1. COSPAR 2004, Paris D1.2-0001-04 July 21, 2004 THE HELIOSPHERIC DIFFUSION TENSOR John W. Bieber University of Delaware, Bartol Research Institute, Newark Supported by NSF grant ATM-0000315 Collaborators: W. H. Matthaeus, G. Qin, A. Shalchi Visit our Website: http://www.bartol.udel.edu/~neutronm/

2. PARKER’S TRANSPORT EQUATION

3. DIFFERENT ASPECTS OF DIFFUSION

4. Advances in Heliospheric Turbulence

5. Turbulence Dissipation Range At frequency (ν) ~ 1 Hz, magnetic power spectrum steepens from inertial range value (ν -5/3 ) to dissipation range value of ν -3 or steeper Important for low- rigidity electrons (<30 MeV) Figure adapted from Leamon et al., JGR, Vol 103, p 4775, 1998.

6. Advances in Heliospheric Turbulence Turbulence is inherently dynamic Cosmic ray studies often employ a magnetostatic approximation, but dynamical effects may be important at low rigidities and near 90 o pitch angle, where ordinary resonant scattering is weak.

7. PARALLEL DIFFUSION Geometry resolves discrepancy at intermediate- high rigidity Dissipation explains high electron mean free paths at low rigidity Pickup ions still a puzzle

8. PERPENDICULAR DIFFUSION Key Elements Particle follows random walk of field lines (FLRW limit: K ┴ = (V/2) D ┴ ) Particle backscatters via parallel diffusion and retraces it path (leads to subdiffusion in slab turbulence) Retraced path varies from original owing to perpendicular structure of turbulence, permitting true diffusion

9. NONLINEAR GUIDING CENTER (NLGC) THEORY OF PERPENDICULAR DIFFUSION Begin with Taylor-Green-Kubo formula for diffusion Key assumption: perpendicular diffusion is controlled by the motion of the particle guiding centers. Replace the single particle orbit velocity in TGK by the effective velocity TGK becomes

10. NLGC THEORY OF PERPENDICULAR DIFFUSION 2 Simplify 4 th order to 2 nd order (ignore v-b correlations: e.g., for isotropic distribution…) Special case: parallel velocity is constant and a=1, recover QLT/FLRW perpendicular diffusion. (Jokipii, 1966) Model parallel velocity correlation in a simple way: 

11. NLGC THEORY OF PERPENDICULAR DIFFUSION 3 Corrsin independence approximation Or, in terms of the spectral tensor The perpendicular diffusion coefficient becomes

12. NLGC THEORY OF PERPENDICULAR DIFFUSION 4 “Characteristic function” – here assume Gaussian, diffusion probability distribution After this elementary integral, we arrive at a fairly general implicit equation for the perpendicular diffusion coefficient

13. NLGC THEORY OF PERPENDICULAR DIFFUSION 5 The perpendicular diffusion coefficient is determined by To compute Kxx numerically we adopt particular 2-component, 2D - slab spectra These solutions are compared with direct determination of Kxx from a large number of numerically computed particle trajectories in realizations of random magnetic field models. We find very good agreement for a wide range of parameters. and solve

14. NLGC Theory: λ ║ Governs λ ┴ where

15. APPROXIMATIONS AND ASYMPTOTIC FORMS NLGC integral can be expressed in terms of hypergeometric functions; though not a closed form solution for λ ┴, this permits development of useful approximations and asymptotic forms. Figure adapted from Shalchi et al. (2004), Astrophys. J., 604, 675. See also Zank et al. (2004), J. Geophys. Res., 109, A04107, doi:10.1029/2003JA0 10301.

16. NLGC Agrees with Numerical Simulations

17. NLGC AGREES WITH OBSERVATION Ulysses observations of Galactic protons indicate λ ┴ has a very weak rigidity dependence (Data from Burger et al. (2000), JGR, 105, 27447.) Jovian electron result decisively favors NLGC (Data from Chenette et al. (1977), Astrophys. J. (Lett.), 215, L95.)

18. A COUPLED THEORY OF λ ┴ AND λ ║ (MORE FUN WITH NONLINEAR METHODS)

19. WEAKLY NONLINEAR THEORY (WNLT) OF PARTICLE DIFFUSION λ ║ and λ ┴ are coupled: λ ║ = λ ║ (λ ║, λ ┴ ); λ ┴ = λ ┴ (λ ║, λ ┴ ) Nonlinear effect of 2D turbulence is important: λ ║ ~ P 0.6, in agreement with simulations λ ┴ displays slightly better agreement with simulations than NLGC λ ┴ / λ ║ ~ 0.01 – 0.04 Figures adapted from Shalchi et al. (2004), Astrophys. J., submitted.

20. TURBULENCE TRANSPORT THEORY → TURBULENCE PARAMETERS THROUGHOUT HELIOSPHERE Energy Temperature Correlation Length Cross Helicity

21. SUMMARY Major advances in our understanding of particle diffusion in the heliosphere have resulted from: Improved understanding of turbulence: geometry (especially), dissipation range, dynamical turbulence Nonlinear methods in scattering theory (NLGC, WNLT) Improvements in turbulence transport theory