Spatiotemporal dynamics of ice streams due to a triple-valued sliding law

We show that a triple-valued sliding law can be heuristically motivated by the transverse spatial structure of an ice-stream velocity field using a simple one-dimensional model. We then demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. We find a generation of rapid stream-like solutions within a slow ice-sheet flow, separated by narrow internal boundary layers (shear margins), and analyse numerical simulations in two horizontal dimensions over a homogeneous bed and including longitudinal shear stresses. Different qualitative behaviours are obtained by changing a single physical parameter, a mass source magnitude, leading to changes from a slow creeping flow to a relaxation oscillation of the stream pattern, and to steady ice-stream-like solution. We show that the adjustment of the ice-flow shear margins to changes in the driving stress in the one-dimensional approximation is governed by a form of the GinzburgLandau equation and use stability analysis to understand this adjustment. In the model analysed here, the width scale of the stream is not set spontaneously by the ice flow dynamics, but rather, it is related to the mass source intensity and spatial distribution.