Exact solutions for ice flow across the no-slip to free-slip transition

Daniel Moreno-Parada , Frank Pattyn

The downhill flow of a viscous ice stream across a no-slip to free-slip transition remains elusive to exact solutions due to the existence of singularities. Here we replace the point-wise sharp transition by a functional dependency of parametrised horizontal length. Analytical solutions are found for the steady Navier-Stokes flow with Newtonian properties. Our solution remains smooth and well-behaved across the entire domain. The upper surface deflection is observed if sliding is sufficiently reduced upstream of the transition. Ice flow overturning is possible under sharp basal transitions, illustrating that ice-core dating can be biased by the transition zone effect. The pressure singularity arises in the limit where the basal transition becomes discontinuous. Previous singular solutions are a result of a physically unrealistic domain definition. A simple ice-sheet–shelf junction is constructed from the analytical solutions, and admissible grounding line positions are identified by exploring a range of ice flux values. These solutions satisfy both contact inequalities for multiple flux values, demonstrating that equilibrium positions are not discrete, but rather form a continuos set of solutions. As a result, the grounding line position is not univocally determined by the ice flux in the linearised problem.