5.b) Applications to transport in fusion plasmas Fractional … · 2005. 10. 6. · Fractional...

Fractional calculus:

basic theory and applications(Part III)

Diego del-Castillo-Negrete

Oak Ridge National LaboratoryFusion Energy Division

P.O. Box 2008, MS 6169

Oak Ridge, TN 37831-6169

phone: (865) 574-1127

FAX: (865) 576-7926

e-mail: [email protected]

Lectures presented at the

Institute of Mathematics

UNAM. August 2005

Mexico, City. Mexico

5.b) Applications to transport in

fusion plasmas

Here we present phenomenological models of plasma transport based on

the use of fractional diffusion operators. In particular, we extend the

fractional diffusion models discussed before, by incorporating finite-size

domain effects, boundary conditions, sources, spatially dependent

diffusivities, and general asymmetric fractional operators. These additions

are critical in order to order to go beyond tracers transport calculations.

We show that the extended fractional model is able to reproduce within a

unified framework some of the phenomenology of non-local, non-

diffusive transport processes observed in fusion plasmas, including

anomalous confinement time scaling, “up-hill” transport, pinch effects,

and on-axis peaking with off-axis fuelling. The discussion presented here

is based on [del-Castillo-Negrete-etal,2005a] and [del-Castillo-Negrete-

etal,2005b] where further details can be found.

Beyond the standard diffusive transport paradigm

•Experimental and theoretical evidence suggests that transport in

fusion plasmas deviates from the diffusion paradigm:

!tT = !

x" x,t( )!

xT[ ] + S(x, t)

•Anomalous confinement time scaling

• Fast propagation and non-local transport phenomena

• Inward transport observed in off-axis fueling experiments

•To overcome the limitations of the diffusion paradigm, we haveproposed transport models based on fractional derivative operatorsthat incorporate in a unified way non-locality, memory effects, andnon-diffusive scaling.

Examples:

Towards an effective transport model for non-diffusive

turbulent transport

!

dr r

dt= ˜ V =

1

B2" ˜ # $

r B

! P

! t+ ˜ V " #P = 0

•Individual tracers follow the turbulent field

•The distribution of tracers P evolves according to

•The idea is to construct a model that “encapsulates”

the complexity of the turbulence field in an

effective flux !, and reproduces the observed pdf

homogeneous

isotropic

turbulence! = "# $

xP

! P

! t="

! #

! x

Gaussian

closures

diffusive

transport

transport in pressure driven

plasma turbulence

˜ V

Non-Gaussian

distributionsP

x

!" #$=%

Riemann-Liouville derivatives

!

aDx

" # =1

$(n %")

& n

& xn# u( )

x % u( )"%n+1

du

a

x

'Left derivative

Right derivative

!

n = ["]+1

a bx

!

xDb

" # =$1( )

n

%(n $")

& n

& xn# u( )

u $ x( )"$n+1

du

x

b

'

["]=integral part of "

Some limitations of the RL definition

!

L0Dt

" #[ ] = s" ˆ # (s) $ sk

0Dt

"$k$1#[ ]t= 0

k= 0

n$1

%

Although a well-defined mathematical object, the Riemman-Liouville

definition of the fractional derivative has some problems when it comes

to apply it in physical problems. In particular:

!

limx"a a

Dx

# $ =%

unless

!

" (k )(a) = 0!

aDx

"A =

A

# 1$"( )

1

x $ a( )"

The derivative of a

constant is not zero

This might be an issue when

using RL operators for

writing evolution equations

The RL derivative

is in general singular

at the lower limit

This might be an issue

when applying boundary

conditions

The Laplace

transform

of the RL

derivative

depends on

the fractional

derivative at zero

This might be an

Issue when solving

Initial value problems

Behavior near the lower terminal

!

"(x) =" (k )(a)

k!k= 0

#

$ x % a( )kLet

!

aDx

" # =# (k )(a)

k!k= 0

$

% aDx

"x & a( )

k&"

!

aDx

"x # a( )

k

=$ k +1( )

$ k +1#"( )x # a( )

k#"

using

!

aDx

" # = # (k )(a)x $ a( )

k$"

% k +1$"( )k= 0

&

'

!

limx"a a

Dx

# $ = limx"a

$ (k )(a)

% k +1&#( )k= 0

m&1

'1

x & a( )#&k

!

limx"a a

Dx

# $ =% unless

!

