Transfer of Basal Sliding Variations to the Surface of a Linearly Viscous Glacier

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Abstract
The transfer of basal velocity anomalies to the surface of a glacier is investigated using a model of a planar parallel-sided slab (thicknessH) of linear viscous rheology. Surface velocity parallel (us) and normal (vs) to the surface is calculated for various spatial distributions of basal velocity anomalies with components parallel (ub) and normal (vb) to the surface. Four scales of differing behavior can be identified depending on the spatial lengthLof the basal anomalies. At very short scales (L≤ 1H) there is essentially no response at the surface. At short scales (1H≤L≤ 5H), a basal anomalyubinduces a response in bothusandvs. The spatial pattern ofusis such that velocity peaks inuscan be shifted from peaks inub, and may differ in number. The amplitude ofusis up to about 0.3|ub|. The amplitude of the cross-component effectvsmay be greater than the amplitude ofus. A basal anomalyvbinduces a response in bothvsandus. The pattern ofvsis the same as the pattern ofvb, and the amplitude ofvsis up to about 0.7 |vb|. The amplitude of the cross-component effectusis less than the amplitude ofvs. At intermediate scales (5H≤L≤ 10H), results differ from the short scale in two respects; velocity peaks inuscorrespond with peaks inub; and surface amplitudes are increased, except for cross-component effects for which surface amplitudes are of the same order as at the short scale. These cross-component effects at the short and intermediate scales show in particular that substantial anomalous surface-normal motions can be induced by deformation, even though the basal velocity anomaly is parallel to the surface. At long scales (10H≤L), the velocity anomaly at the surface is essentially the same as the anomaly at the bed. For all scales, the longitudinal strain-rate averaged over depth is larger in magnitude than the longitudinal strain-rate at the surface and, at the short scale, it may differ in sign, so thatvscannot be easily estimated from surface strain-rate. Although the simplifications of the model do not allow its rigorous quantitative application to field measurements, the results indicate the need for caution in interpreting surface-velocity variations in terms of basal velocity anomalies. It is important to establish the spatial pattern of surface motions for any chance of a confident interpretation in terms of basal motions.