1. Introduction
In mountainous terrain, the diurnal variations of the surface sensible heat flux can lead to thermally driven upvalley or upslope flow during the daytime and downvalley or downslope flow during the nighttime (Zardi and Whiteman 2013). Topographic effects on radiation can strongly influence these flows by creating large spatiotemporal inhomogeneities in the net radiation, and thus in the surface energy budget. These effects include topographic shading, where direct solar radiation is blocked by surrounding topography, and slope effects (also known as “self shading”), where the local slope angle modifies the incoming solar radiation based on the angle of incidence of the direct solar beam at the surface relative to the local slope.
Previous field work has documented the influence of topographic effects on the surface radiation and energy budgets in mountainous terrain (Whiteman et al. 1989a,b; Matzinger et al. 2003; Katurji et al. 2013). Some studies have extended this work to examine how topographic effects on the surface radiation and energy budgets influence the development of both valley and slope flows. For example, Whiteman et al. (1989b) found that sign reversals in the local surface sensible heat flux corresponded to reversals of both the slope and valley wind systems. Several studies have focused specifically on the development of downslope flows during the evening transition, finding that downslope flow development follows the shadow front. On east-facing slopes, the downslope flow transition has been found to follow the shadow front down the slope (Papadopoulos and Helmis 1999; Lehner et al. 2015), while on west-facing slopes, the downslope flow transition may follow the shadow front up the slope (Nadeau et al. 2013).
With this in mind, it is important to capture topographic effects on radiation in atmospheric models over mountainous terrain. Perhaps the first implementation of a topographic shading algorithm in a major mesoscale atmospheric model was done by Colette et al. (2003) in the Advanced Regional Prediction System (ARPS). Since then, other models have included topographic effects on radiation. These include the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5; Zängl 2005) and the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008), which is used in this study. Although topographic shading improves the representation of surface fluxes over sloped terrain, the terrain-following vertical coordinate used by most mesoscale atmospheric models becomes skewed over moderate slopes, which can result in numerical errors (e.g., Janjić 1977; Mahrer 1984; Schär et al. 2002; Klemp et al. 2003; Zängl 2002, 2003; Zängl et al. 2004). The severity of these errors depends on both the steepness of the terrain and the grid aspect ratio (the horizontal grid spacing divided by the vertical grid spacing), as shown in Fig. 1 of Daniels et al. (2016). This limitation restricts the use of models such as WRF in regions of steep and/or complex terrain where topographic effects on radiation are most important.
The immersed boundary method (IBM), which uses a nonconforming grid and represents the topography by imposing boundary conditions along an immersed terrain surface, provides an alternative to terrain-following vertical coordinates. IBM reduces numerical errors related to the terrain slope, thus extending the range of slopes that can be represented on the computational grid. IBM was originally implemented into WRF (referred to here as WRF-IBM), version 2.2, and coupled to a subset of its atmospheric physics parameterizations by Lundquist et al. (2010). Topographic effects on radiation were not included in the standard WRF release until version 3 (see http://www2.mmm.ucar.edu/wrf/users/wrfv3/updates.html), so they were not coupled to IBM or tested in the Lundquist et al. (2010) study.
Here, WRF-IBM (based on WRF, version 3.6.1) is modified to include topographic effects on radiation, expanding on the implementation of Lundquist et al. (2010). Changes to the model are validated by confirming agreement in radiation and land surface fluxes, as well as temperature and velocity fields, between WRF-IBM and standard WRF when the same initialization and forcing conditions are applied. The validation is performed in a domain with an idealized two-dimensional valley that is forced by incoming solar radiation. The maximum valley slope is relatively small (approximately 15°) such that WRF still performs well with terrain-following coordinates. Thus, WRF-IBM and standard WRF results are expected to agree in this case.
