Reformulating the full-Stokes ice sheet model for a more efficient computational solution

Abstract. The first-order or Blatter-Pattyn ice sheet model, in spite of its approximate nature, is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands. In contrast, the unapproximated Stokes ice sheet model is more difficult to solve and computationally more expensive. This is primarily due to the fact that the Stokes model is indefinite and involves all three velocity components, as well as the pressure, while the Blatter-Pattyn discrete model is positive-definite and involves just the horizontal velocity components. The Stokes model is indefinite because it arises from a constrained minimization principle where the pressure acts as a Lagrange multiplier to enforce incompressibility. To alleviate these problems we reformulate the full Stokes problem into an unconstrained, positive-definite minimization problem, similar to the Blatter-Pattyn model but without any of the approximations. This is accomplished by introducing a divergence-free velocity field that satisfies appropriate boundary conditions as a trial function in the variational formulation, thus dispensing with the need for a pressure. Such a velocity field is obtained by vertically integrating the continuity equation to give the vertical velocity as a function of the horizontal velocity components, as is in fact done in the Blatter-Pattyn model. This leads to a reduced system for just the horizontal velocity components, again just as in the Blatter-Pattyn model, but now without approximation. In the process we obtain a new, reformulated Stokes action principle as well as a novel set of Euler-Lagrange partial differential equations and boundary conditions. The model is also generalized from the common case of an ice sheet in contact with and sliding along the bed to other situations, such as to a floating ice shelf. These results are illustrated and validated using a simple but nontrivial Stokes flow problem involving a sliding ice sheet.