On the Theory of the Advance and Retreat of Glaciers

The main effect on a glacier of a change of climate is to change the rate of supply (accumulation) and removal (ablation) of ice. As a result the end of the glacier changes in position. The differential equations governing this phenomenon, derived in a previous paper, show that the effects of changes of accumulation and ablation are propagated down a glacier as kinematic waves. The present paper examines the effect of diffusion of the waves, since diffusion can have a large effect on quantitative results.The validity of the basic assumptions is re-examined and particular attention is given to the proper choice of boundary conditions. A special model which shows a realistic amount of diffusion allows an explicit solution for the response to any variation in rate of accumulation, in terms of certain averages over past history. The responses to a stepfunction and to a pulse are found, and also the response to a harmonic variation of rate of accumulation as a function of frequency. The inverse problem of calculating the climatic changes from the variation of the position of the end of the glacier has a simple general solution for this model. An interesting asymmetry is then revealed between two inference problems: to calculate the current glacier behaviour requires a record of the climate extending back for many hundreds of years; but to calculate the current climate only requires a record of the glacier behaviour over the very recent past (say 10 yr).For an actual glacier the frequency response can be calculated by numerical methods. The usefulness of the special model considered in this paper is that it indicates the general nature of the results to be expected from numerical analysis.