1. Introduction
Improved simulations of atmospheric flow over complex terrain would benefit a wide range of atmospheric applications, allowing for improvements in the fields of wind energy, wildfire prediction, and air pollution modeling, among others. With this aim, mesoscale weather prediction models, such as the Weather Research and Forecasting (WRF) Model, are increasingly being used for high-resolution simulations, including those resolutions required for large-eddy simulation (LES). While most LES models make use of idealized lateral boundary conditions, LES simulations within the WRF Model can use the available grid nesting framework, which employs multiple telescoping grids of increasing resolution, to provide downscaled lateral boundary conditions (Y. Wang, S. Basu, and L. Manuel 2013, personal communication; Taylor et al. 2016; Muñoz-Esparza et al. 2017). Additionally, LES simulations performed in the WRF Model can make use of an extensive suite of atmospheric physics parameterizations, including an emerging set of parameterizations that are scale aware (Talbot et al. 2012; Powers et al. 2017).
While it is highly desirable to perform LES-scale simulations in models such as WRF, the use of terrain-following coordinates in WRF and most other mesoscale atmospheric models has limited LES to flow over terrain with shallow slopes (Catalano and Moeng 2010; Taylor et al. 2016). Terrain-following coordinates work well at the resolutions used in mesoscale simulation, but numerical errors can arise at fine resolutions where steep terrain slopes can be captured, resulting in a highly skewed grid with large numerical errors (Mahrer 1984; Schär et al. 2002; Zängl 2002; Klemp et al. 2003; Zängl et al. 2004). These numerical errors contaminate the simulation, often leading modelers to apply smoothing filters to the terrain or increase numerical damping constants in an effort to enable the simulation to run (see e.g., Marjanovic et al. 2014).
To address the limitations of terrain-following coordinates in WRF, an immersed boundary method (IBM) has previously been implemented into the WRF Model (Lundquist et al. 2010, 2012; Ma and Liu 2017). Immersed boundary methods use nonconforming grids, often Cartesian, with the terrain boundary immersed within the grid, thereby alleviating numerical errors associated with the terrain-following grid transformation. When IBM is used, boundary conditions at the terrain surface are set through interpolation procedures for grid cells near the immersed surface, rather than at a grid interface aligning with the terrain.
Lundquist et al. (2010) first developed a WRF-IBM implementation and tested it for mountainous and urban terrain, including coupling the IBM to atmospheric physics parameterizations (e.g., the radiation and land surface models). Lundquist et al. (2012) showed that WRF-IBM performed comparably to a code using body-fitted coordinates for urban flow simulations from the Joint Urban 2003 field campaign in Oklahoma City, Oklahoma. A limitation of these IBM simulations was the use of a no-slip boundary condition at the immersed surface. The no-slip boundary condition simplified the implementation and validation of IBM and is commonly used in urban simulations, but is inappropriate for general atmospheric flows, where the surface layer is not resolved (Moeng 1984; Garratt 1994).
We begin with a detailed description of the improved WRF-IBM implementation where a log-law boundary condition is used to parameterize surface stress. The implementation in WRF is described in detail and the method is compared to existing methods in the literature. Then, a neutral atmospheric boundary layer over flat terrain is simulated using both the native terrain-following coordinates in WRF and the new IBM implementation. Next, simulations over Askervein Hill, Scotland, are performed and results are compared to field observations. This test case is chosen for its relatively shallow terrain slopes, which enable direct comparison between WRF simulations using the native terrain-following coordinate and those with the new IBM implementation. Finally, simulations of flow over Bolund Hill, Denmark, are performed and WRF-IBM results are compared to field data. This case includes steep terrain slopes, including a near-vertical escarpment on the west face of the hill, which presents difficulty for simulations using native WRF terrain-following coordinates.
