Instability of thin-film flows with temperature-dependent viscosity

We explore thermo-viscous instabilities in an idealised setting motivated by ice-sheet flow, modelled as a shallow layer of fluid with a temperature-dependent viscosity flowing down an inclined plane and subject to a constant geothermal heat flux. In this setting, multiple steady states are possible for a given slope, and we examine their linear stability to general wave-like perturbations in both the streamwise and transverse directions. We confirm that the lower and upper branches are stable with respect to perturbations that are spatially uniform in the two horizontal directions, but the intermediate state is unstable. For finite-wavenumber perturbations and with heat diffusion into the underlying bedrock, we find that all three branches are unstable. The lower and upper branches exhibit a relatively weak instability with respect to waves with finite streamwise wavelength and long transverse wavelengths, caused by diffusion of heat into the bedrock. Across all branches, the strongest instability occurs when the transverse wavelength is infinite and streamwise wavelengths are order one. Along the intermediate branch, modes remain unstable for all transverse wavenumbers, and growth rates do not decay in the limit of short transverse wavelengths. Motivated by this observation, we also briefly examine a simplified nonlinear problem for transverse perturbations, in order to clarify the implications of this missing short-wavelength cutoff.