Cavitation on Deformable Glacier Beds

The formation of water-filled cavities at the interface between a glacier and its bed can significantly affect the drainage of meltwater along the base of a glacier, which in turn is one of the most important controls on glacier sliding. In this paper, we analyze a mathematical model for cavity formation on deformable glacier beds. By contrast with the case of rigid glacier beds, the cavities described here are the result of an interfacial instability in coupled ice-sediment flow. This instability causes bumps on the ice-sediment interface to grow until normal stress in the lee of bed bumps drops to the local porewater pressure, at which point the ice begins to lose contact with the surface of the sediment. We extend the basic instability model to cover the case of cavity formation, and analyze the corresponding traveling wave problem. This takes the form of a viscous contact problem in which the obstacle on the boundary—the traveling bed bump caused by the initial instability—must be determined as part of the solution. A classical complex variable method allows the traveling wave problem to be cast as an eigenvalue problem which is straightforward to solve numerically. Our results show that solutions for different wavelengths can be obtained from an apparently unique solution to a scaled problem, and that the amplitude of traveling waves increases with wavelength, while their speed decreases with wavelength.