Stress-gradient coupling in glacier flow: III. Exact longitudinal equilibrium equation

Abstract

The "vertically" integrated, exact longitudinal stress-equilibrium equation of Budd (1970) is developed further in. such a way as to yield an equation that gives explicitly and exactly the contributions to the basal shear stress made by surface and bed slope, surface curvature, longitudinal stress deviators, and longitudinal stress gradients in a glacier flowing in plane strain over a bed of longitudinally varying slope. With this exact equation, questions raised by various approximate forms of the longitudinal equilibrium equation can be answered decisively, and the magnitude of errors in the approximations can be estimated. To first order, in the angle δ that describes fluctuations in the surface slope ɑ from its mean value, the exact equilibrium equation reduces to (1 + 2sin^2θ)τ_B = pghsinɑ + 2G + T + B + K where G and T are the well-known stress-deviator-gradient and "variational stress" terms, K is a "longitudinal curvature" term, and B is a "basal drag" term that contributes a resistance to sliding across basal hills and valleys. Except for T, these terms are expressed in simple form and evaluated for practical situations. The bed slope θ (relative to the mean slope) is not assumed to be small, which allows the effects of bedrock topography to be determined, particularly through their appearance in the B term.