The Cryosphere, 6, 21-34, 2012
www.the-cryosphere.net/6/21/2012/ doi:10.5194/tc-6-21-2012 © Author(s) 2012. This work is distributed under the Creative Commons Attribution 3.0 License. |

Research Article

06 Jan 2012

Climate, Ocean, and Sea-Ice Modeling (COSIM) Project, Group T-3, MS B216, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA

Received: 25 May 2011 – Published in The Cryosphere Discuss.: 01 July 2011

Revised: 21 November 2011 – Accepted: 12 December 2011 – Published: 06 January 2012

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.Revised: 21 November 2011 – Accepted: 12 December 2011 – Published: 06 January 2012