## Shallow ice approximation

Most numerical models of ice sheets use what is known as the shallow ice approximation. While the shallow ice approximation was first introduced in glaciology by Fowler and Larson (1978) and in slightly different form by Hutter (1981), our discussion is based on Hutter (1983, p. 256ff). This approximation makes use of the fact that the horizontal extent of an ice sheet is large compared with its thickness. Longitudinal derivatives of stress, velocity, and temperature are thus small compared with vertical derivatives. If the wavelengths of major undulations in the surface and bed topography are relatively long, so the surface and bed elevations are slowly varying functions of x, the longitudinal coordinate can be scaled by using the relation f = ¡xx, where x is small. One logical possibility is to take x as the ratio of the mean thickness to the horizontal extent of the ice sheet. The vertical coordinate is not changed. With this scaling, x is introduced into the momentum balance, energy balance, and continuity equations, and into the boundary conditions. If all terms involving x are then neglected and the resulting equations are solved, the solution is referred to as the zeroth-order solution. In this solution, the stress and velocity fields are calculated as if the ice sheet were a slab of uniform thickness, and the basal shear stress turns out to be pgha. Longitudinal stress gradients are thus not included. If terms involving xl are included in the solution, the result is called a first-order solution. Longitudinal stress gradients are included in this solution, and a flow law that includes a linear term at low stresses is necessary to avoid a singularity in these stresses at the surface. This is the solution used in most existing finite-difference models of ice sheets. As might be expected from the above, a solution including terms in x2 is called a second-order solution. In

Figure 11.4. A possible finite-element discretization of a tapered glacier margin, showing the flexibility of the finite-element method to accommodate such geometries.

modeling ice shelves, longitudinal stress effects first appear in second-order solutions. Solutions including j2 or higher powers of j are called higher-order solutions.

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