Following validation, the WRF-IBM implementation is applied to a semirealistic simulation of Granite Mountain, Utah, where topographic shading is known to affect downslope flow development in the evening (Fernando et al. 2015; Lehner et al. 2015; Jensen et al. 2017). The horizontal grid spacing in this case is 50 m, and at this resolution, local terrain slopes reach roughly 55°. This case does not run in WRF when its native terrain-following coordinates are used, so direct comparison to standard WRF is not possible. To the knowledge of the authors, this is the first application of WRF using physics parameterizations to model flow over steep, nonidealized complex terrain at such high resolution. This application effectively combines the functionality of a traditional computational fluid dynamics model for simulating flow over complex geometries with that of an atmospheric model for including detailed parameterizations for atmospheric physics. The simulation of thermally driven slope flows on Granite Mountain thus demonstrates the applicability of WRF-IBM to investigate atmospheric flows in complex terrain environments.
This work is part of an ongoing effort to enable multiscale simulations over complex terrain within the WRF modeling framework. A major focus of this effort is the use of IBM to overcome the slope limitations associated with WRF’s native terrain-following coordinate. Lundquist et al. (2010) first implemented a two-dimensional IBM in WRF with a no-slip boundary condition and coupling to the MM5 (Dudhia) shortwave radiation scheme, the RRTM longwave radiation scheme, the MM5 surface layer module, and the Noah land surface module [for documentation of these schemes, see Skamarock et al. (2008), and references therein].
Additional WRF-IBM development projects have included extending the method to three dimensions and using WRF’s Smagorinsky turbulence model (Lundquist et al. 2012), as well as the implementation of a surface momentum flux parameterization based on logarithmic similarity theory (Bao et al. 2016, 2018). Because WRF-IBM uses an alternate vertical grid that requires points underneath the terrain, specialized routines must be developed for the IBM domain to receive lateral boundary conditions from analysis data or to nest an IBM domain within a WRF domain using terrain-following coordinates. The capability to use an independent vertical grid on each domain was implemented by Daniels et al. (2016) and Mirocha and Lundquist (2017), and efforts to nest domains using IBM within those using terrain-following coordinates are detailed in Wiersema et al. (2016, 2018). Studies by other groups have also used the immersed boundary method in WRF. For example, Ma and Liu (2017) demonstrated neutral boundary layer flow over Bolund Hill using WRF at horizontal grid spacings of 1 and 2 m.
A final consideration impacting the integration of WRF and IBM is the need for each WRF physics parameterization to be coupled to IBM separately and subsequently validated. Given that the aforementioned WRF-IBM projects are in various stages of development, the present work is focused on extending the original implementation of WRF-IBM in Lundquist et al. (2010, 2012) to include topographic effects on radiation. The roles of turbulence, regional or synoptic forcing, or other physical processes are therefore not investigated here, and are reserved for future studies.
The computational cost of the present WRF-IBM implementation is roughly a factor of 2 larger than that of standard WRF for comparable cases such as the idealized two-dimensional valley validation case. However, computational cost cannot be compared directly in more complex terrain cases, such as Granite Mountain, because standard WRF does not run. Much of this increase in cost is a result of WRF’s pressure-based vertical coordinate system, for which the vertical levels change every time step, thus requiring new interpolation factors for variables at the immersed boundary. In a code with a fixed vertical grid, these factors would only need to be calculated once. Decreasing the computational cost of WRF-IBM is a subject of future work.
2. Implementation
Example grid and topographic effects on radiation for (a) standard WRF and (b) WRF-IBM. The incoming solar radiation at point A is increased by slope effects resulting from the nearly perpendicular angle of incidence of the radiation with respect to the topography. The incoming solar radiation to the east of point B on the slope is blocked by the topography and is therefore shaded.
