## 1. Introduction

Radiation, the main source and sink of energy at the earth’s surface, is significantly influenced by topography. Local slope and aspect angles considerably modify the amount and daily course of downwelling shortwave radiation (e.g., Whiteman et al. 1989). For instance, a time lag for the maximum downwelling shortwave radiation of about 2 h, and intensities increased by 200 W m^{−2}, were observed on inclined surfaces in an alpine valley (Matzinger et al. 2003). Simulated diurnal averages of downwelling shortwave radiation showed variations as large as 450 W m^{−2} for the Tekapo watershed (Oliphant et al. 2003). Sky view restriction increases downwelling longwave radiation and generates spatial variability of diffuse radiation (Matzinger et al. 2003; Dubayah and Loechel 1997). Scherer and Parlow (1994) have thoroughly demonstrated the consequences of terrain-induced modifications of radiation fluxes on the energy balance, and hence on snowmelt and snow hydrology for a drainage basin in Svalbard. In their study they showed that specific topographic conditions led to significantly increased solar radiation on the west-exposed slopes of the studied drainage basin in northwest Spitsbergen, which is the main reason for intensified snowmelt and subsequent meltwater runoff measured in the field. The parameterization scheme used in their spatially distributed local-scale radiation model forms the basis of the one developed in our study, but it has to be noted that many other models for the treatment of radiation in complex terrain have been developed over the last several years (e.g., Dozier 1980, 1989; Duguay 1993; Dubayah and Rich 1995; Kumar et al. 1997). Our parameterization is specially tailored for mesoscale models in the way that it deals with coarse-grid resolutions used in mesoscale weather forecast models, severe computational cost restrictions, as well as ease of implementation and portability.

To the knowledge of the authors, the majority of mesoscale weather forecast models only consider a few or none of the topographic influences on radiation. A survey of considered topographic effects for popular mesoscale models in operational use is given in Table 1. It can be seen that most models only include slope aspect and slope angle, which in fact are the most important effects, as mentioned by Oliphant et al. (2003). However, at typical grid resolutions of a few kilometers, such effects decrease due to the flattening of slopes. Dubayah et al. (1990) analyzed topographic modulation of clear-sky irradiance using a parameterization very similar to that of this study. They found that variance and spatial autocorrelation of simulated radiation changed with sun angle and grid spacing. As grid spacing increased, variance decreased and spatial autocorrelation increased. Recently, shadow effects were included into the Advanced Regional Prediction System (ARPS), and its importance was demonstrated (Colette et al. 2003). Mesoscale weather forecast models are usually running at spatial resolutions of a few kilometers, thus resolving larger valleys on the grid. However, sloping surfaces are flattened, and topographic details are not resolved, so that radiation computation based on the grid-scale topography has little effect. The proposed parameterization scheme considers topographic effects on radiative fluxes by using an arbitrarily fine resolved topography. Thus the scheme’s accuracy is not restricted by the spatial resolution of the weather forecast model grid but only by the one of an available digital elevation model (DEM). If the DEM used for radiation computations has a higher resolution than the model grid, the model grid cell radiation fluxes are representative means based on the higher-resolution fluxes as computed using the DEM grid.

Treatment of radiation in the described way seems to be computationally expensive, but most computations can be done prior to model execution, leaving only a few multiplications for the weather forecast model during time integration, so that computational costs are negligible.

In the first part of this article we derive the parameterization scheme and present the preprocessing steps. In the second part we demonstrate the skill improvement for air temperature forecasts due to the new parameterization scheme under clear-sky summer and winter conditions, under overcast conditions, as well as for a whole month of strongly varying cloud conditions. Therefore, the scheme was implemented into the Nonhydrostatic Mesoscale Model (NMM) of the National Oceanic and Atmospheric Administration (NOAA)/National Centers for Environmental Prediction (NCEP) (Janjic et al. 2001; Janjic 2003).

