In two recent papers the validity of the Shallow Ice Approximation (SIA) and its higher order extension, the Second-Order SIA (SOSIA), is analyzed. In Ahlkrona et al. (2013a) , the full Stokes equations are solved with the ice sheet/ice flow model Elmer/Ice for different aspect ratios (reflecting the shallowness of an ice sheet). By this approach it is possible to determine how the stresses, velocities and pressure depend on the aspect ratio. These dependencies, or scaling relations, are important to know correctly in order to make approximations such as the SIA. It is found that there is a thick boundary layer at the ice surface, altering the scaling relations in a way which is not considered in the classical derivation of the SIA and the SOSIA. These numerical results are consistent with existing boundary layer theory for ice sheets. In Ahlkrona et al. (2013b), we investigate, both by analysis and by numerical simulations using the ice sheet/ice flow model Elmer/Ice, how this boundary layer influences the accuracy of the SIA and SOSIA. We find that due to the boundary layer, the order of accuracy of the SIA is lower than usually assumed, and the SOSIA is in many cases not a significant improvement to the SIA and is also dependent on an ad-hoc auxiliary parameter introduced to evade singularities in the boundary layer.

References:
Ahlkrona, J., N. Kirchner, and P. Lötstedt, 2013a. A Numerical Study of Scaling Relations for Non-Newtonian Thin-film Flows with Applications in Ice Sheet Modelling, Quarterly Journal Of Mechanics And Applied Mathematics, 66(4), 417-435, doi:10.1093/qjmam/hbt009. [link to paper]
Ahlkrona, J., N. Kirchner, and P. Lötstedt, 2013b. Accuracy of the zeroth- and second-order shallow-ice approximation – numerical and theoretical results, Geosci. Model Dev., 6, 2135-2152, doi:10.5194/gmd-6-2135-2013. [link to paper]