## 1. Introduction

The Weather Research and Forecasting model (WRF) is an atmospheric modeling system that can be employed to simulate flow at many scales, from numerical weather prediction (NWP) to large-eddy simulation (LES; Skamarock et al. 2008). WRF has been used primarily in NWP applications, using computational meshes with horizontal spacings Δ*h* of several kilometers. Increasingly, WRF is being run in LES mode, with sufficiently fine mesh spacings (Δ*h* of order 100 m or less) to resolve explicitly the energy-producing scales of atmospheric turbulence (e.g., Moeng et al. 2007).

NWP-scale simulations (Δ*h* of order 10 km or greater) use coarse enough grids that no three-dimensional turbulence can be resolved on the computational mesh. In such cases, the grid cell averages predicted on the mesh represent the slowly varying or “mean” state of the flow (e.g., Stull 1988). Since motions on the scale of atmospheric turbulence cannot be resolved, their influence on the slowly varying resolved component must be modeled. Typically the horizontal mixing is specified using the deformation of the large-scale flow, while the vertical mixing is prescribed in a planetary boundary layer (PBL) parameterization. Such parameterizations utilize assumptions appropriate for large scales to provide a representation of the effects the entire turbulence spectrum on the evolution of the slowly varying component of the flow field represented on the coarse mesh. WRF contains several PBL parameterizations schemes as detailed in the WRF model documentation (see online at http://www.mmm.ucar.edu/wrf/users/docs/arw_v3.pdf).

LES, in contrast, is a technique in which the energy-producing scales of three-dimensional atmospheric turbulence are explicitly resolved, while the smaller-scale portion of the turbulence spectrum is removed from the flow field using a spatial filter. Selecting a filter scale within the inertial subrange separates the flow field into a resolved component, which contains most of the scales responsible for turbulent transport and turbulence kinetic energy (TKE) production, and a subfilter component, consisting of scales within the turbulence cascade, which function primarily to dissipate energy from the resolved scales. The scales removed by the filter, having undergone some cascade, tend to be more isotropic and homogeneous, and are therefore more amenable to general parameterization than the larger resolved scales. The effects of the scales removed by the filter on the resolved component of the flow are modeled with a subfilter-scale (SFS) stress model. The SFS stress model used in LES hence plays a different role than that of a PBL parameterization used in a larger-scale model. Whereas large-scale PBL models are responsible for parameterizing all of the turbulent transport of momentum, heat, and other scalars during a simulation, LES resolves most of those processes explicitly, with the SFS stress model playing a reduced, yet still important role.

The LES technique was first applied to atmospheric boundary layer (ABL) flows in the early 1970s (e.g., Deardorff 1970, 1972, 1974). Because of computational restrictions, those early simulations employed coarse spatial resolutions and were applied to neutral and unstable ABLs, for which characteristic eddy sizes are relatively large. Since those early simulations, the expansion of computational resources has permitted extension of LES to increasingly complicated and computationally intensive flows, including complex geometries and stably stratified conditions, for which characteristic eddy sizes are relatively small. Continuing improvement of the technique has resulted in the application of LES to investigations of fundamental aspects of atmospheric turbulence that are difficult or costly to measure in experimental settings (Stevens and Lenschow 2001; Beare et al. 2004). LES results are routinely used as “data,” and may be employed in the validation as well as the design of atmospheric boundary layer parameterization schemes used in larger-scale models (Cuxart et al. 2006). The validity of such approaches depends crucially on the assumption that the SFS stress model is providing the correct input to the energy-producing scales of the turbulence (e.g., Stevens and Lenschow 2001; Ludwig et al. 2009).

Recent advances in high-performance computing, coupled with the development of atmospheric models utilizing sophisticated software architectures and grid nesting capabilities are extending LES to an ever-widening array of uses. Grid nesting can, in principle, allow successively refined meshes to resolve portions of a domain down to LES scales, permitting LES on a refined inner nest to be driven by time-evolving mesoscale forcing predicted on a coarser outer mesh. Two-way information exchange at the nest interfaces allows both downscale and upscale transfers of information.

While this convergence of technology provides a framework for potentially powerful modeling tools, its development presents a host of challenges. Wyngaard (2004) provides an important discussion regarding some challenges associated with modeling the scales of motion between those for which LES is suitable and those for which a large-scale PBL parameterization is appropriate. In a study of two-way nested LES in WRF, Moeng et al. (2007) uncovered several difficulties involving the communication of information between the nests, requiring the need for blending functions for some variables, and modifications to the SFS stress model.

In addition to these issues, a third component is also required for the successful extension of WRF to LES, that being the development of adequate models for turbulence-resolving flows. Moeng et al. (2007) modified the existing 1.5-order TKE-based WRF SFS stress closure model to ignore the storage and advection terms in the prognostic SFS TKE equation to avoid mismatched values at nest boundaries, and also implemented a two-part eddy-viscosity model to correct the predicted eddy-viscosity profiles near the surface. Their simulations, however, reveal the signature overprediction of dimensionless wind shear near the surface that is characteristic of the linear eddy-viscosity SFS stress models, illustrating the need for improved SFS stress models in WRF.

This paper addresses some of the issues regarding SFS stress modeling within WRF and presents results from simulations of idealized flow using several different approaches for modeling the SFS stresses, including the two standard WRF SFS stress models and two formulations of a new nonlinear SFS stress model implemented into WRF (distributed in WRF version 3.2).

