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A model for polythermal ice incorporating gravity-driven moisture transport

Published online by Cambridge University Press:  23 May 2016

C. Schoof  and
I. J. Hewitt
C. Schoof*
Affiliation:
Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Road, Vancouver, BC, V6T 1Z4, Canada
I. J. Hewitt
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

The flow of ice sheets and glaciers dissipates significant amounts of heat, which can result in the formation of ‘temperate ice’, a binary mixture of ice and small amounts of melt water that exists at the melting point. Many ice masses are polythermal, in the sense that they contain cold ice, below the melting point, as well as temperate ice. Temperature and melt water (or moisture) content conversely affect the flow of these ice masses through their effect on ice viscosity and sliding behaviour. Ice flow models therefore require a component that can solve for temperature and moisture content, and determine the free boundary between the cold and temperate subdomains. We present such a model, based on the theory of compacting partial melts. By contrast with other models, we describe gravity- and pressure-gradient-driven drainage of moisture, while maintaining a divergence-free ice flow at leading order. We also derive the relevant boundary conditions at the free cold–temperate boundary, and find that the boundary behaves differently depending on whether ice enters or exits the temperate region. The paper also describes a number of test cases used to compare with a numerical solution, and investigates asymptotic solutions applicable to the limit of small compaction pressure gradients in the temperate ice regions. A simplified enthalpy-gradient model is finally proposed, which captures most of the behaviour of the full model in this limit.

JFM classification

Geophysical and Geological Flows: Ice sheets Low-Reynolds-number flows: Porous media Phase change: Solidification/melting
Type
Papers
Information
Journal of Fluid Mechanics , Volume 797 , 25 June 2016 , pp. 504 - 535
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Copyright
© 2016 Cambridge University Press 

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References

Aschwanden, A. & Blatter, H. 2009 Mathematical modeling and numerical simulation of polythermal glaciers. J. Geophys. Res. 114, F01027.Google Scholar
Aschwanden, A., Bueler, E., Khroulev, C. & Blatter, H. 2012 An enthalpy formulation for glaciers and ice sheets. J. Glaciol. 58, 441–457.CrossRefGoogle Scholar
Bear, J. & Bachmat, Y. 1990 Introduction to Modeling of Transport Phenomena in Porous Media. Springer.CrossRefGoogle Scholar
Bercovici, D., Ricard, Y. & Schubert, G. 2001 A two-phase model for compaction and damage. 1. General theory. J. Geophys. Res. 106 (B5), 8887–8906.CrossRefGoogle Scholar
Blatter, H. & Hutter, K. 1991 Polythermal conditions in Arctic glaciers. J. Glaciol. 37, 261–269.CrossRefGoogle Scholar
Biot, M. A. 1941 General theory of three-dimensional consoldiation. J. Appl. Phys. 12, 155–164.CrossRefGoogle Scholar
Duval, P. 1977 The role of water content on the creep of poly-crystalline ice. IAHS-AISH 118, 29–33.Google Scholar
Fowler, A. C. 1984 On the transport of moisture in polythermal glaciers. Geophys. Astrophys. Fluid 28, 99–140.CrossRefGoogle Scholar
Fowler, A. C. & Larson, D. A. 1978 On the flow of polythermal glaciers I. Model and preliminary analysis. Proc. R. Soc. Lond. A 363, 217–242.Google Scholar
Greve, R. 1997a A continuum-mechanical formulation for shallow polythermal ice sheets. Phil. Trans. R. Soc. Lond. A 355, 921–974.CrossRefGoogle Scholar
Greve, R. 1997b Application of a polythermal three-dimensional ice sheet model to the greenland ice sheet: response to steady-state and transient climate scenarios. J. Clim. 10, 901–918.2.0.CO;2>CrossRefGoogle Scholar
Hewitt, I. J. & Fowler, A. C. 2008 Partial melting in an upwelling mantle column. Proc. R. Soc. Lond. A 464, 2467–2491.Google Scholar
Holmes, M. H. 1995 Introduction to Perturbation Methods, Texts in Applied Mathematics, vol. 20. Springer.CrossRefGoogle Scholar
Hutter, K. 1982 A mathematical model of polythermal glaciers and ice sheets. Geophys. Astro. Fluid 21, 201–224.CrossRefGoogle Scholar
Lliboutry, L. A. & Duval, P. 1985 Various isotropic and anisotropic ices found in glaciers and polar ice caps and their corresponding rheologies. Ann. Geophys. 3, 207–224.Google Scholar
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713–765.CrossRefGoogle Scholar
Nye, J. F. 1953 The Flow law of ice from measurements in glacier tunnels, laboratory experiments and the Jungfraufirn borehole experiment. Proc. R. Soc. Lond. A 219, 477–489.Google Scholar
Nye, J. F. & Mae, S. 1972 The effect of non-hydrostatic stress on intergranular water veins and lenses in ice. J. Glaciol. 11 (61), 81–101.CrossRefGoogle Scholar
Nye, J. F. & Frank, F. C. 1973 Hydrology of the Intergranular veins in a temperate glacier. IASH 95, 157–161.Google Scholar
Ockendon, J., Howison, S., Lacey, A. & Movchan, S. 2003 Applied Partial Differential Equations. Oxford University Press.CrossRefGoogle Scholar
Robin, G. Q. 1955 Ice movement and temperature distribution in glaciers and ice sheets. J. Glaciol. 2, 523–532.CrossRefGoogle Scholar
Schoof, C., Hewitt, I. J. & Werder, M. A. 2012 Flotation and free surface flow in a model for subglacial drainage. Part 2. Distributed drainage. J. Fluid Mech. 702, 126–156.CrossRefGoogle Scholar
Spiegelman, M. 1993 Flow in deformable porous media. Part 2. Numerical analysis – the relationship between solitary waves and shock waves. J. Fluid Mech. 247, 39–63.CrossRefGoogle Scholar
Suckale, J., Platt, J. D., Perol, T. & Rice, J. R. 2014 Deformation-induced melting in the margins of the West Antarctic ice streams. Geophys. Res. Earth Surf. 119, 1004–1025.CrossRefGoogle Scholar
Terzaghi, K. 1923 Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Akad. Wiss. Wien 132, 125–132.Google Scholar
Turcotte, D. L. & Ahern, J. L. 1978 A porous flow model for magma migration in the asthenosphere. J. Geophys. Res. 83, 767–772.CrossRefGoogle Scholar
Zwinger, T., Greve, R., Gagliardini, O., Shiraiwa, T. & Lyly, M. 2007 A full Stokes-flow thermo-mechanical model for firn and ice applied to the Gorshkov crater glacier, Kamchatka. Ann. Glaciol. 45, 29–37.CrossRefGoogle Scholar
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