# Stress-gradient coupling in glacier flow: II. Longitudinal averaging of the flow response to small perturbations in ice thickness and surface slope

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## Abstract

As a result of the coupling effects of longitudinal stress gradients, the perturbations Δu in glacier-flow velocity that result from longitudinally varying perturbations in ice thickness Δh and surface slope Δɑ are determined by a weighted longitudinal average of ФhΔh and Ф_ɑΔɑ, where Фh and Ф_ɑ are "influence coefficients" that control the size of the contributions made by local Δh and Δɑ to the flow increment in the longitudinal average. The values of Ф_h and Ф_ɑ 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 length ℓ and with Ф_h = n + l and Ф_ɑ + n, where n is the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practice Ф_ɑ differs little from n, whereas Ф_h can differ considerably from n + 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" µ and σ, where µ is 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/µ is 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 coordinate x, 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 indicate n to be about 2.5, if interpreted on the basis of a response factor ψ + 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. Ф_ɑ/Ф_h + 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 in x.

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