The starting point for calculating insolation at the surface is the solar radiation flux at the top of the atmosphere, Stoa, 99.9% of which lies in the spectral band from 0.2 to 100 pm. Stoa may easily be calculated as a function of geographical location, season, and time (see, e.g., Iqbal, 1983). The part of the solar spectrum up to 4.0 pm is usually called shortwave radiation, representing about 99.2% of Stoa. However, solar radiation is absorbed by some atmospheric constituents and is reflected and scattered by clouds and aerosol. Accurate calculation of incoming shortwave radiation at the earth's surface, from Stoa thus requires the use of a radiative transfer model, but the data required for driving such models are rarely available in practical snow-cover applications. Incoming shortwave radiation is the most important energy source for snow cover in most situations (see Section 3.5). The net flux of shortwave radiation, SN, at the snow surface is given by:
where Sf is the reflected shortwave radiation and a is the albedo, i.e. the spectrally integrated reflectance as discussed in Section 2.5.2. As snow albedo ranges between 0.50 for old, wet snow and 0.95 for new snow, changes in snow albedo lead to substantially different amounts of energy absorbed by the snowpack. Moreover, the interaction between the surface and the atmosphere is substantially altered over seasonal snow covers because albedo is quite different over snow-covered ground as opposed to bare ground.
If measurements of are not available, it is generally necessary to parameterize this flux in terms of solar zenith angle and easily observed quantities such as total cloud cover. Key et al. (1996) discuss and evaluate a number of such parameteri-zations. Generally, is calculated first evaluating the insolation under clear sky conditions and then applying a correction for cloud cover. Clouds are reflective; hence, increasing cloud cover tends to reduce the magnitude of the downward component of shortwave radiation Over high-albedo surfaces, such as snow, it is important to include the effect of multiple reflections between the snow surface and the cloud base in any parameterization used (Shine, 1984; Gardiner, 1987).
A non-negligible part of downward shortwave radiation reaches the ground as diffuse, nearly isotropic radiation. Its percentage depends primarily on cloudiness and isotropy greatly simplifies its parameterization. As Varley et al. (1996) pointed out, many areas in heterogeneous terrain (e.g. alpine topography) receive little or no direct radiation in wintertime, making the separate modeling of direct and diffuse radiation an important issue there. In addition, diffuse radiation reflected from the surrounding topography has to be taken into account (e.g. Dozier, 1980; Pluss, 1997).
3.3 The fluxes involved in the energy balance 3.3.2 Longwave radiation
Longwave or terrestrial radiation encompasses wavelengths from approximately 4 to 100 |^m. Downward longwave radiation at the surface, L|, results from thermal emission from both atmospheric gases (notably water vapor and carbon dioxide) and clouds while upward longwave radiation, Lf, is thermally emitted from the surface.
While an accurate evaluation of L\ once again requires the use of a radiative transfer model, the relatively strong absorption of infrared radiation by water vapor means that L\ is largely determined by conditions in the lowest few hundred meters of the atmosphere (Ohmura, 2001) and, consequently, can be parameterized sufficiently well for many practical applications in terms of near-surface variables. In analogy with the Stefan-Boltzmann equation, parameterizations generally take the form:
where ctsb is the Stefan-Boltzmann constant, Ta is a near-surface air temperature (in K) and seff is an "effective" emissivity for the atmosphere. seff is usually specified as a function of cloud cover only, cloud cover and near-surface humidity (Konzelmann et al., 1994) or near-surface humidity only under clear sky (Brutsaert, 1975). Clouds are very efficient infrared emitters and have effective emissivities close to unity, while, under clear skies, seff is typically around 0.75 but may reach values as low as 0.55 in alpine regions (Marty, 2000). Thus, for fixed Ta, increasing cloud cover will increase the magnitude of the downward component of longwave radiation L\. Key et al. (1996) as well as Konig-Langlo and Augstein (1994) give further examples of such parameterizations.
Upward longwave radiation can be calculated from the snow surface temperature, To (in K), and snow infrared emissivity s (see Section 2.5.3) as:
In the stably stratified conditions that often prevail in the surface boundary layer over snow-covered surfaces, To may be several degrees colder than Ta and the use of Ta instead of To in (3.8) may result in a significant overestimate of Lf. Over a melting snow cover, To may be set to 273.15 K (0 °C). It should be further noted that both downward and upward longwave radiation are usually assumed to be isotropic, resulting in negligible errors only (Pluss, 1997).
To account for longwave radiation in heterogeneous terrain, it is necessary to calculate the incoming fluxes from the sky and the surrounding terrain separately (Pluss and Ohmura, 1997). They showed that in snow-covered environments, where the surface temperature is usually below the air temperature, neglecting the effects
due to air temperature leads to an underestimation of the incoming longwave radiation flux on inclined slopes.
3.3.3 Net radiation
In summary, the net radiative flux at the surface is:
The value of a is crucial to the sign of net radiation. Under clear sky, e.g., albedo needs often to be below 0.75 before RN becomes negative and hence represents a positive energy gain to the snowpack (see Equation 3.1). However, RN is mostly negative under overcast conditions, independent of the value of a.
Furthermore, as we have seen above, cloud cover has opposite effects on Si and Li. So it is not immediately clear how an increasing cloud cover will effect net radiation. Over a high-albedo surface such as snow, the increase in magnitude of Li with increasing cloud cover can more than outweigh the reduction in net shortwave radiation, leading to an increased energy gain to the snowpack (Ambach, 1974). This situation of positive cloud radiative forcing is most likely to arise in the polar regions, where both surface albedo and cloud transmissivities in the shortwave region are high and the sun is low. However, the effect of cloud radiative forcing depends critically on surface and cloud properties and both positive and negative forcing has been observed over polar snow surfaces (Bintanja and van den Broeke, 1996).
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