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
When the WRF namelist option “slope_rad” is activated, the incoming solar radiation is corrected based on the solar zenith angle relative to the local surface normal vector. The slope correction increases the solar radiation on surfaces whose slopes approach perpendicularity with the incoming solar beam [
When the namelist option “topo_shading” is activated, WRF searches within a radius “slp_dist” (another namelist option) around each grid point to determine if any topography intersects a line drawn between the sun and the given grid point. If so, a topographic shadow is cast on that grid point (the WRF variable “shadowmask” is set to 1) and the direct component of the incoming solar radiation is set to 0. Effectively, D is set to 1 in Eq. (1) such that 
In standard WRF, the terrain height is used to define both the elevation of the surface terrain and the lower extent of the computational domain. In WRF-IBM, however, the elevation of the surface terrain is represented by an immersed boundary, while the lower extent of the computational domain is beneath the immersed boundary. Coupling WRF’s algorithm for topographic effects on radiation to IBM therefore involves changing the relevant terrain height variable to that representing the immersed boundary surface (see Fig. 1). The terrain slope and solar zenith angle 
It should be noted that the immersed boundary method (including the implementation discussed here and in other references) is not yet included in the standard WRF release. However, in the process of implementing topographic effects on radiation in WRF-IBM, an error was discovered in the standard WRF implementation that caused the diffuse component of solar radiation to be incorrect when the terrain slope was zero. This error was fixed in the WRF code used here, and in the standard WRF, version 3.7, release (see http://www2.mmm.ucar.edu/wrf/users/wrfv3.7/updates-3.7.html).
3. Validation
The validation of land surface coupling in WRF-IBM with topographic effects on radiation follows the approach of Lundquist et al. (2010), who originally validated land surface coupling without including topographic effects. Radiation quantities, surface fluxes, and temperature and velocity fields are compared between WRF-IBM and standard WRF in a domain with an idealized two-dimensional valley. This case was used previously in Lundquist et al. (2010) and Schmidli et al. (2011), and was chosen because it combines the use of atmospheric physics parameterizations with an idealized terrain and atmosphere, making the setup suitable for validation cases. Although the results are not included herein, an additional validation case with a three-dimensional Witch of Agnesi hill was completed to confirm agreement in topographic shadow location between WRF-IBM and standard WRF for a variety of seasons and latitude–longitude locations.
In validating land surface coupling, it is necessary to turn on several atmospheric physics modules (in addition to the aforementioned shortwave radiation module) that were originally implemented in WRF-IBM by Lundquist et al. (2010), including the RRTM longwave module, MM5 surface layer module, and Noah land surface module. These routines required minor modifications to update the coupling to the immersed boundary method from WRF version 2.2 (used in Lundquist et al. 2010) to version 3.6.1 (used here). Validation of their updated implementations is thus included in the present study.
a. Model configuration
The location of the valley is set to 36°N, 0°, and the model is run between 0600 and 1800 UTC 21 March 2007. Certain land surface properties that are used as inputs in WRF’s radiation, land surface, and surface layer models are also initialized following Lundquist et al. (2010). These include spatially uniform values of soil type (sandy loam), land use (savannah), and vegetation fraction (0.1). The initial soil temperature is equal to the initial surface atmospheric temperature, and the soil moisture is set to 0.0868 m3 m−3 (a 20% saturation rate for sandy loam).
Lateral boundary conditions are periodic for all variables. At the immersed boundary, either Dirichlet or Neumann boundary conditions can be used. Boundary conditions at the immersed topography are set as in Lundquist et al. (2010), where a Dirichlet boundary condition is used for velocity and a Neumann boundary condition is used for potential temperature and moisture.
In the validation cases, a constant eddy viscosity 
b. Model results
Over the course of the day, solar heating of the valley topography leads to thermally driven flow near the surface (Fig. 2). Air flows upslope on both sides of each hill, and a buoyant plume of warm air rises at each peak. Return flow develops within the valley and on the outside of each hill, completing the circulation. Such flow is typical of valley topography with surface heating, and is similar to the flow modeled in Schmidli et al. (2011).
Comparison of horizontal velocity u profiles between WRF-IBM and standard WRF for the two-dimensional valley test case at 1200 UTC. The color scale shows the potential temperature θ field from the WRF-IBM simulation.
