Linear analysis of ice-shelf topography response to basal melting

Floating ice shelves in Antarctica and Greenland limit land-ice contributions to sea level rise by resisting the flow of grounded ice. Melting at the surface and base of ice shelves can lead to destabilisation by promoting thinning and fracturing. Basal melting often results in channelised features that manifest as surface topography due to buoyancy. The assumption of hydrostatic flotation commonly underlies estimates of basal melting rates. However, numerical simulations and ice-penetrating radar data have shown that narrow topographic features do not necessarily satisfy the local flotation condition. Here, we introduce a linearised model for ice-shelf topographic response to basal melting perturbations to quantify deviations from hydrostatic flotation and the stability of topography. While hydrostatic flotation is the dominant behaviour at wavelengths greater than the ice thickness, ice elevation can deviate from the perfect flotation condition at smaller wavelengths. The linearised analysis shows that channelised features can be stable when the timescale of extensional thinning is small relative to the timescale of viscous flow towards the channel. When extension is non-negligible, channels can break through the ice column. We validate the linearised analysis by comparing numerical solutions to a nonlinear ice-flow model with steady-state solutions obtained via a Green's function.