Stochastic Upscaling for Snow Accumulation and Melt Processes with PDF Approach

Abstract

Although the importance of subgrid variability in the snow model has been recognized, only a few modeling studies on the parameterization of subgrid effects have been performed. This study proposes a new upscaling approach from a single-layered point-scale snow model toward a finite-scale model, using the probability density functions (PDF). The Fokker-Planck equation (FPE) for the snowmelt process is formulated in a two-dimensional probability domain for snow temperature versus snow depth. This Fokker-Planck equation, governing the evolution of the probability density of the snow temperature–snow depth bivariate state variable, can describe the subgrid effects of the snow process. The numerical solutions of the FPE are validated with Monte Carlo simulations. It is demonstrated that the derived FPE can express the spatial variability of the snow process sufficiently well once the necessary information on the spatial variation in shortwave radiation and snowfall are given. The validation results are encouraging and point toward potential use of this PDF model as an engine for large-scale grid-based modeling to capture the subgrid variability of snow processes.