Numerical convergence of viscous-plastic sea ice models

We investigate the convergence properties of the nonlinear solver used in viscous-plastic (VP) sea ice models. More specifically, we study the nonlinear solver that is based on an implicit solution of the linearized system of equations and an outer loop (OL) iteration (or pseudo time steps). When the time step is comparable to the forcing time scale, a small number of OL iterations leads to errors in the simulated velocity field that are of the same order of magnitude as the mean drift. The slow convergence is an issue at all spatial resolution but is more severe as the grid is refined. The metrics used by the sea ice modeling community to assess convergence are misleading. Indeed, when performing 10 OL iterations with a 6 h time step, the average kinetic energy of the pack is always within 2% of the fully converged value. However, the errors on the drift are of the same order of magnitude as the mean drift. Also, while 40 OL iterations provide a VP solution (with stress states inside or on the yield curve), large parts of the domain are characterized by errors of 0.5-1.0 cm s(-1). The largest errors are localized in regions of large sea ice deformations where strong ice interactions are present. To resolve those deformations accurately, we find that more than 100 OL iterations are required. To obtain a continuously differentiable momentum equation, we replace the formulation of the viscous coefficients with capping with a tangent hyperbolic function. This reduces the number of OL iterations required to reach a certain residual norm by a factor of similar to 2.