Superdiffusive Transport at Shocks in Space Plasmas

1. Superdiffusive Transport at Shocks in Space Plasmas International Astrophysics Forum Alpbach IAFA 2011 Frontiers in Space Environment Research Alpbach, 20-24 June 2011 Gaetano Zimbardo and Silvia Perri Universita’ della Calabria, Cosenza, Italy

2. Plan of presentation: • Overview of diffusive and anomalous transport regimes • Propagators for normal and superdiffusive transport • Evidence of superdiffusion from analysis of energetic particle profiles at CIR shocks, at the termination shock, and at CME driven shocks • Influence of superdiffusion on the acceleration time at shocks

3. Transport regimes Understanding the transport of energetic particles in the presence of magnetic turbulence is relevant both to cosmic ray acceleration and transport and to laboratory plasma confinement. Two regimes are usually considered: Normal diffusion: Ballistic transport: Are there transport regimes intermediate between normal diffusion and scatter-free propagation? … yes! Both sudiffusive and superdiffusive regimes can be found.

4. Early warnings of anomalous diffusion … The divergence of the integral of the velocity autocorrelation function … … is precisely what is required for anomalous (super) diffusion! For normal diffusion, a diffusion coefficient is obtained as D = l2/t Superdiffusion is related to long range correlations in the velocity, i.e., to a non local process. Further, no characteristic length scale can be defined for the random walk.

5. Normal, Gaussian diffusion • The normal diffusion equation is • … which leads to a mean square deviation growing as • If we choose a localized initial condition • … the solution for the particle density is a Gaussian: • This is also called the propagator P(x,t) of the transport process. • The diffusion coefficient D is given by

6. Sketch of superdiffusion from Lévy random walk …(see Klafter et al., Phys. Rev. A, 35, 3081 (1987). • The essential point is to use a space-time coupled jump probability: • Consider the probability density of being in r at time t …  Montroll – Weiss equation (J. Math. Phys. 1965)

7. The mean square derivation is obtained as … and the non Gaussian propagator as (Zumofen and Klafter, 1993) small x : large x :

8. Gaussian versus Lévy random walk

9. Anomalous transport depends on the turbulence anisotropy: (from Zimbardo et al., ApJL, 2006) Quasi-2D Quasi-slab z x,y

10. Perpendicular subdiffusion considered by Qin, Matthaeus, and Bieber by computing the running diffusion coefficients: Composite turbulence (mostly 2D), recovery of diffusion (Qin et al., ApJL, 2002) Slab turbulence, perpendicular subdiffusion (Qin et al., GRL, 2002)

11. A useful parameter is the Kubo number R = (dB/B)(l z /l x) From Pommois et al., Phys. Plasmas, 2007

12. The transport regime also depends on the ratio r/l(Pommois et al., Ph. Pl., 2007)

13. Parallel superdiffusion and perpendicular subdiffusion also found by Shalchi and Kourakis, Astr. Astroph, 2007 Test particle simulation with 20% slab and 80% 2D composite turbulence model

14. Parallel superdiffusion and perpendicular subdiffusion also found by Tautz, PPCF, 2010 • Perpendicular subdiffusion for magnetostatic slab turbulence (solid lines) • The inclusion of time dependent electric and magneti fields leads to superdiffusion (dashed lines) with <(Dz)2> = K t1.31 • The reported results are obtained with independent realizations of magnetic turbulence

15. How to detect anomalous diffusion from observations? Superdiffusion from analysis of energetic particle profiles measured by spacecraft • The flux of energetic particle can be expressed by means of the probability of propagation from (x’, t’) to (x, t): • We consider particles emitted at an infinite planar shock moving with velocity V_sh:

16. Normal diffusion, Gaussian propagator At some distance from the shock the turbulence level and the diffusion coefficient can be assumed to be constant; integration over the propagator yields: (e.g., Fisk and Lee, ApJ, 1980; Lee, JGR, 1983)

17. Power-law particle profile with a = 4 - m = 2 - g (Perri and Zimbardo, ApJL 2007, JGR 2008) Superdiffusion, power-law propagator valid for large (x- x’) and 2<m<3 (Zumofen and Klafter, PRE, 1993) At some distance upstream of the shock:

18. Fast and slow streams in the solar wind …

19. Shock crossing of October 11, 1992 • Both protons and electrons are accelerated at CIR shocks: • (data from CDAweb, thanks to Lancaster and Tranquille, PI D. McComas, L. Lanzerotti) The pitch angle diffusion rate depends on the wave magnetic field power:

20. Event of October 11, 1992 Dt=|t-tsh| Power law J=A(Dt)-g Exponential J=K exp(-GDt) Electron transport is superdiffusive! Proton transport is normal diffusive (from Perri and Zimbardo, Adv. Spa. Res. 2009)

21. A further event observed by Ulysses …

22. Electron superdiffusive transport at the reverse shock of May 10, 1993, with a = 2-g (from Perri and Zimbardo, ApJL 2007) Superdiffusion with a = 1.15-1.38

23. The same technique was applied to CME driven shock waves by Sugiyama and Shiota, ApJL, 2011: ACE data for December 13, 2006 CME shock

25. As foreseen, the termination shock has been observed by Voyager 1 and Voyager 2 at about 94 and 84 AU From Decker et al., Science, 2005 Everything all right?

26. Magnetic field and plasma data from Voyager 2 at the TS (Burlaga et al., ApJ, 2009)

27. We considered a number of ion energy channelsupstream of the termination shock (PLS data from space.mit.edu/pub/plasma/vgr/v2/dailyLECP data from sd-www.jhuapl.edu/VOYAGER/v2_data)

28. Ion superdiffusion at the solar wind termination shock The power-law fits better than the exponential for all energy channels observed by Voyager 2, with g = 0.68-0.71 corresponding to superdiffusion: Perri and Zimbardo, Astrophys. J., 693, L118 (2009)

29. Influence of superdiffusion on the acceleration time at shocks • Superdiffusive shock acceleration assumes an acceleration time based on -> • In the case of superdiffusion, we can equate … • and … • from which …

30. We can compare the above cycle times … Normal diffusion: Superdiffusion:

31. Anomalous Cosmic Rays do not peak at the TS (Stone et al., Nature, 2008) The acceleration of anomalous cosmic rays (ACRs) has stimulated a number of studies (see Lazarian and Opher, 2009); but why are ACRs not accelerated at the TS ?

32. Particle acceleration at the Termination Shock • The particles accelerated at the shock have energies in the range 40 keV – 4 MeV, lower than ACRs. • For a number of energy channels we have superdiffusion with a = 1.3, from which a/(2-a) = 1.85 is found; • For 1 MeV protons this yields a factor 20 increase in the acceleration time; this can explain why termination shock particles do not reach the energies expected for ACRs.

33. Summary • Superdiffusive transport in magnetic turbulence is obtained from a number of numerical simulations; • Electron superdiffusive transport from Ulysses data; • Ion superdiffusive transport from ACE and Voyager 2 data; • Ion superdiffusion at the termination shock is consistent with the observations that ACRs are accelerated elsewhere; • Superdiffusion allows a quicker escape from the shock region, and a new scaling of the acceleration time with the shock speed is proposed.