Chapter 15

River flow

The Chezy and Manning laws lead to the basic one-dimensional model for river flow, and the flood hydrograph is analysed on this basis. If fluid momentum is included as well, the St. Venant equations are derived, and the (linear) analysis of these shows the distinction between tranquil flow at Froude number less than 1, and rapid flow flow at Froude number greater than 1. For nonlinear waves, the equations are shown to be hyperbolic, and will thus support shocks, but there is in fact little discussion of this.

There is brief discussion of sediment transport, dune formation, river meanders, and the formation of drainage networks. These are interesting because they are formed by a continuous process, but give a structure which has a fractal characteristic. The Amazon river basin on page 265, also reproduced on the front cover, illustrates this. One continuum model of erosion and sediment transport is given and partially analysed: it is shown that a uniform substrate is unstable to the formation of channels. There is a deeper conceptual modelling problem here, with wide ramifications: how can a continuous model predict a solution with fractal characteristics?

At least in the present case, the structure of the answer may be the following: the continuous model is of singular perturbation type, with the (outer) solution admitting `shocks' which are in fact the channels of the network. The description of the fractal structure relies on a local description of channel density as a function of channel width (or depth) and time, which evolves locally using the `inner' description of the model. This kind of idea (which has not been carried out so far) has large scale implications for modelling heterogeneous processes; the most obvious example is fluid turbulence, but there are plenty of others: dendritic structure in solidification (chapter 17) is just one.