" (k )(a) = 0

!

k =1,Lm "1!

m "1#$ < m

Caputo fractional derivativeThese problems can be resolved by defining the fractional operators in the

Caputo sense. Consider the case

!

aDx

" # $#(a)

%(1$")

1

(x $")"$

#'(a)

%(2 $")

1

(x $")"$1=

# (k+2)(a) x $ a( )

k+2$"

% k + 3$"( )k= 0

&

'

regular termssingular terms

!

aDx

" #(x) $#(a) $# (1)(a)[ ] =# (k+2)

(a) x $ a( )k+2$"

% k + 3$"( )k= 0

&

'

!

0

CDx

"# =aDx

" #(x) $#(a) $# (1)(a)[ ]

!

1<" < 2

Define the Caputo

derivative by subtracting

the singular terms

!

" " # (u)

x $ u( )%$1

du

a

x

& =# (k+2)

(a)

k!

x $ a( )k

x $ u( )%$1

a

x

&k= 0

'

( duusing

!

a

CDx

" # =1

$(2 %")

& & # (u)

x % u( )"%1

du

a

x

'

!

a

CDx

" # =1

$(n %")

&u

n#

x % u( )"%n+1

du

a

x

'In general

!

x

CDb

" # =($1)n

%(n $")

&u

n#

x $ u( )"$n+1

du

a

x

'

!

L0Dt

" #[ ] = s" ˆ # (s) $ sk

0# ("$k$1)[ ]

t= 0k= 0

n$1

%

!

limx"a a

Dx

# $ = 0

!

a

CDx

"A = 0and as expected:

Note that in an infinite domain

!

"#

CDx

$ % = "#Dx

$ %

!

x

CD"

# $ =xD"

# $

Nonlocal transport model

!

"T

" t=#

"

" xq

l+ qr + qG[ ] + P(x, t)

!

ql

= " l 2 "#( )$ l 0Dx

#"1T

!

qr = r 2 "#( )$ r xD1#"1

T

!

qG = " #G $x T

!

0Dx

"#1T =

1

$ 3#"( )x # y( )

2#"% % T dy

0

x

&

!

x D1

"#1T =

1

$ 3#"( )y # x( )

2#"% % T dy

x

1

&

Diffusive flux

Left fractional flux

Right fractional flux

Regularized finite-size left derivative

Regularized finite-size right derivative

10 x!

qr

!

qlNonlocal fluxes

Nonlocal transport model

!

"T

" t=#

"

" xq

l+ qr + qG[ ] + P(x, t)

!

ql

= " l 2 "#( )$ l 0Dx

#"1T

!

qr = r 2 "#( )$ r xD1#"1

T

!

qG = " #G $x T

!

x " 0,1( )

!

" T (0) = T(1) = 0

Finite size

domain

Boundary conditions

Diffusive flux

10

“core”

!

"G

Diffusive

transport

Non-diffusive

transport

!

"G," l ,"g

!

ql (0) + qr (0) + qG (0) = 0

Nondiffusive

fluxes

Local and non-local transport

!t" + !

xQ = 0 Q = !D "

x#

Local

diffusive

transport

!L(x) =

1

2L! x + s( ) ds

L

"

!t" = D!

x

2" = 6D

"L# "

L2

as L! 0

x

#2L

L=“radius of influence”

Q (x,t) = ! D G(x ! x' ) " x' #(x' ,t) dx'$

Non-local

diffusive

transport

Perturbative transport experiments have shown evidence of

non-local diffusion in fusion plasmas

Nonlocal transport

!(x, t) = dx' P x " x' ;t( )"#

#

$ !(x' ,t = 0)

!(x' ,t = 0)P(x ! x' ,t)

x'

!(x' ,t = 0)P(x ! x' ,t)

x'

local transport nonlocal transport

!(x, t) " 0 !(x, t) " 0

Due to the algebraic tail of the propagator, fractional diffusion

leads to nonlocal transport

limx!"

P ~ x#( 1+$)lim

x!"

P ~ e# x 2

T (r = a)

Nonlocal transport

Edge temperature rise

Fractional

diffusion

Gaussian

diffusion

Nonlocal transport and global diffusive coupling

!x

"# = lim

h$ 0

%h

"#

h"

! h

" #(x) = $1( )j "

j

%

& ' ( ) j= 0

N

* # x $ j h( )

! x"#k $

%h

"#( )

k

h" =

1

h" wj

"

j= 0

k

& #k ' j

w0

!=1 wj

!= 1 "

! +1

j

#

$ % &

' wj"1

!

