Articles | Volume 6, issue 6
Geosci. Model Dev., 6, 2135–2152, 2013
Geosci. Model Dev., 6, 2135–2152, 2013

Model evaluation paper 19 Dec 2013

Model evaluation paper | 19 Dec 2013

Accuracy of the zeroth- and second-order shallow-ice approximation – numerical and theoretical results

J. Ahlkrona1,2, N. Kirchner2,3, and P. Lötstedt1 J. Ahlkrona et al.
  • 1Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden
  • 2Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden
  • 3Department of Physical Geography and Quaternary Geology, Stockholm University, Stockholm, Sweden

Abstract. In ice sheet modelling, the shallow-ice approximation (SIA) and second-order shallow-ice approximation (SOSIA) schemes are approaches to approximate the solution of the full Stokes equations governing ice sheet dynamics. This is done by writing the solution to the full Stokes equations as an asymptotic expansion in the aspect ratio ε, i.e. the quotient between a characteristic height and a characteristic length of the ice sheet. SIA retains the zeroth-order terms and SOSIA the zeroth-, first-, and second-order terms in the expansion. Here, we evaluate the order of accuracy of SIA and SOSIA by numerically solving a two-dimensional model problem for different values of ε, and comparing the solutions with afinite element solution to the full Stokes equations obtained from Elmer/Ice. The SIA and SOSIA solutions are also derived analytically for the model problem. For decreasing ε, the computed errors in SIA and SOSIA decrease, but not always in the expected way. Moreover, they depend critically on a parameter introduced to avoid singularities in Glen's flow law in the ice model. This is because the assumptions behind the SIA and SOSIA neglect a thick, high-viscosity boundary layer near the ice surface. The sensitivity to the parameter is explained by the analytical solutions. As a verification of the comparison technique, the SIA and SOSIA solutions for a fluid with Newtonian rheology are compared to the solutions by Elmer/Ice, with results agreeing very well with theory.