Can a fractional derivative diffusion equation model Laboratory scale fluvial transport

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

2. 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 A simple problem described by a diffusion model [area/time] solution [length/s] Piston subsidence of base

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

4. 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. Our first attempt is based on the left hand Caputo derivative With Note The divergence of a non-local fractional flux Solution

5. Clearly Not a good solution

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

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

8. Right

9. 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 a

10. The solution of the transient fractional diffusion equation on the infinite domain with the initial condition of Dirac delta function at x = 0 Right Is a stable a Levy PDF distribution maximally skewed to the Left Skew factor Left Is a stable a Levy PDF distribution maximally skewed to the Right • Can we associate the “long tails” • non-local movements (jumps) of sediment down slope? (a left derivative) or • 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

11. 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 And now an element of confusion We consider the steady sate fractional diffusion equations in a fixed domain [0,1] Left Right

12. A Monte Carlo Solution CLAIM: If steps are chosen from a Levy distribution maximum negative skew, This numerical approach will also recover Solutions to 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’ Nright Nleft Tpoint = fraction of walks that exit on Left Caputo

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

14. 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 !!!

15. 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.