Fig. 2.40.4. Computed electron spectra in the equatorial plane at 1, 5, 10, 50, 70, 80, and 90 AU, corresponding to the function shown for k// in Fig. 2.40.3 and based on the assumption that K__ ^ K//. Data are at ~5AU for 1997 from the Ulysses/KET experiment. From Potgieter et al. (1999).
Potgieter et al. (1999) came to conclusion that the analysis of electron modulation illustrates how important k_r, but especially k_|_q and its rigidity dependence is to electron modulation below 100 - 300 MeV. It was illustrated that although k__r and k_q was only 5% and 15% of the value of K// respectively, perpendicular diffusion dominates electron modulation below ~ 100 MeV. It was argued that if the increasing intensity with decreasing energy below ~ 100 MeV in the observed electron spectra in the inner Heliosphere were taken as a characteristic of modulated CR electron spectra, as in Fig. 2.40.4, then, to assure reasonable compatibility with data below ~ 100 MeV, K_r and certainly k_q must be nearly independent of kinetic energy below ~ 100 MeV.
2.41. Comparison of 2D and 3D drift models for galactic CR propagation and modulation in the Heliosphere
Ferreira et al. (1999a) note that the propagation and modulation of galactic CR in the Heliosphere is described successfully by Parker's (1965) transport equation (see Eq. 2.39.1 in Section 2.39). This equation has been solved with increasing complexity over the years. However, to solve it numerically for three spatial dimensions (3D), a rigidity and a time-dependence is rather complex and has not yet been done successfully. By assuming an axisymmetric CR distribution one can neglect the equation's azimuthal dependence which leads to 2D models which have been used widely for modulation studies (le Roux and Potgieter, 1990). The main difficulty in 2D models is how to emulate the effect of the wavy Heliospheric current sheet (HCS) because it cannot be done directly. This was done successfully for the first time by Potgieter and Moraal (1985). The technique was improved by Burger and Potgieter (1989). Hattingh (1993) developed a refined 2D model, which was called the WCS model, and after several years also a 3D model which includes an actual wavy HCS (Burger and Hattingh, 1995; Hattingh 1998). This 3D model was compared carefully to the first 3D model developed by Kota and Jokipii (1983) with excellent results. For a review and detail of the different models, see le Roux and Potgieter (1990), Hattingh and Burger (1995a,b), Burger and Hattingh (1995) and for an application of the 3D model, see Burger and Hattingh (1998). An obvious next step was to compare the 2D and 3D models to establish how reliable the 2D models are, and to establish to what extent they can be used for modulation studies. This was done by Hattingh (1998) for CR protons and it was found that the agreement between the 3D and the 2D WCS model varied between good to excellent. At Earth, the largest variation of ~16% in the ratio of the two sets of solutions was found at low rigidities for a tilt angle a = 20° during an A < 0 (e.g. ~1980 to ~1990) solar polarity cycle. At 60 AU, the largest variation was ~26% at low energies during the A < 0 cycle. This comparative study was continued by Ferreira (1999) who concentrated on the modulation of CR electrons in the
Heliosphere because electrons may have a different diffusion tensor than protons, experience less adiabatic energy losses than protons at energies of interest to modulation, and for which drifts become less significant with decreasing energy.
The paper of Ferreira et al. (1999a) reports on the comparative study of the 2D and 3D models using electron modulation, with emphasis on the tilt angle dependence because the computation of the wavy HCS and its effects on modulation are the important difference between the 2D and 3D numerical models. In Ferreira et al. (1999a) the modulation of galactic CR electrons in the Heliosphere was used to compare solutions of a 2D and 3D drift model, both developed by the Potchefstroom Modulation Group. These steady-state models are based on the numerical solution of Parker's transport equation and include the main modulation mechanisms: convection, diffusion, gradient, curvature and neutral sheet drifts. Examining computed electron spectra, with identical modulation parameters in both models, as a function of the Heliospheric neutral sheet tilt angle yielded no qualitative differences and insignificant quantitative differences between the solutions of the 2D and 3D models. Taking into account the large amount of resources needed for the 3D model, the use of a 2D model for modulation studies is well justified.
A short description of the 2D WCS model is given by Ferreira et al., 1999b (see Section 2.39) with detail given by Burger and Hattingh (1995). The 3D model is based on the numerical solution of Parker's (1965) equation:
f = -(u + (v*) ))/ + V-(s-V/) + 3 (V-u )/, (2.41.1)
where u is the solar wind velocity and / (r, p, t) is the CR distribution function where p is rigidity, r is position, and t is time. The symmetric part of the diffusion tensor ks consists of a parallel diffusion coefficient K// and a perpendicular diffusion coefficient k__ .The antisymmetric part Ka describes gradient and curvature drifts in the large scale Heliospheric magnetic field (HMF). The pitch angle averaged guiding centre drift velocity for a near isotropic CR distribution is given by
with eb = B/B, where B is the magnitude of the background HMF and h(r) is a transition function which varies from 1 to -1 across the HCS and is zero in the HCS. This transition function modifies Ka across the wavy HCS which is positioned at
with — the angular velocity of the Sun and 6, <, and r the heliocentric spatial coordinates. The solar wind speed u was assumed to change from 450 km.s-1 in the equatorial plane (6 = 90°) to a maximum of 850 km.s-1 when 6 < 60°. The HMF was modified according to Jokipii and Kota (1989) and the outer boundary of the simulated Heliosphere was assumed at 100 AU. The galactic electron spectrum based on COMPTEL results (Strong et al., 1994) was assumed as the local interstellar spectrum. To produce spectra compatible to both Ulysses and Voyager 1 measurements (Potgieter et al., 1999), the following parallel diffusion coefficient was used:
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