This work is one piece of several ongoing efforts to enable simulations spanning the meso- to microscale using the WRF framework. This log-law boundary condition implementation enables high-resolution simulations over complex terrain, contributing to the overall purpose to develop a model that can include mesoscale input and atmospheric physics parameterizations in finescale complex terrain simulations. To include real atmospheric physics for WRF-IBM, Lundquist et al. (2010) coupled the WRF-IBM to the MM5 shortwave radiation scheme, the RRTM longwave radiation scheme, the MM5 surface layer model, and the Noah land surface model. Arthur et al. (2018) coupled the topographic shading effect with WRF-IBM and validated it with idealized simulations and field data. Daniels et al. (2016) developed and tested a new vertical-refinement capability allowing the grid on each domain to be optimized and thus improving WRF’s multiscale simulation capabilities. Mirocha and Lundquist (2017) further assessed the effects of vertical refinement and grid aspect ratios at fine resolution in WRF using large-eddy simulation. Finally, Wiersema et al. (2016) described work on nesting WRF-IBM within terrain-following WRF simulations. With these combined efforts, including the implementation of the log-law boundary condition for WRF-IBM detailed in this paper, WRF-IBM will enable complex terrain simulations at a new level: allowing steep terrain to be represented with much finer resolution and simultaneously including mesoscale forcing and the full atmospherics capabilities in WRF.
2. Background and methods
a. Immersed boundary method
The implementations of the log law with the immersed boundary method in the literature can be divided into three categories. Chester et al. (2007) developed a shear stress reconstruction IBM (SR-IBM), which modifies several layers of shear stress in the vicinity of the immersed boundary. Shear stress values on the immersed surface are calculated according to the log law. The shear stress values within a defined distance, external to the immersed boundary, are then set to the surface stress value. Stress values interior to the immersed boundary are linearly extrapolated from the surface value and a value outside the band of nodes being reconstructed. Velocity is set to zero interior to the immersed boundary. Diebold et al. (2013) used this SR-IBM with large-eddy simulation to simulate flow over Bolund Hill. The results obtained are among the best results of the modeling efforts for the Bolund case summarized in Bechmann et al. (2011). Cheng and Porté-Agel (2013, 2015) used this SR-IBM to investigate a turbulent boundary layer flow over a two-dimensional cube and uniform arrays of cubes. The simulations were validated with wind tunnel experimental data with good agreement. Ma and Liu (2017) followed this SR-IBM method and implemented it into WRF, testing it with large-eddy simulations over Bolund Hill compared with observations. Simulations with fine vertical resolution and the Lagrangian-averaged scale-dependent Smagorinsky model showed improvement over standard Smagorinsky for the mean speedup error and for capturing the recirculation zone. Fang and Porté-Agel (2016) performed an intercomparison of terrain-following coordinates and the SR-IBM in large-eddy simulation for flow over an idealized and moderately sloped 3D hill. They found that the SR-IBM results predicted a larger recirculation zone in the lee side of the hill compared to benchmark simulations. The SR-IBM method was also tested in WRF by Bao et al. (2016), where similar issues were found in the recirculation zone.
Another approach was taken by Anderson (2013) who used a canopy-like drag model to characterize surface stress. In this method, a canopy stress model is used to impose a momentum sink for cells near the surface. Cells where the extra drag is applied are determined based on surface geometry and can include single or multiple points, interior, and exterior to the immersed surface. Nodes internal to the immersed boundary are set to have zero velocity. Validation cases performed by Anderson (2013) included flow over a single cube and multiple cubes. Results using very fine 1-m resolution showed good agreement with benchmark data from previous studies. The canopy-like method was also tested in WRF by Bao et al. (2016), where the model predicted smaller velocities and a larger recirculation zone in the lee of a hill, compared to benchmark simulations and other immersed boundary options.
