The Cryosphere www.the-cryosphere.net 1994-0416 1994-0424 2 2 2008 10.5194/tc-2-77-2008 http://www.the-cryosphere.net/2/77/2008/ http://www.the-cryosphere.net/2/77/2008/tc-2-77-2008.html http://www.the-cryosphere.net/2/77/2008/tc-2-77-2008.pdf 77 93 2008-07-16 Analytical solutions for the surface response to small amplitude perturbations in boundary data in the shallow-ice-stream approximation G. H. Gudmundsson British Antarctic Survey, High Cross, Madingley Rd., Cambridge CB3 0ET, UK New analytical solutions describing the effects of small-amplitude perturbations in boundary data on flow in the shallow-ice-stream approximation are presented. These solutions are valid for a non-linear Weertman-type sliding law and for Newtonian ice rheology. Comparison is made with corresponding solutions of the shallow-ice-sheet approximation, and with solutions of the full Stokes equations. The shallow-ice-stream approximation is commonly used to describe large-scale ice stream flow over a weak bed, while the shallow-ice-sheet approximation forms the basis of most current large-scale ice sheet models. It is found that the shallow-ice-stream approximation overestimates the effects of bed topography perturbations on surface profile for wavelengths less than about 5 to 10 ice thicknesses, the exact number depending on values of surface slope and slip ratio. For high slip ratios, the shallow-ice-stream approximation gives a very simple description of the relationship between bed and surface topography, with the corresponding transfer amplitudes being close to unity for any given wavelength. The shallow-ice-stream estimates for the timescales that govern the transient response of ice streams to external perturbations are considerably more accurate than those based on the shallow-ice-sheet approximation. In particular, in contrast to the shallow-ice-sheet approximation, the shallow-ice-stream approximation correctly reproduces the short-wavelength limit of the kinematic phase speed given by solving a linearised version of the full Stokes system. In accordance with the full Stokes solutions, the shallow-ice-sheet approximation predicts surface fields to react weakly to spatial variations in basal slipperiness with wavelengths less than about 10 to 20 ice thicknesses. Baral, D R. and Hutter, C.: Asymptotic Theories of Ice Sheets and Ice Shelves, in: Geomorphological Fluid Mechanics, edited by Balmforth, N J. and Provenzale, A., Lecture Notes in Physics, 11, 227–278, Springer, 2001. Fowler, A C.: Waves on glaciers, J. Fluid Mech., 120, 283–321, 1982. Gudmundsson, G H.: Transmission of basal variability to a glacier surface, Journal of Geophysical Research, 108(B5), 2253, doi:10.1029/2002JB002107, 2003. Hindmarsh, R. C A.: A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling, J. Geophys. Res., 109, F01012, \doi10.1029/2003JF000065, 2004. Hutter, K.: Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets, D. Reidel Publishing Company/Tokyo, Terra Scientific Publishing Company, 1983. Jóhannesson, T.: Landscape of temperate ice caps, Ph.D. thesis, Univ. of Washington, 1992. MacAyeal, D R.: Large-scale ice flow over a viscous basal sediment: Theory and Application to Ice Stream B, Antarctica, J. Geophys. Res., 94, 4071–4078, 1989. Morland, L W.: Thermomechanical balances of ice-sheet flows, Geophysical and Astrophysical Fluid Dynamics, 29, 237–266, 1984. Muszynski, I. and Birchfield, G E.: A coupled marine ice-stream-ice – shelf model, J. Glaciology, 33, 3–14, 1987. Nye, J F.: The response of glaciers and ice-sheets to seasonal and climatic changes, Proceedings of the Royal Society of London, Ser. A, 256, 559–584, 1960. Raymond, C F.: Shear margins in glaciers and ice sheets, Journal of Glaciology, 42, 90–102, 1996. Raymond, M. and Gudmundsson, G H.: On the relationship between surface and basal properties on glaciers, ice sheets, and ice streams., Journal of Geophysical Research, 110, B08411, \doi10.1029/2005JB003681, 2005. Reeh, N.: Steady-state three-dimensional ice flow over an undulating base: first-order theory with linear ice rheology, J. Glaciology, 33, 177–185, 1987. Schoof, C.: A note on inverting ice-stream surface data, J. Glaciology, 51, 181–182, 2005. Schoof, C.: A variational approach to ice stream flow, Journal of Fluid Mechanics, 556, 227–251, \doi10.1017/SS0022112006009591, 2006.