A calculation on the sliding of ice over a wavy surface using a Newtonian viscous approximation

A glacier slides over its irregular rock bed by a combination of regelation and plastic deformation (Weertman 1957). An exact calculation for this combined process is made possible by using a model in which the flow properties of the ice are simplified as Newtonian viscous, rather than obeying a more realistic nonlinear flow law. The bed is represented as a smooth plane on which there are perturbations of general three-dimensional form but small slope, and the ice is assumed to maintain contact with the bed everywhere. The first-order solution for the velocity field leads to an expression for the drag, which is a second-order effect. It is found that the velocities due to regelation and to viscous flow are additive only when the bed consists of a single sine wave. In the general case the total drag is a summation of the drags due to each of the Fourier components of the bed relief taken separately. The total drag is expressible in terms of a single average property of the bed relief, namely, the product of its mean square amplitude and its autocorrelation function, or, alternatively, its power spectrum. Numerical illustrations are given for a Gaussian autocorrelation function.