Fractional diffusion models of anomalous transport:theory and applications D. del-Castillo-Negrete...
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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
( )[ ]