The Use of a Rational Model in the Mathematical Analysis of a Polythermal Glacier

Abstract

AbstractWe here describe the process by which a complex model set of equations and boundary conditions may be rationally reduced to a simpler and more manageable set by the. processes of non-dimensionalization and asymptotic approximation. Such a reduced model (derived elsewhere) is then presented for an incompressible, two-dimensional ice flow. It consists of two coupled equations for the stream function and enthalpy variable, together with a complex set of boundary conditions.The important dimensionless parameters which arise are given, and various limiting values of these are commented on. Nye’s (1960) equation for kinematic waves may be reproduced, and a non-linear analysis of this reveals that disturbances reach the glacier snout in finite time, and are uniformly bounded there: in the particular case considered here, one can also show that the temperature field is stable.It is shown that the effect of introducing a (realistic) sliding law which is continuously dependent on the temperature has a major effect on the bedrock temperature profile.Lastly we consider seasonal waves using a kinematic wave equation based on a plausible form of the sliding law when cavitation is present. The main observed features are qualitatively reproduced.