## 1. Introduction

The Weather Research and Forecasting Model (WRF; Skamarock et al. 2008) is a leading atmospheric simulation framework, enabling a broad spectrum of applications in both basic science and operational forecasting, ranging from global to microscales. One emerging application is multiscale simulation, which can involve downscaling large-scale weather or climate simulations to better resolve regional features, or providing microscale or turbulence-resolving large-eddy simulations (LES) with boundary conditions more realistically depicting larger-scale weather and other environmental forcing than is typical of commonly used idealized setups. Multiscale simulation in WRF utilizes grid nesting, whereby one or more subset(s) of a computational domain can be resolved at higher resolution, with variables exchanged at the nest’s lateral boundaries. Nesting can be sequentially applied, permitting refinement to very fine scales. WRF supports several nesting options, including concurrent, for which all domains are run within the same integration cycle, and serial, which requires independent simulation of each nested domain using input postprocessed from a completed bounding-domain simulation. Serial simulation permits only one-way nesting, for which information is provided to the nest, but nest information does not impact the bounding domain. Concurrent nesting supports both one- and two-way nesting, the latter permitting feedback from the nest to the bounding domain and also permits the high-frequency exchange of information between nests, with state variables communicated between domains at each model time step, and also includes the exchange of vertical velocity, which is ignored with serial simulation.

While concurrent nesting provides the greatest ease of use, a limitation in previous versions of WRF is that mesh refinement is restricted to the horizontal directions only, with serial nesting required to refine the vertical coordinate and with integer refinement being the only available option. For applications using multiple nests, or involving high resolution, which requires frequent lateral boundary updates, the use of serial nesting can become cumbersome, requiring management of multiple files and processing steps. For this reason, the more convenient concurrent nesting option is often employed. However, not refining the vertical coordinate has consequences, including the requirement of large grid aspect ratios, α = Δ*x*/Δ*z*, where Δ*x* = Δ*y* are the horizontal and Δ*z* the vertical grid spacings, on bounding domain(s). Previous LES studies have shown strong sensitivity to α values near the surface (e.g., Brasseur and Wei 2010; Mirocha et al. 2010), with an inability to maintain uniform α values across domain boundaries resulting in differences of near-surface velocity profiles on nested versus bounding domains (e.g., Mirocha et al. 2013). The impacts of large α values further degrade the accuracy of WRF’s numerical integration over slopes of just a few degrees, with increasing degradation with increasing slopes (e.g., Schar et al. 2002; Lundquist et al. 2008, 2010).

To address the drawbacks of refinement in the horizontal directions only, vertical refinement for concurrent simulation, permitting arbitrary vertical grid spacing on the nested domain, was recently implemented into WRF and demonstrated in both mesoscale and LES applications (Daniels et al. 2016), involving both real and idealized setups. Their LES application involved neutral flow over flat, rough terrain using the simple Smagorinsky subfilter-scale stress (SFS) model (Smagorinsky 1963; Lilly 1967), simulated with a fine LES domain nested within a coarser LES. They showed that independent control of α values on both domains improved the agreement between the simulated wind speed profiles and the theoretical logarithmic similarity solution [log-law (LL)] on both domains simultaneously, an impossibility with horizontal-only mesh refinement. However, the distance required for wind speed profiles on nested domains to reach equilibrium values increased slightly with vertical refinement added. The delayed equilibration is hypothesized to have resulted from the sensitivity of the Smagorinsky closure to the stepwise reduction of the grid-dependent length scale *l* = (Δ*x*Δ*y*Δ*z*)^{1/3} (the isotropic formulation) at domain interfaces. As the Smagorinsky closure formulates stresses as the product of *l* and a time and space invariant constant *C*_{S}, the addition of vertical mesh refinement increases the stepwise reduction of *l* at the nest boundary, thereby further reducing the magnitude of the SFS stresses at the nested domain’s inflow. Additionally, the bounding domain contains more coarsely resolved turbulence in vertically nested configurations.

Mirocha et al. (2013) demonstrated that the sensitivity of near-surface wind speed profiles resulting from changes to α also depends upon the SFS models. Two dynamic SFS models, each varying *C*_{S} in time and space based on the state of the flow and thereby relaxing the dependence of the SFS stresses on *l*, improved the agreement between the simulated mean wind speed and the log-law, and reduced equilibration times for flow parameters on nested domains, relative to constant-coefficients formulations. The present work extends that of Daniels et al. (2016) and Mirocha et al. (2013) by investigating the impacts of vertical mesh refinement on nested LES performance using multiple SFS models with simulation and configuration choices. Further, new procedures for calculating several near-surface parameters, including (i) the extrapolation of horizontal velocity components at the nested domain lateral boundaries to grid points at heights below the lowest bounding-domain grid index [following the recommendations of Daniels et al. (2016)] and (ii) velocity deformation elements impacting the SFS stresses, were also examined.

