Nonlinear simulation of shape-preserving delta growth

In this paper, we demonstrate the existence of shape-preserving (self-similar) delta. We focus our study on a Gilbert-type delta growing by the sediment discharged at the river mouth. The evolving velocity of the shoreline depends on its local geometrical configuration including the water depth, repose angle, and curvature. Linear analysis of this local model suggests that when the sediment supply to the shoreline front can be maintained at a constant value, unstable growth occurs for adverse bathymetry (back-tilted basement) and there exists a critical water depth that leads to a self-similar evolution. A novel nonlinear analysis based on a rescaling idea reveals that there exists a critical flux J_c at the shoreline such that a desired shoreline shape can be achieved and maintained self-similarly independent of the water depth information, though the critical flux J_c may depend on the water depth. We then develop a semi-implicit numerical method to investigate the nonlinear dynamic of the shoreline. Our numerical results are in excellent agreement with the linear theory when the shape perturbation is small, and confirm that in the nonlinear regime the shoreline may evolve self-similarly under J_c. In particular, we demonstrate that a prescribed shoreline morphology can be achieved by a well-designed J_c. The existence of shape-preserving growing delta goes beyond the well known dynamical patterns and highlights the feasibility of shape control.