Conservation laws for free-boundary fluid layers

TL;DR

This paper develops mathematical models for thin fluid layers with free boundaries, analyzing conservation laws and proposing methods to estimate conservation errors in numerical simulations relevant to climate dynamics.

Contribution

It introduces a framework for analyzing conservation laws in free-boundary fluid models and proposes computable error bounds for numerical schemes.

Findings

Free boundaries with negative sources hinder exact conservation.

Well-posedness of models is established in several cases.

A posteriori quantities enable conservation-error estimation.

Abstract

Time-dependent models of fluid motion in thin layers, subject to signed source terms, represent important sub-problems within climate dynamics. Examples include ice sheets, sea ice, and even shallow oceans and lakes. We address these problems as discrete-time sequences of continuous-space weak formulations, namely (monotone) variational inequalities or complementarity problems, in which the conserved quantity is the layer thickness. Free boundaries wherein the thickness and mass flux both go to zero at the margin of the fluid layer generically arise in such models. After showing these problems are well-posed in several cases, we consider the limitations to discrete conservation in numerical schemes. A free boundary in a region of negative source -- an ablation-caused margin -- turns out to be a barrier to exact conservation in either a continuous- or discrete-space sense. We then propose computable a posteriori quantities which allow conservation-error bounds in finite volume and finite element schemes.