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Perpendicular transport/diffusion

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### Subdiffusive regime transport of energetic particles

### Recovery of diffusion transport of energetic particles

### How much transverse structure is sampled by particles ? transport of energetic particles

### Turbulence Geometry

Turbulence, magnetic field complexity and perpendicular transport of energetic particles in the heliosphereW H MatthaeusBartol Research Institute, University of DelawareCollaborators: G. Qin, J. Bieber, D. Ruffolo, P. Chuychai

Perpendicular transport/diffusion transport of energetic particles

- Field Line Random Walk (FLRW) limit is a standard picture.
- K (v/2) D

Fokker Planck coefficient

for field line diffusion

“BAM” theory of perp diffusion transport of energetic particles

Bieber and Matthaeus, 1997

TGK formula for diffusion

Ansatz #1: form of v-v correlation

Ansatz #2: effective collision time

BAM result

Puzzling properties of perpendicular transport/diffusion transport of energetic particles

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

Importance of turbulence geometry and structure transport of energetic particles

- In parallel scattering, the distribution of turbulence energy over wave vector direction, as well as the wave vector magnitude, is very important.
- In perpendicular scattering, the spectral characteristics are also important, especially the transverse complexity and structure of the magnetic field.

Bieber et al, 1994. Schlickesier et al, 1994,

Droege 2002

Correlation/Spectral anisotropy transport of energetic particles

MHD Anisotropy: dynamical development oftransverse complexity

Shredding of flux tubes in 3D transport of energetic particles

Two component

Model is three-dimensional:

20% slab + 80% 2D

1D slab only

Slab or near-slab models:

very little transverse complexity

Numerical Results: 0.9999 slab fluctuations: transport of energetic particles Parallel and perpendicular transport in the same simulation!

- Running diffusion coefficient: K = (1/2) d<2>/dt
- Parallel: free-streaming, then diffusion ( QLT)
- Perpendicular: initially approaches FLRW, but is thwarted…behaves as t-1/2
i.e., subdiffusion

Qin et al, 2002

IH Urch, Astrophys. Space Sci., 46, 389 (1977). transport of energetic particles

J. Kota and J.R. Jokipii Ap. J. 531, 1067 (2000)

Perpendicular Subdiffusion- In evaluating K~z/t Dinstead of using z/t =v, assume that the parallel motion is diffusive, and z =(2 K t )1/2
- K = D (K / t)1/2
- For this to occur, nearby field lines must be correlated. If the transverse structure sampled by the particle becomes significant, can diffusion be restored?

Turbulence with adequate

transverse complexity

Numerical simulation using 2-component turbulence: transport of energetic particles 80% 2D + 20% slab

Qin et al, 2002

Running diffusion coefficients

vs. time

- Parallel: free stream, then diffuse, but at level < QLT
- nonlinear effect of 2D fluctuations

- Perpendicular: evolves towards FLRW, subsequent decrease, and then a regime of “second diffusion” emerges.

Perpendicular diffusion: Loss and recovery transport of energetic particles

- Subdiffusion (Urch-type) should be taken seriously
- Subdiffusion always occurs in slab geometry
- FLRW rarely if ever works (but may be the “target”)
- BAM theory doesn’t work either
- Composite geometry simulations provide insight:
- FLRW and BAM tend to bracket computed results
- Kperp/Kpar ~ constant as function of rigidity (0.02-0.05) in some parameter regimes

- Numerical evidence points to need for a nonlinear theory
- FLRW + parallel scattering not enough
- Require an additional source of randomization
Interaction between transverse complexity of the field and particle gyro-orbits

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

Recovery by stochastic instability transport of energetic particles

Rechester and Rosenbluth PRL, 1978

- Field lines exponentially separate
- Field line mapping is reversible diffusion is destroyed
- Add scattering restore diffusion
- Weak collisions destroy reversibility and restore FLRW
- Stronger scattering balance between stretching and scattering
- Chandran and Cowley (PRL, 1998) modify R+R for collisonless case

See also Kadomtsev and Pogutse, 1979

A new theory of nonlinear perpendicular transport of energetic particles (guiding center) 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