Grunwald-Letnikov definition

Finite difference

approximation

Global coupling

coefficients

For " =1 and 2 usual nearest neighbor coupling of Laplacian operator

For 1 < " < 2 fractional diffusion implies global coupling

Steady state and anomalous confinement time scaling

Uniform heating On axis heating

Blue "=2

Green " =1.75

Red " =1.5

Black " =1.25

!

" =T dx

0

L

#P dx

0

L

#

!

T = P0

L"

#l

$

% &

'

( )

1

* " +1( )1+

x

L

$

% & '

( ) ",

- .

/

0 1

Steady state solution

!

" =#

$ 2 +#( )

L#

%l

Confinement time

Anomalous scaling

!

"r

= 0

"l# 0

1 < ! < 2

Confinement time scaling

!

~L

L*

"

# $

%

& '

2

!

~L

L*

"

# $

%

& '

2 !

~L

L*

"

# $

%

& '

(

!

~L

L*

"

# $

%

& '

2!

~L

L*

"

# $

%

& '

(

!

T*

="d

#

"a

2

$

% &

'

( )

1

2*#

!

L*

="d

"a

#

$ %

&

' (

1

2)*

!

!

L = Domain size

Standard diffusivity

!

"d

=

!

"a

= Anomalous diffusivity

!

" = Confinement time Diffusion +

Right fractional

Diffusion +

Symmetric fractional

Diffusion +

Left fractional

“Up-hill” transport and pinch velocity

Gaussian

Flux

Fractional

Flux

!

r > 0

l = 0

Standard diffusion Fractional diffusion

Up-hill

Transport

zone

!

qG = " #G $x T

!

qr = r xD1"#1

T

Fractional flux asymmetry and pinch velocity

Flux

!

r > l

Symmetric fractional

diffusionAsymmetric fractional

diffusion

!

r = l

Right flux

Left flux

Right flux

Left flux!

qr

!

qr

!

ql

!

ql

“Cold-pulse” fast propagation


x x

tim

e

tim

e

Steady state

Pulse

Off-axis heating

Flux Flux


!

"R

!

qG

!

qR

!

qT = qG + qR

!

qG

5.c) Application to reaction-

diffusion systems

Here we discuss the role of fractional diffusion in reaction-diffusion

systems. In particular, we present a numerical and analytical study of

front propagation in the fractional Fisher-Kolmogorov equation. The

discussion presented here is based on [del-Castillo-Negrete-etal,2003]

where further details can be found.

Reaction diffusion models of front dynamics

( )!!"!#$!$ %+= 12

xt

Fisher-Kolmogorov equation

diffusion reaction

! = 0

! =1

!"2=c

c

unstable

stable

•Plasma physics

•Model of genes dynamics

•Model of logistic population growth with dispersion

•Other applications include combustion and chemistry.

The fractional Fisher-Kolmogorov equation

( )( )!

"#

#"###$

=x

xx dyyx

yD

1

2

)2(

1%

% &'

%&

1 < ! < 2( )!!"!#!$ % &+= '& 1xtD

The asymmetry in the fractional derivative leads to an asymmetry in the fronts

Self-similar dynamics, exponential

tails, constant front velocityQuasi-self-similar dynamics,

algebraic tails, front acceleration

“left moving” “right moving”

! = 1.25

Asymmetric front dynamics

!(x, 0) =1

21 + tanh

x " x02W

# $

% &

'

( ) *

+ ,

x

!

Exponential decaying

constant speed frontsAlgebraic decaying

accelerated fronts

x

!

!(x, 0) = 1"1

21 + tanh

x " x02W

# $

% &

'

( ) *

+ ,

! = 1.25

Asymmetric front dynamics

!(x, 0) =1

21 + tanh

x " x02W

# $

% &

'

( ) *

+ ,

Exponential decaying

constant speed frontsAlgebraic decaying

accelerated fronts

!(x, 0) = 1"1

21 + tanh

x " x02W

# $

% &

'

( ) *

+ ,

x

t

x

t

Leading edge calculation for algebraic fronts

At large x, !"!#!$ %+=

xtD

!(x, 0) =1 x < 0

e"#x

x > 0

$ % &

! <<1 implies

!!"

