Basic Concepts

The Energy Budget Method is based on the Utah Energy Balance (UEB) model (Tarboton and Luce, 1996; Luce, 2000; Tarboton and Luce, 2001; You, 2004). The UEB snowmelt model is a physically-based energy and mass balance model. Energy is exchanged between the snowpack, the air above, and the soil below.

$//<![CDATA[ \begin{array}{l}dU/dt = Q_s_n + Q_l_i + Q_p + Q_g - Q_l_e + Q_h + Q_e - Q_m\end{array} //]]>$

where $//<![CDATA[ \begin{array}{l}Q_s_n\end{array} //]]>$ is net shortwave radiation, $//<![CDATA[ \begin{array}{l}Q_l_i\end{array} //]]>$ is incoming longwave radiation, $//<![CDATA[ \begin{array}{l}Q_p\end{array} //]]>$ is advected heat from precipitation, $//<![CDATA[ \begin{array}{l}Q_g\end{array} //]]>$ is ground heat flux, $//<![CDATA[ \begin{array}{l}Q_l_e\end{array} //]]>$ is outgoing longwave radiation, $//<![CDATA[ \begin{array}{l}Q_h\end{array} //]]>$ is sensible heat flux, $//<![CDATA[ \begin{array}{l}Q_h\end{array} //]]>$ is latent heat flux due to sublimation/condensation, and $//<![CDATA[ \begin{array}{l}Q_m\end{array} //]]>$ is advected heat removed by meltwater.

$//<![CDATA[ \begin{array}{l}dW/dt = P_r + P_s - M_r - E\end{array} //]]>$

The surface energy balance is:

Surface heat conduction describe the exchange of heat from the snow surface into the snowpack. Snow surface heating varies dramatically over the course of a day and over longer time periods resulting in a nonlinear temperature profile. Nonlinearity in snowpack temperature profile is largely caused by daily temperature fluctuations at the surface, which have a sinusoidal pattern.

Surface Heat Conduction

HEC-HMS uses the modified force-restore with shallow snow correction method. The force-restore method estimates the driving flux at the surface as sinusoidal (since daily temperature fluctuations follow an approximately sinusoidal pattern). However, the force-restore method may be a poor approximation because the temperature gradient does not cycle on a daily time scale. Temperature variation (and heat fluxes) with depth is caused by lower frequency fluctuations. Therefore, the heat fluxes caused by lower frequency variability are superimposed on the gradient in daily average temperature. 

The low frequency effective depth $//<![CDATA[ \begin{array}{l}d_l_f\end{array} //]]>$ is used to associate a frequency with a distance used in the daily average gradient estimate:

where $//<![CDATA[ \begin{array}{l}k_g\end{array} //]]>$ is the soil thermal diffusivity, $//<![CDATA[ \begin{array}{l}\omega_l_f\end{array} //]]>$ is the frequency of low-frequency temperature variation, and $//<![CDATA[ \begin{array}{l}\omega_l_f = \omega_1/4\end{array} //]]>$

The surface heat flux is computed as:

where $//<![CDATA[ \begin{array}{l}\bar T_s\end{array} //]]>$ is daily average surface temperature and $//<![CDATA[ \begin{array}{l}\bar T_a_v_e\end{array} //]]>$ is daily average depth average snowpack temperature.

The shallow snow correction involves computation of an effective thermal depth of combined snowpack and ground and a weighted thermal conductivity when the thermal damping depth extends into the ground. The shallow snowpack correction is applied when the snow depth is less than the effective depth.