Hamilton-type principles applied to ice-sheet dynamics: new approximations for large-scale ice-sheet flow

Ice-sheet modelers tend to be more familiar with the Newtonian, vectorial formulation of continuum mechanics, in which the motion of an ice sheet or glacier is determined by the balance of stresses acting on the ice at any instant in time. However, there is also an equivalent and alternative formulation of mechanics where the equations of motion are instead found by invoking a variational principle, often called Hamilton's principle. In this study, we show that a slightly modified version of Hamilton's principle can be used to derive the equations of ice-sheet motion. Moreover, Hamilton's principle provides a pathway in which analytic and numeric approximations can be made directly to the variational principle using the Rayleigh-Ritz method. To this end, we use the Rayleigh-Ritz method to derive a variational principle describing the large-scale flow of ice sheets that stitches the shallow-ice and shallow-shelf approximations together. Numerical examples show that the approximation yields realistic steady-state ice-sheet configurations for a variety of basal tractions and sliding laws. Small parameter expansions show that the approximation reduces to the appropriate asymptotic limits of shallow ice and shallow stream for large and small values of the basal traction number.