Supplementary material from "Ice sheet flow with thermally activated sliding. Part 2: the stability of subtemperate regions"

The onset of sliding in ice sheets may not take the form of a sharp boundary between regions at the melting point,  in which sliding is permitted,  and regions below that temperature,  in which there is no slip. Such a hard switch leads to the paradox of the bed naturally wanting to refreeze as soon as sliding has commenced. A potential alternative structure is a region of subtemperate sliding. Here temperatures are marginally below the melting point and sliding velocities slower than they would if the bed was fully temperate. Rather than being controlled by a standard sliding law,  sliding velocities are then constrained by the need to maintain energy balance. This thermal structure arises in temperature - dependent sliding laws in the limit of strong sensitivity to temperature. Here,  we analyse the stability of such subtemperate regions,  showing that they are subject to a set of instabilities that occur at all length scales between ice thickness and ice sheet length. The fate of these instabilities is to cause the formation of patches of frozen bed,  raising the possibility of highly complicated cold - to - temperate transitions with spatial structures at short length scales that cannot be resolved in large - scale ice sheet simulation codes.