Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope
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, b...
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Cambridge University Press (CUP)
1986
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Online Access: | http://dx.doi.org/10.1017/s0022143000015604 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022143000015604 |
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crcambridgeupr:10.1017/s0022143000015604 2024-04-07T07:53:42+00:00 Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope Kamb, Barclay Echelmeyer, Keith A. 1986 http://dx.doi.org/10.1017/s0022143000015604 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022143000015604 en eng Cambridge University Press (CUP) Journal of Glaciology volume 32, issue 111, page 267-284 ISSN 0022-1430 1727-5652 Earth-Surface Processes journal-article 1986 crcambridgeupr https://doi.org/10.1017/s0022143000015604 2024-03-08T00:36:29Z 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 ... Article in Journal/Newspaper Journal of Glaciology Cambridge University Press Journal of Glaciology 32 111 267 284 |
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Cambridge University Press |
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crcambridgeupr |
language |
English |
topic |
Earth-Surface Processes |
spellingShingle |
Earth-Surface Processes Kamb, Barclay Echelmeyer, Keith A. Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
topic_facet |
Earth-Surface Processes |
description |
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 ... |
format |
Article in Journal/Newspaper |
author |
Kamb, Barclay Echelmeyer, Keith A. |
author_facet |
Kamb, Barclay Echelmeyer, Keith A. |
author_sort |
Kamb, Barclay |
title |
Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
title_short |
Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
title_full |
Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
title_fullStr |
Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
title_full_unstemmed |
Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope |
title_sort |
stress-gradient coupling in glacier flow: i. longitudinal averaging of the influence of ice thickness and surface slope |
publisher |
Cambridge University Press (CUP) |
publishDate |
1986 |
url |
http://dx.doi.org/10.1017/s0022143000015604 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022143000015604 |
genre |
Journal of Glaciology |
genre_facet |
Journal of Glaciology |
op_source |
Journal of Glaciology volume 32, issue 111, page 267-284 ISSN 0022-1430 1727-5652 |
op_doi |
https://doi.org/10.1017/s0022143000015604 |
container_title |
Journal of Glaciology |
container_volume |
32 |
container_issue |
111 |
container_start_page |
267 |
op_container_end_page |
284 |
_version_ |
1795669777586323456 |