In a third approach, Fadlun et al. (2000) and Senocak et al. (2004) used velocity reconstruction near the surface assuming a linear or a log profile, respectively. Fadlun et al. (2000) first used a linear interpolation between a point in the fluid (the second point from the boundary) and a no-slip boundary condition at the surface to reconstruct the velocity at a layer of fluid points outside of the immersed surface (the first point from the boundary). For atmospheric flows, the fine resolution required to use a linear interpolation scheme (to resolve the viscous sublayer) would not be practical. Senocak et al. (2004) extended this method by using a log-law reconstruction scheme. The tangential flow is reconstructed at the first fluid node using the log law. Choi et al. (2007) extended this to reconstruction of a power law between the surface and a reference velocity at a fluid node 
Here, we implement the method from Senocak et al. (2004) into WRF, with some minor differences (Bao et al. 2016). This method was selected for this study from among the three IBM options implemented and explored in detail in Bao et al. (2016). The method from Senocak et al. (2004) is most straightforward in directly setting the velocity boundary condition to satisfy the log velocity profile near the surface. Details of the minor differences between the method from Senocak et al. (2004) and our method implemented into WRF are provided in section 2c. Notably, WRF’s pressure-based terrain-following coordinate system presents challenges in an IBM implementation that are unique compared to traditional incompressible computational fluid dynamics codes. We compare our results to simulations using a simple log-law condition with WRF’s native terrain-following coordinate (see section 2b). This allows the surface stress in both WRF and WRF-IBM to be parameterized with a surface roughness length scale 
b. WRF’s surface boundary condition
c. Velocity reconstruction IBM implementation
Schematic showing the velocity reconstruction IBM (VR-IBM) as described in the text. The dashed and solid grid lines indicate the staggered grid used by WRF. The red line indicates the immersed boundary.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
The slight difference between our method and Senocak et al. (2004) is with respect to setting the eddy viscosity. Senocak et al. (2004) modify the near-surface eddy viscosity based on a hybrid RANS–LES approach (see also Senocak et al. 2015; Umphrey et al. 2017). Prandtl’s mixing length hypothesis is adopted near the surface and a dynamic Smagorinsky model is used away from the surface. We tested this approach and found no significant effect on the near-surface velocities. We, therefore, retained WRF’s native eddy viscosity parameterization to simplify comparisons between native WRF and the VR-IBM implementation.
3. Validation
a. Neutral boundary layer simulation over flat terrain
For initial validation of the new IBM method in WRF, simulations are carried out for neutral atmospheric boundary layer flow over flat terrain, where the mean velocity profile in the surface layer should follow the log law. The following section details the domain setup and results for simulations using the native WRF and WRF-IBM grids. Comparisons are made between the simulation results when using the native WRF boundary condition and both the no-slip and velocity reconstruction IBM techniques (NS-IBM and VR-IBM).
1) Simulation setup
The neutral boundary layer simulation setup in WRF is similar to the standard setups used in Chow et al. (2005) and Mirocha et al. (2013, 2014). Flow is driven by a pressure gradient that would balance a geostrophic wind of 
All cases are initialized with a neutral and dry sounding with a uniform 10 m s−1 wind in the x direction. The surface is set to have a constant surface roughness of 
2) Results
Figure 2 shows the time evolution of the inertial oscillations using the domain-averaged U and V velocities. Inertial oscillations are sufficiently damped after 48 h so that the solution can be considered to have reached a steady-state condition and time averaging is appropriate. It can be seen here that while all three methods have similar oscillation periods, the NS-IBM results in a different domain-averaged velocity. The cause of this can be seen in Fig. 3, which shows the time- and planar-averaged U and V velocities for simulations using the original WRF terrain-following coordinate and the two IBM methods. Results are time averaged over hours 48–72 of simulation time, with data at 15-min intervals. As seen in the figure, the VR-IBM method does an excellent job in recreating the original WRF solution for this case. The no-slip boundary condition, shown for contrast, results in slow velocities and additional stress near the surface.
Domain-averaged wind velocities U and V as a function of time for terrain-following WRF, VR-IBM, and NS-IBM.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
Horizontal velocity components U and V, averaged in time and horizontally over the domain, for terrain-following WRF, VR-IBM, and NS-IBM.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
This initial validation for idealized neutral boundary layer flow verifies that our VR-IBM implementation can recreate the native WRF surface stress boundary condition and the velocity profile over flat terrain. Additionally, the VR-IBM method has the flexibility to handle complex terrain, as demonstrated in the following sections.
b. Askervein Hill simulation
1) Field campaign description
The Askervein Hill field campaign, which took place in 1982 and 1983, studied flow over a low-amplitude and moderately sloped hill near the west coast of South Uist in Scotland (Taylor 1983). Askervein is a 116-m-high hill and elliptical in plan view, with a 2-km major axis along a general northwest–southeast line and a 1-km minor axis. During the field campaign, there were more than 50 towers, which were placed in three arrays (A, AA, and B lines in Fig. 4), as well as an “upstream” reference site (RS). These tower measurements generated a detailed dataset of the surface wind field. The dominant wind directions during the campaign were from the southwest, nearly perpendicular to the major axis of the hill. A detailed description of the field campaign including the instrumentation and measurements are given in Taylor (1983) and Taylor and Teunissen (1985, 1987). We select Askervein Hill to evaluate the implementation of the log-law boundary condition for WRF-IBM because the small terrain slope makes it possible to compare to a reference simulation using standard WRF with terrain-following coordinates.
