Fractional diffusion models of anomalous transport:theory

and applications

D. del-Castillo-Negrete

Oak Ridge National Laboratory

USA

Anomalous Transport:Experimental Results and Theoretical Challenges

July 12 to 16, 2006,Bad Honnef Bonn, Germany

Remembering Radu Balescu

•Radu Balescu had a prolific scientific career that touched many aspects of theoretical physics. Plasma physics was among his main interests

“I have been ‘in love’ with the statistical physics of plasmas over my whole scientific life…..” R. Balescu, in Transport processes in plasmas, North Holland (1998)

Some of his contributions to plasma physics include:

•In 1960, Balescu derived a new kinetic equation (the Balescu-Lenard equation) that incorporates the collective, many-body character of the collision processes in a plasma.

•In 1988, Balescu published Transport Processes in Plasmas, a comprehensive set of 2 volumes describing the classical and the neoclassical theories of transport in plasmas

•In more recent years, Balescu pioneered the use of novel statistical mechanics techniques, including the continuous time random walk model, to describe anomalous diffusion in plasmas.

Fusion plasmas

Fusion in the sun

Controlled fusion on earth

• Understanding radial transport is one of the key issues in controlled fusion research

•This is a highly non-trivial problem!

•Standard approaches typically underestimate the value of the transport coefficients because of anomalous diffusion

Magnetic confinement

Beyond the standard diffusive transport paradigm

•The standard theory of plasma transport in based on models of the form

€

∂t T = ∂x χ ∂x T[ ] + S(x, t)

•Our goal is to use fractional diffusion operators to construct and test transport models that incorporate these anomalous diffusion effects.

•The main motivation is plasma transport, but the results should be of interest to other areas.

•However, there is experimental and numerical evidence of transport processes that can not be described within this diffusive paradigm.

•These processes involve non-locality, non-Gaussian (Levy) statistics, non-Markovian (memory) effects, and non-diffusive scaling.

Outline

I Fractional diffusion models of turbulent transport

II Fractional diffusion models of non-local transport in finite-size domains

I will focus on two applications of fractional diffusion to anomalous transport in general and plasma physics in particular

D. del-Castillo-Negrete, B.A. Carreras, and V. Lynch: •Phys. Rev. Lett. 94, 065003, (2005).•Phys. of Plasmas,11, (8), 3854-3864 (2004).

D. del-Castillo-Negrete, •Fractional diffusion models of transport in magnetically confined plasmas, Proceedings 32nd EPS Plasma Physics Conference,Spain (2005).•Fractional models of non-local transport. Submitted to Phys. of Plasmas (2006).

Turbulent transport

δr r (t)=

r r (t)−

r r (0)

=ensemble average

M(t)= δr r =mean

σ 2(t)= δr r − δ

r r [ ]

2=variance

P(δr r ,t) =probability distribution

r r (t)

homogenous, isotropic turbulence

Brownian random walk

M(t)=Vt

σ 2(t)=Dt

P(δr r ,t) =Gaussian

limt→ ∞

V= transport velocity D=diffusion coefficient

Coherent structures can give rise to anomalous diffusion

trapping region

exchange region

transport region

flightevent

trapping event

t

x

Coherentstructures correlations

€

σ 2(t) ~ tγ

P(x, t) = non - Gaussianlimt→ ∞

Coherent structures in plasma turbulence

ExB flow velocity eddiesinduce particle trapping

Tracer orbits

Trapped orbit

“Levy”flight

“Avalanche like” phenomena induce largeparticle displacements that lead to spatial non-locality

Combination of particle trapping and flights leads to anomalous diffusion

φ

ρ θ

Anomalous transport in plasma turbulence

δr2 ~t4/ 3

Levy distribution of tracers displacements

Super-diffusive scaling3-D turbulence model

∂t +˜ V ⋅∇( ) ∇⊥

2 ˜ Φ =B0

min0rc

1r

∂˜ p ∂θ

−1

ηmin0R0

∇ ||2 ˜ Φ +μ∇ ||

4 ˜ Φ

∂ p∂ t

+1r

∂∂r

r ˜ V r ˜ p =S0 +D1r

∂∂r

r∂ p∂r

⎛ ⎝ ⎜ ⎞

⎠

∂t +˜ V ⋅∇( ) ˜ p =

∂ p∂r

1r

∂˜ Φ ∂θ

+χ⊥∇⊥2 ˜ p +χ||∇||

2 ˜ p

Tracers dynamics

dr r

dt=

r V =

1B2 ∇ ˜ Φ ×

r B

xn ~tnν ν ~2/3ν =1/2Diffusive scaling

Anomalous scalingsuper-diffusion

tν P

x/ tνMoments

Probability density function

€

∂t P = ∂x χ ∂x P[ ] ??????????