## 2. Large-eddy simulation using WRF

*ũ*are the resolved velocities; with

_{i}*i*,

*j*= 1, 2, 3 denoting the zonal

*u*, meridional

*υ*, and vertical

*w*components, respectively;

*p̃*is the resolved pressure;

*ρ̃*is the resolved density;

*x*,

_{i}*x*refer to the spatial coordinates; with

_{j}*i*,

*j*= 1, 2, 3 indicating zonal

*x*, meridional

*y*, and vertical

*z*components, respectively; and

### a. Standard WRF SFS stress models

*ν*is the eddy-viscosity coefficient, and

_{T}*S̃*= ½(∂

_{ij}*ũ*/∂

_{i}*x*+ ∂

_{j}*ũ*/∂

_{j}*x*) is the resolved strain-rate tensor.

_{i}*C*is the Smagorinsky coefficient, which, in WRF, is given a default value of 0.25. The length scale is given by

_{S}*l*= (Δ

*x*Δ

*y*Δ

*z*)

^{1/3}, Pr is the turbulent Prandtl number with a value of 0.7, and

*N*

^{2}is the Brunt–Väisälä frequency. Equation (5) includes a modification to the magnitude of the stress depending upon static stability, reducing the stresses with increasing stability to a limit of no mixing when

*N*

^{2}is large relative to the deformations. This limiting behavior might not be appropriate during strongly stable conditions, for which turbulence, though weakened, does not vanish completely (e.g., Stull 1988).

*e*is the SFS TKE, and

*C*= 0.15. When this subfilter option is selected, the WRF model integrates a prognostic SFS TKE equation.

_{e}Each of these models is well known, simple to understand and to implement, and widely used (e.g., Ludwig et al. 2009). Applications using the Smagorinsky model (e.g., Cai 2000; Camelli et al. 2006; Nakayama et al. 2008) have tended to be of a more applied nature than those using the TKE-1.5 closure (e.g., Dosio et al. 2003; Kim et al. 2005; Khanna and Brasseur 1998; Foster et al. 2006). Despite their popularity, each of these models suffers from fundamental deficiencies. First, each uses constants, for which a wide range of values is reported to work optimally across a range of stabilities, forcing scenarios, and proximities to boundaries (e.g., Ciofalo 1994). Both models also assume that turbulence dissipation balances production locally. This assumption, while explicit in the derivation of the Smagorinsky constant, is implicit in the dissipation term used in the TKE equation. Assuming local balance between TKE production and dissipation is clearly inappropriate in the presence of terrain heterogeneity or instabilities, for instance, or at high spatial resolutions, in which cases advection of TKE can be significant (Lundquist and Chan 2007). Another deficiency is failure to include backscatter, the upscale transfer of energy from small to large scales. Backscatter has been shown to be particularly important near solid boundaries and in regions of shear or stability (Mason and Thompson 1992; Kosović 1997). Additionally, each of the SFS stress models relates the stress linearly to the strain rate through an eddy-viscosity coefficient. This approach is fundamentally at odds with observations that the eigenvectors of the stress and strain rate tensors are generally not aligned at the scales at which LES SFS stress models are applied (e.g., Sullivan et al. 2003). Finally, these approaches fail to correctly predict normal SFS stress components in the case of sheared, homogeneous turbulence (Kosović 1997).

To address the above discussed shortcomings of the standard WRF SFS stress models and to improve WRF’s LES performance, two nonlinear SFS stress models have been implemented into the ARW core.

### b. Nonlinear SFS stress models

*R̃*= ½(∂

_{ij}*ũ*/∂

_{i}*x*− ∂

_{j}*ũ*/∂

_{j}*x*) is the resolved rotation-rate tensor,

_{i}*C*= [8(1 +

_{S}*C*)/27

_{b}*π*

^{2}]

^{1/2},

*C*= (8

_{e}*π*/27)

^{1/3}

*C*

_{S}^{4/3},

*C*

_{1}=

*C*

_{2}= 960

^{1/2}

*C*/7(1 +

_{b}*C*)

_{b}*S*,

_{k}*S*= 0.5, and

_{k}*C*= 0.36.

_{b}All model parameters depend upon only one free parameter, the backscatter coefficient *C _{b}*. This coefficient, which accounts explicitly for backscatter, is given a value of 0.36, as determined by Kosović (1997). The other constants that depend on

*C*are formulated such that proper normal stresses are obtained for sheared homogeneous turbulence. In the limit

_{b}*C*→ 0, the NBA models given by Eqs. (7) and (8) revert to the linear eddy-viscosity Smagorinsky and 1.5-order TKE SFS stress models, respectively (although with different values for the constants than the WRF default values).

_{b}The backscatter parameter in the NBA model represents an average value, which reduces the overall dissipation rate, leading to improved near-surface stress and overall spectral characteristics. However, Chen et al. (2003, 2005), Chen and Tong (2006), and Chen et al. (2009) have argued that it is the conditional backscatter (rather than instantaneous or random backscatter) that is a necessary condition for the correct evolution of the resolved-scale velocity joint probability density function (JPDF), due to the direct influence of the conditional energy transfer rate on the JPDF transport equation.

The NBA model has shown considerable improvements over linear eddy-viscosity models in convective, neutral and stably stratified ABL simulations, including superior agreement with similarity profiles near the surface, improved spectral characteristics, and improved anisotropy (Kosović 1997; Kosović and Curry 2000). In simulations of a convective ABL, Chen and Tong (2006) showed that the NBA1 model improved predictions of both the conditional SFS stress and the conditional SFS stress production rate, as well as the level of anisotropy, relative to the Smagorinsky model. However, both the magnitude of the conditional SFS stress (when the mean energy transfer is matched) and the level of anisotropy were underpredicted relative to observations.