## 2. Methodology

Figure 1 depicts the computational setup utilized herein. Following Mirocha et al. (2013), we employ both a two-domain “nested” configuration and a single “stand alone” (SA) domain. Periodic lateral boundary conditions (LBCs) are applied to both the outer domain (d01) of the nested configuration and to the SA domain. The SA domain is identical to the nested domain (d02) of the nested configuration, except for the LBCs, with the nested domain (d02) receiving its LBCs from the d01 solution. Both d01 and SA are initialized identically and forced with the same geostrophic winds and surface boundary conditions. Since d02 and SA utilize identical meshes, and both configurations (nested and SA) use identical large-scale forcing, differences between the d02 and SA solutions can be attributed to the grid nesting methodology.

The nested simulations use horizontal mesh resolutions of Δ*x* = Δ*y* = 33 and 11 m on d01 and d02, respectively, while the SA domains also use Δ*x* = Δ*y* = 11 m. Here and throughout, *x*, *y*, and *z* refer to the zonal, meridional, and vertical directions, respectively, beginning at 0 in the domain’s bottom-left corner (Fig. 1a), while *i*, *j*, and *k* refer to the grid indices in those directions, respectively, beginning with 1 in the bottom-left corner. Following Mirocha et al. (2013), who showed that WRF achieves the closest agreement between the mean wind speed profiles and the log-law for α ≅ 4 (for flow over flat terrain with a roughness length of *z*_{0} = 0.1 m), two different vertical mesh spacings Δ*z* are examined, using three configurations, as shown in Fig. 2. These configurations allow comparison of using α ≅ 4 on either the outer or the inner domain, but not both, as occurs with horizontal-only (HN) nesting, versus using α ≅ 4 on both domains simultaneously, as enabled by the addition of vertical nesting (VN).

For the horizontal grid spacings utilized herein, Δ*z* values of approximately 8.25 m on all domains, 2.75 m on all domains, and 8.25 m on d01 and 2.75 m on d02 are specified, as shown by the horizontal dotted lines in Fig. 2, which depict cell interface heights, relative to the horizontal grid spacing, within the lowest 20 m. The vertical velocity *w* is computed at interface heights, while the zonal and meridional velocities, *u* and *υ*, are computed at the midpoints between these heights. All domains have a height of approximately 1400 m. Due to WRF’s use of a pressure-based vertical coordinate, Δ*z* values cannot be precisely specified or maintained during a simulation; however, the values remain within a few percent during the simulations conducted herein. These Δ*z* values apply to the first grid cell above the surface, with stretching of approximately 2.58% per grid index applied above. Details of the domain setups are summarized in Table 1.

Computational domain setups, with Δ*h* and Δ*z* the horizontal and vertical mesh spacings, respectively; α the grid aspect ratio; and *nx*, *ny*, and *nz* the numbers of grid points in the *x*, *y*, and *z* directions, respectively. Values of α and Δ*z* denote the first grid point above the surface.

Following previous studies (e.g., Andren et al. 1994; Chow et al. 2005), we simulate neutral flow over flat, rough terrain with *z*_{0} = 0.1 m, forced by a geostrophic wind *U*_{g} with components [*u*_{g}, *υ*_{g}] = [10, 0] m s^{−1}. Flow is initialized using constant values of *U*_{g} with height, with potential temperature *θ* = 300 K up to *z* = 500 m, surmounted by an increase of *θ* by 10 K km^{−1}. The capping inversion constrains the growth of the boundary layer, while also preventing turbulence from reaching the model top. Rayleigh damping is applied for *z* > 1000 m, with a magnitude of 0.003 s^{−1}, which assists in maintaining a boundary layer height of *H* ≅ 1000 m. Small perturbations *δ* ∈ [±0.25] K, taken from a pseudorandom uniform distribution and applied as a decreasing cubic function of height up to *z* = 500 m, are added to the initial *θ* field, to instigate turbulence formation. Other upper boundary conditions are *w* = *τ*_{ij} = 0 and are free slip for *u* and *υ*.

The surface boundary condition uses Monin–Obukhov similarity theory (Monin and Obukhov 1954) to prescribe the vertical fluxes of horizontal momentum at the surface, *τ*_{i3} = *C*_{D}*U*(*z*_{1})*u*_{i}(*z*_{1}). Here, *i* = 1, 2 indicate the zonal and meridional directions, respectively; *C*_{D} = *κ*^{2}[ln(*z*_{1}/*z*_{0})]^{−2}; *κ* = 0.4 is von Kármán’s constant; *z*_{0} is the surface roughness length; and *U*(*z*_{1}) and *u*_{i}(*z*_{1}) are the wind speed and velocity components, respectively, at their first computed heights above the surface *z*_{1}.

