First-Order Stresses and Deformations in Glaciers and Ice Sheets

Abstract

In this article the distribution of stress and velocities in glaciers and ice sheets is reinvestigated. We first derive the general equations governing non-linear viscous flow under plane deformations and formulate the relevant boundary conditions, including, in particular, a proper treatment of the accumulation–ablation mechanism. It is then shown how the emerging set of non-linear equations for the established boundary-value problem can be separated into a system covering steady-state problems on the one hand, and transient, time-dependent processes on the other hand. This separation is performed under the assumption that steady-state stresses are larger than the corresponding transient counterparts, suggesting a linearization of the transient equations with regard to the stresses. The steady-state equations are then analysed for the special case of an infinitely long, nearly parallel-sided slab. With the assumption that bottom undulations are small as compared to the glacier thickness it is shown that the original non-linear boundary-value problem can be decomposed into an infinite hierarchy of boundary-value problems defined on the simpler domain of the exactly parallel-sided slab, all of which are linear except for the lowest order one. Since its solution is readily available, the determination of the velocities and stresses due to bedrock protuberances is basically a linear problem, even though the constitutive response may be non-linear. Assuming harmonic bedrock undulations we show for a Navier–Stokes fluid that the transfer of the bedrock undulations to the surface strongly depends on the mean inclination of the slab, but, more importantly, does now show a maximum when plotted as a function of wavelength λ. This result is contradictory to the corresponding results of Budd (1970[a]) and implies serious drawbacks to his calculations of longitudinal stresses and strain-rates in his subsequent article (Budd, 1970[b]). Yet, it is not true that for maximal transfer of bottom protuberances to the surface a distinct wavelength would not exist. The calculations of Budd must rather be extended to include non-linear constitutive behaviour, variations of temperature with depth, and sliding at the bed. It then turns out that under certain circumstances maximal transfer of bottom undulations to the surface in a distinct wavelength domain (3 < λ < 5) may indeed exist. Sliding at the bed and vertical temperature variation thereby play a decisive role. Equally important is the stress distribution at the base, in particular the influence of the longitudinal strain effects on the latter. Rheological non-linearities, vertical temperature variations, and the sliding law at the bed play an important role and are investigated in detail. For non-linear constitutive behaviour and spatially dependent temperature-variation solutions must be sought numerically. The finite-difference scheme used suggests a generalization of Glen’s flow law so as to account for a nearly linear behaviour at low strain-rates. We conclude with a perspective of possible extensions of the general theory to various other time-dependent and time-independent problems.