Models of icemass motion

Spatially distributed numerical models of glacier and ice-sheet evolution are based on calculating changes in ice thickness (H) through time (t) according to the mass continuity equation

dt where Ac is accumulation, Ab is ablation, b is basal melt, u is the vertically integrated two-dimensional horizontal velocity and V is the two-dimensional horizontal divergence operator. It follows that an ice mass' geometric response (dH/dt) to a given change in its surface mass balance boundary condition (Ac-Ab) is critically dependent on ice redistribution by motion (-). The most advanced of these models are three-dimensional and, where necessary, thermomechanically coupled, and predominantly, although not exclusively, are solved on fixed, predefined arrays of cells (finite differences) where mass-balance, temperature, stress, strain and ultimately the change in ice thickness are computed itera-

tively. Over the past decade, as numerical time-evolving ice-sheet models have been applied at greater resolutions to a wider variety of scenarios, a significant effort has been expended in developing more sophisticated ice-motion algorithms, which attempt the realistic incorporation of physical processes known to influence specific motion components at appropriate scales and conditions. Such algorithms, however, are often prohibitively expensive in computational resources when incorporated into time-evolving ice-sheet models. Thus, simplified models with reduced ice physics are still widespread, especially in solving problems at large spatial and temporal scales, such as the Antarctic ice sheet, where computational resources are at a premium. For example, a model may include only motion by internal ice deformation, and not that focused near the glacier bed (so-called basal motion). Furthermore, in the most simple and widespread case, the zero-order approximation considers motion to be driven only by local shear stress (also known as the driving stress,t) calculated from local slope (a), ice thickness (H), density (p) and gravity according to t = -pgHa (2)

However, at fine scales where there are large changes in slope or basal motion, significant stresses can be transferred longitudinally or laterally. In order to model these inherited stresses, higher order terms from the Navier-Stokes equations need to be included and solved, requiring more complex derivations and numerical algorithms. The more common of these higher order models solve the first-order terms (i.e. those raised to the power of 1, neglecting those raised to higher powers). Such first-order approximation models of grounded ice flow thereby account for longitudinally transferred stresses. For the vertical or bridging stresses to be calculated, however, higher order terms again need to be solved. Such full solution models are extremely demanding computationally and, to date, have been solvable only on fixed boundary, finite-element glacier geometries to investigate specific small-scale issues (e.g. Gudmundsson, 1999; Cohen, 2000) rather than to long-term studies of ice mass response to climate change, to which the finite-element method is not easily applicable. Neglecting higher order terms is not necessarily a major setback, however, depending on the horizontal resolution of the model in question and the specific conditions under which it is applied. For example, Hindmarsh (personal communication, 2003) has demonstrated, using analytical solutions to a parallel-sided slab geometry, that second-order terms only become important in the ice-flow solution when dealing with grounded ice masses with significant basal motion when the horizontal grid scale approaches the scale of ice thickness.

In reality, inherited stresses are important where variations in traction at the ice margins represent a significant proportion of a glacier's total stress field. Such situations arise at smaller ice masses, where the ratio of boundary area to ice volume is high, and partly- or wholly-temperate ice masses, where marginal traction can vary with the presence of basal meltwater (Clarke & Blake, 1991; Murray et al., 2000b). Thus, inherited stresses are particularly important at temperate valley glaciers. Evidence from Haut Glacier d'Arolla, Switzerland, for example, indicates that such stresses account for up to half of the total stress field in places (Hubbard, 2000). Importantly, ice streams have certain critical features in common with temperate valley glaciers and therefore also demand higher-order modelling to reproduce lateral shear margins and zones of reduced basal traction.

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