Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

The formulation of a physical problem in terms of a variational (or action) principle conveys significant advantages for the analytical formulation and numerical solution of that problem. One such problem is ice-sheet dynamics as described by non-Newtonian Stokes flow, for which the variational principle can be interpreted as stating that a measure of heat dissipation, due to internal deformation and boundary friction, plus the rate of loss of total potential energy is minimized under the constraint of incompressible flow. By carrying out low-aspect-ratio approximations to the Stokes flow problem within this variational principle, we obtain approximate dynamical equations and boundary conditions that are internally consistent and preserve the analytical structure of the full Stokes system. This also allows us to define an action principle for the popular first-order or 'Blatter-Pattyn' shallow-ice approximation that is distinct from the action principle for the Stokes problem yet preserves its most important properties and elucidates various details about this approximation. Further approximations within this new action functional yield the standard zero-order shallow-ice and shallow-shelf approximations, with their own action principles and boundary conditions. We emphasize the specification of boundary conditions, which are problematic to derive and implement consistently in approximate models but whose formulation is greatly simplified in a variational setting.