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Paradoxes and problems with the longitudinal stress approximation used in glacier mechanics
Author(s) -
Hindmarsh Richard C.A.
Publication year - 2006
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201490023
Subject(s) - biharmonic equation , mathematical analysis , mathematics , mechanics , classical mechanics , physics , boundary value problem
Abstract The biharmonic equation in two dimensions is the governing equation for the slow plane flow of a Newtonian fluid. Where the flow domain is thin, and the aspect ratio small, the biharmonic equation can be expanded using the aspect ratio as the expansion parameter. The one‐term approximation to the biharmonic equation is the lubrication approximation. In glaciology, the lubrication approximation is extended to deal with special glaciological features, and is known as the shallow ice approximation. The next order term in glaciology is known as the longitudinal stress approximation, but this is not a higher‐order expansion (two‐term representation) of the biharmonic equation. The first part of this paper deals with this paradox. It shows how that longitudinal stress approximation relates to the truncated biharmonic equation, and carries out a perturbation expansion in the free surface, comparing the decay spectra for surface perturbations of the shallow ice approximation, the longitudinal stress approximation, the truncated biharmonic expansion and the full system of equations. In most glaciological situations, the longitudinal stress approximation is a better approximation to the truncated biharmonic equation, and even where it is not, the loss in accuracy is relatively small. Moreover, the truncated biharmonic equation introduces spurious instabilities, while the longitudinal stress approximation does not. The second part of this paper deals with the perturbation expansion in terms of the widthto‐length ratio of a perfectly slippery streaming flow of ice, where flow is retarded at the lateral margins. The case of a linear rheology is considered. The zeroth order equation for the thickness is a non‐linear diffusion equation, whereas the first‐order equation is linear, even though a finite amplitude perturbation is being considered. The equation is second‐order differential in the flow direction, and integral in the transverse direction. Some numerical examples show finite amplitude stability. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)