Can a fractional derivative diffusion equation model Laboratory scale fluvial transport Vaughan Voller * and Chris Paola Confusion on the incline * Responsible for all math and physical interpretation errors

A simple problem described by a diffusion model Diffusion models have been widely applied to describing fluvial long profiles. But experimental fluvial systems with induced aggradation (through subsidence and/or sea-level rise) typically display much less curvature than would be expected from a diffusional solution [area/time]  length/s] Piston subsidence of base solution

Essentially 100% of the supplied sand is deposited upstream of the break in slope visible around x = 3 m. Braided System=fractal=fractional Diffusion solution “too-curved” ~3m

First we will just blindly try a pragmatic approach where we will write down a Fractional derivative from of our test problem, solve it and compare the curvatures. With Our first attempt is based on the left hand Caputo derivative The divergence of a non-local fractional flux Note Solution

Clearly Not a good solution

With Our second attempt is based on the right hand Caputo derivative Note Solution On [0,1]

Looks like this Has “correct behavior” When we scale to The experimental setup We get a good match

Right

But the question du jour Is this physically meaningful Can observed fluvial surface behaviors Be related to the right-hand Caputo derivative Can the “statistics of behaviors identify 

Is a stable a Levy PDF distribution maximally skewed to the Right The solution of the transient fractional diffusion equation on the infinite domain with the initial condition of Dirac delta function at x = 0 Left Is a stable a Levy PDF distribution maximally skewed to the Left Right Skew factor Can we associate the “long tails” (i)non-local movements (jumps) of sediment down slope? (a left derivative) or (ii)non-local control of upslope by down slope events (a right derivative) Above results suggest that the second may be correct But next result confuses this a bit

And now an element of confusion We consider the steady sate fractional diffusion equations in a fixed domain [0,1] Left Right The Left solution is The Right solution is To demonstrate/understand the connection with the Levy pdf we propose To use a Monte-Carlo Solution

A Monte Carlo Solution N left N right T point = fraction of walks that exit on Left It is well know (and somewhat trivial) that a Monte Carlo simulation originating from a ‘point’ and using steps from a normal distribution will after multiple realizations recover the temperature at the ‘point’ CLAIM: If steps are chosen from a Levy distribution maximum negative skew, This numerical approach will also recover Solutions to Caputo

Left hand Right hand Points MC solution Lines Fractional Eq. Analytical sol. Thus on this closed Interval the association of the Levy is switched The right hand Caputo Is associated with the positive long tail The left hand Caputo Is associated with the neagative long tail

Conclusions So Far We can produce a solution to a fractional diffusion equation that matches the observed fluvial shape Still not clear how we can associate this with a physical model or measurement? Help !!!

An interesting aside A non-linear model of our steady sate problem can be envisioned If diffusivity is Proportional to The absolute slope A contention that can be supported via semi-physical arguments Then solution has form This matches the “best” fractional solution.