A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling

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A computational analysis of the accuracy of different approximations to the Stokes equations for momentum balance used in ice sheet modeling is performed by solving a particular tractable form of the equations appropriate for small perturbations of the ice surface, describing the uniform flow of ice with a Glen rheology on an infinitely long and broad section. The approximants comprise the shallow ice approximation and various schemes for incorporating longitudinal stresses and, in one case, the horizontal gradient of the horizontal plane shear stresses. The simplifications lead to a vertically one‐dimensional numerical problem, whose solution can be computed rapidly. The relaxation rate of perturbations as well as other response descriptors for the stable full system and approximants are compared. Compared with the shallow ice approximation, the inclusion of longitudinal stresses increases accuracy at shorter wavelengths, but accuracy is poor at wavelengths around or less than the ice sheet thickness. Even though analysis shows that the horizontal gradients of the horizontal plane shear stresses are of similar magnitude to longitudinal stress effects, computations show, in agreement with glaciological belief, that longitudinal stress effects are more significant and need to be corrected for first in practice. Two schemes, a multilayer scheme and a one‐layer scheme, are particularly good and should be investigated further in cases where perturbations from uniformity are large. Some other apparently plausible approximations introduce nonphysical instabilities. New schemes need to be assessed in the way described in this paper before being used in real ice sheet models.