Nonlinear longitudinal stress coupling in glacier and ice sheet flow

The Greenland and Antarctic Ice sheets exhibit high variability in flow speed, over multiple orders of magnitude. Faster flow in ice streams, marine terminating glaciers, and ice shelves is described by the Shallow Shelf/Shelfy-Stream Approximation (SSA), which requires a nonlocal balance between driving stress, friction at the ice-bed interface, and longitudinal/membrane stresses. Nonlocal stresses mediate spatial variability in ice flow, over a finite length scale, known as the longitudinal coupling length, which determines how far viscous stresses may be transmitted in glaciers and ice sheets. The strongly nonlinear rheology of ice complicates the stress transmission, but previous work has relied on either linear-Newtonian models or linearized, small-perturbation models, to determine the coupling length. Here, we derive new exact solutions to SSA, which explain how nonlinear feedbacks between the stress state and the nonlinear rheology of ice determine the coupling length. For complex and multi-scale flow fields, these exact solutions provide a foundation for approximations that can reconstruct the effects of a nonlocal stress balance, given input data for the driving stress and friction fields.