The Taylor‐least‐squares time integrator scheme applied to tracer equations of a sea ice model

Abstract

The viscous‐plastic sea ice model based on [2] describes the motion of sea ice for scales of several thousand kilometers. The numerical model for the simulation of sea ice circulation and evolution over a seasonal cycle includes the consideration of the sea ice thickness and sea ice concentration. Transient advection equations describe the physical behavior of both thickness and concentration with the velocity of the sea ice as the coupling field. Recent research on a finite element implementation of the sea ice model is devoted to formulations based on the (mixed) Galerkin variational approach, compare to [1] and [6] for instance. Here, particular treatments are necessary regarding the stabilization of the complex numerical scheme, especially for the first‐order advection equation. It is therefore suggested to utilize the mixed least‐squares finite element method (LSFEM), which is well established in the branch of, e.g., fluid mechanics. A significant advantage of the method is its applicability to first‐order systems. Thus, this method results in stable and robust formulations also for not self‐adjoint operators like the tracer equations. Moreover, in [7], the authors provide a promising higher‐order time integration scheme for transient advection equations denoted as Taylor‐least‐squares scheme, which is investigated in this article. The presented least‐squares finite element formulation is based on the unsteady sea ice equations including two tracer equations of transient advection type. The numerical problem of a Box test case is investigated.

Large Scale Sea Ice Modeling – Problems and Perspectives.

“…Previous studies on the implementation of the sea ice model have shown that the LSFEM is a promising solution to the numerically stiff problem, cf. [24] and [21]. The resolution of the large gradients in the viscosity function, which can be characterized as localization effects, has to be investigated for convergent meshes.…”

“…The time-dependent values of the tracer equations need to be discretized with a suitable time integration scheme. In previous investigations different time integration schemes are elaborated, such as the higher-order Taylor-least-squares scheme based on [22], see [21], and the well known Backward-Euler and Crank-Nicolson schemes, compare to [24] and [21]. In the case of the Taylor-least-squares time integration, higher continuity requirements are imposed on the tracer quantities.…”

Simulating sea ice drift in the Southern Ocean incorporating real wind data using the LSFEM

“…We consider no further boundary conditions for the problem, which is fully time-dependent and, therefore, is sufficiently determined. For this problem we consider the Backward-Euler time discretization approach for the balance of momentum residual and the Crank-Nicolson scheme for both tracer residuals in (7).…”

“…Subsequent to the discretization in time, the space discretization can be implemented. We obtain the problem to be solved by the first variation of the minimization problem (7) with respect to all unknowns G = {σ, v, A ice , H ice }, such that…”

“…Previous studies on sea ice model implementations have shown that the LSFEM is a promising solution to the numerically difficult problem, cf. [9] and [7]. In this work, another essential aspect of sea ice research comes into play: embedding real data into the numerical solution is an important field of research in sea ice and ocean modeling.…”