PARTICLE TRANSPORT THEORY

1. PARTICLE TRANSPORT THEORY John W Bieber Bartol Research Institute, University of Delaware Presentation for Workshop on Particle Effects in MHD Turbulence Santa Fe, October 4-8, 2004

2. DIFFERENT ASPECTS OF DIFFUSION John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

3. THE BOLTZMANN EQUATIONFOR COSMIC RAY TRANSPORT t: time, f: cosmic ray phase space density, μ: cosine of pitch angle, v: particle speed, z: distance parallel to mean field, Ф(μ): Fokker-Planck coefficient for pitch angle scattering, q: source This describes one-dimensional transport of a gyrotropic distribution parallel to the mean field. More complex versions include adiabatic focusing, convection, adiabatic deceleration, perpendicular transport, gyro-anisotropy. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

4. DERIVATION OF THE BOLTZMANN EQUATION:FOKKER-PLANCK APPROACH John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

5. FOKKER-PLANCK APPROACH TO BOLTZMANN EQUATION Jokipii1 cites Chandrasekhar2 as the source of the Fokker-Planck formalism • Jokipii, J. R., Cosmic-ray propagation. I. Charged particles in a random magnetic field, Astrophys. J., 146, 480-487, 1966. • Chandrasekhar, S., Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15, 1-89, 1943. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

6. Chandrasekhar describes a distribution function W changing in accord with a transition probability function ψ John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

7. Chandrasekhar expands the distribution function and transition probability in a Taylor series … • … then recasts the transition probability in terms of Fokker-Planck coefficients, <Δui>, <Δui2>, <Δui Δuj >, John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

8. This yields what Chandrasekhar terms the Fokker-Planck equation John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

9. In terms of the cosmic ray distribution function f and Fokker-Planck coefficients <Δμ>/Δt and <Δμ2>/<Δt> (where μ is cosine of pitch angle), the equation is John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

10. Why Boltzmann? From F. C. Jones, The generalized diffusion-convection equation, Astrophys. J., 361, 162-172, 1990. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

11. NOW COMPUTE THE F-P COEFFICIENT <Δμ2>/<Δt> John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

12. QUASILINEAR APPROXIMATION: INTEGRATE FORCES ALONG UNPERTURBED PARTICLE ORBIT John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

13. TAKE THE SQUARE AND ENSEMBLE-AVERAGE John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

14. APPLY THE AXISYMMETRY CONDITION John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

15. FINALLY TAKE THE LIMIT Δt → ∞ John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

16. DERIVATION OF THE BOLTZMANN EQUATION:ENSEMBLE AVERAGING APPROACH John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

17. ENSEMBLE-AVERAGING DERIVATION OF THE BOLTZMANN EQUATION:START WITH THE VLASOV EQUATION The equation is relativistically correct John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

18. ENSEMBLE AVERAGETHE VLASOV EQUATION John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

19. SIMPLIFY THE ENSEMBLE-AVERAGED EQUATION WITH A TRICK For gyrotropic distributions, only ψ1 matters! John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

20. Digression: The functions <δBiδf> are measurable and potentially useful John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

21. SUBTRACT THE ENSEMBLE-AVERAGED EQUATION FROM THE ORIGINAL EQUATION… THEN LINEARIZE Why “Quasi”–Linear? 2nd order terms are retained in the ensemble-averaged equation, but dropped in the equation for the fluctuations δf John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

22. AFTER LINEARIZING, IT’S EASY TO SOLVE FOR δf BY THE METHOD OF CHARACTERISTICS In effect, this integrates the fluctuating force backwards along the particle trajectory. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

23. MULTIPLY BY δB AND ENSEMBLE AVERAGE John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

24. RECAST <δBXδf> AND <δBYδf> IN TERMS OF ψ1 AND ψ1 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

25. FORTUNATELY A MAJOR SIMPLIFICATION IS POSSIBLE FOR AXSYMMETRIC TURBULENCE John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

26. NOW THE ψ FUNCTIONS ARE … John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

27. ... AND <δBZδF> IS John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

28. FINALLY, COMPARE RESULT NTO FOKKER-PLANCK APPROACH John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

29. FROM PITCH ANGLE DIFFUSIONTO SPATIAL DIFFUSION John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

30. RECENT ADVANCES IN PARTICLE TRANSPORT THEORY John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

31. Advances in Heliospheric Turbulence John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

32. Advances in Heliospheric Turbulence 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. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

33. 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 90o pitch angle, where ordinary resonant scattering is weak. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

34. Observational Constraints: Solar Energetic Particles and Jovian Electrons • Solar Energetic Particles (SEP) provide detailed information on parallel particle transport in the inner heliosphere • Jovian Electrons provide strong constraints on perpendicular transport, as particles diffuse from the Sun-Jupiter to the Sun-Earth field line John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

35. PARALLEL DIFFUSION • Geometry resolves discrepancy at intermediate-high rigidity • Dissipation explains high electron mean free paths at low rigidity • Pickup ions still a puzzle John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

36. PERPENDICULAR DIFFUSION Key Elements • Particle followsrandom 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 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

37. 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 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

38. NLGC THEORY OF PERPENDICULAR DIFFUSION 2 • Simplify 4th order to 2nd 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:  John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

39. NLGC THEORY OF PERPENDICULAR DIFFUSION 3 • Corrsin independence approximation The perpendicular diffusion coefficient becomes Or, in terms of the spectral tensor John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

40. 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 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

41. 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 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

42. NLGC Theory: λ║Governs λ ┴ where John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

43. 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:10.1029/2003JA010301. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

44. NLGC Agrees withNumerical Simulations John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

45. 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.) John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

46. A COUPLED THEORY OF λ┴ AND λ║(MORE FUN WITH NONLINEAR METHODS) John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

47. 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║ ~ P0.6, 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. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

48. TURBULENCE TRANSPORT THEORY →TURBULENCE PARAMETERS THROUGHOUT HELIOSPHERE Energy Correlation Length Cross Helicity Temperature John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

49. 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 John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004

50. John W Bieber, Particle Transport Theory, Santa Fe Workshop, October 2004