The impacts of vertical mesh refinement are examined using six different SFS models, spanning simple to sophisticated approaches. The two simplest models are linear, constant coefficient (*C*_{S}) eddy-viscosity models (Smagorinsky 1963; Lilly 1967), with the eddy viscosity coefficient formulated either using the strain-rate (SMAG) or a 1.5-order prognostic equation for SFS turbulence kinetic energy (TKE). The Nonlinear Backscatter and Anisotropy (NBA) model (Kosović 1997; Kosović and Curry 2000), which utilizes a nonlinear constitutive relation, is also examined. The NBA model contains both an eddy viscosity component, similar to SMAG and TKE, and another term involving nonlinear products of the strain and rotation rate tensors. The NBA model is examined using formulations based on the strain and rotation rates only (NBA1), and incorporating a 1.5-order SFS TKE equation (NBA2). Two dynamic models that base the SFS stresses on the smallest-scale resolved stresses occurring within the flow are also examined. The Lagrangian-averaged scale-dependent (LASD) model (Bou-Zeid et al. 2005) utilizes the SMAG formulation, but specifies the value of the model constant dynamically, using a two-level filtering procedure that determines how the resolved stresses change across scales. As dynamic schemes can become numerically unstable, the LASD eddy viscosity coefficients are averaged along fluid pathlines for stabilization. The Dynamic Reconstruction Model (DRM; Chow et al. 2005) is a mixed model, using explicit filtering and reconstruction for the scale similarity component and a one-level filtering procedure to dynamically specify the eddy viscosity component (Wong and Lilly 1994). The DRM is stabilized via local filtering. Herein, the lowest-order (0) reconstruction of the scale similarity term is used and is denoted DRM0.

Both dynamic models examined herein have been shown to improve the flow characteristics relative to simpler constant-coefficient formulations; however, implementation into atmospheric LES codes using finite-difference numerical solvers required augmentation of the SFS stresses near the wall using an additional parameterization (e.g., Chow et al. 2005; Kirkil et al. 2012) to recover the log-law. Herein, following those above studies, additional near-wall stresses are applied following Brown et al. (2001). Further details regarding these models can be found in the references.

The rates of equilibration of flow parameters within the nested domains are examined as functions of distance from the primary inflow boundary (*x* = 0; *i* = 1). All parameters are averaged over time and space to yield mean values for comparison. Because of the dependence on distance from the inlet planes (see Fig. 1), parameters from nested domains (d02) are averaged in the *y* direction only. Averaging at each grid index (*i*) in the *x* direction shows the evolution of parameters within the nested domain. The *y*-direction averaging excludes the first 100 and final 20 grid points in the *y* direction to eliminate artifacts from the *y*-direction LBCs. These time- and *y*-averaged data are then evaluated against values obtained from SA domains using the same mesh and SFS model. Data from both SA and d01 are averaged over all grid points in the *x* direction, as the flow in these domains is, on average, horizontally homogenous, because of the periodic LBCs. Data from d01 are also averaged over all grid points in the *y* direction, while those on SA are restricted to the same set of grid points as d02, to facilitate comparison between the d02 and SA quantities. All quantities are averaged over 4 h, using 240 instantaneous values in 1-min increments. Model time steps of 1/4 and 1/12 s were used for d01 and d02 (and SA), respectively.

## 3. Results

### a. Mesh refinement in the horizontal directions only

To establish a baseline against which to assess the impacts of the new vertical nesting capability, simulations using the traditional nesting in the horizontal directions only are first performed, using configurations with the optimal value of α ≅ 4 on either the nested or the bounding domain.

Figure 3 shows profiles of horizontal wind speed *U* = (*u*^{2} + *υ*^{2})^{1/2}, scaled by friction velocity at the surface *z*) to the ABL depth *H*, the height at which the model stresses nearly vanish, which, for these computational setups, coincides approximately with the base of the Rayleigh damping layer, *H* = 1000 m. Displaying the results this way permits comparison with the log-law similarity solution.

Figure 3 shows results using all six SFS models and two different horizontal nesting configurations (all without vertical nesting), with the left and right columns using α ≅ 4 on the nested (d02) and bounding (d01) domains, respectively, as shown in Figs. 2a,b. Each row shows results using SFS models that utilize similar formulations and exhibit similar characteristics; SMAG and TKE (top), NBA1 and NBA2 (middle), and DRM0 and LASD (bottom). The legend in each figure panel contains the line styles showing the evolution of the profiles as a function of distance from the inflow boundary, from their inflow profiles (given by the bounding domain averages, denoted d01), at four *i* locations within d02 (locations shown in Fig. 1), toward their expected equilibrium solutions (given by the SA domain averages). As the resolved flows develop smaller-scale motions during their transit through d02, the profiles (dotted and short-dashed lines) are seen to gradually transition from their d01 toward their SA distributions.

Examining the d01 and SA profiles (solid and long-dashed lines) reveals the impacts of the use of different α values on plane-average profiles. For simulations using the SMAG and TKE (Figs. 3a,b), and NBA1 and NBA2 (Figs. 3c,d) SFS models, domains using α ≅ 4 [green and gray lines (SA) in Figs. 3a,c; pink and black lines (d01) in Figs. 3b,d] agree better with the log-law than those with larger [green and pink lines (d01) in Figs. 3a,c] or smaller [black and pink lines (SA) in Figs. 3b,d] values. Dynamic SFS models (DRM0 and LASD; Figs. 3e,f) permit mitigation of dependence on α via adjustment of the near-wall stress model used by those schemes (see Mirocha et al. 2013), thereby achieving good agreement with the log-law irrespective of α.