The two formulations of the NBA model provide flexibility for different applications. The formulation including TKE [Eq. (8)] is recommended for flows with significant buoyancy effects, while for nested simulations, the formulation based upon the strain rate only [Eq. (7)] is recommended, due to the mismatch of the WRF model’s prognostic TKE at nest interfaces. Transformation of the WRF model’s prognostic TKE equation, which is used by the formulation given by Eq. (8), to a diagnostic one, as was done by Moeng et al. (2007), would be sufficient to extend that formulation to nested applications.

## 3. Results

*u*,

_{g}*υ*] = [10, 0] m s

_{g}^{−1}, at a latitude of 45°. The surface stresses

*C*=

_{D}*κ*

^{2}{ln[(

*z*

_{1}+

*z*

_{0})/

*z*

_{0}]}

^{−2}is the coefficient of drag formulated for neutral flow, for which roughness length of

*z*

_{0}= 0.1 m was used, and subscript 1 denotes the first model grid point above the surface for which

*u*and

*υ*are computed. This configuration is similar to other neutral boundary layer LES studies that have appeared in the literature (e.g., Andren et al. 1994; Moeng et al. 2007; Chow et al. 2005). Each simulation utilized constant horizontal grid spacing Δ

*h*while the vertical grid spacing increased by approximately 5% per nodal index above the surface. The upper boundary condition is

*τ*=

_{ij}*w*= 0, with free slip for

*u*and

*υ*. The lateral boundaries are periodic in each direction. The computational domain was 4096 m in each horizontal direction and 1024 m in the vertical.

### a. General considerations

First, a few general results are presented to illustrate some important considerations for conducting a successful LES. The LES technique is predicated upon simulating explicitly and resolving well the largest, most energetic scales of turbulence within the flow. In order for these scales to be adequately resolved, the filter width should be well within the inertial subrange. Sufficiency of the mesh spacing for a given set of flow parameters can be established by examination of the velocity spectra, which should contain scales of turbulence production, including a low-frequency spectral peak, as well as some scales within the inertial subrange. Inertial subrange scales are indicated by a −⅔ slope of the compensated spectra in log–log coordinates. Compensated spectra depict the relationship between the product of the power *F _{ij}* and the wavenumber

*k*and the wavenumber.

_{j}Figure 1 shows a log–log plot of time- and plane-averaged spectra of the vertical (Fig. 1a) and zonal (Fig. 1b) velocity components at approximately 96 m above the surface, from simulations using horizontal mesh sizes successively decreased by a factor of 2, from 128 to 16 m, and a vertical mesh spacing of approximately 8 m near the surface (the heights of the vertical grid points cannot be specified exactly in WRF because of its use of a pressure-based vertical coordinate). Each of these simulations used the TKE SFS stress model and was integrated for 26 h prior to averaging. The spectra were computed in the *x* direction at each *y* grid point within the plane. Those spectra were then averaged over the plane, and in time, at 1-min intervals, until the averages remained nearly constant. The height of 96 m is approximately the top of the surface layer, 0.1*H*, where *H* is the boundary layer depth. This height contains a broad range of eddy sizes and, while strongly influenced by the surface, is a sufficient number of grid points above that that most of the stress field is resolved.

The spectra shown in Fig. 1 demonstrate that, for this particular flow, Δ*h* = 128-m horizontal grid spacing is insufficient for LES, as a small fraction of the turbulence-producing scales are captured for either velocity component. As the mesh spacing is decreased, the simulations capture increasingly smaller energy-producing turbulence structures, as reflected in the increased higher-frequency spectral power. As the vertical velocity component has a shorter integral length scale than the zonal component, the vertical velocity spectra indicate more clearly discernable spectral peaks for the simulations using mesh spacings of 64 m or less. The spectral peak for the zonal spectra are less discernable due to the tendency of many SFS stress models, such as the TKE model used here, to produce elongated correlations oriented in the streamwise direction near the surface and in the lower portion of the ABL.

The sensitivity of the largest of these elongated structures to mesh spacing can be seen in the variability of the low-frequency portions of the zonal velocity spectra. For the TKE SFS stress model (and many others), decreasing the mesh spacing results in a redistribution of kinetic energy and a reduction of the relative proportion of kinetic energy in large-scale structures within the flow. While the specific proportions of resolved kinetic energy will vary among simulations using different mesh spacings and SFS stress models, and based on different flow parameters, spectra can be used to identify the sufficiency of a given mesh spacing for LES.

The adequacy of a given mesh spacing for LES can likewise be inferred from examination of the modeled and resolved stress components. Figure 2 shows profiles of time- and plane-averaged vertical-streamwise stress *τ*_{13} partitioned into total, resolved and SFS components, plotted against height scaled by *H*, from the same four simulations as in Fig. 1. Examination of the stress profile from the Δ*h* = 128-m simulation (Fig. 2a) shows that only a small proportion of the total stress is resolved; almost all is SFS. This means that the SFS stress model is prescribing stresses based primarily on the mean vertical shear, rather than on well-resolved three-dimensional turbulence, and as such yields an incorrect stress profile. For smaller values of Δ*h* (Figs. 2b,c,d), most of the stress is resolved, except near the surface, where the SFS stress model dominates. Since the vertical scales of turbulence are damped more strongly than the horizontal approaching the surface, reducing only Δ*h* still improves resolution of turbulence near the surface.