Standard diffusion Plasma turbulence

Model

Gaussian

Non-Gaussian

Continuous time random walk model

= jumpζn

ζnτn

τn = waiting time ψ τ( ) = waiting time pdf

λ ζ( )= jump size pdf

€

∂t P = dt ' φ(t − t') dx ' λ (x − x ') P(x', t) − λ (x − x') P(x, t)[ ]−∞

∞

∫0

t

∫

ψ (τ)~e−μτ

λ(ζ )~e−ζ 2 / 2σ

No memory

Gaussiandisplacements

€

∂t P = χ ∂x2 P

MasterEquation(Montroll-Weiss)

ψ (τ)~τ−(β+1)

λ(ζ )~ζ−(α +1)

Long waiting times

Long displacements (Levy flights)

€

0cDt

β φ = χ D|x|α φ

€

˜ φ (s) = s ˜ ψ /(1− ˜ ψ )

Standard diffusion

Fractional diffusion

Fluid limit

Riemann-Liouville fractional derivatives

€

a Dxα φ =

1

Γ(n −α )

∂ n

∂ x n

φ u( )

x − u( )α −n +1 du

a

x

∫Left derivative

Right derivative

€

n = [α ] +1a bx

€

x Dbα φ =

−1( )n

Γ(n −α )

∂ n

∂ x n

φ u( )

u − x( )α −n +1 du

x

b

∫

€

0 Dxμ x λ =

Γ(λ +1)

Γ(λ − μ +1)x λ −μ

€

−∞Dxμ e ikx = ik( )

μe ikx

€

0 Dxm =

∂m

∂ xm

Fractional diffusion model

€

∂P

∂ t=−

∂

∂ xql + qr[ ]

•Non-local effects due to avalanches causing Levy flights modeled with fractional derivatives in space.

•Non-Markovian, memory effects due to tracers trapping in eddies modeled with fractional derivatives in time.

€

l = −(1−θ)

2cos απ /2( )

€

r = −(1+ θ)

2cos απ /2( )

“Left” flux

€

ql =− l χ l 0Dtβ −1

aDxα −1 P

“Right” flux

€

qr =r χ r 0Dtβ −1

xDbα −1 P

€

0cDt

β P = l χ l aDxα + r χ r xDb

α[ ] P

Flux conserving form

Comparison between fractional model and turbulenttransport data

Turbulencesimulation Fractional

model

~x−(1+α)

Levy distribution at fixed time

Turbulence

~tβ

model

Pdf at fixed point in space

~t−β

α =3/4 β =1/ 2 x2 ~t2β /α ~t4/3

€

θ =0

Effective transport operators for turbulent transport

€

dr r

dt= ˜ V =

1

B2∇ ˜ Φ ×

r B

∂ P∂ t

=−˜ V ⋅∇ P

Individual tracers move following the turbulent velocity field

The distribution of tracers P evolves according the passive scalar equation

The proposed model encapsulates the spatio temporal complexity of the turbulence using fractional operators in space and time

Fractional derivative operators are useful tools to construct effective transport operators when Gaussian closures do not work

∂t +˜ V ⋅∇( ) ⇔ ∂t −χ ∂x

2( )

€

∂P

∂ t=−

∂

∂ xql + qr[ ]

€

∂t + ˜ V ⋅∇[ ] ⇔ 0 Dtβ − χ a Dx

α +xDbα

( )[ ] Fractionalapproach

Gaussianapproach

II Fractional diffusion models of non-local

transport finite-size domains

Finite size domain transport problem

€

T(L) = 0

Boundary conditions

€

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

φ

ρ θ

r

T

LDiffusive transport

Non-diffusive transport

0

•Fractional diffusion in unbounded domains with constant diffusivities is well understood

•However, this is not the case for finite size domains with variable diffusivities andboundary conditions

•Understanding this problem is critical forapplications of fractional diffusion

•One-field, one-dimensionalsimplified model of radialtransport in fusion plasmas

Singularity of truncated fractional operators

€

φ(x) =φ(k )(a)

k!k= 0

∞

∑ x − a( )k

€

a Dxα x − a( )

k=

Γ k +1( )Γ k +1−α( )

x − a( )k−α

€

a Dxα φ = φ(k )(a)

x − a( )k−α

Γ k +1−α( )k= 0

∞

∑

€

limx →a aDxα φ = limx →a

φ(k )(a)

Γ k +1−α( )k= 0

m−1

∑ 1

x − a( )α −k

€

limx →a aDxα φ = ∞ unless

€

φ(k )(a) = 0

€

k =1,L m −1€

m −1≤ α < m

•Finite size model are non-trivial because the truncate (finite a and b) RL fractional derivatives are in general singular at the boundaries

