A material coordinate treatment of the sea–ice dynamics equations

Abstract

A finite–element algorithm is constructed for a material coordinate formulation of the equations of sea–ice dynamics, using quadratic elements and fully implicit time steps. The material coordinate description allows the nodes of a fixed finite–element mesh to define the same material elements as time proceeds, which avoids interpolation of nodal values on a changing spatial mesh as the pack evolves, quadratic elements preserve continuity of second derivatives, and this time stepping is stable and accurate in standard problems. An earlier finite–element study of a wind–driven pack with two free boundary sections, using spatial coordinates and implicit time steps without iteration, gave rise to numerical instability when the constitutive law for the ice stress induced by floe interactions imposes zero stress in diverging flow. The present more accurate study of the same problem, using a material description and fully implicit time steps, with a smoothed transition to zero stress in diverging flow, significantly extends the time over which a stable solution is obtained. Stability and accuracy of the present algorithm is first demonstrated by comparison with a class of exact solutions to specific problems using linearly viscous relations in converging flow and abrupt transition to zero stress in diverging flow, for which an expanding region of diverging flow is initiated after a finite time at an interior point, either following convergence everywhere, or following an expanded region of neutral flow. The previous problem with two free boundary sections is then solved with the same rheology to demonstrate a stable solution over an extended time period. Next, a general nonlinearly viscous relation is constructed which ensures that the stress lies close to a yield envelope during strongly converging flow, to reflect the commonly used viscous–plastic model without the disjoint stress relations in different regimes. This is applied to a pack flow with a dramatically deforming free boundary driven by a vortex wind, which demonstrates how well the present material formulation can capture large deformations.