A Mathematical Approach to the Theory of Glacier Sliding

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Abstract
Previous theories of glacier sliding are subject to the criticism that they are not properly formulated. Here we describe how the basal ice flow may be related to the bulk ice flow by means of the formal mathematical method of matched asymptotic expansions. A complete model of the basal sliding (involving coupled problems in ice, water film, and bedrock) may be rationally reduced by a dimensional analysis to a consideration of the ice flow only, and regelation may be neglected provided roughness is absent on the finest scales (<c. 1 mm). If the viscosity is supposed to be independent of the moisture content, then complementary variational principles exist which allow bounds on the drag to be obtained. In particular, these determine the magnitude of the basal velocity in terms of two crucial dimensionless parameters. Arguments are presented as to why realistic sliding laws should be taken as continuous functions of the temperature, and a (major) consequence of this assumption is mentioned. Finally the effect of cavitation is discussed, via an (exact) leading-order solution of the ice flow in the particular case of a Newtonian fluid and a “small” bedrock slope.

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