Improving numerical ice sheet models is a very active field of research. In part, this is because ice sheet modelling has gained societal relevance in the context of predictions of future sea level rise. Ice sheet modelling is however also a challenging mathematical and computational subject. Since the exact equations governing ice dynamics, the full Stokes equations, are computationally expensive to solve, approximations are crucially needed for many problems. Shallow ice approximations are a family of approximations derived by asymptotic expansion of the exact equations in terms of the aspect ratio, epsilon. Retaining only the zeroth order terms in this expansion yields the by far most frequently used approximation; the Shallow Ice Approximation (SIA). Including terms up to second order yields the Second Order Shallow Ice Approximation (SOSIA), which is a so-called higher order model. Here, we study the validity and accuracy of shallow ice approximations beyond previous analyses of the SIA. We perform a detailed analysis of the assumptions behind shallow ice approximations, i.e. of the order of magnitude of field variables. We do this by using a numerical solution of the exact equations for ice flow over a sloping, undulating bed. We also construct analytical solutions for the SIA and SOSIA and numerically compute the accuracy for varying epsilon by comparing to the exact solution. We find that the assumptions underlying shallow ice approximations are not entirely appropriate since they do not account for a high viscosity boundary layer developing near the ice surface as soon as small bumps are introduced at the ice base. This boundary layer is thick and has no distinct border. Other existing theories which do incorporate the boundary layer are in better, but not full, agreement with our numerical results. Our results reveal that neither the SIA nor the SOSIA is as accurate as suggested by the asymptotic expansion approach. Also, in SOSIA the ice rheology needs to be altered to avoid infinite viscosity, though both our analytical and numerical solutions show that, especially for high bump amplitudes, the accuracy of the SOSIA is highly sensitive to this alternation. However, by updating the SOSIA solution in an iterative manner, we obtain a model which utilises the advantages of shallow ice approximations, while reducing the disadvantages.
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