The shallow ice approximation for anisotropic ice: Formulation and limits

Ice sheet flow modeling is generally based on the shallow ice approximation (SIA) developed for isotropic ice. We extend this approximation to anisotropic ice and check its validity for both the isotropic and anisotropic cases by comparing the results of up to second-order SIA with those obtained from a model solving the full set of mechanical equations. The theory is developed for a constitutive relation describing Newtonian and non-Newtonian behavior, but numerical results are shown only for a Newtonian behavior. The results are compared for plane flow of isothermal ice under steady state conditions. SIA gives an excellent representation of the Newtonian flow of an isotropic ice sheet and a good description for the anisotropic case. The zero-order approximation is sufficient to describe ice flow precisely over flat bedrock, for the anisotropic and isotropic cases, even close to an ice divide. For uneven bedrock, the second-order SIA gives excellent results for isotropic ice and acceptable results for anisotropic ice. Finally, we quantify the error introduced by using an enhancement factor to represent the anisotropy when longitudinal stresses are taken into account.