Elements of future snowpack modeling – Part 2: A modular and extendable Eulerian–Lagrangian numerical scheme for coupled transport, phase changes and settling processes

Anna Simson,Henning Löwe,Julia Kowalski

doi:10.5194/tc-15-5423-2021

Abstract

Abstract. A coupled treatment of transport processes, phase changes and mechanical settling is the core of any detailed snowpack model. A key concept underlying the majority of these models is the notion of layers as deforming material elements that carry the information on their physical state. Thereby an explicit numerical solution of the ice mass continuity equation can be circumvented, although with the downside of virtual no flexibility in implementing different coupling schemes for densification, phase changes and transport. As a remedy we consistently recast the numerical core of a snowpack model into an extendable Eulerian–Lagrangian framework for solving the coupled non-linear processes. In the proposed scheme, we explicitly solve the most general form of the ice mass balance using the method of characteristics, a Lagrangian method. The underlying coordinate transformation is employed to state a finite-difference formulation for the superimposed (vapor and heat) transport equations which are treated in their Eulerian form on a moving, spatially non-uniform grid that includes the snow surface as a free upper boundary. This formulation allows us to unify the different existing viewpoints of densification in snow or firn models in a flexible way and yields a stable coupling of the advection-dominated mechanical settling with the remaining equations. The flexibility of the scheme is demonstrated within several numerical experiments using a modular solver strategy. We focus on emerging heterogeneities in (two-layer) snowpacks, the coupling of (solid–vapor) phase changes with settling at layer interfaces and the impact of switching to a non-linear mechanical constitutive law. Lastly, we discuss the potential of the scheme for extensions like a dynamical equation for the surface mass balance or the coupling to liquid water flow.

Highlights

The snow density is probably the most important prognostic variable of any snowpack model as, e.g., reflected by a focus on snow water equivalent in past snow model intercomparison projects (Krinner et al, 2018, and references therein)
Settling proceeds the fastest just after the start of the simulation, when the snowpack is at maximum height, and correspondingly its snow density is the lowest
The model accounts for transport of heat and vapor, phase changes, and mechanical settling processes

Summary

Introduction

The snow density is probably the most important prognostic variable of any snowpack model as, e.g., reflected by a focus on snow water equivalent in past snow model intercomparison projects (Krinner et al, 2018, and references therein). A more detailed inspection reveals how mass conservation is accounted for, namely rather indirectly by stating a settling law for individual layers and resorting to a “Lagrangian coordinate system that moves with the ice matrix” (Bartelt and Lehning, 2002) to translate the ice-phase deformation into a thickness evolution of the layers (Brun et al, 1989; Vionnet et al, 2012) While this procedure has been well established for a long time, it is without numerical ambiguities only in the absence of phase changes. Simson et al.: Mixed Eulerian–Lagrangian approach for snow modeling – Part 2 of the most important conservation law in snow makes it virtually impossible to isolate and advance the numerical core of a snowpack model as an encapsulated numerical scheme comprising all involved coupled non-linear partial differential equations

Objectives

Results

Conclusion