Anomalous overland flow on hillslopes: A fractional kinematic wave model, its solutions and verification with data from laboratory observations

Abstract

This paper analyses a fractional kinematic wave equation (fKWE) for overland flow and evaluates its solutions for applications using data from overland flow flumes with simulated rainfall in the laboratory. Solutions of fKWE presented have been derived for large time or when the Laplace transform variable s→0, which is one of the most important situations for overland flow. The solutions include expressions for the depth, velocity, and unit discharge, of overland flow. Fitting the approximate solution for the depth of overland flow to the data yields the values of the order of space-fractional derivatives, ρ, which is around ρ=1.5. It is found that ρ increases with the slope gradient and the rainfall intensity. The approximate order of ρ=1.5, which is about the average of order 1 for the advection equation and 2 for the diffusion equation, implies that diffusive mechanisms manifest in the overland flow. The findings mean that either an fKWE is used or a diffusion term is needed to account for dynamic forces in overland flow as the fKWE captures more physical mechanisms.