Basal-flow characteristics of a linear medium sliding frictionless over small bedrock undulations

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Abstract
The basal deformation of a gravity-driven linear creeping flow sliding frictionless over slowly varying bed undulations in two dimensions is analysed analytically, using results from second-order perturbation theory. One of the key results is that, close to sinusoidal bedrock undulations, up to two different spatial regions of local extrusion flow may arise. The offset and onset of extrusion flow is controlled primarily by the amplitude-to-wavelength ratio. Above the crest of a sinusoidal bed line, a local maximum of the surface-parallel velocity develops for ε : =ak< 0.138, whereais the amplitude andkis the wave number. Asεincreases from zerо to this critical value, the vertical position of the velocity maximum moves fromkz= 1 tokz≈ 1.98, wherezis the vertical distance above the mean bed line. Within and above the trough of a sinusoid, a region of local minimum of the surface-parallel velocity component develops, which shifts fromkz= 1 towards the bed line asεincreases front zero to 1/2. Below this velocity minimum, and for some distance above the velocity maximum, the surface-parallel velocity increases with depth. This type of extrusion flow will cause a reversal of borehole-inclination profiles close to the bedrock.

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