Another factor influencing the accuracy of a large-eddy simulation is the grid aspect ratio, *α* = Δ*h*/Δ*z*. Figure 3 shows vertical profiles of horizontal wind speed *U* scaled by the surface friction velocity *u*_{*} on the ordinate, against height scaled by the boundary layer depth. Aspect ratios of 1, 2, 4, and 8 are shown against the “log law,” the expected similarity profile for neutral boundary layer flow over flat, rough terrain (indicated by the gray line). Simulations were conducted using three different horizontal mesh spacings, (Fig. 3a) Δ*h* = 16 m, (Fig. 3b) Δ*h* = 32 m, and (Fig. 3c) Δ*h* = 64 m. Results of simulations using Δ*h* = 128 m are omitted, as that mesh spacing was shown to be insufficient for LES of this flow.

Figure 3 indicates that, over a range of Δ*h*, 2 < *α* < 4 provides the closest agreement with the similarity solution. These simulations used the default values for the SFS stress model constants, the isotropic length scale formulation, and third- and fifth-order schemes for the vertical and horizontal advections, respectively. No additional diffusion or damping was prescribed. Each of these above factors influences the optimal grid aspect ratio value. While these values of *α* were found to be optimal to flow over flat terrain, because of the WRF model’s terrain-following vertical coordinate and subsequent use of coordinate transformation metrics in the computation of horizontal derivatives, truncation errors arising from those metric terms also become a consideration over nonuniform terrain. Hence, the optimal aspect ratio determined over flat terrain might not be applicable over significant slopes.

Examination of spectra, stress components, and similarity profiles (in the cases for which those exist) can be used to verify the adequacy of a given mesh spacing for successful LES. These principles apply irrespective of the particular SFS stress model being used. However, the choice of SFS stress model can significantly impact the accuracy of the solution for a given mesh size, and can even alter the mesh size required to achieve a particular level of accuracy. The following section examines the performance of the two standard WRF SFS stress closures relative to two formulations of a new nonlinear SFS stress model.

### b. Influence of subfilter-scale stress models

In this section, the nonlinear SFS stress models are compared to WRF’s standard SFS stress models using idealized large-eddy simulations of neutral boundary layer flow over flat terrain. All results constitute averages in time and space taken from statistically steady simulations. Steadiness was achieved in two steps. First, a 24-h simulation using the Smagorinsky closure was conducted to develop nearly-steady turbulence statistics. Following the initial 24-h simulation, additional 2-h simulations were conducted using each SFS stress model to allow the flow to equilibrate to the new SFS stress model physics. After the 26-h period, simulations were continued for lengths of time sufficient for the averages to achieve nearly steady values, ranging from 1 to 8 h depending on the mesh spacing, with coarser-resolution simulations requiring longer sampling periods because of the smaller number of data points in the horizontal averages.

#### 1) Mean profiles

Figure 4 shows the profiles of scaled wind speed versus height predicted by the Smagorinsky (SMAG), 1.5-order TKE (TKE), and the two Nonlinear Backscatter and Anisotropy models given by Eq. (7) (NBA1) and Eq. (8) (NBA2). Results are shown for three horizontal mesh sizes, Δ*h* = 16 m (Fig. 4a), Δ*h* = 32 m (Fig. 4b), and Δ*h* = 64 m (Fig. 4c), each using an aspect ratio of *α* = 4. The similarity solution is shown in gray.

Apparent from Fig. 4 is the self-similar shape of each of the profiles across different mesh spacings. While the profile shapes remain similar, decreasing the mesh spacing reduces the magnitudes of the departures from the log law for each of the models. Each of the models exhibit similar overall profile shapes, featuring a velocity deficit immediately above the surface, transitioning to a velocity overshoot further aloft. In each of the simulations, both of the NBA models significantly improve agreement with similarity solution, reducing departures from the expected solution in both directions.

Figure 5 shows the profiles of dimensionless wind shear, Φ(*z*) = (*κz*/*u*_{*})(∂*U*/∂*z*), versus height, from the same simulations as shown in Fig. 3. The expected solution for neutral flow, Φ = 1, is shown in gray. Figure 5 demonstrates more clearly the velocity deficits predicted immediately above the surface, transitioning to velocity overshoots farther aloft. Figure 5 also indicates the extent to which the NBA models reduce the discrepancies from the expected solution near the surface. The improved agreement with Φ = 1 aloft exhibited by the SMAG and TKE models should be interpreted with respect to the information shown in Fig. 4 as well, where it is revealed that the departures from Φ = 1 exhibited by the NBA models are the result of those profiles returning to their expected similarity values with increasing height. While the NBA models improve the agreement with the similarity profile near the surface, those models also improve agreement in bulk differences between the surface and most portions within the profiles above.

#### 2) Resolved velocity structures

Figure 6 shows instantaneous contours of the *u* velocity in the *x*–*z* plane at approximately 40 m above the surface from the Δ*h*, Δ*z* = 32, 8 m simulations conducted with each of the four SFS stress models. The Coriolis-induced rotation of the wind vector approaching the surface is indicated by the orientation of the contours with respect to the geostrophic forcing, which is applied in the *x* direction.

Comparison of the relative sizes and shapes of velocity structures resolved within the flow indicates that the SFS stress models have a considerable impact on those turbulence structures near the surface. Our simulations using the standard WRF Smagorinsky-type SFS stress models reveal well-defined and highly correlated velocity streaks aligned with the mean wind vector near the surface, similar to those observed in other studies (e.g., Ludwig et al. 2009). In contrast, each of the NBA models results in a significant reduction in the size and coherence of those structures, leading to more intricate and convoluted interfaces, improving qualitative agreement with observed turbulent flows, direct numerical simulations, and higher-resolution LES.

