Theoretical analysis of stress perturbations from a partially-lubricated viscous gravity current

We present a theoretical investigation into the dynamics of a viscous gravity current subjected to spatially-finite lubrication (i.e., a `slippery patch'). The work is motivated by grounded ice sheets flowing across patches of basal meltwater which reduce the ice-bed frictional coupling, causing perturbations enhancing ice motion, with implications for increased ice flux into the ocean and sea level rise. The flow is characterized by transitions between shear- and extension-dominated dynamics, which necessitates boundary-layer solutions at the transition points. We develop a depth-integrated analytical model of Newtonian flow which concisely reveals fundamental relationships between ice sheet geometry (thickness, surface slope, and slippery patch length) and the magnitude and spatial extent of resulting horizontal deviatoric stresses. This reduced-order analytical model shows good quantitative agreement with numerical simulations using 2-D Newtonian Stokes equations, which are further extended to the case of a non-Newtonian flow. From the reduced-order model, we rationalize that the slippery patch-induced stress perturbations are exponentially-decaying functions of distance upstream away from the patch onset. We also show that the amplitude of the perturbation scales linearly with the surface slope and patch length while the decay lengthscale scales linearly with ice thickness. These fundamental relationships have implications for the response of the Greenland Ice Sheet to the inland expansion of basal meltwater presence over the coming warming decades.