Instantaneous contours of U velocity (m s−1) for the (a) outer domain and (b) nested domain at 10 m above the surface. The dashed black line indicates the inner nest domain position within the outer grid. Black contours show Askervein Hill elevation at 23-m intervals and lines A, AA, and B are the measurement transects used in the field campaign.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
2) Simulation setup
Topographic data for Askervein are provided by Walmsley and Taylor (1996) at 25-m resolution. The incoming wind direction is 210°, and the terrain is rotated 60° in the clockwise direction, so that the incoming winds align with the x axis of the domain in the simulation (i.e., the incoming wind direction is set to 270° in the simulation). A one-way nested grid setup is used, where the parent domain has flat terrain and uses periodic lateral boundary conditions, and the nested domain contains Askervein Hill and is forced at its boundaries by the parent domain. This setup is similar to simulations in Golaz et al. (2009) where flow over Akservein hill is simulated in the COAMPS mesoscale model. The nesting is one way so that the parent domain is not influenced by the terrain-induced flow in the nested domain but simply provides the boundary forcing for the inner domain. The boundary layer in the parent domain is neutral and driven by a geostrophic wind speed of 
The flow on the outer domain is allowed to spin up for 18 h to allow inertial oscillations in the neutral boundary layer to dampen. At 18 h, the nested domain is spawned and nested boundary conditions are used. The parent and nested domain are run concurrently for an additional 1.5 h. Results presented here are averaged over the last 30 min of the simulation after a 1-h spinup time for the inner domain. Figure 4 shows the parent and nested domain with the topography of Askervein Hill and the field data transects. We use a constant roughness value of 
3) Comparison with observations
Figure 5 shows the simulated time- and planar-averaged velocity profile from the outer domain compared to observation data at the reference site. The simulation results are time averaged over the last 30 min of the outer domain with an output frequency of 1 min. The RMSE between the wind speed observations and WRF results is 0.8 m s−1, while the RMSE is 0.4 m s−1 for VR-IBM. Reasonable agreement with observations is achieved in both cases, indicating that the outer domain provides the necessary forcing to the inner domain. Note that small differences between WRF and VR-IBM are seen here caused by the coarser vertical resolution used in this case, compared to the flat terrain case presented above. Differences between the velocity profiles and observations are comparable to those in several previous studies of Askervein Hill (Castro et al. 2003; Chow and Street 2009; Golaz et al. 2009).
