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three parameters: the total integrated power a , the correlation length lc, and the spectral index y (for all simulations considered in Giacalone et al. (1999) it was set Y = 5/3). It were considered fluctuations which are approximately spatially homogeneous and isotropic.

Particles are injected at a given energy (which remains constant since the field is time stationary) chosen in such a way that the particle gyro-radius is 0.1 lc . For the interplanetary magnetic field with a typical correlation length of 0.01 AU and mean field strength at 1 AU of 5 nT, this would correspond to proton with an energy of 31.6 MeV. The particles are released isotropically in velocity space at a point in space which it was arbitrarily take to be the origin. They are followed for 1000 gyro-periods, which is larger than the scattering time for all runs that it was report here. The numerical scheme is described in detail in Giacalone and Jokipii (1996, 1999). The diffusion coefficients are compute in the following manner: the cross-field and parallel diffusion coefficients are determined by computing the averages over all particles of ^Ax2^ j(lAt) and ^Azj(lAt), respectively.

The anti-symmetric diffusion coefficients are determined from 2.38.8 as the average over all particles of k^ = {Axv^j , and Kyx = (Ayvx), respectively.

In order to compare the numerical results with Eq. 2.38.3 it must vary the particle mean free path. To accomplish this, Giacalone et al. (1999) vary the power in the random fluctuations, a . According to the standard quasilinear theory (e.g.

Jokipii, 1966) the mean free path varies as the inverse of a . Giacalone et al. (1999) emphasize, however, that here they compute the mean free path directly from the simulations from the relationship A// = 3 k/// v (which was divide by the particle gyro-radius to get n).

Shown in Fig. 2.38.1 are the ratios k_|_/k// and Ka/K// as a function of n. The corresponding values of the turbulence variance range from 0.03 < a2/< 30. The curves in Fig. 2.38.1 are from Eq. 2.38.3.

Fig. 2.38.1. Comparison of numerical simulations (solid circles), and analytic theory based on classical scattering (curves, determined by Eq. 2.38.3). From Giacalone et al. (1999).

Fig. 2.38.1 shows that the cross-field diffusion coefficient is considerably larger than the classical scattering result of Eq. 2.38.3. This is due to the fact the k_ is enhanced by the field-line random walk. This result is consistent with obtained in Giacalone and Jokipii (1999). On the other hand, the simulated values of Ka agree nicely with the classical scattering result of Eq. 2.38.3 for large values of n. In order to obtain the smaller values of Ka Giacalone et al. (1999) had to set the power in the random fluctuations considerably larger than the power in the mean field. Consequently, the field becomes almost completely random with no preferential direction. There should be no drifts under such a situation. This is the reason why the simulation results deviate noticeably from the curve. Giacalone et al. (1999) point out however that the statistics were very poor in determining these points and that additional simulations are needed to verify these findings.

2.38.4. Summary of main results

Giacalone et al. (1999) have performed numerical simulations of charged-particles moving in turbulent magnetic fields and compared these with analytic theory. They have concentrated primarily on the drifts associated with these motions and have derived expressions for determining the anti-symmetric diffusion coefficients. Giacalone et al. (1999) have found that the computed anti-symmetric diffusion coefficient agrees well with the classical theory when mean-free path largely exceeds the particle gyro-radius. On the other hand, Ka is significantly smaller than the predicted value when the mean-free path is less than several particle gyro-radiuses, which occurs when the power in the random fluctuations exceeds that in the mean field. These conclusions regarding Ka are restricted to a small range of parameters. Future work will extend this to a more comprehensive range of parameters. The small value of ka at n < 5 is potentially of significance for models of CR transport in the Heliosphere, where drifts play an important role.

2.39. Increased perpendicular diffusion and tilt angle dependence of CR electron propagation and modulation in the Heliosphere

2.39.1. The matter of the problem

It is well-known that the wavy Heliospheric current sheet (HCS) is a very important modulation parameter as were predicted by drift models (Jokipii and Thomas, 1981; le Roux and Potgieter, 1990). The computed effects of the HCS "tilt angle" a which represents the extend to which it is warped is however dependent of other modulation parameters, in particular the parallel k// and perpendicular k__ diffusion coefficients. Concerning k__ it was argued by Kota and Jokipii (1995) that it is not isotropic but seems enhanced in the polar directions. This enhancement has been studied intensively in modulation models (e.g., Potgieter, 1997) and it was illustrated that the enhancement is necessary to make these models compatible with the small latitude effects observed for protons onboard the Ulysses spacecraft.

Ferreira et al. (1999b) note that for numerical solutions of Parker's CR transport equation to be compatible with the small latitudinal gradients observed for protons by Ulysses, enhanced perpendicular diffusion seems needed in the polar regions of the Heliosphere. The role of enhanced perpendicular diffusion was further investigated by examining electron modulation as a function of the tilt angle a of the wavy current sheet, using a comprehensive modulation model including convection, diffusion, gradient, curvature and neutral sheet drifts. Ferreira et al. (1999b) found that by increasing perpendicular diffusion in the polar direction, a general reduction occurs between the modulation differences caused by drifts effects for galactic CR electrons as a function of a for the A > 0 (e.g. ~1990 to ~2000) and A < 0 (e.g. ~1980 to ~1990) solar magnetic polarity cycles. This aspect is also pursued in Potgieter, 1996 (detailed description of the importance of the various parameters in electron modulation) and in Ferreira and Potgieter, 1999 (where the effects are illustrated for spectra and differential intensities as a function of radial distance and polar angle).

2.39.2 The propagation and modulation model

The model for the study of Ferreira et al. (1999b) is based on the numerical solution of the Parker's (1965) transport equation:

where f (r,R, t) is the CR distribution function, R is rigidity, r is position, and t is time. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion and adiabatic energy changes respectively, with u the solar wind velocity. The symmetric part of the tensor K$ consists of a parallel diffusion coefficient K// and a perpendicular diffusion coefficient k__ . The anti-symmetric part Ka describes gradient and curvature drifts in the large scale Heliospheric magnetic field (HMF) with the pitch angle averaged guiding center drift velocity for a near isotropic CR distribution is given by

where eb = B/B, with B the magnitude of the background HMF. Eq. 2.39.1 was solved in a spherical coordinate system assuming azimuthal symmetry, and for a steady-state, that is df / dt = 0.

The HMF was modified according to Jokipii and Kota (1989). Qualitatively, this modification is supported by measurements made of the HMF in the polar regions of the Heliosphere by Ulysses (Balogh et al., 1995). The solar wind speed u was assumed to change from 450 km/s in the equatorial plane (9 = 90°) to a maximum of 850 km/s when 9 < 60°, with 9 the polar angle. The outer boundary of the simulated Heliosphere was assumed at 100 AU which is a reasonable consensus value. The galactic electron spectrum published from the COMPTEL results (Strong et al., 1994) was used as the local interstellar spectrum; see also Potgieter (1996). Solutions for tilt angle a up to 70° were computed for both A > 0 and A < 0 polarity epochs. For the parallel and perpendicular diffusion coefficients, and the 'drift' coefficient, the following general forms were assumed respectively:

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