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Spatiotemporal dynamics of ice streams due to a triple-valued sliding law

Published online by Cambridge University Press:  02 December 2009

ROIY SAYAG  and
ELI TZIPERMAN
ROIY SAYAG*
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
ELI TZIPERMAN
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
*
†Email address for correspondence: sayag@fas.harvard.edu
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Abstract

We show that a triple-valued sliding law can be heuristically motivated by the transverse spatial structure of an ice-stream velocity field using a simple one-dimensional model. We then demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. We find a generation of rapid stream-like solutions within a slow ice-sheet flow, separated by narrow internal boundary layers (shear margins), and analyse numerical simulations in two horizontal dimensions over a homogeneous bed and including longitudinal shear stresses. Different qualitative behaviours are obtained by changing a single physical parameter, a mass source magnitude, leading to changes from a slow creeping flow to a relaxation oscillation of the stream pattern, and to steady ice-stream-like solution. We show that the adjustment of the ice-flow shear margins to changes in the driving stress in the one-dimensional approximation is governed by a form of the Ginzburg–Landau equation and use stability analysis to understand this adjustment. In the model analysed here, the width scale of the stream is not set spontaneously by the ice flow dynamics, but rather, it is related to the mass source intensity and spatial distribution.

JFM classification

Geophysical and Geological Flows: Ice sheets Instability: Instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 483 - 505
Copyright
Copyright © Cambridge University Press 2009

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References

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Sayag and Tziperman supplementary movie

Movie 1. The solution with small mass source intensity M0=30 m/yr (no ice-stream). The large panel on the right shows the velocity field (colors represent dimensionless speed), the top-left panel shows v(x,y=0.5), and the lower-left panel shows the total mass source influx (blue) and the total mass out-flux (magenta) across y=0.5. This mass source is too weak to drive the flow across the bottom shear-stress threshold (τc1) and therefore the velocity is everywhere lower than vc1. The total outflow mass flux converges to the value of the total influx within ~1000 years and the flow settles in a steady state.

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Video 1.6 MB

Sayag and Tziperman supplementary movie

Movie 2. The solution with large mass source intensity, M0=150 m/yr (steady ice-stream). Panels are described in the caption of Movie 1. A wide stream-like pattern is formed near the mass source, and advances downstream. The shear-margins then converge towards the stream center until a steady pattern of ice-stream is reached within 15-20 years.

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Video 1.5 MB

Sayag and Tziperman supplementary movie

Movie 3. The solution with intermediate mass source intensity, M0=100 m/yr (relaxation-oscillation). Panels are described in the caption of Movie 1. A stream-like pattern is formed near the mass source, and advances downstream as the stream margins move away from the center (x=0.5). The margins then converge back towards the stream center, and the outflow mass flux declines simultaneously. The distance between the margins declines significantly so that they overlap. Consequently, the stream pattern collapses to a no-stream flow (time=~6.5 years) and a new cycle initiates. The motion of the shear-margins with respect to the ice-stream center (x=0.5) is related to the presence of ice thickness gradients, as explained in the text.

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Sayag and Tziperman supplementary movie

Movie 4. The solution with intermediate mass source intensity, M0=100 m/yr (relaxation-oscillation-mode). Top panel shows the thickness field, h(x,y) (aspect ratio 1:140) and lower panel shows the velocity field (u,v) as in Movie 3. This shows how the evolution in the thickness distribution and the velocity are coupled. The flow accelerate as the ice surface slopes increase upstream. At t=3 years, a stream-pattern is formed and the out-flow mass flux sharply inclines. As a result, the ice thickness diminishes rapidly, and the slopes get smoothed. Once the stream collapses, surface slopes build up again up-stream, driving a new cycle.

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Video 1.4 MB