Conservation laws for free-boundary fluid layers

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 or balance in numerical schemes. A free boundary in a region of negative source -- an ablation-caused margin -- turns out to be a barrier to exact balance for a numerical scheme (in either a continuous- or discrete-space sense). We propose computable \emph{a posteriori} quantities which allow conservation-error accounting in finite volume and element schemes.