Examination of the dotted and short-dashed lines indicates differences in how the profiles evolve from their inflow values (d01) toward the expected (SA) solution, depending upon the SFS model. All simulations show a rapid adjustment from the inflow within the first 40–140 grid cells, followed by a more gradual transition toward the expected solution. Both the greatest departures from the log-law and the slowest equilibration rates are observed for SFS models using static values of the model constants (SMAG, TKE, NBA1, and NBA2). The dynamic models, which adjust their constants based on local flow parameters, improve both the magnitudes of the errors and their persistence within the nested domain. Comparison of the dotted and dashed lines within each panel shows that, for these simple, neutral flows over flat terrain, the impacts of incorporating SFS TKE (TKE versus SMAG and NBA2 versus NBA1) are small, slightly delaying equilibration. Negligible differences are observed between the different dynamic SFS models (DRM0 versus LASD) except at the location closest to the inflow.

While *U*/*u*_{*} is one measure of flow equilibration, comparison of Figs. 3 and 1 shows that *U*/*u*_{*} can appear to have equilibrated while turbulence structures are still evolving. To better understand the mechanisms controlling the flow equilibration process, and how equilibration may vary for different parameters, two turbulence quantities are also examined: resolved turbulence kinetic energy *K* = (*u*_{*} = [(^{2} + (^{2}]^{1/4}. Here, *a*′ = *a* − *a* representing a velocity component and overbars indicating averaging in space and time. Spatial averaging was performed over the same footprints as described above, over half-hour time intervals.

Figure 4 shows profiles of the resolved component of *K*, from the two horizontal-only nesting configurations, from all six SFS models. The panel layout, line styles, and color scheme are identical to those in Fig. 3. All simulations show significant increases of *K* in the lower portion of the boundary layer (*z* < 0.3*H*) as the flow advects through the first 40–140 grid points within d02 (dotted and short-dashed lines) relative to the inflow (green solid and pink long-dashed lines). While values gradually approach the SA values (gray solid and black long-dashed lines), large departures are still observed at *i* = 340. The differences between the inflow and SA values can be attributed to the finer horizontal mesh spacing on the SA (and d02) than on the d01 domains, resulting in peak *K* values occurring closer to the surface on SA (and d02). Differences in the magnitudes of the peak *K* values depend on both the SFS model and α. For the SMAG, TKE, NBA1, and NBA2 solutions (Figs. 4a–d), larger peak *K* values are observed on d01 than on SA for the *α*_{d01} ≅ 12 cases (Figs. 4a,c), whereas the peak magnitudes are similar on both d01 and SA for the *α*_{d01} ≅ 4 cases (Figs. 4b,d). The incorporation of SFS *K* has only a small impact on resolved *K* values, again slightly delaying equilibration.

Despite the close agreement of *U* among the dynamic models, considerable differences are observed in their predictions of resolved *K* (Figs. 4e,f). These differences are due primarily to the DRM model’s SFS contribution being slightly larger than other closures, reducing the resolved-scale component (see, e.g., Chow et al. 2005; Kirkil et al. 2012). As with the wind speeds shown in Fig. 3, the dynamic SFS models result in both smaller overpredictions of *K* on d02 and more rapid equilibration toward the SA values, for both grid configurations.

Figure 5 shows resolved *u*_{*} values, as in Figs. 3 and 4, revealing similarities with *K*, including rapid overpredictions on d02 (dotted and short-dashed lines) followed by slow decays toward SA values (gray solid and black long-dashed lines), peak values occurring closer to the surface with finer Δ*h* (SA and d02) and generally smaller overpredictions and more rapid equilibration with the dynamic SFS models (Figs. 5e,f).

Explanations for the observed characteristics of the evolution of flow parameters during transit through d02 involve the consequences of the immediate reduction of the SFS stresses upon entry into the nested domain (via the stepwise reduction of the length scale, *l*), relative to the time scale required for the smallest scales of turbulence resolvable on the finer mesh to develop. Over the distance required for the smaller scales to form, the absence of those scales within the energy cascade allows the resolved scales to decouple from dissipation (parameterized by the SFS model), causing *K* and *u*_{*} to increase. The similar behavior of *K* and *u*_{*} on d02 suggests that the overprediction of both parameters is due to the intensification of correlated structures comprising the turbulence field (rather than numerical noise, which would be expected to increase *K* but not *u*_{*}).

Simultaneously, the downward momentum transport near the surface, which is governed by the smallest resolvable scales due to eddy size scaling with height above the surface within the surface layer (e.g., Garratt 1994), decreases, resulting in near-surface deceleration. The dynamic SFS models, via their ability to adjust their constants to local flow conditions, encourage more rapid formation of the smallest resolvable scales, reducing the magnitudes and extents of the anomalies.

The scale gaps in the resolved velocity field near the inflow planes of the nested domains are evident in Fig. 6, which shows time-averaged compensated spectra of the *u*-velocity component. Spectra are shown at *z* ≅ 30 m, corresponding to the heights of the peak overpredictions of resolved *K* and *u*_{*} (Figs. 4–5). Here, *k* indicates the vertical index above the surface, which is larger for the *α* ≅ 4 domain (see Fig. 2). Spectra are taken in the *y* direction, over the same range of grid points (100 ≤ *j* ≤ 220) on both nested (dotted and short-dashed lines) and SA domains (solid gray and long-dashed black lines). As in Figs. 3–5, the left and right columns compare solutions with *α* ≅ 4 on the nested and bounding domains, respectively. The d01 spectra are taken at *i* = 1 on d02, which are the d01 velocities projected onto the d02 mesh. Dotted and long-dashed lines show spectra as functions of distance from the inflow plane, as in previous figures, from simulations using comparable SFS models, as identified in the panels.

