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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: [email protected]
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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

REFERENCES

Anandakrishnan, S. & Alley, R. B. 1997 Stagnation of Ice Stream C, West Antarctica by water piracy. Geophys. Res. Lett. 24, 265–268.CrossRefGoogle Scholar
Balmforth, N. J., Craster, R. V. & Toniolo, C. 2003 Interfacial instability in non-Newtonian fluid layers. Phys. Fluids 15 (11), 3370–3384.CrossRefGoogle Scholar
Bindschadler, R. A., King, M. A., Alley, R. B., Anandakrishnan, S. & Padman, L. 2003 Tidally controlled stick-slip discharge of a West Antarctic ice stream. Science 301 (5636), 1087–1089.CrossRefGoogle Scholar
Boulton, G. S. & Hindmarsh, R. C. A. 1987 Sediment deformation beneath glaciers – rheology and geological consequences. J. Geophys. Res. 92 (B9), 9059–9082.CrossRefGoogle Scholar
Boulton, G. S., Lunn, R., Vidstrand, P. & Zatsepin, S. 2007 a Subglacial drainage by groundwater-channel coupling, and the origin of esker systems. Part 1. Glaciological observations. Quat. Sci. Rev. 26 (7–8), 1067–1090.CrossRefGoogle Scholar
Boulton, G. S., Lunn, R., Vidstrand, P. & Zatsepin, S. 2007 b Subglacial drainage by groundwater-channel coupling, and the origin of esker systems. Part II. Theory and simulation of a modern system. Quat. Sci. Rev. 26 (7–8), 1091–1105.CrossRefGoogle Scholar
Clark, P. U. & Walder, J. S. 1994 Subglacial drainage, eskers, and deforming beds beneath the Laurentide and Eurasian ice sheets. Geol. Soc. Am. Bull. 106 (2), 304–314.2.3.CO;2>CrossRefGoogle Scholar
DeGiuli, E. & Schoof, C. 2008 Hysteresis and chaos in ice stream behaviour. In Geophysical Research Abstracts, pp. 10, EGU2008-A-00854. EGU.Google Scholar
Fountain, A. G. 1994 Borehole water-level variations and implications for the subglacial hydraulics of South Cascade Glacier, Washington State, USA. J. Glaciol. 40 (135), 293–304.CrossRefGoogle Scholar
Fowler, A. C. 1986 A sliding law for glaciers of constant viscosity in the presence of subglacial cavitation. Proc. R. Soc. Lond. Ser. A – Math. Phys. Engng Sci. 407 (1832), 147–170.Google Scholar
Fowler, A. C. 1987 a Sliding with cavity formation. J. Glaciol. 33 (115), 255–267.CrossRefGoogle Scholar
Fowler, A. C. 1987 b A theory of glacier surges. J. Geophys. Res. 92 (B9), 9111–9120.CrossRefGoogle Scholar
Fowler, A. C. 1989 A mathematical analysis of glacier surges. SIAM J. Appl. Math. 49 (1), 246–263.CrossRefGoogle Scholar
Fowler, A. C. & Johnson, C. 1995 Hydraulic run-away – a mechanism for thermally regulated surges of ice sheets. J. Glaciol. 41 (139), 554–561.CrossRefGoogle Scholar
Fowler, A. C. & Johnson, C. 1996 Ice sheet surging and ice stream formation. Ann. Glaciol. 23, 68–73.CrossRefGoogle Scholar
Fowler, A. C. & Schiavi, E. 1998 A theory of ice-sheet surges. J. Glaciol. 44 (146), 104–118.CrossRefGoogle Scholar
Gagliardini, O., Cohen, D., Raback, P. & Zwinger, T. 2007 Finite-element modelling of subglacial cavities and related friction law. J. Geophys. Res. 112 (F2), F02027.1–F02027.13.Google Scholar
Greenberg, J. M. & Shyong, W. 1990 Surging glacial flows. IMA J. Appl. Math. 45 (3), 195–223.CrossRefGoogle Scholar
Hagberg, A. & Meron, E. 1993 Domain-walls in nonequilibrium systems and the emergence of persistent patterns. Phys. Rev. E 48 (2), 705–708.CrossRefGoogle ScholarPubMed
