Jakobus le Roux (1,2) & Gary Webb (1) New Perspectives on Cosmic Rays in the Heliosphere, South Africa 22-26 March, 2010 Anomalous Perpendicular Diffusion of Cosmic Rays in a Nonuniform Medium Jakobus le Roux (1,2) & Gary Webb (1) Dept. of Physics, University of Alabama in Huntsville, USA Center for Space Plasma & Aeronomic Research, University of Alabama in Huntsville, USA

1. Solar Wind is a Nonuniform Medium Solar wind magnetic field PDF - non-Gaussian tails on short time scales Lots of power in small-scale fluctuations Possible reasons: intermittent patches of intense small-scale Alfven wave (slab) turbulence (2) Intermittent nonlinear coherent structures such as magnetic vortices (islands in 2D) with sharp magnetic boundaries (3) Intermittent nonlinear coherent structures such as shocks, streams, and interaction regions – strong compressive turbulence downstream of shocks

2. Intermittent small-scale slab turbulence in quiet solar wind No turbulence Conventional turbulence Intermittent turbulence Stochastic growth theory [Zank & Cairns, 2000] COSMIC-RAY TRAPPING IN PATCHES In nonuniform plasma medium - Distribution of pitch-angle scattering times QLT for slab turbulence

3. Standard NLGC Theory for Anomalous Perpendicular Diffusion in dominant 2D turbulence with minor slab turbulence component -[Matthaeus et al., 2003] Anomalous diffusion - cosmic rays follow predominantly 2D meandering magnetic field modeled as random linear wave component (no vortices) –quiet solar incompressible wind between vortices Cosmic rays diffuse along 2D meandering magnetic field - gyroresonant interaction with small-scale slab turbulence Cosmic rays diffuse in uniform plasma medium – single scattering time for small-scale slab turbulence Waiting-time (time between scattering) probability distribution sc Anomalous diffusion is classical on intermediate time scales (high energies)

QUESTION: What happens to anomalous diffusion if If cosmic rays are trapped in intermittent patches of intense small-scale slab turbulence because of the reduction in the scattering time in those regions?

4. Generalized NLGC Theory for Anomalous Perpendicular Diffusion in dominant 2D turbulence with minor slab turbulence component [le Roux et al., 2010], submitted to Ap. J.] Cosmic rays diffuse in nonuniform plasma medium – distribution of scattering times for small-scale slab turbulence Patches of intense small-scale slab turbulence (particle trapping) are rare – quiet nearly - incompressible solar wind (2) Averaged over nonuniform medium – the waiting-time (time between scattering) probability distribution is a power law – higher probability of long scattering times Cosmic ray spend most time in patches of intense scattering

Main Result: In a nonuniform medium - anomalous perpendicular diffusion is nonclassical during intermediate times (intermediate energies) If  < 3/2, subdiffusion If  = 3/2, classical diffusion If  >3/2, superdiffusion

5. Different Regimes of Anomalous Diffusion – Generalized NLGC theory at intermediate energies Late times – convection superdiffusion a-1=1/3 a-1=1 Early times – scatterfree superdiffusion Intermediate times – scattering probably subdiffusion

6. Using solar wind to constrain the index  of the distribution of scattering times - deduce anomalous diffusion exponent Burlaga et al. (2004) – fit magnetic field PDF with a q-Gaussian distribution q > 1.5

Interpretation: Patches of intense turbulence closely spaced – turbulent solar wind Because a < 0 – particle bunching (compression) occurring in patches of intense turbulence – suggest turbulence compressive ACE observations in 2003 includes shocks, streams, and interaction regions – suggest cosmic rays mostly affected by patches of intense compressive random wave turbulence associated with shocks, streams and interaction regions

SUMMARY ANOMALOUS PERPENDICULAR DIFFUSION WHEN 2D TURBULENCE DOMINATES IN QUIET SOLAR WIND NEAR EARTH: Original NLGC Theory [Matthaeus, 2003]: Uniform medium – single scattering time - classical diffusion (a = 1) during intermediate times – valid for high energies Generalized NLGC Theory [le Roux et al., 2010]: Nonuniform medium - distribution of scattering times – intermittent patches of intense small-scale slab turbulence – cosmic-ray trapping - possibly subdiffusion (a < 1) during intermediate times – valid for intermediate energies