion depths

*z*

_{i}≫ Δ. Since

*z*≪ Δ ≪

*z*

_{i}, there is a large separation in these three wavenumbers:

*κ*

_{h}≪

*κ*

_{c}≪

*κ*

_{z}. Let us write each factor in the first cross term, (

*c*

^{r}

*u*

^{s}

_{3}

^{r}, as a Fourier–Stieltjes integral, noting the admissible region in the

*κ*

_{1},

*κ*

_{2}plane:

*κ*

^{′}

_{j}

*κ*

_{c}, the restriction |

*κ*

^{′}

_{j}

*κ*

^{"}

_{j}

*κ*

_{c}implies that |

*κ*

^{"}

_{j}

*κ*

_{c}. Thus, only

*u*

_{3}modes of wavenumber magnitude less than 2

*κ*

_{c}can contribute to (

*c*

^{r}

*u*

^{s}

_{3}

^{r}. Figure 3 shows that, as we approach the surface, 2

*κ*

_{c}falls below the range of wavenumbers contributing significantly to

*u*

_{3}so that (

*c*

^{r}

*u*

^{s}

_{3}

^{r}→ 0 as

*z*/Δ → 0. The second cross term in (17), (

*c*

^{s}

*u*

^{r}

_{3}

^{r}, is also eliminated because

*u*

^{r}

_{3}

*z*/Δ → 0.

*z*/Δ ≪ 1 the SGS flux is due entirely to the third term in (17),

*c*

^{s}and

*u*

^{s}

_{3}

*z*much smaller than the scale of the spatial filter (i.e.,

*z*/Δ ≪ 1) the SGS flux

*F̃*

^{r}

_{3}

*z*) ≈ (

*c*

^{s}

*u*

^{s}

_{3}

^{r}is a good surrogate for

*F̃*

^{r}

_{0}

*F̃*

^{r}

_{0}

*u*

^{s}

_{i}

*c*

^{s}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:

*cu*

_{3}

*κ*

_{c}→ 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,

*u*

^{r}

_{i}

*c*

^{r}→ 0, and our decomposition (15) and (16) reduces to

*κ*

_{c}→ 0 the SGS flux budget becomes, under horizontally homogeneous conditions,

## 4. Scaling LES fields in the surface layer

*f*=

*f*

^{r}+

*f*

^{s}(a fluctuating velocity component or scalar) in the surface layer. With wave-cutoff filtering we have

*h*) gradients we have

We will use (29) to determine how the variances of *u*_{h}, *c,* and *u*_{3} for *z*/Δ ∼ *κ*_{c}*z* ≪ 1 are partitioned into their resolvable and SGS parts. We will use (30) to determine how the variances of *u*^{r}_{h,h}*u*^{r}_{3,h}*c*^{r}_{,h}*z* and Δ for *z*/Δ ≪ 1.

### a. Horizontal velocity and scalars

*E*(

*κ*) given in Eq. (19) yields

*l*=

*z,*which makes (

*κl*)

^{2}/

*c*

_{2}a small parameter in (31), and we find

*l*=

*z*

_{i}. For horizontal velocity (

*κ*

_{c}

*l*)

^{2}/

*c*

_{2}is then a large parameter. Equation (31) yields in that limit

### b. Vertical velocity

*E*

_{υ}of vertical velocity

*u*

_{3}in the surface layer:

*E*

_{υ}

*κ*

*T*

*κ*

*E*

*κ*

*T*(

*κ*) is a “continuity transfer function,”

*κz*≪ 1 and ensures that

*E*

_{υ}behaves as required by local isotropy at

*κz*≫ 1.

*l*=

*z*so that

*κ*

_{c}

*z*≪ 1. In this limit we can simplify (37) to

*E*

_{υ}

*κ*

*κz*

^{2}

*E*

*κ*

*l*=

*z*

_{i}and the expressions are

*κ*

_{c}

*z*

_{i}is a large parameter and we have

### c. Vertical gradients

*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

*f*

^{r}as

*f*

^{r}

*x*

_{i}

*t*

*A*

*x*

_{3}

*s*

*x*

_{i}

*t*

*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

*f*

^{r}. It is possible that the vertical spatial scales of

*u*

^{r}

_{3}

*z,*for example. Since

*s*has unit variance, it follows from (44) that

*f*

^{r})

^{2}

*A*

^{2}. In the case of

*u*

^{r}

_{3}

*A*∼

*u*∗(

*z*/Δ)

^{2}.

