[15] The ice velocity is expressed as a velocity averaged over the cross section and includes contributions from basal sliding *U*_{s} and internal ice deformation *U*_{d}. In this model the vertical shear stress is related to strain rate according to Glen's flow law [*Glen*, 1955], which yields [*Paterson*, 1981]

The typical value of the flow law exponent *n* = 3 [*Alley*, 1992] and a rate factor *A* = 1 × 10^{−7} kPa^{−3} yr^{−1}, corresponding to ice near the freezing point [*van der Veen*, 1999, Figure 2.6], are used. The driving stress *S*_{d} is defined as

where ∂*h*/∂*x* is surface slope and *g* = 9.8 m s^{−2} is the gravitational acceleration.

[16] It has been shown that most of the flow resistance on the lower reach of Columbia Glacier is due to basal drag and the rest is mainly due to lateral drag; gradients in longitudinal stress contribute little to the resistance to flow [*van der Veen and Whillans*, 1993; *O'Neel et al.*, 2006]. Lateral drag could be included in the model by introducing a shape factor [*Nye*, 1965; *Bindschadler*, 1983]. However, considering the geometry of Columbia Glacier (with a large width-to-height ratio), these effects are probably too small to be crucial for the large-scale flow of glacier and are therefore ignored in the present model.

[17] The fast flow of Columbia Glacier is primarily due to high sliding velocities [*Meier*, 1994; *Kamb et al.*, 1994]. It has been recognized that subglacial water pressure plays an important role in sliding process [*Weertman*, 1964; *Budd et al.*, 1979; *Iken*, 1981; *Bindschadler*, 1983]. A modified Weertman-type sliding velocity [*Budd et al.*, 1979; *Bindschadler*, 1983] is adopted here

The effective pressure *N*_{eff} is equal to the difference between ice overburden pressure *P*_{i} and subglacial water pressure *P*_{w}. A high subglacial water pressure or a thin glacier front reduces the effective basal pressure which leads to enhanced sliding. Basal drag *S*_{b} is set equal to driving stress *S*_{d}. *Bindschadler* [1983] compared four basal sliding formulations suggested by theoretical and experimental studies and concluded that equation (4) provides the best fit to field measurements. He estimated the empirical parameters *A*_{s} = 84 m yr^{−1} bar^{1−m}, *m* ≈ 3, and *p* = 1 using the best fit between inferred and predicted velocities along Variegated Glacier. In this study we used the observed surface and bed elevation and ice surface velocity of the lower reach of Columbia Glacier (∼15 km) obtained from data during the glacier retreat collected by the U.S. Geological Survey [*Brown et al.*, 1982; *Krimmel*, 1997, 2001; *Meier et al.*, 1985; *Sikonia*, 1982]. There are no direct measurements of sliding velocity available, but the observed surface velocity can be considered as an estimate of the sliding velocity because fast flow in the lower reach is predominantly associated with basal sliding. Using multivariate regression, a best fit between calculated and observed velocities was found for *A*_{s} = 9.2 × 10^{6} m yr^{−1} Pa^{0.5}, *m* = 3, and *p* = 3.5. The basal pressure cannot exceed the ice overburden pressure as this would correspond to a net upward force and

where *ρ*_{i} is the ice density. At the glacier front the terminus may be close to flotation [*Meier et al.*, 1994; *Meier*, 1994], which means that the effective pressure becomes very small, leading to the sliding velocity becoming too large and resulting in numerical instabilities. Therefore a minimum effective pressure of 150 kPa is prescribed. This limit is in the range measured at the lower reach of Columbia Glacier during its retreat [*van der Veen*, 1995]. Another model assumption is that there exists a full and easy water connection between the glacier base and the adjoining sea [*Lingle and Brown*, 1987], so that the subglacial water pressure can be estimated from

where *ρ*_{w} is the water density and *b* denotes height of the ice column below sea level.

[18] Although the sliding formulation is obtained from data during the glacier retreat, it can be also used in the model to simulate the glacier advance. Equation (4) yields high velocities during glacier retreat as has been observed, but during advance, velocities predicted by equation (4) remain moderate because the glacier thickness is large and thus the effective pressure is large. It will be shown that modeled advance does not depend on basal sliding velocity.