The effects of electron drifts on modulation are apparently limited to a relatively narrow energy domain (see Figure 5.1), and constitute another major process involved in the modulation of these particles and their energy distributions. The details of this process and how it is modelled are provided in Section 3.4, whereas the effects of varying drift efficiencies and solar activity on drift-related transport, as well as its latitudinal dependence, are illustrated in Section 4.5 by application of the model to anomalous ions. While Section 5.2 highlights some differences in the modulation and transport of galactic electrons during opposite magnetic polarities, it does not quantify the actual contribution of drifts to electron intensities at different energies or spatial regions. The interplay between electron drifts and diffusion also remains to be explored.
It is endeavoured in the current section to comment on these matters.
Consider first Figure 5.15, where modulated spectra at the Earth and TS are shown, both with
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o 1 AU 94 AU 122 AU
Pk = 0.34 GV
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α = 10o 1 AU 94 AU 122 AU
Pk = 0.45 GV
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o 1 AU 94 AU 122 AU
Pk = 0.60 GV
Figure 5.15:Modelled spectra at the TS (94 AU,θ= 55◦) and Earth (1 AU,θ= 90◦), respectively shown in red and blue, modulated from the HPS at 122 AU in black. Solid and dash-dotted lines respectively represent full-drift solutions for A<0 and A>0, whereas dashed lines represent no-drift solutions.
Each panel corresponds to a configuration in Figure 5.8 for the indicated values ofPk(of Eq. 3.24).
drift effects (for A>0 and A<0) and without, for the three high-energy diffusion scenarios presented in Figure 5.8. It is immediately noticeable that drift effects dramatically raise spectral intensities at energies up to a few GeV and from as low as 100 MeV at the TS and 10 MeV at Earth, whereas the difference between drift solutions during opposite polarities, while signif- icant, is not as spectacular. As far as drifts are affected by diffusion, it follows from the ratios presented in Figures 5.16 and 5.17 that larger MFPs inhibit drift effects and reduce the total con- tribution of drifts to electron intensities. Recall that increasingPk(of Eq. 3.24) in Section 5.4.1 shifted the MFPs at lower rigidities upward. Hence, of the cases presented, MFPs are generally lowest forPk =0.34 GV, which inhibits diffusion while enhancing drift effects. This follows since the increased modulation brought about by smaller diffusion increases intensity gradi- ents, and because the product of the drift term with∇f is contained within the TPE. See also Strauss et al.[2012] for illustrative examples of how increased diffusion can inhibit drift effects.
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Ratio of differential intensities
94 AU, 0.60 GV 94 AU, 0.45 GV 94 AU, 0.34 GV 1 AU, 0.60 GV 1 AU, 0.45 GV 1 AU, 0.34 GV α = 10o
Figure 5.16: Ratios of the drift-enabled model solutions shown in Figure 5.15 with A<0 to that for which A>0. Solutions are shown for different values ofPk corresponding to the configurations illus- trated in Figure 5.8 and as indicated in the legend. The ratios are shown at Earth (1 AU,θ= 90◦) in blue, and the TS (94 AU,θ= 55◦) in red.
These effects, quantified as the ratio of intensities during the negative magnetic polarity cycle to that during the positive cycle, are demonstrated in Figure 5.16 to be more pronounced at the TS than at Earth; intensities during the A<0 cycle are higher with factors of up to nearly 3.5 at the TS than for A>0, whereas at Earth these intensities are doubled at most. The increases at Earth are also more pronounced at lower energies, peaking at roughly 40 to 50 MeV, while the effects at the TS are centred around 200 MeV. For larger values ofPkthese drift effects become consistently smaller and are confined to larger energies.
The total contribution of drifts during both magnetic polarities are represented by ratios of intensities predicted by model solutions for which full drifts (κD,0= 1.0) are assumed to those for which drifts are neglected (κD,0 = 0); these are displayed in Figure 5.17. These ratios display the same qualitative features as discussed above for the ratios illustrated in Figure 5.16: The contributions are confined to higher energies and become smaller for larger values ofPk, and are greater at the TS than at Earth, albeit limited to a narrower energy interval. ForPk =0.34 GV, for instance, it is illustrated in Figure 5.17 that intensities would have been up to 13 times smaller at Earth during the negative polarity cycle, and up to 15 times smaller at the TS, if there were no drifts. Also note that these contributions are consistently smaller for A>0 than for A<
0; this is likely a result of the associated drift patterns. The global drift patterns of electrons and their consequences for intensities in different regions are qualitatively discussed in Section 5.2.