Hagberg, A. & Meron, E. 1994 Pattern-formation in nongradient reaction–diffusion systems – the effects of front bifurcations. Nonlinearity 7 (3), 805–835.CrossRefGoogle Scholar
Hindmarsh, R. C. A. 2006 Stress gradient damping of thermoviscous ice flow instabilities. J. Geophys. Res. 111 (B12409), 1–8, doi:10.1029/2005JB004019.CrossRefGoogle Scholar
Hindmarsh, R. C. A. & Payne, A. J. 1996 Time-step limits for stable solutions of the ice-sheet equation. Ann. Glaciol. 23, 74–85.CrossRefGoogle Scholar
Hutter, K. 1982 a Dynamics of glaciers and large ice masses. Annu. Rev. Fluid Mech. 14, 87–130.CrossRefGoogle Scholar
Hutter, K. 1982 b Glacier flow. Am. Scientist 70 (1), 26–34.Google Scholar
Iken, A. 1981 The effect of the subglacial water pressure on the sliding velocity of a glacier in an idealized numerical model. J. Glaciol. 27, 407–421.CrossRefGoogle Scholar
Joughin, I., MacAyeal, D. R. & Tulaczyk, S. 2004 Basal shear stress of the Ross ice streams from control method inversions. J. Geophys. Res. 109 (B09405), 1–20, doi:10.1029/2003JB002960.CrossRefGoogle Scholar
Kamb, B. 1970 Sliding motion of glaciers – theory and observation. Rev. Geophys. Space Phys. 8 (4), 673–728.CrossRefGoogle Scholar
Kamb, B. 1987 Glacier surge mechanism based on linked cavity configuration of the basal water conduit system. J. Geophys. Res. 92 (B9), 9083–9100.CrossRefGoogle Scholar
Kamb, B. & Engelhardt, H. F. 1991 Antarctic ice stream B: conditions controlling its motions and interactions with the climate system. Intl Assoc. Sci. Hydrol. Publ. 208, 145–154.Google Scholar
Kamb, B., Raymond, C. F., Harrison, W. D., Engelhardt, H., Echelmeyer, K. A., Humphrey, N., Brugman, M. M. & Pfeffer, T. 1985 Glacier surge mechanism – 1982–1983 surge of Variegated Glacier, Alaska. Science 227 (4686), 469–479.CrossRefGoogle ScholarPubMed
Lliboutry, L. 1968 General theory of subglacial cavitation and sliding of temperate glaciers. J. Glaciol. 7 (49), 21–58.CrossRefGoogle Scholar
Lliboutry, L. 1969 Contribution la thorie des ondes glaciaires. Can. J. Earth Sci. 6 (4), 943–953.CrossRefGoogle Scholar
Lliboutry, L. 1979 Local friction laws for glaciers: a critical review and new openings. J. Glaciol. 23 (89), 67–95.CrossRefGoogle Scholar
Lliboutry, L. 1987 Realistic, yet simple bottom boundary-conditions for glaciers and ice sheets. J. Geophys. Res. 92 (B9), 9101–9109.CrossRefGoogle Scholar
MacAyeal, D. R. 1989 Large-scale ice flow over a viscous basal sediment – theory and application to Ice Stream B, Antarctica. J. Geophys. Res. 94 (B4), 4071–4087.CrossRefGoogle Scholar
MacAyeal, D. R., Bindschadler, R. A. & Scambos, T. A. 1995 Basal friction of Ice Stream E, West Antarctica. J. Glaciol. 41, 247–262.CrossRefGoogle Scholar
McMeeking, R. M. & Johnson, R. E. 1986 On the mechanics of surging glaciers. J. Glaciol. 32 (110), 120–132.CrossRefGoogle Scholar
Nye, J. F. 1969 A calculation on sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc. R. Soc. Lond. Ser. A – Math. Phys. Sci. 311 (1506), 445–467.Google Scholar
Nye, J. F. 1970 Glacier sliding without cavitation in a linear viscous approximation. Proc. R. Soc. Lond. Ser. A – Math. Phys. Sci. 315 (1522), 381–403.Google Scholar
Papanastasiou, T. C., Malamataris, N. & Ellwood, K. 1992 A new outflow boundary-condition. Intl J. Numer. Methods Fluids 14 (5), 587–608.CrossRefGoogle Scholar
