Fractional diffusion models of anomalous transport:theory and applications

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Fractional diffusion models of anomalous
transporttheory and applications

• D. del-Castillo-Negrete
• Oak Ridge National Laboratory
• USA

Anomalous TransportExperimental Results and
Theoretical Challenges July 12 to 16, 2006, Bad
Honnef Bonn, Germany
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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.

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Fusion plasmas
Fusion in the sun
Magnetic confinement

• 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

Controlled fusion on earth
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Beyond the standard diffusive transport paradigm

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

• 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.

• 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.

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

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

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

• 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).

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Turbulent transport
homogenous, isotropic turbulence
Brownian random walk
V transport velocity Ddiffusion coefficient
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Coherent structures can give rise to anomalous
diffusion
trapping region
transport region
exchange region
Coherent structures
correlations
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Coherent structures in plasma turbulence
Avalanche like phenomena induce large particle
displacements that lead to spatial non-locality
Combination of particle trapping and flights
leads to anomalous diffusion
ExB flow velocity eddies induce particle trapping
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Anomalous transport in plasma turbulence
3-D turbulence model
Super-diffusive scaling
Levy distribution of tracers displacements
Tracers dynamics
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Standard diffusion
Plasma turbulence
Probability density function
Gaussian
Non-Gaussian
Moments
Anomalous scaling super-diffusion
Diffusive scaling
??????????
Model
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Continuous time random walk model
Master Equation (Montroll-Weiss)
Fluid limit
Standard diffusion
Long waiting times
Fractional diffusion
Long displacements (Levy flights)
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Riemann-Liouville fractional derivatives
Left derivative
Right derivative
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Fractional diffusion model

• 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.

Flux conserving form
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Comparison between fractional model and
turbulent transport data
Levy distribution at fixed time
Turbulence simulation
Fractional model
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Effective transport operators for turbulent
transport
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
Fractional approach
Gaussian approach
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II Fractional diffusion models of non-local
transport finite-size domains
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Finite size domain transport problem

• Fractional diffusion
• in unbounded domains
• with constant diffusivities
• is well understood
• However, this is not the case for finite size
  domains with variable diffusivities and
• boundary conditions
• Understanding this problem is critical for
• applications of fractional diffusion

• One-field, one-dimensional
• simplified model of radial
• transport in fusion plasmas

Boundary conditions
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Singularity of truncated fractional operators

• 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
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Regularization
These problems can be resolved by defining the
fractional operators in the Caputo sense.
Consider the case
regular terms
singular terms
We define the regularized left derivative as
Similarly, for the right derivative we write
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Fractional model in finite-size domains
Boundary conditions
Diffusive flux
Non-local fluxes
Left fractional flux
Regularized finite-size left derivative
Regularized finite-size right derivative
Right fractional flux
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Steady state numerical solutions
Right-asymmetric
Right-asymmetric
Regular diffusion
Symmetric
Symmetric
Left-asymmetric
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Confinement time scaling
Diffusive scaling
Experimentally observed anomalous scaling
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Non-local heat transport
Fractional diffusion
a1.5
T
S
Standard diffusion
Flux
T
S
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Up-hill anti-diffusive transport
Right fractional diffusion
Standard diffusion
Fractional Flux
Gaussian Flux
Up-hill Non-Fickian anti diffusive zone
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Fast propagation of cold pulse
Left asymmetric
Right asymmetric
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Conclusions

• 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 a2/3, b1/2,
  q0.
• 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.