*x*

_{3}, squaring, and averaging gives

*f*

^{r}

_{,3}

^{2}

*A*

^{2}

*s*

_{,3})

^{2}

*s*

^{2}

_{,3}

*A*

_{,3}

*A*

*A*

_{,3}

^{2}

*A*

^{2}

*s*

_{,3})

^{2}

*A*

_{,3}

^{2}

*s*

^{2}

*z.*The gradient variances

*s*

_{,h})

^{2}

*s*

_{,3})

^{2}

*s*

^{2}

*λ*

_{h}and

*λ*

_{z}(Tennekes and Lumley 1972):

*λ*

^{2}

_{h}

*s*

^{2}

*s*

_{,h})

^{2}

*λ*

^{2}

_{z}

*s*

^{2}

*s*

_{,3})

^{2}

*x*

_{h}, squaring, averaging, and combining with (47) yields

*λ*

_{h}is Δ, the smallest surviving horizontal scale in the filtered fields. We will assume that

*λ*

_{z}is of this order as well so that (48) becomes

*u*

^{r}

_{3}

*z*/Δ so that our assumption that

*λ*

_{z}∼

*λ*

_{h}∼ Δ is not critical; we obtain the result (50) for

*λ*

_{z}∼

*z*as well. This is the case for almost all variables.

From (33), (35), and (51) we see that *u*^{r}_{1,1}*u*^{r}_{2,2}*u*^{r}_{3,3}

## 5. Simplifying the resolvable-scale surface-flux budget

### a. The “constant-SGS-flux” layer

*F̃*

^{r}

_{3}

*F*

_{0}

*f*

_{3}

*F̃*

^{r}

_{3,3}

*F̃*

^{r}

_{j,j}

*c̃*

^{r}

_{,j}

*ũ*

^{r}

_{j}

^{r}

*z*/Δ → 0 because the forcing terms in the resolvable-scale scalar equation vanish in that limit.

*F̃*

^{r}

_{0}

*z,*

*F̃*

^{r}

_{3}

*z*), scaled by the mean surface flux

*F*

_{0}, is of order

*z*≪ Δ. Estimating the rms value of the fluctuating SGS flux

*F*

^{′}

_{3}

*σ*

_{u}is the rms fluctuation in the streamwise wind, typically in the range (0.1–0.5)

*U*

_{1}.

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 *F*_{0} (*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 *x*_{1} direction to be that of the mean wind so that *U*_{i} = (*U*_{1}, 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.

### 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 *z*_{1}/Δ.

#### 1) *z*_{1}/Δ ≪ 1

*z*/Δ ≪ 1 we need retain only the

*O*(1) terms in the resolvable-scale surface-flux conservation equation (24), and it reduces to

*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.

*z*

_{0c}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 *T*_{s} ∼ *z*/*u*∗ and *z*/*u*_{f} 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 *T*_{s}. 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.”

*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:

*C*

_{H}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,

*z*

_{1}/

*L̃,*with

*L̃*the local Monin–Obukhov length,

*ũ*∗ and

*Q̃*

_{0}the local SGS friction velocity and surface temperature flux. Based on Eq. (58) we would expect Δ

*C*rather than Δ

*c̃*

^{r}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) *z*_{1}/Δ small but not negligible

*z*

_{1}such that

*z*

_{i}= 10

^{3}m this occurs at

*z*

_{1}= 1 m. The same arguments hold for the time-change term, Eq. (24). We conclude that in application to fine-mesh LES with

*z*

_{1}on the order of a few meters, the time-change and horizontal LSA terms in the surface-flux budget (24) can be significant.

*U*

_{1}+

*u*

^{r}

_{h}

*ũ*

^{r}

_{h}

*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.

*c̃*

^{r}

_{,3}

_{eq}is the local-grid-volume-equilibrium value of the resolvable-scale scalar gradient in the vertical. Because

*z*

_{1}/Δ is small, there are many coupling eddies in the grid volume and to first approximation we can take (

*c̃*

^{r}

_{,3}

_{eq}to be nonrandom and to have a similarity form identical to that of the traditional M–O function for temperature,

*ϕ*

_{h}:

*u*

^{s}

_{3}

*u*

^{s}

_{3}

^{r}is needed; it could be obtained through a local similarity model

*u*

^{s}

_{3}

*u*

^{s}

_{3}

^{r}

*ũ*

^{2}

_{∗}

*f*

*z*

_{1}

*L̃*

*f*is the M–O function for vertical velocity variance. With such a model for (

*u*

^{s}

_{3}

*u*

^{s}

_{3}

^{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.