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Because surface forcing ultimately drives the flow in the valley test case, the validation of land surface coupling is first focused on comparing the relevant surface flux quantities calculated by the radiation and surface schemes in WRF-IBM and standard WRF. These quantities include incoming longwave and shortwave radiation, as well as surface sensible heat and moisture fluxes. Instantaneous across-valley plots of surface flux quantities (Fig. 3) show overall good agreement between WRF-IBM and standard WRF. Plots are shown at three times (0900, 1200, and 1500 UTC) to demonstrate the evolution of surface fluxes over the course of the day, and to highlight the importance of topographic effects.
Instantaneous (a) incoming longwave radiation, (b) incoming shortwave radiation, (c) surface sensible heat flux, and (d) surface moisture flux as functions of x at 0900, 1200, and 1500 UTC for the idealized two-dimensional valley test case using WRF-IBM and standard WRF. In (b)–(d) WRF-IBM results are shown both with and without topographic effects on radiation.
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Although the incoming longwave radiation values (Fig. 3a) are nearly equal when the simulation is initialized, they diverge over time, leading to a maximum instantaneous difference of 1.73 W m−2 (0.61%) at the end of the simulation (1800 UTC). This divergence is caused by slight differences in atmospheric temperature and moisture structure, which affect the absorption and emission of incoming longwave radiation, between WRF-IBM and standard WRF.
Very good agreement is found for the incoming shortwave radiation (Fig. 3b). The maximum instantaneous difference between WRF-IBM and standard WRF is −1.27 W m−2 (−1.20%) at 1745 UTC, when the magnitude of shortwave radiation values is low. At the time of peak shortwave radiation, 1200 UTC, the maximum difference is similar, but the relative difference is reduced to 0.08%. As noted by Lundquist et al. (2010), the differences in radiation quantities between WRF-IBM and standard WRF are quite small relative to differences caused by elevation and/or topographic effects. They are due in part to the differences in the vertical grid, which affects the numerical integration over a vertical column that is used in the radiation schemes.
The surface sensible heat flux 
Time-averaged differences in (a) potential temperature 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Differences between the WRF-IBM and standard WRF simulations are concentrated at the surface, where land surface forcing and the implementation of boundary conditions affect the flow. The potential temperature field in WRF-IBM is slightly cooler near the surface, with 
Some of the disagreement between WRF-IBM and standard WRF shown in Fig. 4 is due to the difference in the grids used for each simulation. Because WRF balances the initial atmospheric pressure, temperature, and moisture profiles onto the model grid, grid differences prevent the initial profiles from being identical, and these differences persist throughout the simulations. Despite minor disagreements, the same overall flow structure occurs in both models, as demonstrated in Fig. 2. The temperature and velocity differences between WRF-IBM and standard WRF in this study are also similar in magnitude to those reported by Lundquist et al. (2010) for the same case without topographic effects on radiation included. Their minimum/maximum differences [also calculated using Eq. (6)] were 
4. Application to realistic complex terrain
Following the successful implementation and validation of topographic effects on radiation in WRF-IBM, the model is applied to a real complex terrain environment. The chosen site is Granite Mountain, a relatively isolated desert mountain in Utah that experiences thermally driven downslope flows during the evening. These flows were a focus of the Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) program and are strongly affected by topographic shading (Fernando et al. 2015; Lehner et al. 2015; Jensen et al. 2017). Because of errors associated with terrain-following vertical coordinates in regions of steep slopes and WRF’s choice of numerical schemes, it is not possible to model Granite Mountain at high resolution (less than roughly 100 m) in standard WRF without modifications to account for steep terrain. With the 50-m horizontal grid spacing used here, local slope values can be as large as 55°.