Price, S. F., Bindschadler, R. A., Hulbe, C. L. & Blankenship, D. D. 2002 Force balance along an inland tributary and onset to Ice Stream D, West Antarctica. J. Glaciol. 48 (160), 20–30.CrossRefGoogle Scholar
Raymond, C. F. 1987 How do glaciers surge – a review. J. Geophys. Res. 92 (B9), 9121–9134.CrossRefGoogle Scholar
Raymond, C. F., Echelmeyer, K., Whillans, I. & Doake, C. 2001 Ice stream shear margins. In The West Antarctic Ice Sheet – Behaviour and Environment, Antarctic Research Series, vol. 77, pp. 137–155. AGU.Google Scholar
Rempel, A. W. 2008 A theory for ice-till interactions and sediment entrainment beneath glaciers. J. Geophys. Res. 113 (F01013), 1–20, doi:10.1029/2007JF000870.CrossRefGoogle Scholar
Retzlaff, R. & Bentley, C. R. 1993 Timing of stagnation of Ice Stream-C, West Antarctica, from short-pulse radar studies of buried surface crevasses. J. Glaciol. 39 (133), 553–561.CrossRefGoogle Scholar
Rommelaere, V. & Ritz, C. 1996 A thermomechanical model of ice-shelf flow. Ann. Glaciol. 23, 13–20.CrossRefGoogle Scholar
Rothlisberger, H. 1972 Water pressure in intra- and subglacial channels. J. Glaciol. 11, 177–203.CrossRefGoogle Scholar
Sayag, R. & Tziperman, E. 2008 Spontaneous generation of pure ice streams via flow instability: role of longitudinal shear stresses and subglacial till. J. Geophys. Res. 113 (B12), B05411, 1–17.CrossRefGoogle Scholar
Schoof, C. 2004 a Bed topography and surges in ice streams. Geophys. Res. Lett. 31 (6), L06401.CrossRefGoogle Scholar
Schoof, C. 2004 b On the mechanics of ice-stream shear margins. J. Glaciol. 50 (169), 208–218.CrossRefGoogle Scholar
Schoof, C. 2005 The effect of cavitation on glacier sliding. Proc. R. Soc. A – Math. Phys. Engng Sci. 461 (2055), 609–627.Google Scholar
Schoof, C. & Rempel, A. 2008 Ice stream shear margin migration. In Geophysical Research Abstracts, pp. 10, EGU2008-A-11577. EGU.Google Scholar
Stephenson, S. N. & Bindschadler, R. A. 1988 Observed velocity fluctuations on a major Antarctic ice stream. Nature 334, 695–697.CrossRefGoogle Scholar
Stokes, C. R., Clark, C. D., Lian, O. B. & Tulaczyk, S. 2007 Ice stream sticky spots: a review of their identification and influence beneath contemporary and palaeo-ice streams. Earth-Sci. Rev. 81 (3–4), 217–249.CrossRefGoogle Scholar
Walder, J. S. 1982 Stability of sheet flow of water beneath temperate glaciers and implications for glacier surging. J. Glaciol. 28 (99), 273–293.CrossRefGoogle Scholar
Walder, J. S. & Fowler, A. 1994 Channelized subglacial drainage over a deformable bed. J. Glaciol. 40 (134), 3–15.CrossRefGoogle Scholar
Weertman, J. 1964 The theory of glacier sliding. J. Glaciol. 5, 287–303.CrossRefGoogle Scholar
Weertman, J. 1972 General theory of water flow at base of a glacier or ice sheet. Rev. Geophys. Space Phys. 10 (1), 287–333.CrossRefGoogle Scholar
Weertman, J. 1979 The unsolved general glacier sliding problem. J. Glaciol. 23, 97–115.CrossRefGoogle Scholar
Whillans, I. M. & van der Veen, C. J. 1997 The role of lateral drag in the dynamics of Ice Stream B, Antarctica. J. Glaciol. 43 (144), 231–237.CrossRefGoogle Scholar

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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Video 2.3 MB

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