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

Ahlkrona, Josefin, Kirchner, N. and Lötstedt, P. (2013) Accuracy of the zeroth- and second-order shallow-ice approximation – numerical and theoretical results Geoscientific Model Development, 6 (6). pp. 2135-2152. DOI 10.5194/gmd-6-2135-2013.

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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 epsilon, 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 epsilon, 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 epsilon, 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.

Document Type: Article
Research affiliation: Kiel University
Refereed: Yes
DOI etc.: 10.5194/gmd-6-2135-2013
ISSN: 1991-9603
Date Deposited: 09 May 2017 09:28
Last Modified: 08 Feb 2018 12:32

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