1. 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

2. 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

3. 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

4. 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)

5. 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?

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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