a. Model configuration
Granite Mountain is modeled using the nested ideal WRF setup shown in Fig. 5a. The inner domain (d02) covers the topography of interest, as shown in Fig. 5b. It is a 15 × 15 km2 grid with 
(a) The full WRF-IBM domain used to study Granite Mountain, including both the outer (d01) and inner (d02) domains. (b) A zoomed-in view of the inner domain (d02), including the location of MATERHORN tower ES5, the location of surface flux measurements shown in Fig. 8. In (b), the box indicates the region shown in Fig. 9. (c) Land-use and (d) soil-type data included in the inner nest of the WRF-IBM domain, based on the USGS classification system. The topography of Granite Mountain is shown by black contour lines between 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Realistic land surface data around Granite Mountain are used in the model setup. For the inner domain, the topography of Granite Mountain is read from ⅓-arc-s (approximately 10 m) data from the U.S. Geological Survey (USGS) National Elevation Dataset. Realistic land-use and soil-type data are also included on the inner domain, as shown in Figs. 5c and 5d. Land-use data are read from a 1-arc-s (approximately 30 m) resolution dataset created for 4DWX (Liu et al. 2008), an operational model used at Dugway Proving Ground, where Granite Mountain is located. Soil-type data are read from the 30-arc-s (approximately 1 km) resolution State Soil Geographic (STATSGO)/Food and Agriculture Organization (FAO) dataset, which is standard in WRF and is also used in 4DWX. For the outer domain, a constant land use of “shrubland” and a constant soil type of “silt loam” are used.
The model is initialized using a single input sounding that contains potential temperature θ, vapor mixing ratio 
(a) Potential temperature θ and (b) vapor mixing ratio 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
The model is forced by incoming solar radiation, moderated by slope and topographic shading effects. The model soil moisture is initialized as 0.15 m3 m−3 based on the average seasonal values presented by Jensen et al. (2016). A constant eddy viscosity is used in both domains. The value used on the inner nest is 
b. Model results
1) Surface energy budget
As seen in field observations, model results suggest that on the east side of Granite Mountain, the evening reversal of the surface sensible heat flux 
Modeled shadow propagation (black contours) and surface sensible heat flux 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Modeled 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
Since the latent heat flux 
The model soil moisture, which was initialized at a constant value of 0.15 m3 m−3, likely plays a role in this discrepancy because it affects the soil thermal conductivity and thus the ground heat flux 
Ultimately, because this is only a semirealistic WRF run, detailed agreement between the modeled and observed surface energy budget is not expected. With continued development of WRF-IBM, future simulations using a nested setup with more realistic model initialization and forcing should allow for improvements in this regard. However, the comparisons in Fig. 8 show that the relevant components of the surface energy budget are generally well captured by the present model.
2) Downslope flow development
Downslope flows are known to develop on sloped terrain after the “evening transition” of the surface sensible heat flux 
Near-surface potential temperature (θ at 0.5 m AGL) and horizontal velocity vectors (u, υ at 30 m AGL), as well as the location of the shadow front (black contours). The topography of Granite Mountain is shown by gray contour lines between 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
The application of WRF-IBM to Granite Mountain allows the spatial heterogeneity of downslope flows to be resolved over a broad range of physical scales. Downslope flow develops primarily within the shaded area, while convective and/or upslope flow remains outside of the shaded area where 
The drainage flow can be visualized in three dimensions in the form of downslope-flowing plumes of cold air (Fig. 10). Figure 10 shows isosurfaces of 
A three-dimensional view of the downslope flow on the east side of Granite Mountain at 1855 MST. Blue isosurfaces are shown for the downslope velocity 
Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-18-0108.1
The present simulation of thermally driven downslope flow on Granite Mountain is intended as a proof of concept for WRF-IBM with topographic effects on radiation, not as a fully realistic atmospheric simulation. A simplified physical setup is employed to show that WRF-IBM is able to capture the basic surface forcing mechanisms of thermally driven flows over complex terrain. For this reason, although the modeled downslope flow is qualitatively representative of what was observed during MATERHORN Spring IOP 4 by Lehner et al. (2015) in that it develops behind the shadow front, direct comparisons of flow speed, direction, and vertical structure are reserved for future work.
