Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

Abstract

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter $0⩽R_{ab}⩽1$—the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, $0⩽\theta \to \infty$, the similarity solution shows that when $R_{ab}<0.4$ there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., $0⩽\theta \sim 2$, reasonable agreement in predictions extents up to $R_{ab}\sim 0.7$. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.