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

10,22

2

xdxhd

[area/time]

length/s]

Piston subsidenceof base

dxdhq 2

2

0h2)1(12 xxxh

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.

10,10,2)(0 xhDD xC

0)1(,2)0(0 hqhDxC

With

Our first attempt is based on the left hand Caputo derivative

dxd

dfddxxfD

x

xCo

011

The divergence of a non-local fractional flux

xxDC 21

1 xDC

0cDC

Note

Solution

)1()1(2

)2()1(2)(

1

xxxh

10,10,2)(0 xhDD xC

0)1(,2)0(0 hqhDxC

)1()1(2

)2()1(2)(

1

xxxh

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Clearly Not a good solution

5.0

10,10,2)( 1 xhDD Cx

0)1(,2)0(1 hqhDCx

With

Our second attempt is based on the right hand Caputo derivative

NoteSolution

dy

ddxxyD

x

Cx )()(

)1(1)(

1

1

)()( 101 xyDxyD xCC

x

On [0,1] 10

1)1()2(

2)(

xxh

10,10,2)( 1 xhDD Cx

0)1(,2)0(1 hqhDCx

1)1()2(

2)(

xxh

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

slope, right .1

right, 0.1

right, .5

left, .5

Looks like this Has “correct behavior”

When we scale toThe experimental setupWe get a good match

5.0

1

Right

But the question du jour

Is this physically meaningfulCan observedfluvial surface behaviors

Be related to the right-handCaputo 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

)0(),0(,0)(

xCCDDtC

x

Left 1

)0(),0(),(

xCCDDtC

x

Is a stable a Levy PDF distribution maximally skewed to the Left

Right

1

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 correctBut 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]

10,10,0)(0 xhDD xC

10,10,0)( 1 xhDD Cx

Left

Right

0)1(1)0( hh

xh 1

)1( xh

The Left solution is

The Right solution is

To demonstrate/understand the connection with the Levy pdf we proposeTo use a Monte-Carlo Solution

A Monte Carlo Solution

NleftNright

Tpoint = fraction of walks that exit on Left

It is well know (and somewhat trivial) that a Monte Carlo simulation originating froma ‘point’ and using steps from a normal distribution will after multiple realizationsrecover the temperature at the ‘point’

CLAIM: If steps are chosen from aLevy distribution

maximum negative skew, 25.0,1

This numerical approach will also recover Solutions to

0)1(,1)0(

10,0

hhdxhd

dxd

0)1(,1)0(

0

TTdxdT

dxd

Caputo

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1

25.0,5.0,1

Left hand

Right hand

1

1

Points MC solutionLines Fractional Eq. Analytical sol.

Thus on this closedInterval 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 matchesthe observed fluvial shape )1(~ xh

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

10,2

xdxdh

dxd

dxdhq 2

20h

If diffusivity is Proportional to The absolute slope

dxdh~

A contention thatcan be supported viasemi-physical arguments

Then solution has form5.1)1(~ xh

This matches the “best” fractionalsolution.