Sliding motion of glaciers: Theory and observation

The sliding motion of glacier ice over bedrock, which contributes about half the flow velocity of temperate glaciers, is analyzed for arbitrary bedrock topography of low roughness. Fourier‐analyzed topography is represented by a roughness spectral function ζ( h, k ) defined in terms of the mean square topographic amplitude. From an essentially exact solution of the sliding problem for linear ice‐flow rheology, an approximate solution for the actual nonlinear rheology is built on the assumption that the second strain‐rate invariant depends only on distance from the ice‐bedrock contact. The transition wavelength λ 0 between regelation and plastic flow, constant in the linear theory, is replaced in the nonlinear theory by a velocity‐ and roughness‐dependent parameter λ α that plays a similar role. Detailed results are given for three special types of ζ( h, k ): (1) white roughness (|ζ| constant); (2) truncated white roughness (|ζ| constant for all wavelengths above a certain lower limit); (3) a single wavelength; and (4) cross‐corrugated sinusoidal waves. The results are tested against field observations of sliding. Given sliding velocity υ, basal shear stress τ, and rheological parameters, the theory predicts roughness values ζ for the different types of ζ( h, k ). When compared with ζ values inferred from observed bedrock outcrops, predicted values for white roughness are somewhat too small, whereas for white roughness truncated at 3.53 meters, they are of the expected size (ζ ∼ 0.05). Predicted λ α values range from 3 to 112 cm; high υ (>20 m yr −1 ) generally gives λ α in the range 10–40 cm, and low υ (<6 m yr −1 ) 30–70 cm. The predicted thickness of the regelation layer (1–10 mm) agrees with observation, but the predicted λ α values appear to be somewhat too small. Extensive separation of the ice sole from bedrock, due to tensile stresses set up in sliding, is predicted in icefalls, whereas for valley glaciers little separation is predicted, unless meltwater under a head of pressure comparable to half the glacier thickness has access to the bed. Extensive separation is not needed to account for typical sliding velocities, provided that the roughness spectrum is truncated. Observed features of glaciated bedrock indicate truncation, which results from glacial abrasion. For the truncated spectrum, the predicted dependence of υ on τ is much more highly nonlinear than for the full white spectrum; this implies a relatively high sensitivity of sliding velocity to changes in glacier thickness or surface slope.