1. 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

2. 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.

3. 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

4. Beyond the standard diffusive transport paradigm The standard theory of plasma transport in based on models of the form 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.

5. 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).

6. Turbulent transport homogenous, isotropic turbulence Brownian random walk V= transport velocity D=diffusion coefficient

7. Coherent structures can give rise to anomalous diffusion trapping region exchange region transport region flight event trapping event t x Coherent structures correlations

8. Coherent structures in plasma turbulence ExB flow velocity eddies induce particle trapping Tracer orbits Trapped orbit “Levy” flight “Avalanche like” phenomena induce large particle displacements that lead to spatial non-locality Combination of particle trapping and flights leads to anomalous diffusion

9. Anomalous transport in plasma turbulence Levy distribution of tracers displacements Super-diffusive scaling 3-D turbulence model Tracers dynamics

10. Diffusive scaling Anomalous scaling super-diffusion Moments Probability density function ?????????? Standard diffusionPlasma turbulence Model Gaussian Non-Gaussian

11. Continuous time random walk model = jump = waiting time = waiting time pdf = jump size pdf No memory Gaussian displacements Master Equation (Montroll-Weiss) Long waiting times Long displacements (Levy flights) Standard diffusion Fractional diffusion Fluid limit

12. Riemann-Liouville fractional derivatives Left derivative Right derivative a bx

13. 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. “Left” flux“Right” flux Flux conserving form

14. Comparison between fractional model and turbulent transport data Turbulence simulation Fractional model Levy distribution at fixed time Turbulencemodel Pdf at fixed point in space

15. 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

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

17. Finite size domain transport problem Boundary conditions r T L Diffusive 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 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

18. Singularity of truncated fractional operators unless 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

19. Regularization These problems can be resolved by defining the fractional operators in the Caputo sense. Consider the case regular termssingular terms We define the regularized left derivative as Similarly, for the right derivative we write:

20. Fractional model in finite-size domains Diffusive flux Left fractional flux Right fractional flux Regularized finite-size left derivative Regularized finite-size right derivative L0x Non-local fluxes Boundary conditions

21. Steady state numerical solutions Right-asymmetric Left-asymmetric Symmetric Regular diffusion Right-asymmetric Symmetric

22. Confinement time scaling Domain size Confinement time L Diffusive scaling Experimentally observed anomalous scaling Fractional model

23. Non-local heat transport Flux Fractional diffusion S DIII-D Luce et al., PRL (1992) Standard diffusion S xxx T T 

24. “Up-hill” anti-diffusive transport Gaussian Flux Fractional Flux Standard diffusion Right fractional diffusion Up-hill Non-Fickian “anti diffusive zone”

25. Fast propagation of cold pulse Left asymmetricRight asymmetric Standard diffusion Fractional diffusion

26. 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. Conclusion s