# A multilayer ice flow model generalizing the Shallow Shelf Approximation

**Guillaume Jouvet
**

Institut für Mathematik

Freie Universität Berlin

Berlin, Germany

### Abstract:

A new hybrid model for the dynamics of glaciers, ice sheets and ice shelves is introduced. Unlike the traditional ice flow models that are simplified by neglecting higher-order terms in the aspect ratio, here, the simplifications rely on an empirical approximation of the streamlines. In this "multilayer" model the domain of ice consists of a pile of thin layers, which are aligned with the streamlines and which can spread out, contract and slide over each other such that the two most relevant types of stresses are accounted for: the membrane ones and the vertical shear ones. Assuming the velocity field to be vertically piecewise-constant in each layer, the model derives from local depth-integrations of the hydrostatic approximation of the Stokes equations. These integrations give rise to interlayer tangential stresses, which are redefined by keeping the vertical shear stress components in the local frame of the interface. The final model consists of a tridiagonal system of two-dimensional non-linear elliptic equations, the size of this system equal to the number of layers. By construction, the model is a multilayer generalisation of the Shallow Shelf Approximation (SSA). Like the SSA, the multilayer model can be advantageously reformulated as a minimisation problem. However, in contrast to the SSA, the functional to be minimised involves a new penalisation-like term representing the vertical shear stresses, which is added to the main non-linear term representing the membrane stresses. Taking advantage of this decoupling, numerical techniques developed for the SSA can be easily extended. When running the model for prognostic benchmark experiments ISMIP-HOM, the multilayer solutions show very good agreements with the Stokes solution if no severe depression occurs in the bedrock. The multilayer approach, which is of mathematical 2D complexity, offers to glacier and ice sheets modellers a mechanically exhaustive and computationally efficient alternative to 3D models when a good approximation of the streamlines is available as it is the case in many practical applications.