Stress-gradient coupling in glacier flow: IV. Effects of the "T" term

Abstract

The "T term" in the longitudinal stress equilibrium equation for glacier mechanics, a double y-integral of ∂^2T_(xy)/∂x^2 where x is a longitudinal coordinate and y is roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stress T_(xy) at any depth to the local shear stress T_B at the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from the T term are as follows: 1. The longitudinal coupling length I is increased by about 5%. 2. The asymmetry parameter σ is altered by a variable but small amount depending on longitudinal gradients in ice thickness h and surface slope ɑ. 3. There is a significant direct modification of the influence of local h and ɑ on flow, which represents a distinct "driving force" in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂T_(xy)/∂_x. For longitudinal variations in h, the "T force" varies as d^2h/dx^2 and results in an in-phase enhancement of the flow response to the variations in h, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in ɑ, the "force" varies as dɑ/dx and gives a phase-shifted flow response. Although the "T force" is not negligible, its actual effect on T_B and on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ ≾2.5h), where the flow response to variations in h does not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the "T force", 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 the T term can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging of T_B, one over which the T term 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 the T term on the flow perturbations Δu that result from perturbations Δh and Δɑ 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 λ ≳ h. For short wavelengths (λ ≾ h), effects of the T term become important in longitudinal coupling, as will be shown in a later paper in this series.

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