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

The first-order or Blatter-Pattyn ice sheet model is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands, in spite of an approximate stress tensor and a limitation to small basal boundary slopes. In contrast, the full unapproximated Stokes ice sheet model is more difficult to solve and computationally more expensive. This is due to the fact that while both models arise from a variational principle, the Blatter-Pattyn variational functional is positive-definite and involves just the horizontal velocity components, while the Stokes functional is indefinite and involves all three velocity components, as well as the pressure. These unfavorable properties arise because Stokes flow is treated as a constrained minimization problem where the pressure acts as a Lagrange multiplier that enforces incompressibility or zero velocity divergence. To alleviate these problems we reformulate the full-Stokes problem as an unconstrained, positive-definite minimization problem, quite analogous to the Blatter-Pattyn model but without the associated approximations, by introducing a velocity field that is already divergence-free and satisfies appropriate boundary conditions, thus dispensing with the need for a pressure. Such a velocity field is obtained by vertically integrating the continuity equation to obtain the vertical velocity as a function of the horizontal velocity components, as is 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. We thus obtain not only a reformulated action principle, which itself is sufficient for obtaining an efficient discrete model, but also a novel set of Euler-Lagrange partial differential equations and boundary conditions that specify the Stokes problem in terms of just the horizontal velocities. The derivations are performed not only for the common case of an ice sheet in contact with and sliding along the bed, which again is analogous to the Blatter-Pattyn model, but also for more general situations, such as for a floating ice shelf.