Non-local behavior in Geomorphology: Vaughan R Voller, [email protected] NCED Summer Institute,...

Non-local behavior in Geomorphology: Vaughan R Voller, [email protected]

NCED Summer Institute, 4:30-6:00 pm, August 19, 2009

Objectives

-- Define anomalous diffusion

-- Illustration of normal, sub and super diffusion

--Outline some very basic elements of factional calculus

-- Provide an explicit physical connection between the order of fractional derivatives and sub and super diffusion processes



Despite our best efforts we can not be in two places at once

Hence our intuitive physical sense of the world is based on LOCAL information

Local

At a point

dx

dq

slope

At an instant

dt

dv

rate



DEPENDENT PATH

dx

dq

dt

dv

But many physical process are NON-LOCAL

Holdup – releaseDepends on time scale

Extensive developing literature that argues that these non local processCan be described by Fractional Derivatives

dx

dq

dx

dv 1,0

!!! How on earth do we associate these constructs with our intuitively locally?



A physically incomplete but meaningful analogy to describe anomalous diffusion

How will drop of colored water spread out on tissue after time t ?




rsubrn

rsup

If blot radius grows as

21

~ trn

Diffusion is said to be normal

If the time exponent differs from ½ Diffusion is said to be anomalous

Supper Diffusion

21

sup ;~ tr

Sub Diffusion

21;~ trsub




rsubrn

rsup

21

~ trnnormalSupper Diffusion

21

sup ;~ tr

Sub Diffusion

21;~ trsub

Described by

)( Ct

C

Transientchange in volume

Divergence of flux(volume balance)

1,0),(

Ct

C

1,0 How do the exponents in the frac. Dervs. relate to sub or supper diffusion

non zero waitnon local flux



1,0 How do the exponents in the frac. Dervs. relate to sub or supper diffusion ?

To answer we consider a well known limit problem for flow in porous media

fixed head 00 hh

0x

0)( shfixed head

)(tsx sharp Moving Front between saturated and dry

Water supply saturated dry

Gov. equ.(mass con + Darcy)

)(0,0 tsxx

h

x

Note transient limits

Extra volume balance condition at front

dt

ds

x

h

ts

)(



01h

0x

0)( sh

)(tsx

saturated dry

)(0,0 tsxx

h

x

Extra volume balance condition at front

dt

ds

x

h

ts

)(

solution satisfying conditions

s

xh 1

sub here

21

20)0(,1 tssdt

dss

Darcy Flux assumes normal diffusionThis results in advance of saturated region with characteristic time scale 2

1

~ t



01h

0x

0)( sh

)(tsx

saturated dry

)(0,0 tsxx

h

x

Now let us look at problem with fractional derivatives

dt

sd

x

h

ts )(

If our fractional derivatives are Caputo derivatives (see below) then we can easily solve this problem. The result is a wetting front that moves as

1,0,11

0,~

ts

So choice of fractional exponents allows us to Move from a sub to super diff. behavior



01h

0x

0)( sh

)(tsx

saturated dry

To solve fractional derivative version of problem we use Caputo derivatives

10,)(

1

1)(

0

dd

dfx

dx

xfd x

a

Can evaluate using Lapalce transform

xfLsdx

xfdL

)( 0)(limand 10

0

xf

xIF

For solution of problem all you need to know is that

1or 1if,

1

1 and 0constant)(

xx

x

x

gamma function

1



01h

0x

0)( sh

)(tsx

saturated dry

)(0,0 tsxx

h

x

Now let us look at problem with fractional derivatives

dt

sd

x

h

ts )(

s

xh 1

solution satisfying conditions

sx

h )1(

Atssdt

sds 0)0(,1

11

)1(

)1()1(

A1

10



01h

0x

0)( sh

)(tsx

saturated dry

)(0,0 tsxx

h

x

dt

sd

x

h

ts )(

s

xh 1 Ats

11

)1(

)1()1(

A1

10



01h

0x

0)( sh

)(tsx

saturated dry

)(0,0 tsxx

h

x

dt

sd

x

h

ts )(

Ats 11

0

Observations: For appropriate choices of fractional derivativesSuper >.5, normal = .5 , or sub <.5 diffusion can be realized

If only flux fractional derivative is used 0 < < 1, = 1 then ONLYSuper diff can be realized

If only trans. fractional derivative is used 0 < < 1, = 1 then ONLYSub diff can be realized



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

time(t)

dist

ance

s

=0.95

=0.5

=0.75

=0.05

=0.25

)1,1.(05.0 ),2.,3.(25.0 ),5.,75.(5.0 )052632.,1(95.0

Figure 1: Movement of the liquid/solid interface for choices of time fractional and flux fractional derivatives

.

)1333.,85.(75.0

This simple problem has allowed for a “trivial” solution to aFractional PDE. A solutionthat has provided a clear physicalconnection between the orderof the fractional derivative and thenature of the anomalous diffusion

Non-local behavior in Geomorphology: Vaughan R Voller, [email protected] NCED Summer Institute,...

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