Stress-gradient Coupling in Glacier Flow: IV. Effects of the “T” Term

Abstract

AbstractThe “Tterm” in the longitudinal stress-equilibrium equation for glacier mechanics, a doubley-integral of ∂2τxy/∂x2wherexis a longitudinal coordinate andyis roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stressτxyat any depth to the local shear stressτBat the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from theTterm are as follows: 1. The longitudinal coupling lengthis increased by about 5%. 2. The asymmetry parameterσis altered by a variable but small amount depending on longitudinal gradients in ice thicknesshand surface slopeα. 3. There is a significant direct modification of the influence of localhandαon flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x.For longitudinal variations inh, the “Tforce” varies as d2h/dx2and results in an in-phase enhancement of the flow response to the variations inh, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dxand gives a phase-shifted flow response. Although the “Tforce” is not negligible, its actual effect onτBand on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ2.5h), where the flow response to variations inhdoes not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “Tforce”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of theTterm can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging ofτb, one over which theTterm is negligible and one over which it is not.Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of theTterm on the flow perturbations Δuthat result from perturbations Δhand Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.The foregoing conclusions apply for long wavelengths λh.For short wavelengths (λh), effects of theTterm become important in longitudinal coupling, as will be shown in a later paper in this series.