Theory of delayed response in river morphodynamics: Applicability and limitations

Abstract

The delayed response of fluvial rivers to external disturbances has been described by many researchers. To simulate such behavior, the rate law model (or the delayed response model) was developed by previous researchers, and has been applied to a series of river morphological problems. However, to date, the applicability of the rate law model has not been fully understood. In the current paper, a physically-based analysis of the rate law model is presented to assess the responses of bed elevation and the grain size of surface sediment, using the response of the Shi-ting River, China, after the 2008 Wenchuan Earthquake as an example. First, a physically-based river morphodynamic model is implemented to reproduce the post-earthquake adjustments of the Shi-ting River. Next, the mathematical properties of the Hirano–Exner equation, which describes the dynamics of both bed elevation and surface texture, are analyzed. It is shown that the dynamics of bed elevation are dominated by a diffusion process and the analytical solution for bed elevation has a similar mathematical formulation to that of the rate law model. Thus, the rate law and morphodynamic models similarly predict the adjustment of bed elevations. In contrast, the dynamics of bed surface texture are controlled by both advection and diffusion processes. The advection and diffusion processes dominate adjustments over the short (annual to decadal scale) and long (century to millenial scale) terms, respectively. These physics produce the multi-scale and non-monotonic character of the adjustment of bed surface texture. As a result, the rate law model is incapable of describing the adjustment of bed surface texture. These findings highlight the applicability and limitations of the rate law model in simulating river morphodynamic processes.