Stress-Gradient Coupling in Glacier Flow: II. Longitudinal Averaging in the Flow Response to Small Perturbations in Ice Thickness and Surface Slope

Abstract

AbstractAs a result of the coupling effects of longitudinal stress gradients, the perturbations ∆uin glacier-flow velocity that result from longitudinally varying perturbations in ice thickness ∆hand surface slope ∆α are determined by a weighted longitudinal average ofϕh∆handϕα∆α, whereϕhandϕαare “influence coefficients” that control the size of the contributions made by local ∆hand ∆α to the flow increment in the longitudinal average. The values ofϕhandϕα depend on effects of longitudinal stress and velocity gradients in the unperturbed datum state. If the datum state is an inclined slab in simple-shear flow, the longitudinal averaging solution for the flow perturbation is essentially that obtained previously (Kamb and Echelmeyer, 1985) with equivalent values for the longitudinal coupling lengthland withϕh=n+ 1 andϕα+n, wherenis the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practiceϕα differs little fromn, whereasϕhcan differ considerably fromn+ 1. The weighting function in the longitudinal averaging integral, which is the Green’s function for the longitudinal coupling equation for flow perturbations, can be approximated by an asymmetric exponential, whose asymmetry depends on two “asymmetry parameters”μand σ, whereμis the longitudinal gradient of ℓ(= dℓ/dx). The asymmetric exponential has different coupling lengths ℓ+and ℓ−for the influences from up-stream and from down-stream on a given point of observation. If σ/μis in the range 1.5–2.2, as expected for flow perturbations in glaciers or ice sheets in which the ice flux is not a strongly varying function of the longitudinal coordinatex, then, when dℓ/dx> 0, the down-stream coupling length ℓ+is longer than the up-stream coupling length ℓ−, and vice versa when dℓ/dx< 0. Flow-, thickness- and slope-perturbation data for Blue Glacier, obtained by comparing the glacier in 1957–58 and 1977–78, require longitudinal averaging for reasonable interpretation. Analyzed on the basis of the longitudinal coupling theory, with 4ℓ + 1.6 km up-stream, decreasing toward the terminus, the data indicatento be about 2.5, if interpreted on the basis of aresponse factor Ѱ+ 0.85 derived theoretically by Echelmeyer (unpublished) for the flow response to thickness perturbations in a channel of finite width. The data contain an apparent indication that the flow response to slope perturbations is distinctly smaller, in relation to the response to thickness perturbations, than is expected on a theoretical basis (i.e.ϕα/ ϕh+ (n/n+ 1) for a slab). This probably indicates that the effective ℓ is longer than can be tested directly with the available data set owing to its limited range inx.