  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.
