Transmission of basal variability to a glacier surface

Transmission of basal variability to a glacier surface is investigated using analytical models for a linearly viscous medium. The three-dimensional transient response of the surface to both bedrock undulations and spatial variations in basal slipperiness for perturbations of arbitrary wavelengths is determined using perturbation methods. Both information transfer toward the surface and lateral transmission of horizontal stresses are strongly affected by the slip ratio, that is, the ratio of basal sliding to deformational velocity. For any mean bedrock slope, and above a minimum value of slip ratio, the amplitude transfer of bedrock undulations toward the surface has a local maximum at undulation span corresponding to about 3-8 times the mean ice thickness. The transmission of basal variability to a glacier surface increases quite significantly with increasing slip ratio. This explains why the surfaces of fast flowing ice streams are more undulating than the slower moving bordering areas. At slip ratios higher than about 100, the flow of glaciers and ice sheets becomes nonlocal in the sense that surface velocities and buildup and propagation of surface undulations cannot be calculated accurately on the basis of local thickness and slope. Using linearized long-wave theories at these slip ratios, instead of the more accurate arbitrary wavelength theory, gives estimates of decay times that are an order-of-magnitude too small and phase velocities several times too large. The problem of the propagation and decay of small-amplitude surface undulations on glaciers in three dimensions is solved. Small-amplitude surface waves on glaciers are strongly diffusive and dispersive. Redistribution of mass on ice sheets and glaciers is a diffusion process, and it is misleading, albeit not mathematically incorrect, to describe the reaction of glaciers to surface perturbations in terms of a wave propagation.