40000 80000 120000 160000

Integration time t

Fig. 2.22.1. Examples of simulated k± versus the integration time t. The power-law fits of the cross-field particle dispersion (ax2) are presented for the model A and for the initial part of the curve for the model B2 (A, B and C are constants). A constant fit is provided for the model B1. From Michalek and Ostrowski (1999).

For the one dimensional plane wave model A one can note that (as required by Giacalone and Jokipii, 1994) the cross-field diffusion coefficient falls off as ^ t-1, as expected for a particle dispersion constant in time. An important feature seen in Fig. 2.22.1 is that the value of k± depends substantially on the assumed shape of the magnetic field perturbations. For the same amplitude and a 'similar' form of modulation applied in models B1 and B2 the diffusion coefficient values can differ by more than an order of magnitude. For the model B2 (sharp-edge modulated waves), the regime of sub-diffusive transport across the mean magnetic field is discovered on a short time scale, with kl slowly decreasing in the beginning and it approaches a constant value at large t. This time evolution of particle spatial cross-field dispersion differs from the one expected for the ordinary diffusion, with a short initial free-streaming followed by the phase with k± fluctuating near some constant value. The observed behavior reflects the restraining influence of stochastic particle trapping by large amplitude magnetic waves. When inspecting the initial part of the curve for the model B1, one observes in a narrow time range an analogous sub-diffusive evolution of particle distribution. In this case the ordinary particle cross-field diffusion is much larger than in the case B2 and particles decorelate from any given 'trap' much earlier. Numerical experiments performed by Michalek and Ostrowski (1997, 1998, 1999) proved that the phenomenon is caused by the long distance correlations introduced by the longest waves. In Fig. 2.22.2 are presented the simulated values of kl versus the wave amplitude SB.

Fig. 2.22.2. The simulated values of k_l versus SB for the wave models B1 and B2. Solid lines join the results obtained using fitting procedure. The adjacent dashed lines provide information about errors as they join the maximum (or respectively minimum) values of the quantity measured within the range used for fitting. From Michalek and Ostrowski (1999).

For the 3-D turbulence models considered Michalek and Ostrowski (1997, 1998, 1999) proved the possibility of substantial (by more than one order of magnitude at the same SB) difference in kl between at first glance similar turbulence models. Such a difference does not disappear for SB > 1. The reason for this difference is a more uniform modulation pattern in model B2 with respect to B1. The value of kl is closely related to the value of the magnetic field line diffusion coefficient Dm (Michalek and Ostrowski, 1997), but the growth of the wave amplitude is accompanied by a slight increase in the ratio of kl/Dm . This corresponds to the relative increase of the particle cross-field scattering owed to particle-wave interactions relative to the diffusion caused by magnetic field line wandering.

2.22.5. Simulations for oblique MHD waves models C-AF, C-AK, C-MF, and C-MK

The derived values of kl for different wave-cone opening angles and for different turbulence amplitudes are presented in Fig. 2.22.3. For the flat spectrum turbulence a systematic increase of kl with amplitude occurs and the rate of this increase roughly scales as SB2. The value of kl at any given SB is a factor ~ 10

larger for the fast-mode waves in comparison to the AlfVen waves. It grows substantially with the increasing wave cone opening a, i.e. with increasing power of waves perpendicular to the mean magnetic field. For the Kolmogorov spectrum a dependence of kl on the perturbation amplitude is flatter, the values of the cross-field diffusion coefficient at small SB are larger and there is a smaller difference between the fast-mode waves and the Alfven waves.

Flat spectrum

Kolmogorov spectrum

Flat spectrum

Kolmogorov spectrum

----40° 90°



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