A finite-element framework to explore the numerical solution of the coupled problem of heat conduction, water vapor diffusion, and settlement in dry snow (IvoriFEM v0.1.0)

Abstract

The poor treatment (or complete omission) of water vapor transport has been identified as a major limitation suffered by currently available snowpack models. As vapor and heat fluxes are closely intertwined, their mathematical representation amounts to a system of nonlinear and tightly coupled partial differential equations that are particularly challenging to solve numerically. The choice of the numerical scheme and the representation of couplings between processes are crucial to ensure an accurate and robust solution that guarantees mass and energy conservation while also allowing time steps in the order of 15 min. To explore the numerical treatments fulfilling these requirements, we have developed a highly modular finite-element program. The code is written in Python. Every step of the numerical formulation and solution is coded internally, except for the inversion of the linearized system of equations. We illustrate the capabilities of our approach to tackle the coupled problem of heat conduction, vapor diffusion, and settlement within a dry snowpack by running our model on several test cases proposed in recently published literature. We underline specific improvements regarding energy and mass conservation as well as time step requirements. In particular, we show that a fully coupled and fully implicit time-stepping approach enables accurate and stable solutions with little restriction on the time step.