Let

using

Regularization

These problems can be resolved by defining the fractional operators in the Caputo sense. Consider the case

€

a Dxα φ =

φ(a)

Γ(1−α )

1

(x −α )α+

φ'(a)

Γ(2 −α )

1

(x −α )α −1+

φ(k +2)(a) x − a( )k +2−α

Γ k + 3−α( )k= 0

∞

∑

regular termssingular terms

€

a Dxα φ(x) − φ(a) − ′ φ (a)(x − a)[ ] =

φ(k +2)(a) x − a( )k +2−α

Γ k + 3 −α( )k= 0

∞

∑

€

0C Dx

α φ = aDxα φ(x) − φ(a) − ′ φ (a)(x − a)[ ]

€

1 < α < 2

€

aCDx

α φ =1

Γ(2 −α )

′ ′ φ (u)

x − u( )α −1 du

a

x

∫

€

xC Db

α φ =1

Γ(2 −α )

φ' '

x − u( )α −1 du

a

x

∫

We define the regularized left derivative as

Similarly, for the right derivative we write:

Fractional model in finite-size domains

€

∂T

∂ t=−

∂

∂ xql + qr + qG[ ] + S(x, t)

€

ql = − l χ l 0ˆ D x

α −1 T

€

qr = r χ r xˆ D L

α −1 T

€

qG = − χ G ∂x T

€

0ˆ D x

α −1 T = 0Dxα −1 T − T(a) − ′ T (a)(x − a)[ ]

Diffusive flux

Left fractional flux

Right fractional flux

Regularized finite-size left derivative

Regularized finite-size right derivative

L0 x€

qr

€

ql

Non-local fluxes

€

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

€

T(L) = 0

Boundary conditions

€

xˆ D L

α −1 T = xDLα −1 T − T(b) − ′ T (b)(b − x)[ ]

Steady state numerical solutions

€

∂t T =0

€

α =1.5

Right-asymmetric

Left-asymmetric

Symmetric

Regular diffusion

Right-asymmetric

Symmetric

€

∂∂ x

−l χ l 0ˆ D x

α −1 + r χ r xˆ D L

α −1 − χ g ∂x[ ] T = S0

Confinement time scaling

€

T* =χ d

α

χ a2

⎛

⎝ ⎜

⎞

⎠ ⎟

1

2−α

€

L* =χ d

χ a

⎛

⎝ ⎜

⎞

⎠ ⎟

1

2−α

€

€

L =Domain size

€

τ =Confinement time

€

τ =T dx

0

L

∫S dx

0

L

∫

L

€

′ L

€

τ (L) < τ ( ′ L )

€

τ ~ L2

€

τ ~ Lα

€

α < 2

Diffusive scaling

Experimentally observed anomalous scaling

€

τ ~ L2€

τ ~ Lα

€

τ /T*

€

L /L*

Fractional model

€

α =1.5 θ = 0

Non-local heat transport

Flux

Fractional diffusion

€

χR

€

qG

€

qR

€

qT = qG + qR

S

DIII-DLuce et al., PRL (1992)

Standard diffusion

S

x

x

x

T

T

α

€

χ∂x2 T = S

“Up-hill” anti-diffusive transport

GaussianFlux

FractionalFlux

€

r > 0

l = 0

Standard diffusion Right fractional diffusion

Up-hillNon-Fickian

“anti diffusive zone”

€

qG = − χ G ∂x T

€

qr = r xD1α −1 T

€

V ≈α +1

2α

⎛

⎝ ⎜

⎞

⎠ ⎟α 1/α tan

α π

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

xm =V θ χ 1/α t β /α

€

α =1.3 θ =1

Fast propagation of cold pulse

Left asymmetric Right asymmetric

€

α =1.25 θ = −1

€

α =1.25 θ =1

Standard diffusion Fractional diffusion

•Fractional calculus is a useful tool to model anomalous transport in fusion plasmas.

•Fractional diffusion operators are integro-differential operators that incorporate in a unified way: nonlocality, memory effects, non-diffusive scaling, Levy distributions, and non-Brownian random walks.

•We have shown numerical evidence of “fractional renormalization” of turbulent transport of tracers in plasma turbulence with

•We have studied anomalous transport in finite size domains using a regularized fractional diffusion model. We used the model to describe, at a phenomenological level, anomalous diffusion in magnetically confined fusion plasmas.

Conclusions

€

∂t + ˜ V ⋅∇[ ] ⇔ 0 Dtβ − χ a Dx

α +xDbα

( )[ ]