#### 3) Spectra

While the snapshots depicted in Fig. 6 provide a basis for qualitative comparisons, spatial spectra can be used to quantify the relative proportion of different length scales occurring within a turbulent flow. Figure 7 depicts time- and plane-averaged spectra of the *u* velocity in the *x* direction at two heights above the surface from simulations using each of the four SFS stress models. The spectra in the left and right columns are shown at heights of approximately 40 (left) and 120 m (right) above the surface. Because of the different vertical mesh spacings used to maintain a constant aspect ratio of 4 at the surface across simulations using different Δ*h*, the vertical index nearest to 40 and 120 m occur at slightly different heights above the surface. The vertical gridpoint index nearest to 40 and 120 m during each simulation is indicated by the value of *nz* shown in each panel.

Several considerations must be accounted for when interpreting the spectra shown in Fig. 7. First, in general, the differences among the distributions of eddy sizes arising from the use of different SFS stress models will be greatest nearer the surface, because of the decrease of the size of eddies approaching the surface. The reduction in eddy size near the surface leads inevitably to an increasing proportion of the energy-producing eddies becoming unresolved, hence requiring parameterization. Therefore, for each of the three resolutions shown in Fig. 7, the spectra closer to the surface (Figs. 7a,c,e) show greater variability among the SFS stress models than those farther aloft (Figs. 7b,d,f).

The influence of the SFS stress model likewise increases with increasing mesh spacing, as the smallest resolvable structures in the flow are determined ultimately by the filter, which in the WRF model is implicitly provided by the mesh spacing (in combination with the accuracy of the numerical integration). Coarser numerical meshes resolve less of the energy-producing turbulence directly, requiring increased input from the SFS stress model. Hence, for each of the two heights shown in Fig. 7, the spectra from the coarser-resolution simulations (Figs. 7e,f) show greater variability than those from the higher-resolution simulations (Figs. 7a,b).

The combined effects of mesh spacing and proximity to the surface on the influence of the SFS stress models are evident in Fig. 7, with the largest differences noted at the coarsest-resolution simulations at the lowest height (Fig. 7e). The differences are much smaller yet still discernable among the highest-resolution simulation occurring farther aloft (Fig. 7b). The differences among the spectra near the surface (Figs. 7a,c,e) confirm the qualitative differences observed in the snapshots of flow structures shown in Fig. 6 regarding the relative proportions of the sizes of structures resolved within the flow. The SMAG and TKE spectra contain considerably more low-frequency power and less high-frequency power than the NBA models. These differences in spectral power distributions indicate that the NBA models reduce the relative occurrences of larger structures within the flow, while increasing the proportion of smaller structures.

Figure 8 depicts time- and plane-averaged spectra of the *u* velocity in the *y* direction at the same three heights above the surface from the same simulations as depicted in Fig. 7. The energy maxima in the *y*-direction spectra are observed to occur at higher frequencies than those in the *x* direction. This pattern arises from the geostrophic forcing, which was applied in the *x* direction only, resulting in alignment of the largest coherent structures more closely with the *x* axis. Hence, while the *x*-direction spectra (Fig. 7) reflect the characteristic lengths of the resolved velocity structures, the *y*-direction spectra (Fig. 8) reflect their widths. Because of the widths of the structures being much smaller on average than their lengths, spectral peaks are easily discerned in each of the spectra shown in Fig. 8. As with the spectra computed in the *x* direction, the NBA models again produce less spectral power at the lower frequencies, and more spectral power at higher frequencies, than the SMAG and TKE models. This behavior indicates that, as with the characteristic lengths, the NBA models likewise reduce the characteristic widths of turbulence flow structures near the surface.

The differences in the proportions of large and small structures predicted by the NBA versus the SMAG and TKE models reflect that the NBA models incorporate additional physics represented by the addition of the second-order terms in Eqs. (7) and (8). The increase of higher-frequency power in particular is also attributable to their less dissipative nature relative to the linear eddy-viscosity SMAG and TKE models, which are absolutely dissipative. The spectra are provided not to assess model accuracy but to compare the effects of the models under consideration and make connections between these effects and modeled physics. The NBA models are formulated explicitly in terms of a backscatter coefficient, which reduces the overall dissipation of TKE both via a reduction in the magnitude of the constant multiplying the forward-scattering linear eddy-viscosity terms in Eqs. (7) and (8), and through the actions of the second-order strain rate tensor product terms, whose net effects are to backscatter energy. The other second-order product terms involving the strain rate and rotation rate tensors do not contribute to the TKE balance, however, they do influence other aspects of the turbulence, in particular the anisotropy of the normal SFS stress components.

### c. Flow over a symmetric transverse ridge

Simulations of neutral flow over a transverse Gaussian ridge were conducted to assess the performance of the SFS stress models over nonuniform terrain. The ridge is symmetric in the *x* direction, and uniform in the *y* direction, with a height of 50 m and width specified such that the maximum slope (the ratio of the change in height to the horizontal distance between model gridpoints) is 0.3. Figure 9 shows a contour of the terrain surface in the *x* direction. The forcing parameters used for these simulations are identical to those used for the flat-terrain simulations.

Direct comparison between these simulations and observations is beyond the scope of this study; here we simply intend to demonstrate differences among the SFS stress models by qualitative comparisons between simulations conducted at different resolutions, with the higher-resolution simulations hypothesized to be more accurate.

