An optimization approach to modeling sea ice dynamics. Part 1: Lagrangian framework

A new model for the dynamics of sea ice is proposed. The pressure field, instead of being derived from a local rheology as in most existing models, is computed from a global optimization problem. Here the pressure is seen as emerging not from an equation of state but as a Lagrange multiplier that enforces the ice's resistance to compression while allowing divergence. The resulting variational problem is solved by minimizing the pressure globally throughout the domain, constrained by the equations of momentum and mass conservation, as well as the limits on ice concentration (which has to stay between 0 and 1). This formulation has an attractive mathematical elegance while being physically motivated. Moreover, it leads to an analytic formulation that is also easily implemented in a numerical code, which exhibits marked stability and is suited to capturing discontinuities. In order to test the theory, the equations for a one-dimensional model are cast in terms of Lagrangian mass coordinates. The solution to the minimization problem is compared to an exact analytic solution derived using jump conditions in a simple test case. Another case is examined, which is somewhat more complicated but still allows our physical intuition to verify the qualitative results of the model. Good agreement is found. A final validation is performed by a comparison with a particle-based model, which tracks individual ice floes and their inelastic interaction in a one-dimensional domain.