(a) Time and planar-averaged wind speed vs elevation for terrain-following WRF and VR-IBM for the outer domain. Field observations with error bars are located at the Askervein reference site (RS). (b) As in (a), but with a logarithmic vertical axis.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
The 10-m wind speedup along lines A, AA, and B is shown in Fig. 6. Along lines A and AA, the maximum speedup occurs at the top of the hill. The standard WRF Model underpredicts the speedup at the hill top especially along line A, which passes over the peak of Askervein Hill. The greatest difference between WRF and VR-IBM again occurs in the lee of the hill, which relates to the discrepancies in the prediction of flow separation and recirculation. Both of WRF and VR-IBM successfully predict the flow deceleration, while the location, size, and strength are different. The standard WRF Model tends to overestimate the flow deceleration along line AA and line A. The poor prediction of leeside flow separation could be due to the grid resolution chosen (Silva Lopes et al. 2007), turbulence closure scheme (Chow and Street 2009; Golaz et al. 2009), and/or the use of the log-law model (Silva Lopes et al. 2007). Silva Lopes et al. (2007) noticed the maximum difference in speedup along line A can be as big as 0.4 on the lee side of the hill between their coarse resolution (
Wind speedup at 10 m AGL over Askevein Hill for WRF and VR-IBM, compared with field observations with error bars (squares): (a) along line AA, (b) along line A, and (c) along line B.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
Next, a direct comparison of the time-averaged velocity profiles between standard WRF and VR-IBM is shown in Fig. 7. The velocity profiles are averaged for the last 30 min (at 1-min intervals) and are shown at several locations along line A. It can be seen that the VR-IBM results are in good agreement with standard WRF, and that the differences between WRF and VR-IBM decrease with height. The difference at the domain top can be seen as due to the difference in bulk inflow velocity, which is about 1 m s−1. The maximum difference of 2.53 m s−1 is found near the surface, in the lee side of the hill (the fifth profile). Considering the difference in bulk inflow velocity, which is about 1 m s−1, the difference between WRF and VR-IBM due to the surface representation is about 1.5 m s−1. As discussed in Bao et al. (2016), the VR-IBM method has some shortcomings in the representation of the near-surface velocity field when compared to WRF, particularly in the lee of the hill. This may be due to sensitivity of the flow to interpolation errors in the flow separation region. The simulation of the lee side of the hill is always very challenging due in part to intermittent flow separation, and results can be influenced by different parameters such as grid resolution and turbulence closure schemes (Chow and Street 2009; Ma and Liu 2017). For example, Chow and Street (2009) found differences of up to 5 m s−1 in the lee of Askervein Hill for simulations using different closure models. As another example, Marjanovic et al. (2014) compared WRF simulations to observations over a different complex terrain site using several closure models; the root mean squared errors between the WRF results and observations improved from 5 to 3.5 m s−1 simply by choosing a different closure model. Furthermore, WRF uses a vertical gradient and IBM uses a surface-normal gradient so there is no guarantee that the WRF result is better than the IBM result over sloped terrain (Lundquist et al. 2010). Given the relatively straightforward ease of implementation of VR-IBM and the points discussed above, the RMSE differences between WRF and VR-IBM seem acceptable, yet remain the subject of further work, as discussed in Bao et al. (2016).
Time-averaged U profiles at various locations along line A over Askervein Hill for terrain-following WRF and VR-IBM in m s−1.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
c. Bolund Experiment simulation
1) Field campaign description
The Bolund Hill field campaign, conducted in the winter of 2007–08 over Bolund Hill, was designed to study atmospheric flow over steep terrain. A description of the field campaign can be found in Berg et al. (2011). The hill is 130 m long (west–east direction) and 75 m wide (north–south direction) with a maximum height of 12 m, as shown in Fig. 8. The hill is covered with grass and is surrounded by water with a long uniform fetch over the sea in the westward direction, which is the origin of the incoming flow for the case simulated here. The geometry of the hill makes atmospheric flow simulations challenging because the western (windward) face of the hill is a steep 90° slope and the lee side of the hill creates a recirculation zone with flow separation. As shown in Røkenes and Krogstad (2009), the details of the crest geometry representation can strongly affect the flow recirculation.
Tower locations over Bolund Hill. The dashed line is along the 242° wind direction.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
During the field campaign, 38 anemometers were deployed over the hill at 10 meteorological tower locations as shown in Fig. 8 (Bechmann et al. 2011), including 26 sonic anemometers, 12 cup anemometers, and 2 lidars. Mast M9 is located east of the hill and not shown. Instruments were placed at multiple heights on the observational towers, frequently located at 2, 5, and 9 m AGL, though each tower did not have instruments at all three heights, and some towers used alternate heights. At each tower, turbulent wind velocities and fluxes were measured. The simulation presented here is case 3 as listed in Bechmann et al. (2011), which details results of a blind simulation intercomparison project. The mean wind speed during the selected measurement period approached 10 m s−1 at about 20-m height, with a wind direction from the southwest at 242°. The observation data are averaged over 10 min.
Results using NS-IBM and VR-IBM are included in this section. Standard terrain-following WRF simulations fail because of the nearly 90° steep slope, which leads to large grid distortion and truncation errors, which lead to model instability.