## 2. Parameterization scheme and data

### a. Mathematical description

The parameterization affects five radiation fluxes at ground level, namely direct downwelling shortwave ^{↑}*E _{s}*

_{,dir}, diffuse downwelling shortwave

^{↓}

*E*

_{s}_{,diff}, upwelling shortwave

^{↑}

*E*, upwelling longwave

_{s}^{↑}

*E*, and downwelling longwave

_{l}^{↓}

*E*. These fluxes are computed by the mesoscale forecast model considering the altitude of each grid cell as provided by a DEM of same spatial resolution as the model grid. The same symbols but marked by * are used for the parameterized radiation fluxes. Depending on the details of the respective model, radiative transfer computations consider the actual atmospheric conditions, but often assuming flat terrain—that is, the surface of each grid cell is horizontally oriented—and also disregarding neighborhood effects like sky view restrictions and shadowing. The NMM, used in this study, computes shortwave radiation absorption, reflection, and scattering in the model atmosphere based on Lacis and Hansen (1974). Absorption for water vapor, O

_{l}_{3}, and CO

_{2}are computed separately and over single bands in the UV/visible and near-infrared (NIR) portions of the solar spectrum, each representing 50% of incoming solar energy. The basic surface albedo taken from climatology is modified to take into account the actual local conditions at the surface. The parameterization for longwave radiative transfer in clear-sky conditions was developed at the Geophysical Fluid Dynamics Laboratory (GFDL) (Fels and Schwarzkopf 1975; Schwarzkopf and Fels 1985, 1991). Radiation interactions with clouds are computed for each vertical layer (Harshvardhan et al. 1989; Hong et al. 1998; Slingo 1987; Xu and Randall 1996). It has to be noted that the proposed parameterization of topographic effects takes radiative fluxes as externally computed input, which allows easy implementation in different mesoscale models.

^{↓}

*E*

_{s}_{,dir}| on slope angle

*α*was formulated by Kondratyev (1977) with the help of the geometry factor cos

*(α)*:where

*α*is the angle between the unit vector of the solar direct beam and the normal vector of the surface,

*ϕ*is the slope aspect, and

_{N}*θ*is the slope angle. The position of the sun is given by the sun elevation angle

_{N}*θ*and sun azimuth angle

_{S}*ϕ*

_{S}.*f*

_{cor}is derived for

^{↓}

*E*

_{s}_{,dir}combining the effects of slope angle and aspect, geometric enlargement of a sloping surface, and shadowing. By using (1), the effective direct downwelling shortwave radiation

^{↓}

*E**

_{s,}_{dir}on an inclined surface can be computed asThe geometric surface enlargement—that is, the ratio between the actual area of a sloping surface and its projected area as given by the spatial resolution of the underlying DEM (Scherer and Parlow 1994)—is given by the third factor in (2). Note that surface enlargement can only be applied for the computation of

^{↓}

*E**

_{s,}_{dir}, since it is the only flux showing directional dependence by definition.

*θ*

_{h,ϕS}is the horizon angle toward the sun azimuth angle

*ϕ*. The great advantage of this approach is its high computational performance, since only a logical operator has to be applied to each grid cell instead of ray tracing or radiosity computations. The time-consuming part of the computation—that is, the derivation of the horizon angles—is carried out as part of the preprocessing.

_{S}^{↓}

*E**

_{s,}_{dir}by simply multiplying

^{↓}

*E*

_{s,}_{dir}with a correction factor

*f*

_{cor}:where, after simplification of (2),Here

^{↓}

*E*

_{s,}_{dir}is computed by the mesoscale model and is representative for a single model grid cell usually covering an area of several square kilometers. Computing topographic parameters from the mesoscale model topography, as needed by (2), would be a crude approximation producing values for

*f*

_{cor}close to 1. A better approach is to compute the radiative fluxes based on a high-resolution DEM covering the whole model domain. Because of the multiplicative nature of (4), it is possible to simply compute a mean

