John W. Bieber Bartol Research Institute, University of Delaware, Newark

1. APS Division of Plasma Physics Nov 15-19, 2004 Savannah, GeorgiaTRANSPORT AND MODULATIONOF COSMIC RAYS IN THE SOLAR WIND John W. Bieber Bartol Research Institute, University of Delaware, Newark Supported by NSF grant ATM-0000315 Visit our Website: http://www.bartol.udel.edu/~NeutronM/

2. A LONG-STANDING PUZZLE:HOW DO ENERGETIC CHARGED PARTICLES SCATTER AND DIFFUSE IN THE SOLAR WIND?

3. TRANSPORT OF ENERGETIC PARTICLES IN THE SOLAR WIND: SOLAR COSMIC RAYS The plot shows traces recorded by neutron monitors viewing in different directions. Why do the solar particles start out anisotropic, then become increasingly isotropic?

4. TRANSPORT OF ENERGETIC PARTICLES IN THE SOLAR WIND: SOLAR MODULATION Solar modulation refers to the influence the Sun exerts upon the intensity of Galactic cosmic rays. As solar activity rises (top), the cosmic rays decrease (bottom). What links solar activity to Galactic cosmic rays? This plot is updated regularly and is available at http://www.bartol.udel.edu/~NeutronM/modplot.html

5. TRANSPORT OF ENERGETIC PARTICLES IN THE SOLAR WIND: JOVIAN ELECTRONS Jovian electrons are injected onto the Sun-Jupiter field line. How do they migrate to other field lines? Data from Chenette et al., Astrophys. J. Lett., L95-L99, 1977.

6. ADVANCES IN PARTICLE TRANSPORT THEORY RECENT ADVANCES HAVE RESULTED IN PART FROM IMPROVED UNDERSTANDING OF TURULENCE IN THE SOLAR WIND, ESPECIALLY TURBULENCE GEOMETRY

7. DIFFERENT ASPECTS OF DIFFUSION

8. PARALLEL SCATTERING IN THE QUASILINEAR LIMIT Particles resonate with parallel wave modes having wavenumber kRES: Fokker-Planck coefficient Φ(μ) is related to power spectrum Pxx(k) evaluated at kRES

9. PARALLEL MEAN FREE PATHAND DIFFUSION COEFFICIENT A Rule of Thumb: (where δBX2 includes only slab mode turbulence) NB: For Kolmogoroff (k -5/3) spectrum λ║ ~ (Rigidity)1/3

10. Advances in Heliospheric Turbulence

11. TWO-COMPONENT TURBULENCEAND THE “MALTESE CROSS” 2-dimensional correlation functions are displayed as contour and surface plots. Pure slab: correlations decay only parallel to the mean magnetic field. 2D/Slab: correlation contours decay in all directions Solar wind1: Observed contours resemble “Maltese cross” similar to 2D/slab (1) Adapted from Matthaeus, Goldstein, & Roberts, JGR, 95, 20673, 1990.

12. PARALLEL SCATTERING AND OBSERVATIONS • For a long time, QLT was disrespected because the slab model prediction disagreed spectacularly with observations … • ... but actually QLT works pretty well when the geometry of the turbulence is taken into account. • Reason: Resonant pitch angle scattering is maximum for slab geometry (because wavevectors are aligned with axis of particle orbit), but is exactly zero! for 2D geometry in the quasilinear limit.

13. ADVANCES IN PARTICLE TRANSPORT THEORY RECENT ADVANCES HAVE ALSO RESULTED FROM THE USE OF NONLINEAR METHODS

14. 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

15. NONLINEAR GUIDING CENTER (NLGC) THEORY OF PERPENDICULAR DIFFUSION[Matthaeus et al., Astrophys. J. (Lett.), 590, L53-L56, 2003] • 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

16. 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: 

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

18. 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

19. 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

20. NLGC Theory: λ║Governs λ ┴ where

21. 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.