Figure 5.18, accompanying Figure 5.3, compliments that discussion; it demonstrates for both polarities, and the standard configuration of Section 5.2, the factors by which drifts increase intensities across the heliosphere from the levels of the corresponding configuration where drifts are disabled. Via the drift patterns explained in Section 5.2 for A >0, intensities are raised inside the heliosphere at all latitudes by factors ranging from roughly100.9 (≈ 8)near the Sun to as low as100.6(≈4)just inside the TS. Intensities are raised in this region by a much
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Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV, A > 0 0.34 GV, A < 0 0.45 GV, A > 0 0.45 GV, A < 0 0.60 GV, A > 0 0.60 GV, A < 0 α = 10o
θ = 90o r = 1 AU
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0 2 4 6 8 10 12 14 16 18
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV, A > 0 0.34 GV, A < 0 0.45 GV, A > 0 0.45 GV, A < 0 0.60 GV, A > 0 0.60 GV, A < 0 α = 10o
θ = 55o r = 94 AU
Figure 5.17:Ratios of the model solutions shown in Figure 5.15 with full drift efficiency to that for which drifts are disabled. Solutions are shown for both polarities and different values ofPk (corresponding to the configurations illustrated in Figure 5.8) as indicated in the legend. The top and bottom panels respectively show the ratios at Earth (1 AU,θ= 90◦) and the TS (94 AU,θ= 55◦).
larger average factor of about101.25 (≈ 18)when A<0, hence displaying the same tendency seen earlier of the negative polarity cycle yielding larger drift contributions for electrons. It also appears from Figure 5.18 that drifts bolster heliosheath intensities somewhat in the equatorial region and reduces them at the poles for A>0, while the converse is observed for A<0. The reductions are represented by colours for which the quantity shown on the logarithmic colour scale is negative, i.e. for factors in the order of10x withx < 0. In general though, intensities in the heliosheath seem largely unchanged from no-drift conditions for both polarities, with an average factor change of more or less100(≈1)in this region.
It is important to note that the drift-related features outlined above are subject to change should the spatial configuration of other processes such as diffusion also change. Consider e.g. that the drift contribution to intensities near Earth may be larger than at some further distance inside the TS (see e.g. Figure 5.18; A >0), because electrons transported to Earth via e.g. drifts are confined there due to smaller diffusion coefficients. Moreover, the enhanced polar diffusion
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Radial distance (AU)
Radial distance (AU)
α = 10o A > 0
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α = 10o A < 0
200 MeV
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Figure 5.18: Contour plots for 200 MeV electron intensity ratios, illustrating the factor by which solu- tions with full drift efficiency is larger (or smaller) than corresponding solutions without drifts in the meridional plane of the heliosphere for A>0 (left) and A<0 (right). The colour bars follow a log- arithmic scale (e.g. 1.5≡ 101.5). The dashed white halfcircle indicates the TS position at 94 AU, the dash-dotted lines indicate the polar extent of the HCS forα= 10◦, and the dashed black line shows the trajectory alongθ= 55◦at which the corresponding ratios in Figures 5.16 and 5.17 are shown. The Sun is located at the origin.
at high latitudes facilitates particle transport through those regions, while the low-diffusion barrier in the heliosheath may hinder transport - this is touched on in the discussion on drift patterns in Section 5.2. It should also be noted that the drift features discussed above, although indicative of basic trends also predicted by 3-D models [e.g.Strauss et al., 2012], are presented within the context of the 2-D model utilised in this study. While CR modulation studies con- ducted using 2-D models with drift capabilities usually scale down the drift efficiency based on scattering assumptions (see Section 3.4), full drift efficiency (κD,0 = 1.0) is assumed for the illustrations in this section. See also the work ofNndanganeni[2012, 2015]. Recently,Ngobeni [2015] andNgobeni and Potgieter[2015], also using a 2-D model, questioned if drifts could be large in the outer heliosphere - an issue that requires further investigation with full 3-D models.