*C̃*

_{H}from its mean value

*C*

_{H}due to the effects of local time change and horizontal advection of SGS flux.

#### 3) *z*_{1}/Δ ⩽ 1

*T*

_{i}represents the remaining terms. We estimate the rms contribution to

*F̃*

^{r}

_{0}

*T*

_{i}as

*δ*

_{i}

*F̃*

^{r}

_{0}

*τT*

_{i}. As

*z*

_{1}/Δ → 1 these contributions increase since, according to Table 1, the

*T*

_{i}increase. In fact, however, (24) describes the budget at

*z*

_{1}, not at the surface, and we showed in (55) that as

*z*

_{1}/Δ increases the rms difference between the fluxes at the surface and at height

*z*

_{1}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

*z*

_{1}:

*δ*

_{i}

*F̃*

^{r}

_{0}

*F̃*

^{r}

_{0}

*F̃*

^{r}

_{3}

*z*

_{1}

*F̃*

^{r}

_{0}

*F̃*

^{r}

_{3}

*z*

_{1}) and taking

*τ*∼

*z*

_{1}/

*u*∗ at neutral and

*τ*∼

*z*

_{1}/

*u*

_{f}in free convection then yields

*z*

_{1}/Δ. We therefore also include the vertical part of RSSP. The surface flux budget then is

*u*

^{r}

_{3,3}

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 *z*_{1}/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 *z*_{1}/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 *z*_{1}/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 *z*_{1}/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

*z*

_{1}/Δ ⩽ 1 form of the conservation equations for resolvable-scale surface temperature flux

*Q̃*

^{r}

_{0}

*τ̃*

^{r}

_{0k}

*Q̃*

^{r}

_{0}

*L̃*is given by (61), and

*ũ*∗ is given by

*ũ*

^{2}

_{∗}

*τ̃*

^{r}

_{01}

^{2}+ (

*τ̃*

^{r}

_{02}

^{2}

*u*

^{s}

_{k}

*u*

^{s}

_{3}

^{r}=

*S̃*

^{r}

_{0k}

*k*= 1, 2. The local-grid-volume-equilibrium value of the resolvable-scale velocity gradient is given by

*ϕ*

_{m}is taken as

*C̃*

^{r}

_{0k}

*u*

^{r}

_{k}

*u*

^{s}

_{3}

*u*

^{s}

_{k}

*u*

^{r}

_{3}

^{r}has zero mean but an rms value of order

*u*

^{r}

_{k}

*u*

^{r}

_{3}

*C̃*

^{r}

_{0k}

*u*

^{r}

_{k}

*u*

^{′}

_{3}

*u*

^{′}

_{k}

*u*

^{r}

_{3}

*n*+ 1 is given by

*t*is the time step, and

*R*

_{Q}(

*n*) is the right-hand side of (76) at time step

*n.*

Since *z*_{1}/Δ is a critical parameter in our analysis, we carried out three sets of simulations: 64^{3}, 128^{2} × 64, and 192^{2} × 64, with *z*_{1}/Δ of 0.13, 0.27, and 0.4, respectively, for a moderately convective boundary layer (−*z*_{i}/*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 *S̃*^{r}_{0k}*C̃*^{r}_{0k}

Figures 4–6 show the nondimensional mean shear *ϕ*_{m}, mean potential temperature gradient *ϕ*_{h}, and vertical velocity variance, respectively, from 64^{3} and 128^{2} × 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) *ϕ*_{m} 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 64^{3} simulation with surface-exchange coefficients, *ϕ*_{h} is somewhat lower than the measured value at the first grid point; above that it is slightly overpredicted. Increasing the horizontal resolution to 128^{2}, while keeping the vertical resolution the same, limits the overprediction of *ϕ*_{h} 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 64^{3} and the 128^{2} × 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 *z*_{i}-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 −*z*_{i}/*L* ≈ 10 using a 128^{3} mesh covering a domain of 5*z*_{i} × 5*z*_{i} × 2*z*_{i}. The next level of refinement in the surface layer was attained by an effective 256^{3} mesh covering a domain of 5*z*_{i} × 5*z*_{i} × 0.125*z*_{i}, and the final level was attained by an effective 512^{3} mesh covering 5*z*_{i} × 5*z*_{i} × 0.06125*z*_{i} 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 512^{3} simulation resolved approximately 90% of the vertical fluxes and variances (the sixth grid level, at *z*/*z*_{i} ≈ 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 96^{2} 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.

