APS Division of Plasma Physics Nov 15-19, 2004 Savannah, Georgia TRANSPORT AND MODULATION OF COSMIC RAYS IN THE SOLAR WIND John W. Bieber Bartol Research Institute, University of Delaware, Newark Supported by NSF grant ATM Visit our Website:

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

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?

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

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.

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

DIFFERENT ASPECTS OF DIFFUSION

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

PARALLEL MEAN FREE PATH AND DIFFUSION COEFFICIENT A Rule of Thumb: (where δB X 2 includes only slab mode turbulence) NB: For Kolmogoroff (k -5/3 ) spectrum λ ║ ~ (Rigidity) 1/3

Advances in Heliospheric Turbulence

TWO-COMPONENT TURBULENCE AND 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 wind 1 : Observed contours resemble “Maltese cross” similar to 2D/slab (1) Adapted from Matthaeus, Goldstein, & Roberts, JGR, 95, 20673, 1990.

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.

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

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

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

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: 

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

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

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

NLGC Theory: λ ║ Governs λ ┴ where

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: /2003JA

NLGC Agrees with Numerical Simulations

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, ) Jovian electron result decisively favors NLGC (Data from Chenette et al. (1977), Astrophys. J. (Lett.), 215, L95.)

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

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.

TURBULENCE TRANSPORT IN THE SOLAR WIND AND APPLICATION TO SOLAR MODULATION OF GALACTIC COSMIC RAYS

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

PHENOMENOLOGICAL MODEL FOR TRANSPORT AND EVOLUTION OF SOLAR WIND TURBULENCE Transport equations for turbulence energy Z 2, energy- containing scale λ, temperature T, [and cross-helicity σ c ]. 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.]

TURBULENCE TRANSPORT: EQUATORIAL PLANE Model Compared with Observation

TURBULENCE TRANSPORT: HIGH LATITUDE Model Compared with Observation

AB INITO MODELS OF THE SOLAR MODULATION OF GALACTIC COSMIC RAYS

PARKER’S TRANSPORT EQUATION

DRIFT PATTERNS Positive Solar Polarity Negative Solar Polarity

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

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

THE DIFFUSION TENSOR THROUGHOUT THE HELIOSPHERE Diffusion tensor computed from first principles using turbulence transport solutions Drifts competitive or dominant in outer heliosphere in this model Recall λ ik ≡ 3 K ik / V Note that λ rr = λ ║ cos 2 ψ + λ ┴ sin 2 ψ where ψ is spiral angle

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.

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.

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.

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