22. NLGC Agrees withNumerical Simulations

23. 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.)

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

25. 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.

26. TURBULENCE TRANSPORTIN THE SOLAR WIND AND APPLICATION TO SOLAR MODULATION OF GALACTIC COSMIC RAYS

27. SOLAR MODULATION FROM FIRST PRINCIPLES:THE CHALLENGE OF AB INITIO MODELLING

28. PHENOMENOLOGICAL MODELFOR TRANSPORT AND EVOLUTIONOF SOLAR WIND TURBULENCE • Transport equations for turbulence energy Z2, energy-containing scale λ, temperature T, [and cross-helicity σc]. • Effect of large scale wind shear ΔV ~100 km/sec • Wave generation by pickup ions • Anisotropic decay and heating phenomenology Matthaeus et al, 1996; Zank et al, 1996; Matthaeus et al, PRL, 1999; Smith et al, JGR, 2001; Isenberg et al, 2003 [Recent modifications for cross-helicity not shown. See Matthaeus et al., GRL, 2004.]

29. TURBULENCE TRANSPORT: EQUATORIAL PLANEModel Compared with Observation

30. TURBULENCE TRANSPORT:HIGH LATITUDEModel Compared with Observation

31. AB INITO MODELS OF THE SOLAR MODULATION OF GALACTIC COSMIC RAYS

32. PARKER’S TRANSPORT EQUATION

34. DRIFT PATTERNSPositive Solar Polarity Negative Solar Polarity

35. TRANSPORT OF ENERGETIC PARTICLES IN THE SOLAR WIND: SOLAR MODULATION Solar modulation refers to the influence the Sun exerts upon the intensity of Galactic cosmic rays. As solar activity rises (top), the cosmic rays decrease (bottom). What links solar activity to Galactic cosmic rays? This plot is updated regularly and is available at http://www.bartol.udel.edu/~NeutronM/modplot.html

36. EVIDENCE FOR CHARGE SIGN DEPENDENT SOLAR MODULATION NOTES • Ratios change abruptly at each polarity reversal • Negative charge has generally higher intensity during negative solar polarity • Negative/positive ratio varies little during positive solar polarity, but exhibits an “M” shape during negative solar polarity

37. THE DIFFUSION TENSORTHROUGHOUTTHE HELIOSPHERE • Diffusion tensor computed from first principles using turbulence transport solutions • Drifts competitive or dominant in outer heliosphere in this model • Recall λik ≡ 3 Kik / V • Note that λrr= λ║ cos2ψ + λ┴ sin2ψ where ψ is spiral angle

38. Ab Initio Solar Modulation:Energy Spectrum • The Good News: Correct shape, correct ordering of positive and negative polarity • The Bad News: Too little modulation in positive polarity Positive Polarity: Solid line (theory) and stars (data) Negative Polarity: Dashed line (theory) and diamonds (data) Results from Parhi et al., in preparation, 2004.

39. Ab Initio Solar Modulation:Radial Gradients • The Good News: Correct +/- ordering; overall good fit … • The Bad News: Positive polarity too high, as before, in inner heliosphere. (But doesn’t look as bad on a linear scale.) Positive Polarity: Solid line (theory) and stars (data) Negative Polarity: Dashed line (theory) and diamonds (data) Results from Parhi et al., in preparation, 2004.

40. Ab Initio Solar Modulation:Latitude Gradients • The Good News: Correct shape in positive polarity • The Bad News: Peak too high in magnitude, too low in rigidity • Data from Ulysses fast-scan in negative polarity will be very useful Positive Polarity: Solid line (theory) and stars (data) Negative Polarity: Dashed line (theory) Results from Parhi et al., in preparation, 2004.

41. SUMMARY Major advances in our understanding of cosmic ray transport and modulation in the solar wind have resulted from: • Improved understanding of turbulence, especially turbulence geometry • Nonlinear methods in scattering theory • Improvements in turbulence transport theory • Ab Initio models of the solar modulation of cosmic rays