# Convex duality and glacier momentum balance

# Convex duality and glacier momentum balance

ABSTRACT: The momentum balance equation for glacier flow can be expressed as a convex optimization problem for the ice velocity. The geometry of real glaciers, however, evolves in time as the terminus advances and retreats. The orthodox approaches to solving this free boundary problem are to either (1) set a minimum thickness and accept the resulting mass balance errors, or (2) apply the level-set method or related approaches that "turn off" the physics where the ice thickness is zero. Both of these approaches have their drawbacks. Here, I'll show an alternative formulation of the momentum balance equation that I derived using the principle of convex duality. This dual form adds the stresses as unknowns, but it remains solvable even when the ice thickness is equal to 0. Solvability at zero thickness allows us to simulate terminus advance and retreat -- including iceberg calving -- in a simple way. I'll show some numerical demonstrations and discuss what solution methods were viable. I have a hunch that elements of this approach could be applied to other thin-film flows including variably-saturated porous media and overland water drainage.

BIO: Daniel Shapero is a Senior Research Scientist at the Applied Physics Laboratory of the University of Washington who received PhD in applied mathematics, also at UW. Shapero develops a software package called icepack (see https://icepack.github.io) for simulating the flow of glaciers and ice sheets.