Abstract
The reduced model for the flow of a large ice sheet is uniformly valid when the bed topography is flat or has slopes relative to the horizontal no greater than the magnitude e of the surface slope, where ε 2 defines a very small dimensionless viscosity based on the geometry and flow parameters. The reduced model is given by the leading order balances of an asymptotic expansion in e. Real ice sheet beds will have much greater slopes, of order unity in places, but commonly of moderate magnitude δ over large regions, such that ε ≪ δ ≪ 1. The length s over which a moderate slope extends may be as small as the sheet thickness, or considerably greater subject to the restriction that the amplitude a = δs of the local topography does not exceed the sheet thickness. Then, in addition to ε, there are two further independent parameters from the trio δ, s and a. An asymptotic expansion is derived for steady plane non-linearly viscous flow with a prescribed temperature field over moderate slope bed topography. The leading order balances define an enhanced reduced model, which is constructed explicitly, and the first order correction terms have magnitude δ/s when s is dimensionless with unit the sheet thickness. The first and second order correction problems are derived. It is shown that their structure is distinct for topography scales s = 1 and s ≫ 1. When s = 1, the differential equations of the correction problems retain an elliptic structure, but when s ≫ 1, explicit depth integration of the momentum balances is possible, the crucial simplifying property of the leading order reduced model. The leading order, enhanced reduced model, is solved in the simpler case of linearly viscous isothermal flow for a particular bed form for a range of the parameters δ, s and a, to illustrate the influence of the enhancement, and also show the distributions with height of the deviatoric stresses and horizontal and vertical velocity in the region of the moderate bed slope.
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References
Cliffe, K. A. and Morland, L. W. 2000. Pull and reduced model solutions of steady axi-symmetric ice sheet flows over small and large topography slopes. Cont. Mech. Thermodyn., 12, 195–216
Drăghicescu, A. 2000. Effects of bed topography on the steady plane flow of ice sheets. PhD thesis, School of Mathematics, University of East Anglia, in preparation.
Greve, R., Mügge, B., Baral, D., Albrecht, O. and Savvin, A. 1999. Nested high-resolution modelling of the Greenland Summit region. In: Hutter, K., Wang, Y. and Beer, H. (eds.), Advances in cold-region thermal engineering and sciences, Springer, Berlin etc., 285–306.
Hutter, K., 1981. The effect of longitudinal strain on the shear stress of an ice sheet: In defense of using stretched co-ordinates. J. Glaciol., 27, 39–56.
Hutter, K. 1982a. A mathematical model of polythermal glaciers and ice sheets. Geophys. and Astrophys. Fluid Dyn., 21, 201–224.
Hutter, K. 1982b. Dynamics of glaciers and large ice masses. Ann. Rev. Fluid Mech., 14, 87–130.
Hutter, K. 1983. Theoretical glaciology. Reidel, Dordrecht.
Hutter, K., Legerer, F. and Spring, U. 1981. First-order stresses and deformations in glaciers and ice sheets. J. Glaciol., 27, 227–270
Huybrechts, P., Payne, A. J. and the EISMINT Intercomparison Group. 1996. The EISMINT benchmarks for testing ice-sheet models. Ann. Glaciol., 23, 1–14
Morland, L. W. 1984. Thermomechanical balances of ice sheet flows. Geophys. Astrophys. Fluid Dyn., 29, 237–266.
Morland, L. W. 1993. The flow of ice sheets and ice shelves. In Hutter, K. (ed.), Continuum mechanics in environmental sciences and geophysics, CISM Lectures 1992, no. 337, Springer, Berlin etc., 402–446.
Morland, L. W. 1997. Radially symmetric ice sheet flow. Phil. Trans. R. Soc. Lond. A, 355, 1873–1904.
Morland, L. W. 2000. Steady plane isothermal linearly viscous flow of ice sheets on beds with moderate slope topography. Proc. R. Soc. Lond. A, 456, 1711–1739.
Morland, L. W. and Johnson, I. R. 1980. Steady motion of ice sheets. J. Glaciol., 25, 229–245.
Morland, L. W. and Johnson, I. R. 1982. Effects of bed inclination and topography on steady isothermal ice sheets. J. Glaciol., 28, 71–90.
Morland, L. W. and Smith, G. D. 1984. Influence of non-uniform temperature distribution on the steady motion of ice sheets. J. Fluid Mech., 140, 113–133.
Smith, G. D. and Morland, L. W., 1981. Viscous relations for the steady creep of polycrystaline ice. Cold. Reg. Sci. Tech., 5, 141–150
Van Dyke, M. 1975. Perturbation methods in fluid mechanics. Parabolic, Stanford.
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Morland, L.W. (2001). Influence of Bed Topography on Steady Plane Ice Sheet Flow. In: Straughan, B., Greve, R., Ehrentraut, H., Wang, Y. (eds) Continuum Mechanics and Applications in Geophysics and the Environment. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04439-1_15
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DOI: https://doi.org/10.1007/978-3-662-04439-1_15
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