## The velocity field in a glacier

Many problems in glaciology require an understanding of the flow field in a glacier. For example, the way in which flow redistributes mass determines the shape of a glacier, and also the rapidity with which glaciers respond to climatic change. Flow also redistributes energy and thus affects the temperature distribution. This, in turn, has important implications for the nature of the coupling with the glacier bed. Spatial variations in speed, or strain rates, are of concern to structural geologists using glaciers as analogs for deformation of rocks. From a geomorphic perspective, the entrainment of debris and the character of moraines constructed from this debris are dependent upon the flow field. In short, understanding the flow field is fundamental to the analysis of many problems in glacier mechanics.

For a full description of the flow field in a glacier, we need the horizontal and vertical components of the velocity at every point. By making several assumptions, we can obtain approximate solutions to this problem that will give insights into certain characteristics of glaciers and the landforms they produce. Initially, we will limit the analysis to two dimensions and also assume a steady state.

We will begin by studying the distribution of horizontal velocity. Given the pattern of accumulation and ablation over a glacier, we can use conservation of mass to determine the mean (depth-averaged) horizontal velocity. Then, by using conservation of momentum and a simplified version of the flow law (Equation (4.5)), we will consider the variation in horizontal velocity with depth in an ice sheet and in a valley glacier. Differences between these solutions and measured velocity distributions reveal inadequacies of the theory, and draw attention to the need for a better understanding of the basal boundary condition. Finally, by integrating the velocity over depth, we calculate the mass flux, and also obtain an expression for the mean velocity in terms of the glacier thickness.

The vertical velocity field is treated next. Again we will use the steady-state assumption and the pattern of mass balance (conservation of mass) to determine the vertical velocity at the surface. We then use the longitudinal strain rate, or rate of stretching in the longitudinal direction, at the surface to estimate its variation with depth, and thus calculate the variation in vertical velocity with depth. This yields an approximation to the full velocity field.

Next, we discuss the role of drifting on the flow field, and show that drifting patterns at the surface of an ice sheet can be traced at depth using radio echo sounding techniques. Drifting also affects the topography of a glacier surface, and plays an important role in the formation of certain types of moraine. Finally, we will explore inhomogeneous flow in ice sheets, as manifested by ice streams.

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