Basic physical processes in rivers

Abstract

A “one-and-a-half”-dimensional model of a river is developed. It is actually one-dimensional but allows for horizontal curvature using natural curvilinear co-ordinates. The governing long wave equations can be developed with very few limiting approximations, especially using momentum rather than energy. The curvature is then shown to be rarely important and is subsequently ignored. Wave periods, imposed by boundary conditions, are asserted to be fundamental. Long waves have speeds and propagation properties that depend on period, and there is no such thing as a single long wave speed. Examination of dimensionless equations and solution of linearised equations using wave period shows a novel interpretation of terms in the momentum equation: the “kinematic” approximation and wave are misnomers: the approximation lies not in the neglect of inertial terms but is actually a very long period one. The outstanding problem of river modelling, however, is that of resistance to the flow. A large data set from stream-gauging is considered and it is shown that the state of the bed, namely the arrangement of bed grains by previous flows, is more important than actual grain size. A formula for resistance is proposed which contains a parameter representing bed state. As that state is usually changing with flow, one can not be sure what the resistance actually will be. This uncertainty may have important implications for modelling. The momentum principle is then applied also to obstacles such as bridge piers, and a simple approximation gives greater understanding and a practical method for incorporation in river models. Finally, river junctions are considered, and the momentum approach with the very long period approximation shows that they can be modelled simply.