Common features of all spectra include relatively greater low-wavenumber power and narrower inertial subranges on d01 than on the SA and d02 domains. The high-wavenumber oscillations at the inflow (d01 values here projected onto the d02 mesh to facilitate comparison across wavenumber space) are interpolation artifacts that rapidly dissipate with downstream distance. The excess low-wavenumber power on d01 reflects the presence of long streamwise streaks common to both observations (e.g., Hutchins and Marusic 2007) and simulations (e.g., Fang and Porté-Agel 2015) of boundary layer flow, which are often enhanced on coarsely resolved LESs (e.g., Mirocha et al. 2013).

For all simulations, spectra on d02 evolve toward the SA solutions with distance. However, characteristics of the evolution exhibit wavenumber dependence. While power associated with wavenumbers corresponding to the smallest resolvable scales on d02 increases rapidly, because of the downscale cascade, low-wavenumber power increases initially, as a result of the decoupling of the well-resolved scales from dissipation, until the higher-wavenumber spectral content approaches SA values. Thereafter, with TKE production and dissipation recoupled through the inertial scales, low-wavenumber power slowly decreases. The peak overpredictions of *K* values (Fig. 4) correspond closely to the distance at which the higher-wavenumber spectral content reaches SA values.

The spectra in Fig. 6 indicate slightly more rapid convergence toward corresponding SA solutions for configurations with *α* ≅ 4 on d02 rather than d01. The *K* and *u*_{*} profiles shown in Figs. 3–5 exhibit the same tendency; however, the differences in the convergence rate among the different configurations are more readily discernible from the spectra.

### b. Mesh refinement in the horizontal and vertical directions

Figure 7 compares results of nested simulations using refinement in only the horizontal directions [horizontal nesting (HN)] versus refinement in the vertical and horizontal directions [vertical nesting (VN)], the latter permitting meshes with *α* ≅ 4 on both d01 and d02. VN results are compared to HN results from the configuration using *α* ≅ 12 and 4 on d01 and d02, respectively, as this HN configuration shows slightly better convergence overall and also permits comparison of VN and HN to the same SA solution (long-dashed black line). In Fig. 7 and others in this section, each panel compares VN to HN results for one SFS model, as indicated in the legend. The d01 solutions differ as a result of the use of different *α*_{d01} values (solid green and pink short-dashed lines). Dotted and short-dashed lines correspond to the HN and VN solutions, respectively, at the same six locations within d02, as in the previous figures and as indicated in the legend.

The most discernible differences between the VN and HN simulations again involve the d01 profiles using the constant-coefficient SFS models (SMAG, TKE, NBA1 and NBA2; Figs. 7a–d). For these simulations, the anomalous departures from the log-law near the surface for the *α*_{d01} ≅ 12 configurations are ameliorated for the VN simulations. While the VN simulations yield closer agreement with the log-law on d01, on d02 the VN solutions produce slightly larger departures from the log-law and slightly slower equilibration toward the SA solution, likely because of both the coarser turbulence resolved on the VN simulation’s bounding domain, and the larger stepwise reduction in grid size. Differences between the HN and VN solutions decrease with distance within d02. The dynamic SFS models, despite larger initial deviations, again produce smaller departures from the log-law overall and more rapid equilibration to the SA profiles, with only negligible differences between the HN and VN solutions beyond *i* ≅ 140.

Figure 8 shows profiles of the resolved component of *K*, using all six SFS models and using both HN and VN, as in Fig. 7. Again, the VN simulations show slightly larger overpredictions and slightly slower equilibration toward the SA *K* profiles for all SFS models. While the dynamic SFS models yield trivial differences between the HN and VN solutions by *i* = 240, large differences persist throughout d02 for the other SFS models.

Figure 9 shows spectra of the *u* velocity at *z* ≅ 30 m, here comparing VN versus HN simulations using each SFS model. A key difference between the VN and HN solutions is a slight widening of the inertial subrange on d01 from configurations using *α*_{d01} = 12, as a result of the finer vertical mesh resolution capturing energy containing motions up to slightly higher wavenumbers. Simulations using dynamic SFS models show less sensitivity to *α*_{d01} than do the others, with DRM0 showing the least, likely because of contributions from the additional resolvable SFS (RSFS) component (not shown). Impacts of different α values on d01 again diminish rapidly with distance within d02.

