Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

Abstract

AbstractFor a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocityu(“vertically” averaged), ice thicknessh, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) valuesuo,h0, andα0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of∆h(x) and ∆α(x),xbeing the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of localh(x) and α(x) on the flowu(x):where the integration is over the lengthLof the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variablex′, represents the influence of localh(x′),α(x′), and channel-shape factorf(x′), at longitudinal coordinatex′, on the flowuat coordinatex, the influence being weighted by the “influence transfer function” exp (−|x′ −x|/ℓ) in the integral.The quantityℓthat appears as the scale length in the exponential weighting function is called thelongitudinal coupling length. It is determined by rheological parameters via the relationship, wherenis the flow-law exponent,ηthe effective longitudinal viscosity, andηthe effective shear viscosity of the ice profile,ηis an average of the local effective viscosityηover the ice cross-section, and (η)–1is an average of η−1that gives strongly increased weight to values near the base. Theoretically, the coupling lengthℓis generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer,ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations inhor α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called thelongitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length4lthat gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding toℓ/h≈ 2 in both cases.Ifℓvaries withx, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation,ℓin the averaging integral isℓ(x) but is not a function ofx′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.