Figure 10 shows instantaneous contours of the *u* velocity in the *x*–*y* plane at a height of approximately 10 m above the surface from simulated flow over the ridge using each of the SFS stress models. A domain of 1024 m in each direction was used, with a relatively fine computational mesh spacing of Δ*h*, Δ*z* = 8, 2 m. In each panel, flow is from left to right, and contours of negative zonal velocity are depicted by dotted lines. As with the snapshots of flow over flat terrain shown in Fig. 6, the NBA models again produce flows containing generally smaller and more convoluted structures than those of the SMAG and TKE models. Separation and reverse flow in the lee of the hill near the surface are also evident in all of the simulations.

Because of the importance of flow separation and reversal in hilly terrain to many applications, including atmospheric dispersion and wind power production, the relative occurrences of flow reversal occurring in the lee of the hill were selected for a qualitative comparison. Figure 11 shows the relative frequencies of the magnitude of the *u*-velocity component at four heights above the surface downstream from the hill apex, from the higher-resolution simulations (Δ*h*, Δ*z* = 8, 2 m) using each of the four SFS stress models. Histograms were computed from the flow at 160 m downstream from the ridge apex, using velocities that were linearly interpolated from bounding grid points in the vertical direction to the four heights specified in Fig. 11. The velocities were binned into 160 intervals spanning [−4, 10] m s^{−1}. The interpolation and binning were repeated across the entire domain in the *y* direction. The velocity distributions at each of the four heights are depicted in the same panel for each of the SFS stress models. The solid gray vertical lines indicate zero, while the dotted gray vertical lines indicate the mean value from each of the distributions in which they are embedded.

Figure 11 reveals a generally high level of agreement among some aspects of the flow predicted by all four SFS stress models, including both the general pattern of significant flow reversal near the surface transitioning to similar positive velocities at the highest location. However, differences are also apparent. For instance, the NBA models predict more frequent negative velocities nearer the surface, with negative values in the mean at 6 m. The transitions of the flow from predominantly negative velocities near the surface to exclusively positive velocities at 48 m also show considerable differences between the NBA models and the SMAG and TKE models, while at the greatest distance above the surface, the differences between the models have been significantly reduced, with all models showing similar mean values and distributions.

While we cannot comment on the accuracy of these solutions without observational data for comparison, we can assume that simulations conducted at using smaller mesh spacings are more accurate to those conducted using larger mesh spacings. As such, we compare the simulations performed on the finest computational mesh (Δ*h*, Δ*z* = 8, 2 m) to simulations performed on meshes coarsened by factors of 1.5 (Δ*h*, Δ*z* = 12, 3 m) and 2 (Δ*h*, Δ*z* = 16, 4 m), the distributions of which are shown in Figs. 12 and 13, respectively. The domain for the Δ*h*, Δ*z* = 12, 3 m simulations was slightly smaller, 1020 m in each horizontal direction, and the histograms were taken from velocities at 156 m downstream from the hill apex.

Comparison among the distributions among the two higher-resolution simulations, shown in Figs. 11 and 12, reveals that, among the coarser-resolution simulations, the NBA models produce qualitatively better agreement with the higher-resolution simulations in several respects. First, the NBA models predict a relatively larger proportion of negative velocities at the two lowest heights above the surface, heights at which each of the higher-resolution simulations produced significant proportions of negative velocities. A related improvement is that, because of the differences in the velocities near the surface, the NBA models produce a range of velocities within the vertical profile that more closely match the higher-resolution simulations. The NBA simulations also predict lower peak probabilities at the two middle heights than the other models, which again more closely match those of the highest-resolution simulations.

The coarsest-resolution simulations, whose distributions are shown in Fig. 13, agree poorly with the highest-resolution simulations overall; however among those, the NBA models again produce relatively more negative velocities near the surface, a broader range of velocities across the vertical profile, and lower peak probabilities at the intermediate heights, again more consistent with the higher-resolution simulations.

While lacking observation validation, the simulations of flow over the transverse ridge suggest that the more complete physics represented within the nonlinear SFS stress models improve the simulations relative to the SMAG and TKE models. For each of the coarser-mesh domains, the distributions from the NBA models closely match the simulations using the SMAG and TKE models on domains with meshes a factor of 1.5 finer. This correspondence suggests that the improvements afforded by the NBA models are nearly equivalent to a corresponding decrease in mesh spacing when using the simpler models. While the NBA1 and NBA2 models increase computation requirements by approximately 20% relative to the SMAG and TKE models, respectively, the ability to use a mesh coarsened by a factor of 1.5 to achieve similar solution accuracy represents approximately an eightfold reduction in computational effort (including mesh coarsening in three dimensions, and the use of a correspondingly larger time step).

## 4. Summary and conclusions

Two formulations of a new nonlinear SFS stress model have been implemented into the WRF-ARW version 3.0 and compared to the standard WRF linear eddy-viscosity SFS stress models in simulations of neutral, boundary layer flow over both flat terrain, and a symmetric transverse ridge. The nonlinear NBA models improve upon the physics represented by the linear, eddy-viscosity SMAG and TKE models in two significant ways. First, the NBA models explicitly account for backscatter, which is absent in the SMAG and TKE models, and second, the NBA models predict normal stresses, with the correct anisotropy, in the case of sheared homogeneous turbulence. The SMAG and TKE predict no normal stresses for such flows.

Results of several simulations indicate that the new SFS stress models improve the overall behavior of the WRF LES capability in several ways. First over flat terrain, the new SFS stress models reduce deviations from the log law, thereby more closely approximating the expected similarity solution in the surface layer. The new models also reduce the relative proportion of large-scale features while increasing those of smaller-scale structures within the flow. Finally, in flow over a transverse ridge, the new nonlinear SFS stress models improve the representation of qualitatively correct physical flow phenomena, flow separation, and reversal in the lee of the ridge, using larger computational mesh spacings. These results suggest that the increased sophistication of the new SFS stress models is able to represent certain physics that must be resolved using less sophisticated models, enabling potentially significant improvements in computational efficiency.

