## 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.