Abstract
Abstract. 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.
Highlights
Large-scale ice sheet models commonly employ approximations to the momentum equations for increased computational efficiency. These approximations are derived from the full-set of momentum equations through scaling analysis motivated by the size of some geometrical aspect ratios, such as ice thickness and ice-sheet span, and some expectations about relative sizes of various stress terms
For example, from the scalings used in the SSTREAM approximation that the slip ratio, the ratio between mean basal motion and mean forward deformational velocity, must be O(δ−2), where δ is the ratio between typical thickness and horizontal span
I present new analytical solutions to the shallow SSTREAM equations based on small-amplitude perturbation analysis and compare them with corresponding FS analytical solutions given in Gudmundsson (2003) and Johannesson (1992)
Summary
Large-scale ice sheet models commonly employ approximations to the momentum equations for increased computational efficiency. The derivations of the SSHEET and the SSTREAM approximations give some indications about their applicability to different flow regimes It follows, for example, from the scalings used in the SSTREAM approximation that the slip ratio, the ratio between mean basal motion and mean forward deformational velocity, must be O(δ−2), where δ is the ratio between typical thickness and horizontal span (see Appendix A). Focusing on small-amplitude perturbations reduces computational times making direct estimates of absolute errors feasible Another advantage that comes from analysing small-amplitude solutions is the added insight they can give into the nature of the approximations. I present new analytical solutions to the shallow SSTREAM equations based on small-amplitude perturbation analysis and compare them with corresponding FS analytical solutions given in Gudmundsson (2003) and Johannesson (1992). The solutions are valid for linear medium and small-amplitude perturbations in surface topography, bed topography, and basal slipperiness
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