Superdiffusive and subdiffusive transport of energetic particles in astrophysical plasmas: numerical simulations and experimental evidence Gaetano Zimbardo S. Perri, P. Pommois, and P. Veltri Universita della Calabria, Cosenza, Italy Workshop on Nonlinear dynamics and structure formation in complex systems, Frascati, 21-22 September 2009
Plan of presentation: Overview of diffusive and anomalous transport regimes Numerical simulation of particle transport in the presence of magnetic turbulence Evidence of superdiffusion from analysis of energetic particle profiles Application to particle propagation upstream of the termination shock Scherzo
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 space weather predictions. Two regimes are usually considered: Normal diffusion: Ballistic transport: Are there transport regimes intermediate between normal diffusion and scatter-free propagation? Both sudiffusive and superdiffusive regimes can be found.
Superdiffusive transport Superdiffusive transport is related to a broad distribution of jump lengths, with powerlaw tails: The second order moment of the free path lengths, i.e., the mean free path, is diverging: This implies that the Central Limit Theorem does not apply, and a non-gaussian statistics is involved. Superdiffusion is essentially related to long range correlations and non Markovian memory effects, here described as Levy random walks
Gaussian vs Levy random walk
Subdiffusive transport Subdiffusion is related to the presence of dynamic traps and antipersistent behaviour, like tracing back the trajectory as in compound diffusion: Subdiffusion is characterized by a broad distribution of waiting, i.e., trapping times. A very wide range of systems exhibits non-gaussian behaviour, including magnetic turbulence fluctuations in the solar wind, solar flare waiting time distributions, and more (e.g., Klafter et al., PRA, 1987; Metzler and Klafter, Phys. Rep., 2000).
Energetic particle transport in the Heliosphere: Is anomalous transport possible for particle propagating in the solar wind turbulence? Two approachs are presented here: Numerical simulation of particle transport in models of magnetic turbulence. Analysis of energetic particle profiles measured by spacecraft
Numerical Simulation The magnetic field is represented as a superposition of a constant field and a fluctuating field where with
Numerical Simulation Wave vectors on a cubic lattice 128x128x128 Anisotropic power law spectrum: Band spectrum: Here N min = 4, N max = 16. Future simulations with longer spectrum.
Anisotropy in physical and phase space Quasi-slab Quasi-2D
We can study the structure of magnetic flux tubes for different anisotropies: From Isichenko, PPCF, 1991
Magnetic flux tube cross section for axisymmetric anisotropies and δb/b = 0.5 at 1 AU (Zimbardo et al., JGR 2004) Quasi-2D Quasi-slab
Quasi-slab Injecting particles with different Larmor radii ρ... (Pommois et al., Ph.Pl. 2007; Zimbardo et al., IEEE Trans. Plasma Sci., 2008) Quasi-2D
This simulation matches the observations of solar energetic particle dropouts in the solar wind (Mazur et al., ApJ 2000)
Anomalous transport depends on the turbulence anisotropy: (from Zimbardo et al., ApJL, 2006) Quasi-slab z Quasi-2D x,y
Perpendicular subdiffusion considered by Qin, Matthaeus, and Bieber by computing the running diffusion coefficients: Slab turbulence, perpendicular subdiffusion (Qin et al., GRL, 2002) Composite turbulence (mostly 2D), recovery of diffusion (Qin et al., ApJL, 2002)
A useful parameter is the Kubo number R = (db/b)(l z /l x) From Pommois et al., Phys. Plasmas, 2007
The transport regime also depends on the ratio ρ/ λ((pommois et al., Ph. Pl., 2007)
Parallel superdiffusion and perpendicular subdiffusion considered by Shalchi and Kourakis, Astr. Astroph, 2007
Summary of numerical simulations Parallel superdiffusion is found for small ρ / λ and /or quasi-slab turbulence. Superdiffusion also for small turbulence levels. Wave-particle interactions are present, but not so strong as to correspond to a non-gaussian statistics. Rather, the changes in parallel velocity correspond to a Lévy random walk. Perpendicular subdiffusion is possible, especially for quasi-slab turbulence, when particles can trace back their trajectory, as in compound diffusion (e.g., Kota and Jokipii, ApJL, 2000). A number of different regimes can be obtained (look for field line separation )
