A novel approximation for the flow of ice in narrow valleys

Recently, a very efficient numerical scheme for the shallow ice approximation (SIA) was proposed. It technically even allows for 0.25 year time increments at spatial resolutions of 25 m, which would make it a very interesting tool for simulating alpine glaciers. However, the SIA in its simplest form neglects all horizontal stress components, which leads to severe limitations in alpine valleys, where the ice thickness is typically not sufficiently small compared to the valley width. The question of whether there is a chance to extend the new numerical scheme to more complex flow models remained open.

Here, preliminary tests of an approach that somehow builds a new house from the roof are presented. The idea is not to include the additional terms of existing extensions of the SIA (e.g., second-order terms or a combination with the shallow shelf approximation) in the numerical scheme, but to develop a new approximation to the Stokes equations that already harmonizes well with the new numerical scheme. As a key point, the basic structure of the SIA is kept in the sense that ice flow still follows the steepest decline of the ice surface (hydrostatic approximation). In turn, the diffusivity term of the SIA is no longer parameterized directly by the ice thickness and the slope, but described by an additional differential equation. This differential equation is developed explicitly for taking into account the transverse shear stresses, which contribute much to the deficiencies of the SIA in narrow alpine valleys.

The first results, obtained from 2-D simulations of valley cross sections, are very promising. The additional differential equation can be parameterized in such a way that the error in total ice flux is only a few percent for various valley shapes even down to aspect ratios of 2:1. The across-valley profile of the surface velocity is also reproduced quite well as long as sliding is not too strong. As a main limitation, however, longitudinal stress components cannot be included easily.