For 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 velocity u (“vertically” averaged), ice thickness h, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) values uo, h0, and α0, 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), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and α(x) on the flow u(x):
where the integration is over the length L of the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x′, represents the influence of local h(x′), α(x′), and channel-shape factor f(x′), at longitudinal coordinate x′, on the flow u at coordinate x, the influence being weighted by the “influence transfer function” exp (−|x′ − x|/ℓ) in the integral.
The quantity ℓ that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. It is determined by rheological parameters via the relationship , where n is the flow-law exponent, η the effective longitudinal viscosity, and η the effective shear viscosity of the ice profile, η is an average of the local effective viscosity η over the ice cross-section, and (η)–1 is an average of η−1 that gives strongly increased weight to values near the base. Theoretically, the coupling length ℓ is 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 in h or 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 the longitudinal 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 length 4l that 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 to ℓ/h ≈ 2 in both cases.
If ℓ varies with x, 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, ℓ in the averaging integral is ℓ(x) but is not a function of x′. 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.