2) Simulation setup
As in the previous simulation, a domain containing Bolund Hill is nested within an outer domain on which the flow is a fully developed boundary layer over flat terrain. Flow is driven in the parent domain with a uniform pressure gradient chosen to produce a velocity profile that matches observations (see Fig. 10). Boundary conditions are passed from the flat domain to the nested Bolund Hill domain at each outer domain time step, as in the Askervein Hill case above. The outer domain is periodic, with 
Horizontal slice of instantaneous U velocity at 5-m elevation for (a) outer domain and (b) inner domain. The dashed line indicates the inner nest location. The black contours are the Bolund Hill terrain at 1.2-m intervals.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
3) Comparison with observations
Averaged wind speed profiles at M0 for NS-IBM and VR-IBM on both outer (domain 1) and inner (domain 2) nests compared to observations and the idealized log law. Results are time and planar averaged on the outer domain, but only time averaged on the inner domain at the location M0.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
Figure 11 shows time-averaged wind vectors at instrument locations placed 2, 5, and 9 m above the terrain. Observations are shown as red vectors while the VR-IBM and NS-IBM simulations are shown in blue and black, respectively. Qualitatively, it can be seen that the VR-IBM method improves agreement with observations over the NS-IBM method. For example, in the top panel of Fig. 11, the wind direction of the NS-IBM simulation is incorrect on the windward face of the hill, and the wind speed is too small at almost all locations, but especially in the lee of the hill. Flow separation and reattachment is difficult to predict in computational fluid dynamics (CFD) models (including WRF). Piomelli and Balaras (2002) tested an equilibrium log-law model and Piomelli (2008) extended the work to three different surface flux models for large-eddy simulation (equilibrium model, zonal model, and hybrid model). They found that the accuracy of the surface flux models was highly dependent on the grid resolution and turbulence closure, and that there was no single method that was clearly better than others in representing flow separation and reattachment. To quantify simulation error, the RMSE for the nine measurement stations are listed in Table 1 for both NS-IBM and VR-IBM. The RMSE for wind speed decreases with height, with the maximum RMSE at the 2-m wind vector for both NS-IBM and VR-IBM. The VR-IBM method has significantly lower RMSE compared with observations than the NS-IBM results. The maximum differences between simulations and measurements exist in the 2-m field.
Wind vectors for NS-IBM (black), VR-IBM (blue), and observations (red) at tower locations shown in Fig. 8 for three different heights: (a) 2, (b) 5, and (c) 9 m AGL.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
RMSE (m s−1) for 2-, 5-, and 9-m wind vectors compared with observations from the Bolund experiment at all nine meteorological towers.
In Fig. 12, the ratio of LES wind speed to measured wind speed is shown as a function of elevation. It is clear that the largest mismatch occurs at the lowest points (2 m). There are many factors that can influence the mismatch near the surface, perhaps including the need for a more accurate subgrid model to parameterize turbulence near the surface, and a finer grid resolution. In general, the VR-IBM performs better than NS-IBM, leading to a mean ratio close to 1 and a smaller standard deviation than NS-IBM. The results are comparable to Diebold et al. (2013) who reported a mean value of 0.96 with a standard deviation of 0.18.
Ratio (
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
Figure 13 (left) shows a scatterplot of VR-IBM and NS-IBM results versus observation data. In this plot, the wind speeds are normalized by 
(a) Scatterplot of normalized wind speed (
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
where U is the mean wind speed at the sensor location and 
Speedup ratio over Bolund Hill along incoming wind direction of 242° (M1, M2, M3, and M4) showing observations, NS-IBM, and VR-IBM results at (a) 5 m and (b) 2 m, and (c) with topography.
Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0067.1
Statistical performance metrics for VR-IBM and NS-IBM compared to observations at towers M1–M8 (see text for details).
The speedup errors obtained with VR-IBM are comparable to the modeling efforts summarized by Bechmann et al. (2011), Diebold et al. (2013), and Ma and Liu (2017) in Table 3. VR-IBM compares quite favorably to the results (2-m grid resolution) from Diebold et al. (2013) and Ma and Liu (2017), which both used a more sophisticated turbulence model. Both Diebold et al. (2013) and Ma and Liu (2017) showed that increasing grid resolution (1-m horizontal spacing, compared to the 2-m spacing used here) can decrease the speedup error further.
