A MATERIAL COORDINATE TREATMENT OF THE SEA-ICE DYNAMICS EQUATIONS

A finite-element algorithm is constructed for a material coordinate fo rmulation of the equations of sea-ice dynamics, using quadratic elemen ts and fully implicit time steps. The material coordinate description allows the nodes of a fixed finite-element mesh to define the same mat erial elements as time proceeds, which avoids interpolation of nodal v alues on a changing spatial mesh as the pack evolves, quadratic elemen ts preserve continuity of second derivatives, and this time stepping i s stable and accurate in standard problems. An earlier finite-element study of a wind-driven pack with two free boundary sections, using spa tial 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. Th e present more accurate study of the same problem, using a material de scription and fully implicit time steps, with a smoothed transition to zero stress in diverging flow, significantly extends the time over wh ich a stable solution is obtained. Stability and accuracy of the prese nt algorithm is first demonstrated by comparison with a class of exact solutions to specific problems using linearly viscous relations in co nverging flow and abrupt transition to zero stress in diverging flow, for which an expanding region of diverging flow is initiated after a f inite time at an interior point, either following convergence everywhe re, or following an expanded region of neutral flow. The previous prob lem with two free boundary sections is then solved with the same rheol ogy to demonstrate a stable solution over an extended time period. Nex t, a general nonlinearly viscous relation is constructed which ensures that the stress lies close to a yield envelope during strongly conver ging 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 v ortex wind, which demonstrates how well the present material formulati on can capture large deformations.