*f*

_{cor}and multiply it with mean

^{↓}

*E*

_{s,}_{dir}to obtain mean

^{↓}

*E**

_{s,}_{dir}. Figure 1 illustrates the benefits of computing subgrid fluxes based on a 1-km resolved DEM rather than using the 4-km resolved mesoscale model topography. Shown are verifications of 2-m temperature forecasts for clear-sky conditions on 21 June 2003. The grid-scale resolved parameterization lies between the unparameterized control run and the subgrid parameterized run. Thus the sugbrid computation of fluxes doubles the effect of a grid-scale parameterization in the current configuration. Both parameterizations have the same memory requirements and computational costs during time integration. The verification procedure is described later in section 3a.

*f*

_{cor}can be computed diagnostically prior to model execution for each hour

*t*of the model run period based on all

*n*grid points of the DEM that cover a mesoscale model grid cell:The parameterization also considers topographic effects on

^{↓}

*E*

_{s,}_{diff}by restricted sky view. Effective diffuse downwelling shortwave radiation

^{↓}

*E**

_{s,}_{diff}is approximated bywhere

*f*

_{sky}is the sky view factor. The last term in (7), which is the part of downwelling shortwave radiation reflected by adjacent areas, assumes homogenous albedo equal to that of the model grid cell. This is a simplification, since the reflected shortwave radiation may originate from a neighboring grid cell having a different albedo. Furthermore, albedo depends on the relative incident angle, which can vary across the entire range at different locations and times throughout the day. However, the effect of this simplification is usually small, and computational performance would significantly decrease by a more sophisticated treatment.

^{↓}

*E*may be modified by radiation emitted by adjacent areas. In analogy to

_{l}^{↓}

*E**

_{s,}_{diff}, this effect is considered byUsually, the surface is warmer than the sky; thus

^{↓}

*E*in complex topography is increased in most cases by the parameterization given by (7) in areas of restricted sky view.

_{l}### b. Preprocessing

The required topographic parameters are derived as spatially distributed properties from a DEM of arbitrary spatial resolution or coordinate system. Because of their stationarity they need to be computed only once. The computations are following the scheme described by Scherer and Parlow (1994). The only modification was done with respect to DEMs that use nonmetric coordinate systems like the global digital elevation model of 30 arc second grid spacing (GTOPO30) dataset used in our study.

Since *θ _{h}*

_{,}

*and*

_{ϕ}*f*

_{sky}are nonlocal properties, they have to be computed considering a neighboring area for each grid element of suitable horizontal extent. In our study, we used an area of 40 km by 40 km, where the grid element, for which the properties are to be computed, is located in the center. The grid elements of the DEM within this area are resampled from the original grid to a Cartesian grid using a local orthographic projection, also taking into account the earth’s curvature. The procedure is thus applicable for any model domain of the earth.

*ϕ*and

_{N}*θ*are local properties and are derived from the resampled neighborhood by locally fitting a biquadratic surface through the central grid element and its eight neighbors. To save computational and disk storage resources,

_{N}*θ*

_{h}_{,}

*is computed for 24 discrete azimuth angles*

_{ϕ}*ϕ*, that is, using an azimuthal step width of 15°. For other azimuth directions, the corresponding horizon angle can be obtained with sufficient accuracy by linear interpolation, and

*f*

_{sky}is computed from

*θ*

_{h}_{,}

*after Dozier and Marks (1987):For the dark shaded subset in Fig. 2, the spatial distribution of*

_{ϕ}*f*

_{sky}is shown in Fig. 3b. Darker shades indicate smaller values of

*f*

_{sky}. The corresponding topography is contoured in Fig. 3a. In regions of complex terrain,

*f*

_{sky}is significantly reduced and can be as low as 0.7. Note that

*f*, computed from 1-km resolved GTOPO30 data, was aggregated to the 2-km grid of NMM.