*e*= (

*u*

^{s}

_{i}

*u*

^{s}

_{i}

^{r}/2, is calculated explicitly and used as the velocity scale for the SGS eddy diffusivity (

*K̃*):

*S̃*

^{r}

_{ij}

*c*

_{k}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,

*a*

_{i}, and a random temperature source,

*q,*to the resolvable-scale equations:

*a*

_{i}was taken to be the curl of another random vector:

*a*

_{i}=

*ϵ*

_{ijk}

*b*

_{k,j}. As indicated by Mason and Thomson, this random forcing adds a production rate

*a*

_{i}

*a*

_{i}

*t*to the resolved-scale kinetic energy equation and 2

*q*

^{2}

*t*to the equation for resolved-scale temperature variance, where Δ

*t*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

*c*

_{b}

*ϵ,*where

*ϵ*is the ensemble-average dissipation rate and

*c*

_{b}= 1.4; the mean backscatter of resolved-scale temperature variance is 2

*c*

_{bθ}

*χ,*where

*χ*is the destruction rate of one-half temperature variance and

*c*

_{bθ}= 0.45. The random components

*b*

_{i}and

*q*are scaled such that

*c*

_{k}in (86) to (1 +

*c*

_{b})

*c*

_{k}to account for the additional production of resolved-scale kinetic energy. The turbulent Prandtl number in (87) was increased by a factor of (1 +

*c*

_{b})/(1 +

*c*

_{bθ}) 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*/

*c*

_{k}, 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 *ϕ*_{m} profile in the 128^{2} × 64 simulation of the moderately convective boundary layer. The original SGS model and drag coefficients overestimate *ϕ*_{m} at the first grid point (*z*/*z*_{i} ≈ 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 *ϕ*_{m}, 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 128^{2} × 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, *z*_{1}/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.

## REFERENCES

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

Bradshaw, P., 1969: Comments on “On the relation between the shear stress 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: Evaluation of subgrid-scale models using an accurately simulated turbulent flow.

*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 System in the large-eddy-simulation mode: From inhomogeneous surfaces to cirrus clouds.

*Large-Eddy Simulation of Complex Engineering and Geophysical Flows,*B. Galperin and S. Orszag, Eds., Cambridge University Press, 369–398.Deardorff, J. W., 1970: A numerical study of three-dimensional turbulent 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 surface layer.

*J. Atmos. Sci.,***35,**18–24.Khanna, S., and J. G. Brasseur, 1996: Analysis of Monin–Obukhov similarity 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 turbulent flows.

*Advances in Geophysics,*Vol. 18A, Academic Press, 237–248.Lilly, D. K., 1967: The representation of small-scale turbulence in numerical simulation experiments.

*Proc. Tenth IBM Scientific Computing 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 Atmospheric Turbulence.*Interscience, 239 pp.Mason, P. J., and N. S. Callen, 1986: On the magnitude of the SGS eddy coefficient in LES of turbulent channel flows.

*J. Fluid Mech.,***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 simulations of boundary layers.

*J. Fluid Mech.,***242,**51–78.Moeng, C.-H., 1984: A large-eddy-simulation model for the study of planetary 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. Schumann, 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: The characteristics of turbulent velocity components in the surface layer 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-scale backscatter in turbulent and transitional flows.

*Phys. Fluids A,***3,**1766–1771.Schumann, U., 1993: Large-eddy simulation of turbulent convection over flat and wavy surfaces.

*Large-Eddy Simulation of Complex Engineering 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.

*Atmospheric 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 Modeling,*A. Venkatram and J. C. Wyngaard, Eds., Amer. Meteor. Soc., 9–61.——, and L. J. Peltier, 1996: Experimental micrometeorology in an era of turbulence simulation.

*Bound.-Layer Meteor.,***78,**71–86.——, O. R. Coté, and Y. Izumi, 1971: Local free convection, similarity, and the budgets of shear stress and heat flux.

*J. Atmos. Sci.,***28,**1171–1182.