The study has also outlined some general guidelines for conducting successful LES using WRF, including the importance of verifying that the computational mesh resolution, grid aspect ratio, and model parameters are adequate to resolve the spectral energy peak and to produce reasonable near-surface flow distributions.

### a. Future work

An avenue for future research is to better understand why each of the SFS stress models overpredict the stresses above the surface within the WRF model. Such overprediction has not been observed in simulations conducted with the NBA model implemented into a pseudospectral code (Kosović 1997; Kosović and Curry 2000). The backscatter coefficient, about which the NBA model constants are determined, was given a uniform value in the vertical direction during the simulations presented herein; however, because near the surface the filter cutoff is in energy the producing range and not in the inertial range, it should probably be formulated in a height dependent fashion, which could potentially improve the near-surface agreement with the expected solution.

Future work will likewise assess the performance of the SFS stress models in different physical contexts, including over more complex terrain and during nonneutral conditions, where differences in the assumptions underlying the various modeling approaches are likely to more strongly influence the resulting flow. The NBA2 model was shown to significantly improve performance relative to the Smagorinsky model in simulation of weakly stable boundary layer flow over the Arctic Ocean pack ice (Kosović and Curry 2000). Extensions of the LES technique to flow in nonneutral conditions, and over complex terrain have been prioritized within both the research and industry communities, particularly in applications related to wind energy (e.g., Schreck et al. 2008) and atmospheric dispersion (e.g., Gopalakrishnan and Avissar 2000).

The suite of atmospheric processes models (e.g., microphysics, land surface, etc.) contained within WRF permit simulation of many of the time-dependent physical processes that interact strongly with turbulence occurring within the ABL. WRF’s grid nesting and LES capabilities provide a framework for coupling these scales, providing a potentially powerful tool for addressing both fundamental and applied research questions. As such, continued improvement in the LES capability within WRF is strongly motivated. This study of relatively simple flows over idealized surfaces represents a step in this direction.

## Acknowledgments

The authors thank F. K. Chow and K. A. Lundquist for many helpful discussions. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Staff effort at LLNL and UC Berkeley was supported by the LLNL Laboratory Directed Research and Development (LDRD) Program, Projects 06-ERD-026 and 09-ERD-038. Computations at LLNL’s Livermore Computing were also supported by LDRD.

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Vertical profiles of time- and plane-averaged vertical-streamwise stress components *τ*_{13} from simulations using horizontal mesh resolutions of (a) Δ*h* = 128 m, (b) Δ*h* = 64 m, (c) Δ*h* = 32 m, and (d) Δ*h* = 16 m with Δ*z* = 8 m using the TKE SFS stress model.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Vertical profiles of time- and plane-averaged vertical-streamwise stress components *τ*_{13} from simulations using horizontal mesh resolutions of (a) Δ*h* = 128 m, (b) Δ*h* = 64 m, (c) Δ*h* = 32 m, and (d) Δ*h* = 16 m with Δ*z* = 8 m using the TKE SFS stress model.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Vertical profiles of time- and plane-averaged vertical-streamwise stress components *τ*_{13} from simulations using horizontal mesh resolutions of (a) Δ*h* = 128 m, (b) Δ*h* = 64 m, (c) Δ*h* = 32 m, and (d) Δ*h* = 16 m with Δ*z* = 8 m using the TKE SFS stress model.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Vertical profiles of time- and plane-averaged wind speed scaled by friction velocity, as functions of height scaled by boundary layer depth. Simulations conducted at aspect ratios, *α* = Δ*h*/Δ*z* of 1, 2, 4, and 8, with horizontal mesh resolutions of (a) Δ*h* = 16 m, (b) Δ*h* = 32 m, and (c) Δ*h* = 64 m, using the TKE SFS stress model are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Vertical profiles of time- and plane-averaged wind speed scaled by friction velocity, as functions of height scaled by boundary layer depth. Simulations conducted at aspect ratios, *α* = Δ*h*/Δ*z* of 1, 2, 4, and 8, with horizontal mesh resolutions of (a) Δ*h* = 16 m, (b) Δ*h* = 32 m, and (c) Δ*h* = 64 m, using the TKE SFS stress model are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Vertical profiles of time- and plane-averaged wind speed scaled by friction velocity, as functions of height scaled by boundary layer depth. Simulations conducted at aspect ratios, *α* = Δ*h*/Δ*z* of 1, 2, 4, and 8, with horizontal mesh resolutions of (a) Δ*h* = 16 m, (b) Δ*h* = 32 m, and (c) Δ*h* = 64 m, using the TKE SFS stress model are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged wind speed scaled by surface friction velocity vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged wind speed scaled by surface friction velocity vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged wind speed scaled by surface friction velocity vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution (gray).

