Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

Abstract

ABSTRACT. For a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress graruents on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the •vertically" (cross-sectionaUy) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (196g), Linearization of the flow-coupling equation, by treating the flow velocity ii ("vertically" averaged), ice thickness h, and surface slope a in terms of small deviations t:lii, llh, and Aa from overall average (datum) values ii 0 , h 0 , and ~· results in a differential equation that can be solved by Green's function methods, giving t:lii(x) as a function of M(x) and Aa(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and a(x) on the flow u(x):where the integration is over the length L of the glacier. The A operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x' , represents the influence of local h(x' ), a(x' ), and channel-shape factor f(x' ), at longitudinal coordinate x' , on the flow ii at coorrunate x, the influence being weighted by the "influence transfer function• exp(-lx' -xl/1) in the integral. The quantity I that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. lt is deter~ned by rheological parameters via the relationship I = 2h n{Ti/3'fl, where n is the flowlaw exponent, ii the effective longitudinal viscosity, and n the effective shear viscosity of the ice profile. ii is an average of the local effective viscosity n over the ice crosssection, and (l'lr 1 is an average of n-1 that gives strongly increased weight to values near the base. Theoretically, the coupling length I is generally in the range one to three times the ice thickness for vaUey glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer, J -12h. It is distinctly longer for non-linear (n • 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow .The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or a, with wavelength >., are attenuated by the factor 1/(1 + (2nl/ >.) 2 ) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 19gO), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for >. s 21 , 21 s >. s 201 , and >. ;;:; 201.

Numerical modeling of glacial erosion and headwall processes in alpine valleys

“…The pattern of N eff is discussed in the following section. Basal shear stress (τ b ) is a product of ice thickness, ice density, gravity, and slope of the glacier surface, and is modified by a longitudinal ice-coupling rule Kamb and Echelmeyer, 1986), and a cross-sectional shape factor, calculated as a function of the ice thickness/valley width ratio assuming a parabolic valley cross-section (Paterson, 1994, p. 269). The length scale of longitudinal stress coupling is a function of local ice thickness; in temperate valley glaciers with ice thicknesses on the order of 100-300 m, longitudinal coupling lengths are 1-2 km.…”

The verification and significance of three approaches to longitudinal stresses in high–resolution models of glacier flow

“…Alison and Kruss 1977;Oerlemans 1986Oerlemans , 1988Stroeven et al 1989;Huybrechts et al 1989;Zuo and Oerlemans 1997;Oerlemans et al 1998;Wallinga and van de Wal 1998). Although some verification of such models has been carried out (Stroeven 1996;Morten 1996), the ice-physics which underpin the glacier flow component has, with the exception of Greuell (1992) remained essentially unchallenged despite being outdated (Hanson 1995;Blatter 1995;Gudmundsson 1994;Kamb and Echelmeyer 1986;van der Veen 1987) and discredited at a general level (Hutter 1983). Essentially, one of the main problems is that the generalised rheology incorporated in these models is based on the concept of a local driving stress, assumed to balance exactly the local basal shear stress.…”

Inferring ice thickness from a glacier dynamics model and multiple surface data sets

“…Our method relies on good-quality surface observations, and the physical model is very sensitive to small variations in surface elevation; therefore, we pre-smooth the input processes such as the surface elevation and accumulation rate. This pre-smoothing procedure is often performed in practice (Berliner et al, 2008b;Kamb & Echelmeyer, 1986;Pollard & DeConto, 2012) to eliminate small-scale features that would otherwise result in unreasonable variations.…”