Physics of the spatially averaged snowmelt process

It has been recognized that the snowmelt models developed in the past do not fully meet current prediction requirements. Part of the reason is that they do not account for the spatial variation in the dynamics of the spatially heterogeneous snowmelt process. Most of the current physics-based distributed snowmelt models utilize point-location-scale conservation equations which do not represent the spatially varying snowmelt dynamics over a grid area that surrounds a computational node. In this study, to account for the spatial heterogeneity of the snowmelt dynamics, areally averaged mass and energy conservation equations for the snowmelt process are developed. As a first step, energy and mass conservation equations that govern the snowmelt dynamics at a point location are averaged over the snowpack depth, resulting in depth averaged equations (DAE). In this averaging, it is assumed that the snowpack has two layers. Then, the point location DAE are averaged over the snowcover area. To develop the areally averaged equations of the snowmelt physics, we make the fundamental assumption that snowmelt process is spatially ergodic. The snow temperature and the snow density are considered as the stochastic variables. The areally averaged snowmelt equations are obtained in terms of their corresponding ensemble averages. Only the first two moments are considered. A numerical solution scheme (Runge-Kutta) is then applied to solve the resulting system of ordinary differential equations. This equation system is solved for the areal mean and areal variance of snow temperature and of snow density, fbr the areal mean of snowmelt, and for the areal covariance of snow temperature and snow density. The developed model is tested using Scott Valley (Siskiyou County, California) snowmelt and meteorological data. The performance of the model in simulating the observed areally averaged snowmelt is satisfactory.