  Long waves in rivers, estuaries and floods are described by the St Venant and
Boussinesq equations in classical fluid dynamics. Based on the widely used
$k$-$\epsilon$ model for turbulence, we use the techniques of centre manifold
theory to show how these classical models can be derived from a unified
approach to low-dimensional modelling. The model may be refined to an arbitrary
degree of accuracy and predicts interesting interactions between the mean flow
and the mean turbulence.
