Incorporation of basal motion explicitly into ice-sheet models generally is achieved by introducing a sliding term across those cells where basal temperature attains the pressure melting point according to Equation (9), using the local basal shear stress but
Figure 67.5 Deformation profiles in glacier ice (a) above a basal zone of low traction corresponding to the location of a major melt-season basal channel and (b) above adjacent ice characterized by higher basal traction. (After Willis et al. (2003) with the permission of the International Glaciological Society.)
neglecting any effective pressure term (i.e. N = 1). For example, Payne et al. (2000) utilize a sliding term of the form
which relates to Equation (7) with p = 1, q = 0 and local shear stress calculated through Equation (2), with the sliding parameter Z as a constant. More recently though, in a newly developed first-order thermomechanical model, Payne (personal communication, 2005) allows Z to vary with the basal melt flux, which goes some distance to incorporating variations in basal water pressure where there is a closed hydrological system (from which water drains ineffectively). Pattyn (2002) uses a sliding term of a similar form to Equation (7), with p = 3 and q = 1. This follows the work of Bindschadler (1983), who compared different sliding relations against observations and concluded that such a 'Weertman-type' sliding law relation, explicitly corrected for subglacial effective water pressure N, was optimal (although basal water pressure is not predicted by the Pattyn model). Marshall & Clarke (1997b) address the problem slightly differently by introducing a stressfree basal boundary condition (determined by subglacial geology, topography and thermal regime) to represent ice streaming in the Hudson Bay area in their model of the Laurentide Ice Sheet.
In contrast to ice sheets, high resolution velocity data are available to constrain spatial and temporal variations in the basal motion at small valley glaciers. For example, Hubbard et al. (1998) used winter (no-sliding) patterns of surface motion measured at Haut Glacier d'Arolla, Switzerland, to direct a three-dimensional, first-order algorithm of the glacier's ice deformation field. This field was then perturbed in a manner consistent with field-based borehole inclinometry data (indicating systematic temporal and spatial patterns of basal sliding) to model the long-term aggregate pattern of glacier motion (presented in more detail by A. Hubbard, this volume, Chapter 69). Similarly, Sugiyama et al. (2003) and Sugiyama (this volume, Chapter 68) introduced a slippery zone at the base of their finite-element (full solution) models of Unteraargletscher and Lauteraargletscher, to match variations in the glacier's measured surface velocity field.
To date, slippery zones such as these have been imposed as varying boundary conditions and no truly integrated hydrological-motion model has yet been developed at the ice-mass scale. However, integrated hydrological models such as that developed by Flowers & Clarke (2002a) are moving some way towards this end.
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