To better understand the roles of the SFS models in the evolution of flow within the nested domain, Fig. 10 shows profiles of the SFS component of *α*_{d01}, with the *α*_{d01} ≅ 12 domain producing larger SFS stresses immediately above the surface, with smaller values farther aloft. The larger near-surface values are the result of larger mean near-surface vertical shear values resolved within the HN but not in the VN simulations (with values of 0.194 and 0.126 s^{−1}, respectively, for the SMAG simulations) and sufficiently larger to offset the smaller value of *l* on the HN domains. The dynamic SFS models exhibit reduced dependence on α near the surface because of both their ability to alter the value of the constant and also their use of the separate near-wall stress model. While all solutions exhibit shared characteristics on d02 and rapid reductions followed by overshoots before gradually approaching corresponding SA values, equilibration is much more rapid for the dynamic models because of their more rapid formation of the smallest resolvable scales, augmenting downward momentum transport toward the surface, thereby reducing shear.

### c. Algorithm modifications in the near-surface region

With a view toward improving WRF’s near-surface flow physics, and equilibration rates in nested configurations, alternative methods of calculating the model parameters impacting the near-surface flow are examined.

#### 1) Extrapolation for vertical nesting

One-way nesting requires the projection of coarse-domain flow information onto the fine-domain mesh at nest interfaces. In WRF, this projection occurs within a user-specifiable range of grid cells extending from the nest perimeter into the nested domain at all four lateral boundaries. In addition to the horizontal interpolation required for the standard horizontal mesh refinement, vertical refinement also requires vertical interpolation. Hermite interpolation following Steffen (1990) is used in the vertical nesting algorithm; however, as WRF uses a staggered Arakawa C grid, extrapolation is required for any fine-domain grid points *z*_{f,k} extending above or below the range of heights on the coarse domain mesh. In this case, the standard WRF extrapolation method, utilizing a quadratic Lagrange polynomial (LP), is applied. However, when applied to the horizontal velocity components, this extrapolation generally does not result in logarithmic velocity profiles near the surface. A new method was implemented that instead uses the LL to specify near-surface horizontal velocities using *u*_{i}(*z*_{f,k} < *z*_{c,1}) = *u*_{i}(*z*_{c,1})[ln(*z*_{f,k}/*z*_{0})/ln(*z*_{c,1}/*z*_{0})], where *z*_{c,1} is the lowest grid index on the coarse domain, thereby enforcing a logarithmic profile near the surface.

Figure 11 shows the differences between the standard LP (blue) and new LL (red) extrapolation methods used within the nested domain’s projection zone. Instantaneous profiles of *u* and υ at the nested domain inflow boundary are shown relative to their parent domain values (black) at two times during a simulation using the LASD SFS model. When the d01 profile is nearly logarithmic, both methods produce similar results for the extrapolated *z*_{do2,k} value below *z*_{d01,1} (*u* velocity in Fig. 11a). However, when a similarly nearly logarithmic profile contains a very small velocity at the first grid point above the surface, the LP method may project an anomalous larger velocity magnitude of the opposite sign (*υ* velocity in Fig. 11a). Further, based on the slope and curvature of the d01 profile, the LP method often produces larger magnitudes at the extrapolated locations than the LL method (*u* and *υ* velocities in Fig. 11b). The same LP projection method is also applied at every grid point to enforce the kinematic boundary conditions, which may not be appropriate when large velocity fluctuations near the surface are resolved.

While Fig. 11 shows instantaneous velocity profiles, Figs. 12a,b show the time- and *y*-direction-averaged profiles of *U* within the lowest 13 m using both the LP and LL methods, from simulations using both the SMAG and LASD SFS models, which represent the generic behavior of the constant-coefficient and dynamic SFS models, respectively. Here, solid green and gray lines show the d01 and SA solutions, with filled squares showing heights above the surface of the lowest two and four grid points, respectively. Dotted and dashed lines show results using the LP and LL methods, respectively, at the same heights as the SA solution, both at the inflow plane (*i* = 1; purple) and at four distances downstream (different locations than in previous plots). The LL projection matches the SA solution much more closely at the inflow plane; however, the impact of the projection method on the evolving near-surface flow vanishes rapidly with downstream distance.

Changes to the *U* and *K* profiles over a deeper layer resulting from the different projection methods are shown in Figs. 12c,d and Figs. 12e,f, respectively. The LASD simulations show slightly more sensitivity to the projection method, with impacts persisting farther downstream. The different projection methods only negligibly impact the evolution of velocity spectra at 30 m (not shown).

#### 2) Calculation of near-surface deformation elements

Another standard WRF model method potentially impacting near-surface flow characteristics in LESs involves the computation of deformation elements *S*_{13} and *S*_{23}. The standard WRF method computes *S*_{13} and *S*_{23} at grid cell interface levels, for *k* ≥ 2, by vertically differencing the *u* and *υ* velocities from their native cell center locations. The subsequent projection of *S*_{13} and *S*_{23} to the cell centers, as needed by the SFS models, requires values at the surface to be included in the projection stencil for *k* = 1. These surface values are simply set to zero.

