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Assuming Eq. 2.41.4, and apart from the inherent azimuthal dependence of the HCS, no additional azimuthal dependence was incorporated in the 3D model. For the perpendicular diffusion and the 'drift' coefficient the following general forms were assumed respectively:

Here B is the ratio of the speed of the CR particles to the speed of light, Bm is the magnitude of the modified HMF, a = 0.05 is a constant determining the value of k_lr which contributes to perpendicular diffusion in the radial direction, b = 0.15 is a constant determining the value of k__q which contributes to perpendicular diffusion in the polar direction. Diffusion perpendicular to the HMF was therefore enhanced in the polar direction by assuming b > a (Kota and Jokipii, 1995; Potgieter, 1996). The coefficient {ka )O in Eq. 2.41.6 specifies the amount of drifts allowed. According to Ferreira et al. (1999a) it was necessary to take (ka )o = 0.5

which corresponds to medium drift effects.

The effective radial diffusion coefficient is given by

with ythe angle between the radial direction and the averaged HMF direction. Note that w^ 90° when r > 10 AU with the polar angle 6 ^ 90°, and y^ 0° when 6 ^ 0°, which means that K// dominates Krr in the inner and polar regions and k± r dominates in the outer equatorial regions of the Heliosphere. The differential intensity, J ^ p2 f, is calculated in units of particles m 2sr 1s 1MeV

2.41.3. Main results on comparison and discussion

To compare the results of the 3D model with the 2D WCS model, the average of the 3D solutions had to be calculated in Ferreira et al. (1999a) over one solar rotation, i.e. for azimuthal angles ^ = 0 ^ 2n. As a first comparison the modulated electron spectra computed with both models are shown in Fig. 2.41.1 for the polar regions, 6 = 30° (panel a), and for the equatorial regions, 6 = 90° (panel b), at radial distances of 1 AU and 60 AU with tilt angle a = 20°.

Fig. 2.41.1. Panel a: Computed electron spectra produced by the 2D and 3D drift model. Differential intensities are shown for 1 AU and 60 AU at a polar angle of 6 = 30° and a tilt angle a = 20° in units of m-2sr-1s-1MeV-1for the A > 0 polarity cycle. Panel b: Similar to panel a, but for a polar angle of 6 = 90°. Note that the spectra for the two models essentially coincide. From Ferreira et al. (1999a).

Fig. 2.41.1. Panel a: Computed electron spectra produced by the 2D and 3D drift model. Differential intensities are shown for 1 AU and 60 AU at a polar angle of 6 = 30° and a tilt angle a = 20° in units of m-2sr-1s-1MeV-1for the A > 0 polarity cycle. Panel b: Similar to panel a, but for a polar angle of 6 = 90°. Note that the spectra for the two models essentially coincide. From Ferreira et al. (1999a).

Solutions in Fig. 2.41.1 are shown for the A > 0 polarity epoch (e.g. ~1990 to ~2000) only, because during this cycle when electrons are drifting in along the HCS the largest difference between the two models occurs (see discussion below). These spectra can be considered as typical for minimum modulation periods. Evidently, the electron spectra produced by the two models as shown in Fig. 2.41.1 essentially coincide despite the use of a rather complex rigidity dependence for k// and k__ .

The two models obviously differ in the way the HCS is treated. Therefore, an appropriate way to compare the two models is by examining the a dependence of the differential intensities. In Fig. 2.41.2 the ratio of the computed 2D and 3D differential intensities is shown as a function of tilt angle a for both the A > 0 and A < 0 magnetic polarity cycles.

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