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged nondimensional shear, Φ(*z*) = (*κz*/*u*_{*})(∂*U*/∂*z*) vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution, Φ = 1.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged nondimensional shear, Φ(*z*) = (*κz*/*u*_{*})(∂*U*/∂*z*) vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution, Φ = 1.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Profiles of time- and plane-averaged nondimensional shear, Φ(*z*) = (*κz*/*u*_{*})(∂*U*/∂*z*) vs height scaled by boundary layer depth using all four SFS stress models at mesh resolutions of (a) Δ*h*, Δ*z* = 16, 4 m, (b) Δ*h*, Δ*z* = 32, 8 m, and (c) Δ*h*, Δ*z* = 64, 16 m, with *α* = 4. Results are shown against the similarity solution, Φ = 1.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Contours of instantaneous *u* velocity in the *x*–*y* plane at approximately 40 m above the surface from each of the four SFS stress models showing turbulence structures embedded within the flow. Flow is forced with a geostrophic wind in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Contours of instantaneous *u* velocity in the *x*–*y* plane at approximately 40 m above the surface from each of the four SFS stress models showing turbulence structures embedded within the flow. Flow is forced with a geostrophic wind in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Contours of instantaneous *u* velocity in the *x*–*y* plane at approximately 40 m above the surface from each of the four SFS stress models showing turbulence structures embedded within the flow. Flow is forced with a geostrophic wind in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Time- and plane-averaged *u*-velocity spectra computed in the *x* direction at heights of approximately (left) 40 and (right) 120 m using mesh resolutions of (top) Δ*h*, Δ*z* = 16, 4 m, (middle) Δ*h*, Δ*z* = 32, 8 m, and (bottom) Δ*h*, Δ*z* = 64, 16 m, using each of the four SFS stress models. Here *z* and *nz* indicate the height and corresponding vertical gridpoint index, respectively, at which the spectra are computed.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Time- and plane-averaged *u*-velocity spectra computed in the *x* direction at heights of approximately (left) 40 and (right) 120 m using mesh resolutions of (top) Δ*h*, Δ*z* = 16, 4 m, (middle) Δ*h*, Δ*z* = 32, 8 m, and (bottom) Δ*h*, Δ*z* = 64, 16 m, using each of the four SFS stress models. Here *z* and *nz* indicate the height and corresponding vertical gridpoint index, respectively, at which the spectra are computed.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Time- and plane-averaged *u*-velocity spectra computed in the *x* direction at heights of approximately (left) 40 and (right) 120 m using mesh resolutions of (top) Δ*h*, Δ*z* = 16, 4 m, (middle) Δ*h*, Δ*z* = 32, 8 m, and (bottom) Δ*h*, Δ*z* = 64, 16 m, using each of the four SFS stress models. Here *z* and *nz* indicate the height and corresponding vertical gridpoint index, respectively, at which the spectra are computed.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 7, but with spectra computed in the *y* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 7, but with spectra computed in the *y* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 7, but with spectra computed in the *y* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Lowest 100 m of the computational domain with the symmetric transverse Gaussian ridge showing cross section of height in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Lowest 100 m of the computational domain with the symmetric transverse Gaussian ridge showing cross section of height in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Lowest 100 m of the computational domain with the symmetric transverse Gaussian ridge showing cross section of height in the *x* direction.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Instantaneous two-dimensional sections of *u* velocity in the *x*–*y* plane at approximately 10 m above the surface from simulated flow over a symmetric transverse Gaussian ridge using each of the four SFS stress models with a mesh resolution of Δ*h*, Δ*z* = 8, 2 m. Flow is forced by a geostrophic wind in the *x* direction. Dotted contours indicate negative velocities, showing flow separation and reversal in the lee of the ridge.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Instantaneous two-dimensional sections of *u* velocity in the *x*–*y* plane at approximately 10 m above the surface from simulated flow over a symmetric transverse Gaussian ridge using each of the four SFS stress models with a mesh resolution of Δ*h*, Δ*z* = 8, 2 m. Flow is forced by a geostrophic wind in the *x* direction. Dotted contours indicate negative velocities, showing flow separation and reversal in the lee of the ridge.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Instantaneous two-dimensional sections of *u* velocity in the *x*–*y* plane at approximately 10 m above the surface from simulated flow over a symmetric transverse Gaussian ridge using each of the four SFS stress models with a mesh resolution of Δ*h*, Δ*z* = 8, 2 m. Flow is forced by a geostrophic wind in the *x* direction. Dotted contours indicate negative velocities, showing flow separation and reversal in the lee of the ridge.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Histograms of *u* velocity from 160 m downstream of the hill apex at 4 heights above the surface from simulations using mesh resolutions of Δ*h*, Δ*z* = 8, 2 m, using each of the four SFS stress models. Solid gray lines indicates zero, and dotted gray lines indicate the mean values from each of the distributions.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Histograms of *u* velocity from 160 m downstream of the hill apex at 4 heights above the surface from simulations using mesh resolutions of Δ*h*, Δ*z* = 8, 2 m, using each of the four SFS stress models. Solid gray lines indicates zero, and dotted gray lines indicate the mean values from each of the distributions.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

Histograms of *u* velocity from 160 m downstream of the hill apex at 4 heights above the surface from simulations using mesh resolutions of Δ*h*, Δ*z* = 8, 2 m, using each of the four SFS stress models. Solid gray lines indicates zero, and dotted gray lines indicate the mean values from each of the distributions.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but at 156 m downstream of the hill apex, using mesh resolutions of Δ*h*, Δ*z* = 12, 3 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but at 156 m downstream of the hill apex, using mesh resolutions of Δ*h*, Δ*z* = 12, 3 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but at 156 m downstream of the hill apex, using mesh resolutions of Δ*h*, Δ*z* = 12, 3 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but using mesh resolutions of Δ*h*, Δ*z* = 16, 4 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but using mesh resolutions of Δ*h*, Δ*z* = 16, 4 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1

As in Fig. 11, but using mesh resolutions of Δ*h*, Δ*z* = 16, 4 m.

Citation: Monthly Weather Review 138, 11; 10.1175/2010MWR3286.1