_{sky}Temporal discretization of the time-dependent parameters can be achieved with sufficient accuracy using a resolution of 1 h. To save computational resources during time integration of the mesoscale model, the computation of *θ _{S}*,

*ϕ*

_{S}, mask

_{shadow}, and

*f*

_{cor}for each grid cell is carried out prior to model execution in a preprocessor, as well as the subsequent aggregation of

*f*

_{cor}to the model grid. Because of the preprocessor, the only computational task left for the mesoscale model itself is application of Eqs. (4), (7), and (8).

Figure 3c shows the spatial distribution of the direct shortwave correction factor *f*_{cor} for a morning situation. It is evident that eastward-sloping surfaces have larger *f*_{cor} values than westward-sloping surfaces, where *f*_{cor} is smaller than 1. On horizontal surfaces, *f*_{cor} becomes unity. At noon, southward-sloping surfaces have higher *f*_{cor}, and in the evening westward-sloping surfaces are subject to higher values of *f*_{cor}.

### c. Model area and measurement data

Case studies were carried out in a domain shown in Fig. 2, covering Switzerland and surroundings, thus including terrain ranging from very complex in the Swiss Alps with elevations over 4000 m MSL to very flat in the Rhine plane of southern Germany. Figure 3a shows the topography for the dark shaded subset of the model domain indicated in Fig. 2. Data from about 400 automatic weather stations, half of those located in the Swiss Alps, are available for the entire model domain, thus yielding a statistically significant number of observations. From all stations hourly mean temperatures could be obtained, allowing a comparison with model data at a temporal resolution of 1 h.

## 3. Results

The proposed radiation parameterization was implemented into the NMM of Janjic et al. (2001) and Janjic (2003). NMM is used for operational weather forecasts at NOAA/NCEP. Effects of the parameterization are analyzed using modeled and observed air temperatures at 2 m AGL. Air temperature is strongly correlated to radiation and is measured at every observing site. In addition, air temperature is an important forecast variable that is expected to be improved by application of the parameterization scheme.

### a. Verification results

Results for clear-sky conditions on 22 June 2003 and 25 December 2003, as well as completely overcast conditions on 20 October 2003, are presented separately to discuss the different effects of the parameterization. To evaluate the benefits of the parameterization for operational weather forecasts under different synoptic conditions, parallel runs were carried out for September 2004. All situations were simulated with full model physics but with an unparameterized topographic radiation control run (CTL) and with the new radiation parameterization (PAR) at spatial resolutions of 4 and 2 km. Forecast initialization was done using NOAA 1°-resolution GFS data and one-way nesting via a 22-km resolution NMM run covering Europe.

As expected, the effects of the radiation parameterization are most pronounced in complex terrain (cf. Whiteman et al. 1989; Colette et al. 2003; Chow et al. 2004). Thus, available observations are divided into an alpine and a nonalpine group, as shown in Fig. 2. The alpine group consists of all stations within the Alps and represents measurements from the most complex topography. The nonalpine group consists of the remaining stations. Verification scores are computed on an hourly basis by comparing all measured temperatures with the corresponding modeled temperatures. Note that temperature forecasts for stations were taken from the closest model grid point without further processing. Because of the very complex small-scale topography of the Alps, elevation differences between model grid cells and actual station heights are around 400 m for alpine stations, which contributes to the large mean and rms temperature errors. However, these values are typical for temperature forecasts of the Alps. The high-resolution model (aLMo) running at the Swiss national weather service (MeteoSwiss) obtains rms errors for stations located above 1500 m MSL of 2.5–3.5 K in summer and 4–6 K in winter, respectively (Schubiger 2001; F. Schubiger 2004, personal communication).