## APPENDIX A

### Simplifying the Resolvable-Scale Surface Scalar Flux Budget

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

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

^{s}

*c*

^{s}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*c*

^{s}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

^{r}

*c*

^{s}

^{r}

*a*

^{r}

*b*

^{s})

^{r}. The wavenumber components of

*a*

^{r}have magnitudes between 0 and

*κ*

_{c}. The wavenumber components of

*b*

^{s}have magnitudes greater than

*κ*

_{c}. Let us write this symbolically as

*a*

^{r}

*b*

^{s}

^{r}

*κ*

_{c}

*κ*

_{c}

^{r}

*κ*

_{c}

*a*

^{r}

*b*

^{s}are the vector sum of all those of

*a*

^{r}and

*b*

^{s}. Therefore wavenumbers of

*b*

^{s}larger than 2

*κ*

_{c}cannot contribute to (

*a*

^{r}

*b*

^{s})

^{r}, so we can write (A3) as

*a*

^{r}

*b*

^{s}

^{r}

*κ*

_{c}

*κ*

_{c}

*κ*

_{c}

^{r}

*b*is the temperature field, whose dominant wavenumbers have magnitudes of the order of 1/

*z*(Peltier et al. 1996). Thus, as

*κ*

_{c}

*z*→ 0, there are no modes of

*b*in the range (

*κ*

_{c}, 2

*κ*

_{c}] so that (

*a*

^{r}

*b*

^{s})

^{r}vanishes, and we can write

*a*

^{r}

*b*

^{s}

^{r}

*ab*

^{s}

^{r}

*κ*

_{c}

*z*

*scale-separation*argument.

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*c*

^{s}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*c*

^{s}

^{r}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*c*

^{s}

^{s}

^{r}

*u*

^{s}

_{j}

*c*

^{s}. We calculated it for

*j*= 1 and

*j*= 3 from high-resolution (256

^{3}) LES data. The results, shown in Fig. A1, indicate that the spectrum of

*u*

^{s}

_{j}

*c*

^{s}is like that of

*u*

_{3}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

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*c*

^{s}

^{r}

^{r}

*κ*

_{c}

*z*

*u*

^{r}

_{j}

*u*

^{s}

_{3}

*c*

^{s}

_{,j}

^{r}

*u*

^{r}

_{j}

*u*

^{s}

_{3}

*c*

^{s}

^{r}

_{,j}

^{r}

*u*

^{r}

_{j}

*u*

^{s}

_{3}

*c*

^{s}

^{s}

_{,j}

^{r}

*u*

^{r}

_{j}

*u*

^{s}

_{3}

*c*

^{s}

^{r}

_{,j}

^{r}

*u*

^{r}

_{j}

*F̃*

^{r}

_{3,j}

^{r}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

*c*

^{s}

^{r}

_{,j}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

^{s}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

^{r}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{r}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{s}

^{r}

*c*

^{r}

_{,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{r}

^{r}

*U*

_{3,j}(

*u*

^{s}

_{j}

*c*

^{s})

^{r}from (A1) because it vanishes under horizontally homogeneous conditions. The simplified budget reads

## APPENDIX B

### The Cross Components of SGS Flux

*F̃*

^{r}

_{3}

*c*

^{r}

*u*

^{s}

_{3}

*c*

^{s}

*u*

^{r}

_{3}

*c*

^{s}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

*u*

^{s}

_{3}

^{r}and (

*c*

^{s}

*u*

^{r}

_{3}

^{r}as the cross components. By the arguments in section 3 of the paper, the contributing wavenumber ranges of

*u*

^{s}

_{3}

*c*

^{s}here are

*κ*

_{c}<

*κ*⩽ 2

*κ*

_{c}. We can therefore estimate the magnitude of the cross fluxes as

*c*

^{r}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

*δ*

*u*

^{s}

_{3}

*c*

^{s}

*u*

^{r}

_{3}

*δc*

^{s}

*u*

^{r}

_{3}

*δ*

*u*

^{s}

_{3}

*u*

^{s}

_{3}

*κ*

_{c}<

*κ*⩽ 2

*κ*

_{c}. Since, in general, we can write

*δf*

^{s}

^{2}

*f*

^{r}

*κ*

_{c}

^{2}

*f*

^{r}

*κ*

_{c}

^{2}

*δ*

*u*

^{s}

_{3}

*u*

^{r}

_{3}

*δc*

^{s}

*c*

^{r}

*c*

^{r}

*u*

^{s}

_{3}

^{r}

*c*

^{r}

*u*

^{r}

_{3}

*c*

^{s}

*u*

^{r}

_{3}

^{r}

*c*

^{r}

*u*

^{r}

_{3}

## APPENDIX C

### Extension to Resolvable-Scale Surface Stress

#### The resolvable-scale surface stress budget

*R̃*

^{r}

_{ij}

*R̃*

^{r}

_{ij}

*ũ*

^{r}

_{i}

*ũ*

^{s}

_{j}

*ũ*

^{s}

_{i}

*ũ*

^{r}

_{j}

*ũ*

^{s}

_{i}

*ũ*

^{s}

_{j}

^{r}

*ν*

*ũ*

^{r}

_{i,j}

*ũ*

^{r}

_{j,i}

*deviatoric*SGS stress tensor:

