COSPAR 2004, Paris D1.2-0001-04July 21, 2004

THE HELIOSPHERICDIFFUSION 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/

PARKER’S TRANSPORT EQUATION

DIFFERENT ASPECTS OF DIFFUSION

Advances in Heliospheric Turbulence Turbulence Geometry

Slab Geometry: Wavevectors k parallel to mean field B0. Fluctuating field δB perpendicular to B0.

Motivations: Parallel propagating Alfvén waves. Computational simplicity. 2D Geometry: k and δB both perpendicular to B0.

Motivations: “Structures.” Turbulence theory. Laboratory experiments. Numerical simulations. Solar wind observations.

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.

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.

PARALLEL DIFFUSION

• Geometry resolves discrepancy at intermediate-high rigidity

• Dissipation explains high electron mean free paths at low rigidity

• Pickup ions still a puzzle

PERPENDICULAR DIFFUSIONKey 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

• 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 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:

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:10.1029/2003JA010301.

NLGC Agrees withNumerical 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, 27447.)

• 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║ ~ 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.

TURBULENCE TRANSPORT THEORY → TURBULENCE PARAMETERS THROUGHOUT HELIOSPHERE

Energy

Temperature

Correlation Length

Cross Helicity

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