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

• Published in: Journal of Glaciology
• Main Authors: Kamb, Barclay, Echelmeyer, Keith A.
• Format: Article in Journal/Newspaper
• Language: English
• Published: Cambridge University Press (CUP) 1986
• Subjects: Earth-Surface Processes; Journal of Glaciology
• Online Access: http://dx.doi.org/10.1017/s0022143000015604; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022143000015604

Abstract 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 Δ u , Δ h , and ∆α from overall average (datum) values u o , h 0 , and α 0 , results in a differential equation that can be solved by Green’s function methods, giving Δ u ( 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 ...