In Figs. 4a and 4b verification results of 22 June 2003 are shown for alpine and nonalpine stations, respectively. It can be seen that for both resolutions the new parameterization has the most impact at night and of course is more pronounced for alpine stations, located in complex terrain. The positive effect at night is caused by reduced sky view considered in the parameterization scheme. Longwave radiative loss of energy is reduced in valleys, yielding warmer temperatures with less negative mean errors and smaller rms errors. Thus the parameterization weakens the cold bias, which together with an underestimation of nocturnal ^{↓}*E _{l}* is also observed in the operational models, run at NOAA/NCEP (Z. Janjic 2004, personal communication). During daytime the parameterization seems to have little influence. At 4-km resolution the parameterization slightly increases the forecast error during daytime. This is mainly a problem of forecast verification and model resolution. The parameterization computes a representative mean of net radiation for a model grid cell including slopes and plains, which covers 16 or 4 km

^{2}, respectively. Measurement stations, however, are mostly located in flat areas of a grid cell, so that radiation and resulting temperature computed for a flat area, as is done in the unparameterized version, are more accurate for this particular part of the grid cell, but not for the whole area of the grid cell. For example, if a grid cell contains large slopes facing the sun, the parameterization yields more shortwave radiation that is absorbed, and thus higher temperatures for the whole grid cell. But the station located in the flat terrain part of the grid cell measures a lower temperature, which is not representative for the entire grid cell in complex terrain. At higher resolutions, observed and modeled areas correspond better, and the verification of parameterized and unparameterized runs converge, which is evident from the 2-km model resolution. Reduced sky view exists on the valley bottom as well as on sloping surfaces; thus measurement stations in valleys capture the phenomenon, and the above mentioned verification problem is less severe at night.

Another clear-sky situation is shown in Fig. 5. Those wintertime verification results are very similar to the summer situation, with most rms and mean error reduction at night. Again the benefit of the parameterization is the same for both resolutions of the mesoscale model.

Completely overcast conditions on the entire domain prevailed on 20 October 2003. After initialization, the parameterized and control runs diverge only little when compared to the clear-sky situation. The parameterization has only a small positive impact, but both during day- and nighttime, which is caused by inclusion of the sky view factor. The importance of sky view obstruction under overcast conditions and the resulting spatial variability have been demonstrated by Dubayah and Loechel (1997) for a portion of the Rocky Mountains. It has to be noted that this situation is a rarely occurring “worst case” scenario, since for large model domains there are almost always regions and time windows with no or very few clouds, where the parameterization has more effect. The mean error in Fig. 6 shows that during the daytime the improvement is due to cooler temperatures in the parameterized run. This can be explained by looking at Eq. (7), where ^{↓}*E _{s}*

_{,diff}is reduced by multiplication with the sky view factor and where, under overcast conditions, the second term is only dependent on

^{↓}

*E*

_{s}_{,diff}and albedo. Thus the net effect is a decrease of incoming shortwave radiation. Longwave downwelling radiation is increased because of restricted sky view, but the increase is less than it is under clear-sky conditions. This is due to the presence of relatively warm clouds that decrease the contrast between cold clear-sky temperatures and warmer terrain (cf. Matzinger et al. 2003). The shortwave part of the parameterization dominates during daytime, whereas at night, only the longwave part, with its above-mentioned reduced effect, is present. A verification problem associated with direct downwelling shortwave radiation does not appear under cloudy conditions.

Finally, the parameterization was tested under strongly varying synoptic conditions, such as stable high pressure with warm temperatures, many passages of fronts and even snowfall in the Alps, and unstable situations with development of local thunderstorms. The chosen period starts on 25 August and ends on 29 September 2004. Figure 7 shows rms errors computed for day- and nighttime, respectively. The verification is done using hourly data between 0700 and 1900 LST for daytime and the remaining 11 h for nighttime, by comparing all measured temperatures with the corresponding modeled temperatures. As can be seen, the parameterization reduces the temperature error throughout the time period, especially at nighttime. During daytime, the verification is again less representative, and parameterization effects are thus much smaller.

