Comparison of different numerical approaches to the 1D sea-ice thermodynamics problem

The vertical one-dimensional sea-ice thermodynamic problem using the principle of conservation of enthalpy is revisited here using (1) the Bitz and Lipscomb (1999) finite-difference approach (FD), (2) a reformulation of the sigma-level transformation of Huwald et al. (2005b) (FV) and (3) a Finite Element approach also in sigma coordinates (FE). These three formulations are compared in terms of physics, numerics, and performance, in order to identify the best choice for large-scale climate models. The BL99 formulation sequentially treats the diffusion of heat and the changes in the vertical position of the ice-snow layers. In contrast, the FV sigma-level transformation elegantly treats both simultaneously. The original FV formulation suffers however from slow convergence. The convergence can nonetheless be improved significantly with a few simple modifications to the original code. The three formulations are compared following the experimental protocol of the Sea Ice Model Intercomparison Project for ice thermodynamics (SIMIP2). It is found that all formulations converge to the same solution. The FD approach, however, suffers from the added cost of the remapping step at large number of ice layers (we include in the appendix an optimized version of the FD code–written by one of the reviewer–that resolves this issue). Finally the FE formulation results in a sub-surface temperature over-estimation at low resolution, a problem which disappears at high resolution. Hence, only FD and FV are found suitable for climate models.