Least action principle, equilibrium states, iterative adjustment and the stability of alluvial channels

The energy based least action principle (LAP) has proven to be very successful for explaining natural phenomena in both classical and modern physics. This paper briefly reviews its historical development and details how, in three ways, it governs the behaviour and stability of alluvial rivers. First, the LAP embodies the special stationary equilibrium state of motion and so its incorporation with the principle of energy conservation explains why so many optimizing hypotheses have been proposed in fluvial geomorphology. Second, the variational approach underlying the LAP provides a more straightforward and simpler fuzzy-object orientated method for solving river regime problems than do the various complex Newtonian formulations. Third, it is shown that in fluvial systems with surplus energy the surplus can be expended with slope and/or channel geometry adjustments, with the degree of channel geometry adjustment quantified by the dimensionless numbers F for depth dominated adjustment and H for width/depth dominated adjustment. Different planforms are preferred at different energy levels, with H providing a quantitative measure of the flow's efficiency for moving sediment. In rivers with insufficient energy, the interactions of endogenous and exogenous factors are shown to be capable, in certain circumstances, of achieving a stationary equilibrium condition which acts as the attractor state. Importantly, this study describes how iterative changes enable systems to achieve such a stable equilibrium. Copyright (C) 2007 John Wiley & Sons, Ltd.