A sliding law for glaciers of constant viscosity in the presence of subglacial cavitation

A method of solution for the problem of slow flow of a Newtonian viscous glacier slipping over a rough bed is constructed, for the case where cavities form when the lubricating water film pressure reaches that of the local subglacial drainage system. The treatment of Nye (Proc. R. Soc. Lond. A 311, 445-477 (1969)) is reformulated as a Hilbert problem, and the solution presented for the particular case of a periodic bedrock with one cavity per period. For such bedrocks, it is found that the basal stress has a maximum for a finite basal velocity, and the basal stress decreases towards zero as the velocity tends to infinity, in line with the suggestion ofLliboutry (J.Glaciol. 23,67-95 ( I 979)). For more complicated bedrocks, with many different obstacle sizes, direct solution appears impractical and some kind of further approximation seems advisable.