Reply on RC1

Abstract

The incorporation of vapor transport has become a key demand for snowpack modeling where accompanied phase changes give rise to a new, non-linear coupling in the heat and mass equations. This coupling has an impact on choosing efficient numerical schemes for one-dimensional snowpack models which are naturally not designed to cope with mathematical particularities of arbitrary, non-linear PDE's. To explore this coupling we have implemented a stand-alone finite element solution of the coupled heat and mass equations in snow using FEniCS. We solely focus on the non-linear feedback of the ice phase exchanging mass with a diffusing vapor phase with concurrent heat transport in the absence of settling. We demonstrate that different, existing continuum-mechanical models derived through homogenization or mixture theory yield similar results for homogeneous snowpacks of constant density. For heterogeneous situations in which the snow density varies significantly with depth, we show that phase changes in the presence of temperature gradients give rise to a non-linear advection of the ice phase that amplifies existing density variations. Eventually, this advection triggers a wave instability in the continuity equations. This is traced back to the density dependence of the effective transport coefficients as revealed by a linear stability analysis of the non-linear PDE system. The instability is an inherent feature of existing continuum models and predicts, as a side product, the formation of a low density (mechanical) weak layer on the sublimating side of an ice crust. The wave instability constitutes a key challenge for a faithful treatment of solid-vapor mass conservation between layers, which is discussed in view of the underlying homogenization schemes and their numerical solutions.