In the previous two chapters, we investigated the nature of flow in a horizontal channel assuming (except for the study of flash floods in § 21.4) resistance to flow is negligibly small, and we were able to classify flows as either sub- or super-critical and to investigate the smooth transition from sub- to super-critical and the abrupt transition from super- to sub-critical. We focused on the flow of water (as opposed to other fluids) because it has a very low value of viscosity and in many cases may be treated as inviscid. However, all real flows involve resistance which tends to degrade the mechanical energy. Typically channeled flows are maintained by the release of gravitational potential energy as the material moves downslope.

In this chapter we consider the one-dimensional flow of a continuous material down a slope. The material might be water in a river, ice in a glacier, snow in an avalanche, debris in a rock slide, molten rock in a lava stream, mud in a landslide, muddy water in a turbidity current, etc. We will formulate the problem fairly generally in § 22.1, then specialize it to specific types of material in § 22.2.

Formulation

Most natural downslope flows, exemplified by rivers and glaciers, are much wider than they are deep and move along beds that are relatively flat. Typically a river is wider than deep by a factor of 100 (e.g., see Yalin, 1992). Also, rivers, glaciers and like flows typically are contained by banks that are raised and relatively steep. It follows that a first approximation of the flow bed is a shallow rectangle and that the velocity is independent of the cross-stream direction to dominant order.

Let's consider flow of material in a straight, flat-bottomed channel (at z=0) of constant width; as previously, the downstream direction is x, the (unimportant) cross-stream direction is y and the (nearly) vertical direction is z. The height of the surface is represented by z = h(x, t). In this chapter we will focus on flows that are steady and invariant in the x direction, having h constant and v = u(z)1x. Our goal is to determine the vertical structure of the flow, u(z), driven by a down-gradient component of gravity.