Instability of thin-film flows with temperature-dependent viscosity

Using the shallow ice approximation, we explore the steady-state solutions and linear stability of a glacier flowing over an inclined plane subject to a constant geothermal heat flux. This model is derived by applying classical lubrication theory assumptions to the Stokes equations with a temperature and strain-rate-dependent viscosity. We first consider the steady flow of a parallel-sided layer of ice. By plotting the flux against the slope, we identify up to three steady-state solutions for a given slope, each corresponding to a branch of an S-shaped bifurcation diagram. As suggested by this shape, when the heat flux at the bottom of the ice is constant, the lower and upper branches are stable towards perturbations with no spatial structure, but the intermediate state is unstable. For finite streamwise wavenumber and with time-dependent heat diffusion into the bedrock, however, we find that all three branches are unstable. The lower and upper branches present a relatively weak instability towards waves with relatively long streamwise wavenumbers and infinite transverse wavelengths. Along the intermediate branch, there emerges an instability to waves with relatively short transverse wavelengths which is stationary and likely associated with ice stream formation. Finally, across all branches, the strongest instability occurs when the transversal wavelength is infinite and the streamwise wavenumber is relatively moderate. This instability is advective and potentially connected with glacier surges.