Earlier theories of Weertman and the present author are reviewed and compared; both are insufficient to account for the facts observed at the tongue of the Allalingletscher.
A calculation of the stresses and heat flow at the bed of a glacier with a sinusoidal profile is given which takes account of any degree of subglacial cavitation. The sliding due to plasticity and that due to pressure melting are related to this degree of cavitation and it is shown that these two terms are additive. There results an expression for the friction f
in terms of the total sliding velocity u and the height of the bumps a. For a given and large enough value of u, f
(a) exhibits two maxima which are equal and independent of u.
The paper then considers a more realistic model of the bed consisting of a superposition of sine waves all having the same roughness r, and a decreasing in a geometrical progression. The biggest a may be inferred from the overall profile of the bedrock; the resulting frictional force can be regarded either as part of the total frictional force f in an overall view for which f = ρgh sin α holds, or else as a correction to such a value on the small scale (the best point of view for crevasse studies). To a first approximation Coulomb’s law of friction holds provided one takes account of the interstitial water pressure at the ice-rock interface.
This interstitial pressure p is next related to the thickness of the glacier h. If the subglacial hydraulic system is at atmospheric pressure, p is proportional to h. Next, if the sliding velocity is not too large, the surface slope approaches 1.6r ≈ 0.12 and kinematic waves (which move four times as fast as the ice) disappear rapidly. If the hydraulic system is not at atmospheric pressure the surface slope is smaller and flow instabilities can occur.