Can anomalous diffusion describe depositional fluvial profiles?

Diffusion models have been widely applied to describe fluvial long profiles. However, aggrading rivers simulated in laboratory experiments typically display much less profile curvature than a diffusion model would predict, whether aggradation is driven by subsidence or by base-level rise. Here we explore the possibility that this is due to nonclassical or non-Fickian, anomalous sediment transport in braided networks, which are known to have fractal planform geometry. We solve a fractional diffusion equation for a steady state aggrading fluvial profile for fractional exponents in the spatial derivative a in the range 1.1 < alpha + 1 < 2. The domain is bounded at both ends, and a constant sediment sink forces extraction of all of the imposed, constant sediment supply. We assume the fractional behavior is expressed solely by a nonlocal sediment flux term. Using the right-hand Caputo fractional derivative, we are able to construct a fractional diffusion equation that admits an analytical fluvial profile closely matching the laboratory-scale physical observations. We show that this solution is also in good agreement with a Monte Carlo simulation obtained using step lengths drawn from a Levy probability density. However, despite the clear nonlocal behavior, we are unable to establish a direct physical link between the power law statistics of the fluvial transport system and the mathematical ingredients in a fractional diffusion model. In general, we expect the fractional behavior to be most pronounced when the length scale of significant downstream sediment extraction is comparable to the scale range of the fractal channel pattern behavior. This is typically the case for laboratory experiments but not at field scales, which could explain why anomalously flat fluvial long profiles have not been reported from the field. Applying fractional calculus to depositional river profiles, at any scale, exposes problems in applying fractional calculus posed by the bounded domain and the presence of a distributed sink associated with sediment extraction. With present understanding, the benefit of fractional calculus, which is its ability to capture effects of power law statistics in the underlying dynamics, comes at a significant cost in terms of flexibility to handle other physical effects such as complex domains, boundary conditions, and source terms.