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Perpendicular transport/diffusion

SH11C-1112 Theoretical and Observational Aspects of Perpendicular Diffusion of Charged Particles in the Nonlinear Guiding Center Model W H Matthaeus, J. W. Bieber, A. Shalchi Bartol Research Institute, University of Delaware G. Qin Florida Institute of Technology.

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Perpendicular transport/diffusion

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  1. SH11C-1112Theoretical and Observational Aspects of Perpendicular Diffusion of Charged Particles in the Nonlinear Guiding Center ModelW H Matthaeus, J. W. Bieber, A. ShalchiBartol Research Institute, University of DelawareG. QinFlorida Institute of Technology

  2. Perpendicular transport/diffusion • Field Line Random Walk (FLRW) limit is a standard picture. • K (v/2) D …But it does not work very well Fokker Planck coefficient for field line diffusion

  3. Puzzling properties of perpendicular transport/diffusion Computed K’s fall Well below FLRW at low energy, but above hard sphere and BAM theories • Numerical results support FLRW at high energy, but no explanation for reported low energy behavior • K may be involved in explaining observational puzzles as well • Enhanced access to high latitudes • “chanelling” Slab/2D and Isotropic (Giacalone And Jokipii.) Slab (Mace et al, 2000) FLRW BAM

  4. Here is what is wrong with FLRW: influence of parallel scatteringNumerical simulation using 2-component turbulence: 80% 2D + 20% slab Qin et al, 2002 Running diffusion coefficients vs. time • Parallel motions: free stream, then diffuse • Perpendicular: evolves towards FLRW, subsequent decrease due to “compound” effects, and then a regime of “second diffusion” emerges.

  5. A new theory of nonlinear perpendicular (guiding center) diffusion appears to work rather well: • Based upon Green Kubo formula for diffusion • Key physical assumptions: • perpendicular diffusion is controlled by the motion of the particle guiding centers. • Compound diffusion effects are included (parallel scattering) • Sampling of transverse complexity “restores” diffusion • Key mathematical approximations • Gaussian statistics • Diffusive separation of trajectories • Corssin’s indepdendence approximation • NonLinear Guiding Center (NLGC) theory

  6. Shredding of flux tubes in 3D Two component Model is three-dimensional: 20% slab + 80% 2D 1D slab only

  7. How much transverse structure is sampled by particles ? slab • Slab field lines are identical •  No field line separation • 2D field lines are trapped on flux surfaces • ”escape” catalyzed by slab component • Most effective separation when: •  Field is 3D (e.g., slab+2D mixture) •  Gyro-orbits sample transverse structure 2D 2D+slab 50-50 See Ruffolo and Matthaeus, in preparation, 2002

  8. Result of “NLGC” Theory • The perpendicular diffusion coefficient is determined by • To compute Kxx numerically we adopted a 2-component, 2D - slab spectra • These solutions compare well 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. This provides a posteriori justification for the approximations used. and solve

  9. The only particle property that appears in NLGC is l║ • The integral equation can be written as:

  10. NLGC results: Parallel vs. Perpendicular diffusion • Not a purely linear relationship • Kperp << Kpar

  11. NLGC: λ┴vs.λ║ NLGC theory Test particle results

  12. Test particle simulationsappear to be consistent with NLGCfor a range of rL/λc

  13. Compare to other test particle simulations:K┴ / K║= const (as a function of rigidity)in some parameter regimes • Results of Giacalone & Jokipii (left) as well as Qin (right) indicate K┴ / K║ ~ 0.02-0.04 at typical cosmic ray energies

  14. Some observational assessments of NLGC • Various heliospheric studies suggest ratio of perp to parallel m.f.p. consistent with NLGC • Jovian electrons • Modulation results: latitudinal gradients

  15. K┴Determined from Jovian Electrons From Chenette et al., Astrophys. J. Lett., L95-L99, 1977. Concept: Jupiter is a near “point source” of electrons. Longitudinal spread of electrons at 1 AU provides a direct measure of K┴ Curve a: K┴ = 5X1020 cm2/s Curve b: K┴ = 1021 cm2/s Summary of Results for ~2 MV Jovian electrons: K┴ = 7 x 1020 cm2/s; λ┴ = 0.0047 AU K║ = 5 x 1022 cm2/s; λ║ = 0.33 AU K┴ / K║ = 0.014

  16. NLGC: Jovian electrons

  17. Solar modulation of galactic cosmic rays: latitudinal gradients and NLGC NLGC BAM NLGC appears to “fix” latitudinal gradient problem that plagues modulation calculations See Parhi et al (poster)

  18. Analytic solutions to NLGC (Shalchi et al, 2003) Various interesting asymptotic limits can de derived in closed form: See Shalchi et al, 2003 and also Zank et al, 2003

  19. Conclusions • NLGC is an interesting theory of perpendicular diffusion of charged particles in random magnetic fields • More applications are in progress

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