Nonlinear perpendicular diffusion (1) transport of energetic particles

- 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:

Nonlinear perpendicular diffusion (2) transport of energetic particles

- Corrsin independence approximation

The perpendicular diffusion coefficient becomes

Or, in terms of the spectral tensor

Nonlinear perpendicular diffusion (3) transport of energetic particles

- “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

Nonlinear perpendicular diffusion (4) transport of energetic particles

- 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

Comparison of Nonlinear theory with simulations transport of energetic particles

- Low dB/B= 0.2
- Parallel scattering nearly QLT
- NL GC theory accounts well for perpendicular transport

Comparison for strong turbulence: Four theories transport of energetic particles

- db/B = 1
- FLRW never very good
- BAM good at large rL/Lc
- RR-CC good at low rL/Lc
- NLGC theory remains accurate for wide range of rL/Lc

Parallel vs. Perpendicular diffusion transport of energetic particles

- Not a purely linear relationship
- Kperp << Kpar

G. Qin et al, 2002 GRL transport of energetic particles

G. Qin et al 2002 ApJL

Matthaeus et al, 2002

Perpendicular transport- Particle gyrocenters try to follow fields lines (“FLRW” limit)
- Motion along field lines is inhibited by parallel (pitch angle) scattering
- Low transverse complexity subdiffusion
- Strong transverse complexity recovery of diffusion at lower level than FLRW
- A nonlinear theory of guiding center diffusion agrees well with simulations
- Comparisons with observations are underway

Some remaining problems in perpendicular transport transport of energetic particles

- Observational connections
- Transition between subdiffusion and diffusion
- Channeling and sharp boundaries (conditional/inhomogeneous statistics)
- Dynamical effects (random dynamical effects can be incorporated easily in the present formalism!)
- Inhomogenous large scale field, e.g., Fisk field
- Field line separation effects (in progress, with D. Ruffolo)
- Anisotropic FLRW (ICRC proceedings, Ruffolo, Chuychai and Matthaeus

K transport of energetic particles ┴ / 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

K transport of energetic particles ┴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

Comparison for strong turbulence transport of energetic particles

- Db/B = 1
- Parallel scattering differs from QLT
- NL GC theory remains accurate for wide range of rL/Lc

Slab Geometry: transport of energetic particles 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.

Two component model: transport of energetic particles

Spectral method simulations: cross sections of magnetic energy density

B0

X,z cross sections

Transverse

Section x,y

Dmitruk and Matthaeus, 2002

80-20 turbulence labeled by local 2D vector potential energy density

SW turbulence “sees” at least two preferred directions: energy density

radial (expansion)

local mean magnetic field

Several observational studies confirm lack of isotropy

Multicomponent models: each with fixed symmetry

Two/Three component “slab” + quasi-2D + “structures” model seems to cover most of the constraints:

scattering theory

direct observations

“Maltese cross”

Weakly Compressible MHD theory

Slab component: waves/origin of SW

quasi-2D component: consistent with simulations, theory and lab experiments.

Structures: smaller parallel variance piece (phase mixing, compressible simulations, “5:4:1”, NI Theory)

Symmetry/Anisotropy has major impact on transport, heating, couplings to kinetic effects, diffusion, etc...

Anisotropy and symmetryMaltese Cross

Simulations and Theory suggest that perpendicular cascade is much faster than parallel

Recovery by stochastic instability energy density

Rechester and Rosenbluth PRL, 1978

- Field line distribution is an area preserving map
- Field line complexity grows as field lines exponentially separate
- Map is reversible diffusion is destroyed
- Add perpendicular displacements due to scattering restore diffusion
- A small amount of collisional scattering destroys reversibility and restores to QLT/FLRW value
- Stronger scattering balance between stretching and scattering

Scale for decorrelation of random displacements

Rechester and Rosenbluth (1978) form: energy density

Decorrelation determined

by exponential separation of

field lines

Chandran and Cowley (PRL, 1998)

Modify R+R for collisonless

case:

The RR-CC nonlinear perpendicular

Diffusion coefficient becomes:

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