There are several additional caveats to consider when interpreting the downslope flow in the present simulation of Granite Mountain. Regional- and synoptic-scale features that may influence thermally driven flows on Granite Mountain (Fernando et al. 2015) cannot currently be captured in WRF-IBM because of forcing limitations at the lateral boundaries and are therefore not considered at this time. Additionally, the model results are case specific. Seasonal differences in shadow propagation and soil moisture likely influence the location and timing of downslope flow development and should be examined in future studies. Nevertheless, this test case illustrates the capabilities of WRF-IBM to simulate atmospheric flow over realistic complex topography where topographic effects on radiation strongly influence near-surface dynamics.
5. Conclusions
Topographic effects on radiation, including both topographic shading and slope effects, have been successfully implemented into the WRF Model with the immersed boundary method. Validation was performed by comparing surface fluxes, as well as temperature and velocity fields, in a domain with a two-dimensional valley. The validation simulations highlight the spatial variations in incoming solar radiation and surface sensible heat flux that are present in model runs that include topographic effects on radiation. The spatial variation in surface fluxes leads to changes in the resulting velocity structure that would not be captured without topographic effects.
Following validation, the implementation of topographic effects on radiation in WRF-IBM was applied to a semirealistic simulation of the evening transition to downslope flows on Granite Mountain, Utah. It is not possible to model flow over Granite Mountain at high resolution in standard WRF, or in other atmospheric models with a terrain-following vertical coordinate, due to numerical errors associated with large terrain slopes. This work thus represents an important advancement in the use of WRF to model thermally driven flows over complex terrain surfaces.
The use of IBM allowed WRF to capture the strong spatiotemporal variations in incoming solar radiation and surface sensible heat flux on Granite Mountain, and the surface energy budget components in the WRF-IBM model showed good agreement with observations from the MATERHORN program. Furthermore, downslope flows that are qualitatively consistent with the observations of Lehner et al. (2015) developed in the model in response to surface cooling behind the shadow front. While these results demonstrate that the use of IBM enables WRF to successfully capture the driving force of downslope flows over heterogeneous terrain, certain simplifications in the model setup preclude direct comparison to observed downslope flows, which is reserved for future work.
With continued development, WRF-IBM with topographic effects on radiation will provide a computational tool to study many boundary layer flow processes over complex terrain. Future work on thermally driven downslope flows in particular will focus on improving agreement between modeled and observed flow characteristics, including vertical structure and development timing relative to local sunset. The coupling of IBM with WRF’s turbulence closure models and appropriate surface boundary conditions for mesoscale and large-eddy simulations will improve the representation of unresolved physical processes in the model and should lead to better flow prediction over complex terrain. Although previous implementations of surface stress parameterizations (e.g., Senocak et al. 2004; Chester et al. 2007; Diebold et al. 2013; Ma and Liu 2017; Bao et al. 2018) have been shown to perform well with fine grid spacing (roughly 10 m or less), Bao et al. (2016) showed that these methods do not significantly improve upon the no-slip boundary condition with larger grid spacings of tens of meters and greater, as used in this study. A new method developed in Bao et al. (2016) was shown to improve results at these coarser resolutions in two-dimensional simulations, and is undergoing extension to three-dimensional simulations. Additional development of WRF-IBM to allow for nesting within terrain-following WRF domains (e.g., see Daniels et al. 2016; Wiersema et al. 2016, 2018) will enable the study of thermally driven flow interaction with larger-scale meteorological features, including regional or synoptic winds, cold pools, or mountain wakes, all of which were observed around Granite Mountain during the MATERHORN program.
Acknowledgments
A portion of this work was prepared by LLNL under Contract DE-AC52-07NA27344. Arthur and Chow received support from the Office of Naval Research Award N00014-11-1-0709, Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) program. Lundquist and Mirocha received support from the U.S. DOE Office of Energy Efficiency and Renewable Energy and the LLNL Laboratory Directed Research and Development program as Project 14-ERD-024. Some of the simulations used in this work were performed on the University of California, Berkeley, Savio cluster. We acknowledge S. W. Hoch for sharing MATERHORN ES5 surface flux data with us. We also acknowledge A. D. Anderson-Connolly and D. J. Wiersema for their help in setting up the model, and P. D. Scholl for assisting in data visualization.
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