To determine the impacts of this choice, two new methods for calculating *S*_{13} and *S*_{23} near the surface were examined. The new approaches instead compute *S*_{13} and *S*_{23} at the cell centers directly. This requires first projecting *u* and υ from their native cell center locations to the cell interface levels above and below, after which the vertical derivatives defining *S*_{13} and *S*_{23} at the cell centers are computed. The difference between the two new methods involves how the cell-center *S*_{13} and *S*_{23} values at *k* = 1 are computed. One new method (LP) uses the standard Lagrange polynomial method to project *u* and υ to the surface, consistent with the standard WRF method for interpolation during VN, and enforcing kinematic surface boundary conditions. Thereafter, the values of *S*_{13} and *S*_{23} at *k* = 1 at the cell centers are computed directly. Because of the above-described issues with the LP extrapolation method, another new method (LL) also based on the log-law specifies *S*_{13} and *S*_{23} at *k* = 1 using *S*_{i3} = *κz*_{1}), *i* = 1, 2 (Moeng 1984).

We show the impact of the two new methods for computing *S*_{13} and *S*_{23} (LP, Figs. 13a,b; LL, Figs. 13c,d) on *U*/*u*_{*}, relative to the log-law similarity profile, using the SMAG and LASD SFS models. Each new method slightly alters *U* relative to *U* relative to *k* = 1 and because of increasing estimates of *S*_{13} and *S*_{23}. The LP provides smaller values than the LL method, because of its underestimation of vertical gradients relative to the log-law (see Figs. 11 and 12a,b). Simulations using the LASD model show little to no change in the mean vertical distribution of *U* relative to *k* = 1, relative to the additional near-wall stress term.

The large excursion of *U*/*u*_{*} from the log-law near the surface for the SMAG simulations using the LL method (Fig. 13c) can be ameliorated by applying a near-wall damping function, such as that of Mason and Thomson (1992), which reduces the value of *l* approaching the surface as 1/*l*^{2} = 1/(*C*_{S}*l*_{0})^{2} + 1/(*kz*)^{2}, where *l*_{0} = (Δ*x*Δ*y*Δ*z*)^{1/3}. Incorporation of this near-wall damping reduces the near-surface *U* values relative to *U*/*u*_{*} values on α and *z*_{0}, near-wall damping may not always improve the solution, as it does in this case.

For the LASD solution, the additional near-wall stress model dominates *U*/*u*_{*} near the surface. Figure 13f shows the results of increasing the magnitude of the near-wall stress factor by 6.25%, from 0.80 to 0.85 (LLb), the effects of which produce a *U* profile shape near the wall that is almost identical to the standard method using a near-wall stress factor of 0.80 (Fig. 13b).

Figure 14 shows resolved *K* values from each of the simulations shown in Fig. 13. The alternate methods of computing *S*_{13} and *S*_{23} at *k* = 1 impact the SMAG and LASD simulations differently, slightly increasing *K* values for the SMAG simulations while reducing *K* slightly when using LASD. While the LL method (Fig. 14c) increases *K* values when using the SMAG model, adding near-wall damping (LLa; Fig. 14e) reduces those values to slightly below the ST method by a similar amount as the LL method exceeded the ST values. For the LASD simulations, the LL method (Fig. 14d) reduces *K* values slightly throughout the column, above the peak. Increasing the near-wall stress factor from cf = 0.80 to cf = 0.85 (LLb; Fig. 14f) produces nearly identical *K* values throughout the column as the ST method with cf = 0.80 (Fig. 14a), corresponding to the close agreement of the *U* profiles LLb and ST (Figs. 13a,f).

While comparison against the expected log-law similarity solution is helpful for evaluating the accuracy of the shape of the wind speed profile near the surface, scaling *U* by *u*_{*} can disguise changes in *U* that occur between domains, and how those changes are modified by different numerical methods’ nesting strategies, if *u*_{*} changes similarly. To examine *U* independently, Fig. 15 shows profiles of time- and domain-average *U* from both d01 (black and gray) and the corresponding SA domains (green and pink) from simulations using different methods to compute *S*_{13} and *S*_{23} at *k* = 1, as described in the legends The insets highlight results for *z* ≤ 30 m.

The differences between the d01 and SA domains are due to the higher resolution of the SA domains, since the forcing and LBCs are identical. The SMAG results show somewhat larger differences between the d01 and SA domains than the LASD simulations, especially using LLa. Both the SMAG and LASD solutions show kinks in the profiles near the surface when using LL, with the SMAG solution’s larger, as may be observed in Fig. 13. While these kinks near the surface appear significant when plotted against the log-law (Figs. 13c,d), the actual changes to *U* are modest, with the greatest occurring with SMAG using LLa versus ST (Fig. 15e), which increases the *U* values by less than 0.5 m s^{−1}.

Figure 16 compares changes of *U* under VN versus HN only to those occurring under different formulations of *S*_{13} and *S*_{23} at *k* = 1, with the profile on d02 at *i* = 340 shown relative to the d01 and SA solutions. The insets again highlight the results for *z* ≤ 30 m. For the SMAG simulations, the HN profiles exhibit the closest agreement between the d02 and SA solutions near the surface, whereas the VN profiles exhibit the closest agreement farther aloft. The LASD simulations show smaller differences between profiles from all three domains for all configurations, with the smallest differences occurring using the log-law-based projection method for *S*_{13} and *S*_{23} at *k* = 1, also cf increased from 0.80 to 0.85 (LLb; Fig. 16f).