Comparison between the speedup percent errors from the present study (NS-IBM and VR-IBM), results from Ma and Liu (2017) and Diebold et al. (2013), and from the average of models taking part in the blind test (Bechmann et al. 2011).
4. Conclusions
We have developed an improved IBM for the WRF Model with a log-law boundary condition using velocity reconstruction (VR-IBM), which is capable of handling steep terrain in atmospheric flow simulations. This feature will enable the use of WRF-IBM in nested meso- to microscale simulations where the topography is well resolved and therefore can be quite steep on the finest grids, causing WRF with its traditional terrain-following coordinates to fail. The VR-IRM implementation was first validated by simulating a neutral atmospheric boundary layer flow over flat terrain and comparing the solution to results achieved using the native terrain-following coordinate WRF. This ensures that our implementation of VR-IBM performs well over flat terrain, where detailed comparison with theory is possible.
Next, the method was validated for flow over Askervein Hill, which has a slope moderate enough for WRF using its native terrain-following coordinates. Comparison of the Askervein simulations reveals reasonable agreement between VR-IBM and native WRF simulations and compared with observations. The RMSE for fractional wind speedup are comparable to previous LES studies (see e.g., Chow and Street 2009). The maximum differences occur on the lee side of the hill, where models are quite sensitive to intermittent flow separation (Chow and Street 2009; Castro et al. 2003; Golaz et al. 2009). Overall, the performance of the VR-IBM method is well within the expected range, thus showing the model’s suitability for complex flow situations. Differences between WRF and WRF-IBM can be due to truncation errors in the WRF representation caused by steep terrain slopes, errors in WRF because derivatives are taken in the vertical rather than slope-normal direction, and interpolation errors in the VR-IBM approach discussed by Lundquist et al. (2012) and Bao et al. (2016).
Finally, the VR-IBM is further tested against experimental data from the Bolund Hill field campaign, a test case that is too steep for the native WRF coordinates. In general, the VR-IBM method shows good agreement with field measurements with improvement in predicting the points closest to the hill crest and lee side over the NS-IBM results. The points on the lee side of the steep slopes near the ground are the most difficult to predict accurately, as found in previous LES simulations (Bechmann et al. 2011; Diebold et al. 2013). Nevertheless, the results compare quite well to other LES approaches (Diebold et al. 2013; Bechmann et al. 2011; Ma and Liu 2017) and provide further confidence in the implementation of VR-IBM.
Compared to other work, this WRF-IBM implementation has been validated against a large number of test cases, not only to field observations (Bolund Hill), but also to theory (flat terrain), and directly compared to native WRF (flat terrain and Askervein Hill). Test cases are also chosen at different slopes, to demonstrate the robustness of the implementation (the Askervein Hill case with moderate slopes, and the Bolund Hill case with steep slopes). This series of test cases provides rigorous validation of the implementation of the velocity reconstruction method into WRF-IBM by ensuring that the VR-IBM implementation matches results from WRF with its native coordinate system for moderately sloped terrain and that it has the capability to handle steeply sloped terrain. WRF-IBM requires adequate grid resolution to resolve the immersed boundary to limit interpolation errors that lead to flow differences. On the other hand, native WRF has errors due to the skewed grid cells over steep terrain and its use of vertical derivatives instead of surface-normal derivatives to set fluxes. The Bolund test case illustrates the ability of the WRF-IBM implementation to handle very steep complex terrain that cannot be represented in native WRF at all, gaining further confidence in using WRF-IBM by comparing not just with field data but with other numerical studies that have used a variety of IBM or conformal coordinate systems. We therefore have a robust new tool in WRF that can be used for different applications, such as wind energy, complex terrain, urban dispersion studies, etc. Future work will consider additional complex terrain applications of VR-IBM and further improvement of the immersed boundary conditions to consider the findings of Bao et al. (2016).
Acknowledgments
This work was partly 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 U.S. DOE Office of Energy Efficiency and Renewable Energy (EERE) Wind Energy Technologies Office. JB and FKC’s efforts were partially supported by the Office of Naval Research Award N00014-11-1-0709, Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) Program, and by National Science Foundation Grant ATM-1565483.
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