Energy Balance

An energy balance (EB) method is one of the options included in the GSSHA model to estimate snowfall accumulation and melting. This method is admittedly simple and other factors, soil freezing, change in overland roughness, etc, are not yet considered. This is an area of active research and model development at ERDC. As inputs, the EB method requires hourly hydrometeorological data values of air temperature (T_a), relative humidity (r_h), wind speed (U) barometric pressure (P_a) and cloud cover.

Anytime the air temperature is below 0° C during precipitation, the precipitation is assumed to be snow or ice that will accumulate on the land surface. At air temperatures below 0° C, precipitation is nearly always snowfall (Gray and Prowse, 1993). If snow is already present in a cell, the new snow accumulation is added to the existing accumulated snow. While precipitation in the GSSHA model is distributed over the land surface, the effects of vegetation, elevation, and wind on the spatial distribution of snowfall are ignored.

Snowmelt models use either an energy balance or a temperature-index method. Physically-based systems are recommended for short term forecasts (Gray and Prowse, 1993), which are needed for hydrologic modeling. In the energy budget model the amount of heat available is applied to the snowpack and the amount of meltwater is calculated. The simplest representation of the snowpack is used; each 80 calories of heat added to the snowpack results in the release of 1 cm³ of meltwater (Linsley et al., 1982, Gray and Prowse, 1993). This method ignores complex snowpack behavior, such as ripening of the snowpack and refreezing of meltwater. Hourly values of hydrometeorological variables allow both seasonal and diurnal variations in climatic conditions to be included in the heat balance.

The amount of heat, Q (cal cm^-2 hr^-1) available is computed from the components of the energy balance. In GSSHA the following components are accounted for:

Q^* - net radiation (in - out), Q_v - heat in precipitation, Q_e - heat transferred by sublimation and evaporation, and Q_h - sensible heat transfer due to turbulence.

For non-precipitation periods the net radiation is typically the dominant source of energy for melting of the snowpack (Gray and Prowse, 1993). The net radiation is computed using Stephan-Boltzman’s law, with the assumptions that incoming radiation can be computed from the ambient temperature, T_a (C), and outgoing radiation is computed assuming the snowpack is at 0° C (Bras, 1990):

Q* = 49.56 x 10-10(Ta + 273)4 – 27

(92)

Precipitation falling on the snowpack at temperatures above 0° transmits the difference in heat between the raindrop and the snowpack. Assuming the snowpack is at 0° C and the rainfall is a ambient temperature the difference in heat energy is:

Qv = ITa

(93)

where: I is the precipitation intensity (cm/hr). Heats transferred from evaporation, sublimation, and turbulent energy are usually much smaller parts of the heat balance and are ignored in many computations (Gray and Prowse, 1993). However, convective exchange can be significant (Linsley et al., 1982). If the dew point is below the temperature of the snowpack, assumed to be 0° C, then condensation occurs and heat is transferred (Linsley et al., 1982; Gray and Prowse, 1993). Estimates of turbulent and latent heat exchange are usually based on measurements of air temperature, humidity, and wind speed (Gray and Prowse, 1993). During periods of melt, the temperature of the snowpack is 0° C and the saturated vapor pressure (es) is 6.11 mb (Linsley et al., 1982). The latent heat exchange is computed assuming the latent heat of evaporation/condensation is 600 cal g^-1 (Anderson, 1968) and a water density of 1 g cm^-3 as:

(94)

where: r_h is the relative humidity (%), ƒ(V) = 0.0002 U (km/hr) (Anderson 1978), where UU is the wind speed (m s^-1). Employing the Bowen ratio (Bowen, 1926) the sensible heat transfer is computed assuming the snow pack temperature is at 0° C, latent heat of evaporation is 600 cal g^-1, density of water is 1 g cm^-3, and the Bowen ratio coefficient is 0.61 x 10-3 C^-1 (Bras, 1990) as:

(95)

where: P_a is the atmospheric pressure (mb).

For non-precipitation periods, the energy budget is calculated at an hourly time step (same as the standard hydrometeorological data), so diurnal changes in energy inputs are included in the model formulation. During precipitation periods the energy budget is updated each overland flow routing time step (generally less than 5 minutes).

The described snowfall accumulation and melting calculations proceed anytime the LONG_TERM simulation option is chosen and hourly air temperatures are provided. If snowfall occurs, a warning will be printed to the screen and to the Summary file. When snow accumulation occurs the amount of snow in the watershed is reported at the beginning and end of each event summary in the Summary file.