*τ̃*

^{r}

_{13}

*τ̃*

^{r}

_{23}

*τ̃*

^{r}

_{0k}

*S̃*

^{r}

_{0k}

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

*z*to the fluctuation in stress at height

*z.*This ratio is approximately given by

*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

*u*

^{r}

_{k}

*u*

^{s}

_{3}

^{r}∼

*u*

^{r}

_{h}

*δu*

^{s}

_{3}

*u*

^{r}

_{h}

*u*

^{r}

_{3}

*z*

_{i}(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 z_{1}/Δ ≪ 1

*z*/Δ ≪ 1 only the

*O*(1) terms in (C5) are significant so the streamwise stress budget reduces to

*κ*= 0 component of this equation is the Reynolds budget of streamwise stress (Wyngaard et al. 1971).

*z*/Δ ≪ 1 becomes

*κ*= 0 component) of each term in this equation vanishes, due to the symmetry about the

*x*

_{1}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.

*z*

_{1}/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

*S̃*

^{r}

_{0k}

*C*

_{D}

*z*

_{1}

*L̃*

*s̃*

^{r}

*z*

_{1}

*ũ*

^{r}

_{k}

*z*

_{1}

^{r}

*k*

*z*

_{1}

*C*

_{D}is the mean surface-exchange coefficient for momentum and

*s̃*

^{r}is the resolvable-scale horizontal wind speed.

#### The surface stress budget for z_{1}/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.

*u*

^{r}

_{k,j}

*u*

^{s}

_{j}

*u*

^{s}

_{3}

^{r}

^{r}

*u*

^{s}

_{k}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*u*

^{s}

_{k}

^{r}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

*u*

^{s}

_{k}

^{s}

^{r}

*u*

^{r}

_{3,j}

*u*

^{s}

_{j}

^{r}

*u*

^{s}

_{k}

^{r}

*u*

^{r}

_{3,3}

*u*

^{s}

_{3}

*u*

^{s}

_{k}

^{r}

^{r}

*u*

^{r}

_{3,3}

*S̃*

^{r}

_{0k}

^{r}

*S̃*

^{r}

_{0k}

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 *z*_{1}/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 (*u*^{r}_{k}*u*^{s}_{3}^{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 *u*^{r}_{k}*u*^{r}_{3}

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); −*z*_{i}/*L* ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*L* ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*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

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); −*z*_{i}/*L* ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*L* ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*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

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); −*z*_{i}/*L* ≈ 10. Squares: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*L* ≈ 63. Triangles: LES at 12 m (open) and 27 m (solid); −*z*_{i}/*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

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.

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.

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.

Nondimensional mean shear in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)

Nondimensional mean shear in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)

Nondimensional mean shear in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients;curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971)

Nondimensional mean temperature gradient in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Nondimensional mean temperature gradient in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Nondimensional mean temperature gradient in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Nondimensional vertical-velocity variance in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Nondimensional vertical-velocity variance in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

Nondimensional vertical-velocity variance in a moderately convective boundary layer (−*z*_{i}/*L* ≈ 10). Curves 1 and 2 are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-exchange coefficients; curves A and B are from 64^{3} and 128^{2} × 64 simulations, respectively, using surface-flux equations; solid line is the empirical function of Businger et al. (1971).

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.

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.

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 −*z*_{i}/*L* ≃ 65.

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 −*z*_{i}/*L* ≃ 65.

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 −*z*_{i}/*L* ≃ 65.

Scaling of terms in the budget of surface scalar flux.

A comparison of predicted and actual subgrid-scale temperature flux and stress components from an effective 512^{3} simulation, using nested meshes, of the atmospheric boundary layer (−*z*_{i}/*L* ≈ 8). The comparisons are made at *z*/*z*_{i} ≈ 0.04 and *z*/Δ ≈ 0.75; (q̃ = F̃_{i,i}; *ã*_{i} = *τ̃*_{ij,j}).

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