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:
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)
Superdiffusion, power-law propagator (Zumofen and Klafter, PRE, 1993) A similar approach was considered by del- Castillo-Negrete, Carreras, and Lynch, 2004, 2005
Inserting the power-law propagator into the integral At some distance upstream of the shock: Power-law particle profile with α = 4 µ = 2 γ (Perri and Zimbardo, ApJL 2007, JGR 2008)
Scatter-free propagation, δ function propagator (e.g., Webb, Zank, Kaghashvili, le Roux, ApJ 2006) When inserted in the expression for f, a constant level is obtained
Fast and slow streams in the solar wind
We performed analyses of Ulysses data at CIRs and of Voyager 2 data at the Termination Shock Ulysses observed a series of Corotating Interaction Regions in 1992-1993; both protons and electrons are accelerated at CIR shocks:
Energetic particle profiles seen at CIRs shocks:
Electrons superdiffusion detected by Ulysees Power law J=A(Δt) - γ Exponential J=K exp(-γδt) Δt= t-t sh χ 2 =χ 2 /ν reduced chi-square Energy (kev) χ PL 2 χ EXP 2 γ 42-65 1.42 1.76 1.0017 ± 0.0002 65-112 0.90 1.21 0.92 ± 0.02 112-178 0.17 0.33 0.81 ± 0.03
A further event observed by Ulysses Ulysses observed a series of Corotating Interaction Regions in 1992-1993; both protons and electrons are accelerated at CIR shocks: (data from CDAweb, thanks to Lancaster and Tranquille, PI D. McComas, L. Lanzerotti)
178-290 0.07 0.18 0.85 ± 0.08 Electron superdiffusive transport at the reverse shock of May 10, 1993, with α = 2 γ (from Perri and Zimbardo, ApJL 2007) Energy (kev) χ PL 2 χ EXP 2 γ 42-65 0.10 0.03 0.71 ± 0.08 65-112 0.03 0.11 0.62 ± 0.07 More events are shown in Perri and Zimbardo, Adv. Space Res. 2009 112-178 0.03 0.15 0.69 ± 0.08
Voyager 2 at the termination shock: data from PLS and LECP
We considered a number of ion energy channels upstream of the termination shock (PLS data from space.mit.edu/pub/plasma/vgr/v2/daily LECP data from sd-www.jhuapl.edu/voyager/v2_data)
Fit of particle fluxes: power-law versus exponential The power-law fits better than the exponential for all energy channels, with γ = 0.15 0.78 Corresponding to superdiffusion: See Perri and Zimbardo, Astrophys. J., 693, L118 (2009)
These observations show that many of the standard assumptions of the Diffusive Shock Acceleration (DSA) are not satisfied: for instance, the so called Parker equation assumes both isotropic particle distribtion function f, and normal diffusion; both these assumptions are not verified at the Termination Shock. A fractional transport equation in velocity space was proposed by Milovanov and Zelenyi, PRE, 2001: Strange Fermi process: Turbulent acceleration phenomena due to fractal time accelerations and velocity space Levy flights
The spectral index of energetic particles, 1.25, requires, according to DSA, a compression ratio of 3, while about 2 is observed; we do not really understand energetic particle acceleration at the TS! From Decker et al., Nature, 2008.
Summary of data analysis Electron superdiffusive transport from Ulysses data Ion superdiffusive transport from Voyager 2 data A wide range of anomalous diffusion exponents α is found, including nearly scatter free regimes Transport tends to become normal for increasing gyroradius, in agreement with numerical simulations Ion superdiffusion at the Termination Shock is consistent with the observations that anomalous cosmic rays do not exhibit the features expected from diffusive shock acceleration (DSA). Superdiffusion allows a quicker escape from the shock region, decreasing the efficiency of DSA. Look for other acceleration mechanisms!
Scherzo just a joke!
From asymmetric diffusion to subdiffusion Normal diffusion and standard Fick s law give a linear density profile Look for Fractional Fick s Law! My suggestion for α>2 :
Conclusions We have illustrated theoretical descriptions of anomalous diffusion and Levy random walks. Numerical simulations yield, for proper values of the system parameters, parallel superdiffusion and perpendicular subdiffusion. Analysis of Ulysses and Voyager 2 data gives superdiffusion both for electrons and ions. Ion superdiffusion at the termination shock is consistent with the departures of anomalous cosmic rays from diffusive shock acceleration, and with the strongly anisotropic particle fluxes observed by LECP.
That s all, folks!