Modeling of Sub-grid Variability for Snow Redistribution and Ablation Processes using Fokker-Planck Equation

Abstract:

Sub-grid spatial variability of snow still remains a challenge because of complicated snow accumulation and melt processes. In this study, the Fokker-Planck equation (FPE) was derived to simulate the probability distribution in two-dimensional probability space, snow depth versus snow density. This FPE describes the evolution of probability density function of the state variables throughout the snow season. The snow depth and snow density were selected as stochastic state variables while snow temperature was left deterministic. In this equation, the snowmelt and snow accumulation are treated as external sources of stochasticity. This means that both the mean and variance parts of these variables are taken into consideration in the FPE as advection convection and diffusion effects in the probability domain, respectively. The major challenge of the FPE is calculation of the covariance terms appeared in the diffusion terms due to limitation of the data. In this study, several possible simplifications of the FPE will be discussed. We will test the following hypotheses for the simplifications: (1) the convection correction term is small enough to be neglected compared to the mean convection term, (2) snowmelt and snow accumulation are independent each other, and (3) snow accumulation term can be evaluated by measured snow depth data. For the estimation of snowmelt terms, outputs of a distributed snow model may also be used to calculate the spatial and temporal distribution of snowmelt. Finally, the FPE will be solved with a numerical method in the probability domain of snow depth versus snow density.