## Centre for Polar Observation and Modelling School of Geographical Sciences University of Bristol Bristol BS8 1SS UK

In this case study we will illustrate the range of models available to study terrestrial ice masses by concentrating on the study of the ice streams of Antarctica. Ice streams are the focus of a great deal of contemporary research because they discharge the majority (more than 90%) of the ice leaving Antarctica, and their dynamics are therefore likely to affect the volume of ice stored in the ice sheet and, hence, global sea levels.

Ice-flow models can best be classified according to the accuracy with which they depict the true force balance within an ice mass. Based on the ratio between a typical ice thickness and the characteristic length of the ice mass (the aspect ratio), three main sets of approximations can be made. These are zero, first and second-order models, which are appropriate at successively finer spatial scales of investigation. The order of such models is indicative of how many effects are incorporated into the modelled force balance.

All models assume that the balance of forces is static (i.e. that acceleration is not significant in the balance). Zero-order models then simply assume that all the gravitational driving stress is balanced locally by vertical shear stresses and basal drag

where the symbols are defined in Table 81.1 and a similar expression is used for the force balance in the y dimension.

The other components of a coupled ice-flow model are fairly similar in all three types of model. They consist of (see details in Huybrechts, this volume, Chapter 80): a constitutive relation such as Glen's flow law (Glen, 1955) to convert the predicted stress field to a velocity field; models to predict the evolution of ice thickness and internal temperature field through time; possibly a link between the temperature of ice and its rheology; as well as the use of the continuity equation to diagnose vertical velocity once the horizontal velocity field is known from Equation (1) or similar. Other components may be incorporated to account for processes such as snow accumulation and ablation, basal slip (as a function of gravitational driving stress) and isostasy.

Zero-order models of this type are typically used to study the evolution of the Antarctic Ice Sheet (AIS) over long time-scales of tens to hundreds of thousands of years. They are surprisingly

 Symbol Definition x,y Horizontal coordinates (m) z Vertical coordinate (positive upwards, m) s Elevation of upper ice surface (m) H Ice thickness (m) h Elevation of lower ice surface (bedrock, m) g Acceleration due to gravity (ms-2) P Ice density (kg m-3) Xxz, Xyz Vertical shear stresses (Pa) ^xy Horizontal shear stress (Pa) txz(h),Tyz(h) Basal tractions (Pa) a„ay Normal stresses (Pa) u„uy Horizontal velocity components (m yr-1)

realistic in their depiction of the location of ice streams in Antarctica. Figure 81.1 shows the results from such a model. The positions of many of the major ice streams are depicted correctly and can be compared with their locations derived from the use of balance flux calculations (e.g. Budd & Warner, 1996). The latter assumes that the present-day geometry of Antarctica is in equilibrium and routes the ice flux generated by snow accumulation across the surface of the ice sheet by assuming that ice always flows downhill.

The ice streams in Figure 81.1 are generated by the interaction of ice flow, surface topography and temperature. In the model, ice is assumed to slip over its bed only where the bed is predicted to be at melting point, and the amount of slip is a function of the amount of water at the bed. The fact that the location of many ice streams can be predicted accurately by such a simple model implies that many owe their locations to troughs in the underlying bedrock topography. It is likely that the deeper ice in troughs will flow faster (gravitational driving stresses are larger and velocities are further amplified by the non-linear nature of Glen's flow law). In addition, deeper ice is likely to be warmer (and hence softer) because of the enhanced insulation afforded by thicker ice and the larger amounts of frictional heat generated by the flow.

Figure 81.1 A comparison of the predicted ice flux from a numerical model of the present-day Antarctic Ice Sheet (left) with the balance flux determined by routing snow accumulation across the ice sheet (right, after Bamber et al., 2000a). The same scale (log10myr-1) is used in both panels. The numerical model is based on Payne (1998) and uses geothermal heat flux of 0.06 Wm-2 and a slip coefficient B (ux(h) = Btxz(h)) of 1 X 10-2myr-1Pa-1 in West Antarctica and 1 X 10-2myr-1Pa-1 in East Antarctica. (See www.blackwellpublishing.com/knight for colour version.)

Figure 81.1 A comparison of the predicted ice flux from a numerical model of the present-day Antarctic Ice Sheet (left) with the balance flux determined by routing snow accumulation across the ice sheet (right, after Bamber et al., 2000a). The same scale (log10myr-1) is used in both panels. The numerical model is based on Payne (1998) and uses geothermal heat flux of 0.06 Wm-2 and a slip coefficient B (ux(h) = Btxz(h)) of 1 X 10-2myr-1Pa-1 in West Antarctica and 1 X 10-2myr-1Pa-1 in East Antarctica. (See www.blackwellpublishing.com/knight for colour version.)

Figure 81.2 Illustration of the numerical domain (upper panel) of a three-dimensional first-order model applied to Pine Island Glacier, West Antarctica. The type of boundary condition applied to each exterior surface is: zero traction along the upper surface (air interface) and the lower surface of the ice shelf (water interface); a force balance with the weight of displaced ocean water at the ice-shelf front; a specified (zero) velocity on the other (upstream and side) lateral boundaries and on the bed outside of the ice stream; and a viscous traction law applied to the bed of the ice stream itself. Four of the many intermediate model layers are shown to illustrate the terrain-following nature of the model. Example results (lower panel) show the calculated basal traction (in Pa). Note the enhanced tractions immediately adjacent to the margins of the ice stream, which arise through the lateral transfer of stress away from the weak bed underlying the ice stream. (See www.blackwellpublishing.com/knight for colour version.)

Figure 81.2 Illustration of the numerical domain (upper panel) of a three-dimensional first-order model applied to Pine Island Glacier, West Antarctica. The type of boundary condition applied to each exterior surface is: zero traction along the upper surface (air interface) and the lower surface of the ice shelf (water interface); a force balance with the weight of displaced ocean water at the ice-shelf front; a specified (zero) velocity on the other (upstream and side) lateral boundaries and on the bed outside of the ice stream; and a viscous traction law applied to the bed of the ice stream itself. Four of the many intermediate model layers are shown to illustrate the terrain-following nature of the model. Example results (lower panel) show the calculated basal traction (in Pa). Note the enhanced tractions immediately adjacent to the margins of the ice stream, which arise through the lateral transfer of stress away from the weak bed underlying the ice stream. (See www.blackwellpublishing.com/knight for colour version.)

Although simple, zero-order models may be sufficient to correctly predict the locations of ice streams, it is unlikely that they will be able to predict the flow of ice within the streams accurately. Higher order models are therefore used, which do not make so many restrictive assumptions about the force budget of the ice mass. The mostly commonly used model is the first-order one in which all stress components are considered da

dx dt xy dy

0 0