#

"#

#+#

#

#

+=$

$

$

%

$&

%

$'%(&( '''')

/1

/1

/1

)(

,1

)(

,1

)( )()(

tx

t

tx

ttxdLedLee

( )!!"!#!$ % &+= '& 1xtD

),( txet!" #

= !"!# $

xtD=

( ) ( )[ ] !!"#!#$

$ dtxLtx/1

01,),( %= &'

'%

%

1)/(for~~)( /1)1(

,1 >>! "+"

"

###

# $%&& txxetLt

Right moving algebraic decaying,

accelerated front

c ~ e! t /"

! ~ x"#

Right moving algebraic decaying,

accelerated front

! ~ x"#e$ t

Leading edge calculation for exponential fronts

At large x,

!t" = D !

x

#" + $ "

!(x, 0) =e"x

x < 0

1 x > 0

# $ %

! <<1 implies

In this case the calculation is simple because the front is self-similar

and retains the exponential decaying shape.

!x

"e#x= #

"e#xUsing the fact that we conclude

c =

! "

! #1

$ %

& '

! # 1( )D

"

(

) * +

, -

1/!

if 1/ . /Wc

"

.+

D

.1#!if 1/ . 0Wc

1

2

3 3

4

3 3

Left moving exponential decaying,

constant velocity front

c =! "

! #1

$ %

& '

! # 1( )D

"

(

) * +

, -

1/!Fractional generalization of

Kolmogorov front speed.

6. Some references

Fractional calculus:

•[Samko-et-al-1993] S.G. Samko, A.A. Kilvas and O.I. Maritchev,

Fractional Integrals and Derivatives. (Gordon and Breach Science

Publishers, 1993).

•[Podlunbny-1999] I. Podlunbny, Fractional differential

equations.(Academic Press, San Diego, 1999).

Random walks and fractional calculus

•[Montroll-Weiss-1965] E.W. Montroll and G.H. Weiss, Random

Walks on Lattices. II. J. Math. Phys. 6, 167 (1965).

•[Montroll - Schlesinger-1984] E.W. Montroll and M.F. Schlesinger,

The wonderfull world of random walks. In Nonequilibrium Phenomena

II. From Stochastics to Hydrodynamics, edited by J.L. Lebowitz and

E.W. Montroll. (Elsevier, Amsterdam, 1984)

•[Metzler-Klafter-2000] R. Metzler and J. Klafter, The random walk´s

guide to anomalous diffusion: a fractional dynamics approach. Physics

Reports, 339, December 2000.

Stochastic processes:

•[Paul- Baschnagel-1999] W. Paul, and J. Baschnagel, Stochastic

processes: from physics to finance (Springer –Verlag, Berlin 1999).

•[Bouchard-Georges-1990] J.P. Bouchard, A. Georges, Anomalous

diffusion in disorder media. Physics Reports 195 127 (1990).

Anomalous diffusion in fluids and plasmas

•[del-Castillo-Negrete-1998] D. Del-Castillo-Negrete, Asymmetric

transport and non-Gaussian statistics in vortices in shear. Physics of

Fluids 10, 576 (1998).

•[del-Castillo-Negrete-2000] D. Del-Castillo-Negrete, Chaotic

transport in zonal flows in analogous geophysical and plasma systems.

Physics of Plasmas 7, 1702 (2000).

Fractional diffusion models of transport in plasmas

•[del-Castillo-Negrete-etal-2004] D. Del-Castillo-Negrete, B.A.

Carreras and V.E. Lynch, Fractional diffusion in plasma turbulence.

Physics of Plasmas 11, 3854 (2004).


Carreras and V.E. Lynch, Nondiffusive transport in plasma turbulence:

a fractional diffusion approach. Physical Review Letters 94, 065003

(2005).

•[del-Castillo-Negrete-2005a] D. Del-Castillo-Negrete, Fractional

diffusion models of transport in magnetically confined plasmas.

Proceedings of the 32nd European Physical Society, Plasma Physics

Conference. Tarragona Spain (2005).

•[del-Castillo-Negrete-2005b] D. Del-Castillo-Negrete, Fractional

diffusion models of nonlocal transport in magnetically confined

plasmas. Preprint (2005).

Fractional reaction-diffusion systems


Carreras and V.E. Lynch, Front dynamics in reaction-diffusion systems

with Levy flights: a fractional diffusion approach. Physical Review

Letters 91, 018302-1 (2003).

Numerical integration of fractional equations

•[Lynch-etal-2003] V.E. Lynch, B.A. Carreras, D. Del-Castillo-

Negrete, K.M. Ferrerira-Mejias and H.R. Hicks.Numerical methods for

the solution of partial differential equations of fractional order. Journal

of Computational Physics. 192, 406-421 (2003).

•[del-Castillo-Negrete-2005c] D. Del-Castillo-Negrete, Finite

difference methods for the numerical integration of fractional partial

differential equations. Preprint (2005).

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