## 4. Summary, discussion, and conclusions

A recently implemented vertical mesh refinement capability for concurrently nested simulations using the WRF Model was examined for nested LESs of neutral flow over flat, rough terrain. Six different SFS turbulence models, three different vertical mesh configurations, an alternative vertical projection method for the lowest-level velocity components on nested domain lateral boundaries, and two alternative methods of computing deformation elements *S*_{13} and *S*_{23} were utilized to evaluate the benefits, drawbacks, and sensitivities of vertical nesting, relative to other configuration choices.

Both the representation of steady-state flow parameters, as well as characteristics of their evolution upon entry into nested domains, were compared. Both the new vertical nesting (VN) and the standard horizontal-only nesting (HN) methods contributed anomalies over the lowest ≈0.3*H*, consistent with other studies (e.g., Mirocha et al. 2013; Daniels et al. 2016). These anomalies, generally peaking in magnitude at *z* ≅ 0.1*H*, exhibited rapid growth upon entering the nests, before gradually diminishing.

Examination of *K*, *u*_{*}, and velocity spectra, reveals that the anomalies contributed by mesh refinement coincide with a decoupling of the scales of energy production from those of dissipation as flow enters the nest, with the resolved energy producing structures that advect into d02 from d01 suddenly separated from dissipation (as parameterized by the SFS model) by a range of inertial scales that are resolvable on the refined mesh but not the coarser mesh. The rate of formation of the missing inertial scales influences both the magnitudes of the anomalies on the nested domain and their rates of attenuation. Of the various factors examined, the anomalies exhibited the strongest sensitivity to the SFS model, with the dynamic models exhibiting generally superior performance, as a result of their ability to reduce the model constant, allowing more rapid formation of the inertial scales resolvable on the finer mesh. While important for other applications, for the nested cases examined herein, incorporation of SFS TKE, or nonlinear terms (within the NBA models), only negligibly impacted the flow equilibration.

The addition of VN to the standard HN refinement slightly retards the equilibration of flow statistics, relative to HN, likely because of the larger change in the grid cell size across the nest boundary and the more coarsely resolved turbulence on the bounding domain. While VN improves the agreement of *U*/*u*_{*} with the log-law on both domains, as a result of use of the same value of α ≅ 4, profiles of *U* show little change. Beyond these differences, which are generally smaller than the impacts of the different SFS models, no significant deleterious effects of vertical nesting were observed. Despite the modest delay in turbulence equilibration, the ability to coarsen the bounding domain vertical grid with negligible impacts on the nested domain solution affords potentially significant benefits to applications involving complex terrain, for which numerical stability concerns related to large α values have been documented (e.g., Mahrer 1984; Daniels et al. 2016). Additional computation associated with VN is negligible, with a potential for an overall reduction because of the use of fewer grid points and possibly larger time steps on the bounding domain(s).

A new projection method to prescribe velocity values at nested domain gridpoint locations at heights below the lowest bounding domain value using the log-law was also tested. While the new approach showed improved agreement with both the log-law and the corresponding SA solutions near the nested domain inflow boundaries, the impact on equilibration rates within d02 was negligible. New methods for computing *S*_{13} and *S*_{23} were also implemented, using both WRF’s standard Lagrange polynomial extrapolation and a method based on the log-law. Modification of the near-wall stress field was examined as well, using the near-wall damping of Mason and Thomson (1992) for the SMAG simulations and with different parameter values cf of the additional near-wall stress model used with the LASD. The impacts of these changes were generally no larger than the impacts of other configuration choices (different α values or SFS models), and had only negligible impacts on the equilibration rates of the flow statistics on the nested domains.

Grid configurations, SFS models, methods for calculating near-surface flow parameters, and algorithms to augment or damp the stress field near the surface all work in concert with the numerical solver to define the effective model filter. Similar effects on the near-surface distributions of flow parameters (e.g., *U* and *K*) are achievable via changes to α, the method used to compute *S*_{13} and *S*_{23}, or by augmenting or damping the near-surface stresses. Both the optimal value of α ≅ 4 for the simulations conducted herein, using the constant-coefficient SFS models, as well as characteristics of the near-wall stress model utilized by the dynamic SFS models, were determined using the default (in WRF) and, in our view, incorrect method of calculating *S*_{13} and *S*_{23} at *k* = 1. With the log-law method, which is, in our view, the more correct of the two methods examined herein, the optimal α value is likely to be different. The optimal value of α also likely depends upon the value of any SFS model constant (e.g., *C*_{S} for the SMAG model); any near-wall damping *z*_{0}; the numerical discretization; and other factors influencing near-surface flow physics. Further examination is required to determine if some combination of these configuration choices may improve simulation results in various setups.

Given the relative insensitivity of the flow equilibration process to modifications examined herein, acceleration of near-inflow adjustment at mesh refinement interfaces will likely require the addition of missing inertial scales to the resolved structures advecting into the nested domain. Examination of such methods is planned as future work.

## Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and was supported by the DOE Office of Energy Efficiency and Renewable Energy (EERE) and the LLNL Laboratory Directed Research and Development program as Project 14-ERD-024.

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