On the Physical Basis of Rate Law Formulations for River Evolution, and their Applicability to the Simulation of Evolution after Earthquakes

Abstract:

River morphology evolves in response to trade-offs among a series of environmental forcing factors, and this evolution will be disturbed if such environmental factors change. One example of response to chronic disturbance is the intensive river evolution after earthquakes in southwest China’s mountain areas. When simulating river morphological response to environmental disturbance, an exponential rate law with a specified characteristic response time is often regarded as a practical tool for quantification. As conceptual models, empirical rate law formulations can be used to describe broad brush morphological response, but their physically basis is not solid in that they do not consider the details of morphodynamic processes. Meanwhile, river evolution can also be simulated with physically-based morphodynamic models which conserve sediment via the Exner equation. Here we study the links between the rate law formalism and the Exner equation through solving the Exner equation mathematically and numerically. The results show that, when implementing a very simplified form of a relation for bedload transport, the Exner equation can be reduced to the diffusion equation, the solution of which is a Gaussian function. This solution coincides with the solution associated with rate laws, thus providing a physical basis for such formulations. However, when the complexities of a natural river are considered, the solution of the Exner equation will no longer be a simple Gaussian function. Under such circumstances, the rate law becomes invalid, and a full understanding of the response of rivers to earthquakes requires a complete morphodynamic model.