Fig. 2.36.2. The calculated latitudinal CR protons gradients at radial distance 2 AU from the Sun for different values y and n in Eq. 2.36.5 for perpendicular diffusion, and comparison with Ulysses data (open circles, according to Heber et al., 1996). From Burger et al. (1999).
Comparing panels (a), (b), and (c) in Fig. 2.36.2 it is evident that changing the rigidity dependence of K has little effect on the latitudinal gradient as y changes from - 0.4 to + 0.4. It reduces the gradient somewhat at high rigidity, but is does not shift the maximum. In contrast, changing the rigidity dependence of Kqq changes both the magnitude and the position of the maximum as n changes from -0.8 to 0 in each panel. The best values are y = n = -0.4.
2.36.4. Discussion on the nature of CR latitudinal transport
Burger et al. (1999) came to conclusion that to obtain the correct magnitude of the observed near-Earth latitudinal gradient, enhanced latitudinal transport is required. In the described above model, this is accomplished by increasing the cross-field diffusion in the polar direction with respect to that in the other direction perpendicular to the HMF. To obtain the observed rigidity dependence of this gradient, the cross-field diffusion in the polar direction must have a flatter rigidity dependence than parallel diffusion; at rigidities below about 10 GV k_|_/v should be almost independent of rigidity. Before coming to this conclusion, numerous other options were tried. However, looking at the transport Eq. 2.36.1, it is obvious that Kqq , which appears as a coefficient of dfo/dt, should play a dominant role in governing latitudinal transport. At least two other studies support this conclusion. Comparing Ulysses high-latitude data on the rate of change of integral CR intensities with IMP-8 data, Simpson (1998) concludes that if cross-field diffusion (as opposed to direct magnetic field "channeling"; see Fisk, 1996) occurs, it should be independent of rigidity. In an independent study, Potgieter et al. (1999) came to a similar conclusion studying CR electron modulation and using Ulysses electron data (see below, Section 2.40).
There are however other studies that at a first glance appear to contradict the above conclusion. A numerical simulation of Giacalone (1998) predicts r1/2 in the range 40 MV < R < 2 GV. A second different conclusion follows from the interpretation of Voyager anomalous nuclear component data (Cummings and Stone, 1998 and references therein) which suggests that the perpendicular mean free path is proportional to R below about 1 GV, in agreement with quasi-linear theory (e.g., Bieber et al., 1995).
Can all these different results be reconciled? The answer of Burger et al. (1999) is a guarded yes, but only if those that appear to have observational support are considered, i.e. if the numerical simulations reported by Giacalone (1998) is neglected for the moment. One possibility is that perpendicular transport in the ecliptic and in the polar region is different. In the ecliptic region, QLT may apply -this will explain the result reported in Cummings and Stone (1998). The relative insensitivity of the rigidity dependence of perpendicular gradient Gq on k| near
Earth, is an indication (albeit not a strong one) that the results of the discussed study will not necessarily be invalidated if from QLT is used. More problematic is Kqq . The enhanced latitudinal transport may also be due to direct magnetic field "channeling" in the HMF model of Fisk (1996). The question in this case is, if the transport is parallel to the field, should this process not have the same rigidity dependence as parallel diffusion? The first implementation of the Fisk field in a numerical modulation model (Kota and Jokipii, 1997) unfortunately did not address this issue.
2.37. CR drifts in dependence of Heliospheric current sheet tilt angle
According to Burger and Potgieter (1999), the effect of particle drifts, and in particular drift along the wavy current sheet, has long being thought to be responsible for the characteristic shape of the CR intensity profile observed near Earth around times of minimum solar activity (e.g., Jokipii and Thomas, 1981). Positively charged particles, during a positive solar polarity cycle (when the field in the northern hemisphere of the Sun points outward; denoted by A > 0) exhibit a rather flat response to the changing tilt near solar minimum. During alternate cycles, denoted by A < 0, and for the same range of tilt angles, the intensity profile shows a peak around solar minimum. It is by now well-established that driftdominated models can readily explain these different profiles (e.g., Jokipii and Thomas, 1981). While the role of drifts during periods of minimum solar activity appear to be well understood, the same cannot be said for periods when the Sun approaches maximum activity, and the tilt angle becomes large. Previous studies (Potgieter and Burger, 1990; Webber et al., 1990) with steady-state two-dimensional models that simulate the effect of a wavy current sheet, suggest that the flat response of positively charged particles during A > 0 cycles would persist for large values of the tilt angle. Using a newer version of such a two-dimensional simulated wavy current sheet model, Burger and Hattingh (1998) show that the intensity of CR protons during an A > 0 cycle does respond markedly when the tilt becomes larger than about 40°, approaching the intensity for an A < 0 cycle. In the described below study Burger and Potgieter (1999) extend the analysis of Burger and Hattingh (1998) to show what happens when the tilt angle approaches ~ 90° near maximums of solar activity.
2.37.2 CR propagation and modulation model; solar minimum spectra
The two-dimensional, steady-state numerical modulation model that is used in the study of Burger and Potgieter (1999) was described in detail by Burger and Hattingh (1995). A comparison of CR electron spectra from this model and those from a three-dimensional model was considered in Ferreira et al. (1999); it was found good agreement between the two models (see below, in Section 2.41). Therefore, as Burger and Potgieter (1999) note, from a modeling point of view there is no reason to doubt the validity of the results from the two-dimensional model.
In Fig. 2.37.1 are shown resulting solar minimum spectra (tilt angle is taken 15°) for CR electrons, protons and helium at Earth for the two polarity cycles of the solar magnetic field: A > 0 (e.g., 1996) and A < 0 (e.g., 1987). The modulated CR spectra are shown in comparison with corresponding spectra out of the Heliosphere.
Kinetic energy (GeV) Kinetic energy (GeV) Kinetic energy (GeV)
Fig. 2.37.1. Solar minimum spectra for CR electrons, protons and helium at Earth for the two polarity cycles of the solar magnetic field: A > 0 (e.g., 1996) and A < 0 (e.g., 1987). The tilt angle is 15°. By thick curves are shown corresponding spectra out of the Heliosphere. From Burger and Potgieter (1999).
2.37.3. Tilt angle dependence of CR protons at Earth
Fig. 2.37.2 shows how the intensity of CR protons, relative to the corresponding interstellar value, varies as function of tilt angle. From Fig. 2.37.2 can be seen that at all three energies the classic drift behavior, with the intensity-tilt profiles for an A > 0 cycle flatter than for an A < 0 cycle, is evident only for tilt angles up to about 45°. From 45° to about 60°, the intensity-tilt profiles for both cycles have similar slopes. Beyond about 60°, the A > 0 intensity-tilt profile drops, while the A < 0 intensity-tilt profile flattens, both approaching the no-drift value, indicated with a filled circle. The fact that this approach to the no-drift value becomes more evident as the energy decreases is due to numerical boundary effects, which in these cases diminishes as the particle's gyro-radius decreases. Note that Webber et al. (1990) used such intensity-tilt profiles to deduce that drift effects need to be reduced in a rigidity dependent manner, as is done in the study of Burger et al., 1999 (see above, Section 2.36, Eq. 2.36.5).
Kinetic energy (GeV) Kinetic energy (GeV) Kinetic energy (GeV)
A comparison between the two-dimensional model and a three-dimensional model (Hattingh, 1998) is shown in Fig. 2.37.3
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