A nonlocal theory of sediment transport on hillslopes

Hillslopes are typically shaped by varied processes which have a wide range of event-based downslope transport distances, some of the order of the hillslope length itself. We hypothesize that this can lead to a heavy-tailed distribution of displacement lengths for sediment particles. Here, we propose that such a behavior calls for a nonlocal computation of the sediment flux, where the sediment flux at a point is not strictly a function (linear or nonlinear) of the gradient at that point only but is an integral flux taking into account the upslope topography (convolution Fickian flux). We encapsulate this nonlocal behavior in a simple fractional diffusive model which involves fractional derivatives, with the order of differentiation (1 < alpha <= 2) dictating the degree of nonlocality (alpha = 2 corresponds to linear diffusion and strictly local dependence on slope). The model predicts an equilibrium hillslope profile which is parabolic close to the ridgetop and transits, at a short downslope distance, to a power law with an exponent equal to the parameter alpha of the fractional transport model. Hillslope profiles reported in previously studied sites support this prediction. Furthermore, we show that the nonlocal transport model gives rise to a nonlinear dependency on local slope and that variable upslope topography leads to widely varying rates of sediment flux for a given local hillslope gradient. Both of these results are consistent with available field data and suggest that nonlinearity in hillslope flux relationships may arise in part from nonlocal transport effects in which displacement lengths increase with hillslope gradient. The proposed hypothesis of nonlocal transport implies that field studies and models of sediment fluxes should consider the size and displacement lengths of disturbance events that mobilize hillslope colluvium.