  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.