### b. Spatial impacts of the parameterization on modeled air temperature

Local deviations of the temperature field caused by a more detailed treatment of radiation are larger than the spatially integrated verification results suggest. Figures 8 to 10 illustrate spatial impacts of the parameterization for selected conditions and times. Depicted are differences between the parameterized and the control run for clear-sky conditions on 22 June 2003 and 25 December 2003, as well as overcast conditions on 20 October 2003, respectively. For clarity, only a small spatial subset marked as a dark shaded area in Fig. 2 is used here. In the clear-sky situation of Figs. 8b and 8c the differences are mainly caused by ^{↓}*E* _{s}*

_{,dir}. Higher values of

*f*

_{cor}in Fig. 3c correspond well with warmer areas in Fig. 8b. The correlation is, however, not perfect, since temperature is also controlled by dynamic processes such as turbulent exchange and advection. The spatial correspondence between temperature and

*f*

_{cor}exists for the other hours of the day, depicting the sun’s movement.

In Fig. 9 clear-sky winter conditions at 1100 LST 25 December are shown for 2- and 4-km model resolution, respectively. In addition to warmer regions, likewise modeled for the summer situation, significantly cooler temperatures can also be found in valleys. The cooling is in the range of 0.5 to 3 K and caused by shadows, which are much longer in wintertime. It can be seen that the 4-km resolution develops the main patterns of the higher-resolution run. However, a lot of detail in smaller valleys is missing, and the effects of the parameterization are weaker at the coarser resolution.

At night, when shortwave radiation is not present, restricted sky view (Fig. 3b) leads to reduced net loss of longwave radiation, which results in warmer air temperatures (Fig. 8a). As for shortwave radiation, the effects are best seen in the Alps. Mainly along the big valleys, where sky view factors are generally smaller, temperature increased more than 1 K.

For overcast conditions, spatial patterns are caused by the sky view factor. Figure 10 shows temperature differences at 1400, 1600, and 0100 LST on 20–21 October 2003, respectively. As can be seen by comparison with Fig. 3b, the parameterization leads to the already discussed cooling during the day in regions of restricted sky view. At night, a warming in regions of restricted sky view is observed. It is of smaller magnitude than under clear-sky conditions because of the presence of relatively warm clouds, as mentioned before. Note that advection smooths the spatial patterns.

## 4. Conclusions

The proposed parameterization allows simulation of important radiation effects associated with complex topography at negligible computational costs. Verifications with 2-m air temperatures demonstrate a general improvement of 0.5 to 1 K in rms and mean error, respectively. Especially at night, the consideration of restricted sky view leads to higher air temperatures in complex terrain. Along valleys the nighttime warming is between 0.5 and 1.5 K. During clear-sky daytime, this warming is of the same magnitude for grid cells containing slopes exposed to the sun. The parameterization also shows improvement under overcast conditions, where the nighttime warming was about half that of clear-sky conditions. During an overcast day, the parameterization cools the air in areas of restricted sky view by 0.2 to 0.7 K. In wintertime with lower sun elevation, shadows reduce temperatures in valleys by 0.5 to 3 K during daytime. At a resolution of 4 km the parameterization has significant impacts, which are even more pronounced on the finer 2-km grid.

Higher temperatures on slopes facing the sun may play an important role in triggering convective processes, and thus influence precipitation patterns as well. But this will have to be analyzed in the future.

## Acknowledgments

The authors want to express their gratitude to NOAA/NCEP for providing the NMM source code. The scientific discussion with Zavisa Janjic of NCEP and all his help concerning the NMM were crucial for this research. This work substantially benefited from the comments and suggestions made by the anonymous reviewers. The support of Matthew Pyle of NCEP was very helpful. Thanks to Martin Jacquot of the University of Basel for reconfiguring the Beowulf cluster to run NMM. The authors also want to thank SLF and MeteoSwiss for providing measurement data.

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Topographic effects considered by mesoscale models used for numerical weather prediction. The term shadow is used for shadows cast by surrounding terrain, not self-shading of a grid cell, which is considered within slope and aspect.