Before illustrating the features of the numerical model as applied to the acceleration of ACR Oxygen, the standard model configuration is summarised here first. Singly-charged anoma- lous oxygen is assumed, and hence the ratio of mass number to charge isΘ = 16. Also, in this
55
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Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
A > 0 α = 10o θ = 90o
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A < 0 α = 10o θ = 90o
1 AU 30 AU 50 AU 70 AU 90 AU
Figure 4.1:Modelled ACR Oxygen energy spectra for the A>0 (top panel) and A<0 (bottom panel) magnetic polarity cycles. The spectra are shown for radial distances 1 (Earth), 30, 50, 70 and 90 AU (TS) in the equatorial plane (θ= 90◦) for solar minimum conditions (α= 10◦).
chapter, the respective perpendicular and parallel diffusion coefficients ofBurger et al.[2000]
and Burger et al. [2008] are implemented. It is generally assumed here that κ||,0 = 0.9 and κ⊥r,0 = κ0e = 0.02 (see Section 3.3), however these scaling factors may vary in subsequent sections. Furthermore, the HCS tilt angle is taken asα= 10◦to simulate solar minimum condi- tions, with CR drift assumed to be 55% effective (κD,0 = 0.55) in correspondence with previous studies conducted using 2-D models [Langner, 2004; Ngobeni, 2006, 2015;Strauss, 2010]. Fur- thermore, for this chapter, the positions of the TS and HP are chosen to coincide with that used byStrauss[2010] for comparability. The heliocentric position of the TS is thus taken asrT S =90 AU. That of the HP, however, is taken asrHP =140 AU – an overestimate, but similar to that assumed by HD models [e.g. Ferreira et al., 2007a] prior to the Voyager 1 crossing of the HP.
Moreover, although estimates for the TS compression ratio vary appreciably (see Section 4.3), a moderately strong shock withs =2.5 is settled on in this study. Incompressible solar wind flow (∇·V~sw =0) is assumed in the heliosheath, ruling out adiabatic heating, with stochastic acceleration suppressed by settingκP,0 =0 (See Section 3.6). Having generated solutions with
the above configuration, some care must be taken with their intensities. ACRs are modelled in this study by accelerating a monoenergetic source function, representing a population of injected PUIs, into a power law at the TS [see alsole Roux et al., 1996;Langner, 2004;Strauss, 2010]. Since the intensity of the injected PUI source is chosen arbitrarily (see Section 4.6.2), and by virtue of the linearity of the Parker TPE, the resultant solutions are scaled to fit measured intensity levels. See e.g.Strauss and Potgieter[2010],Strauss et al.[2010a] orStrauss et al.[2011a]
for similar modelling approaches, and e.g.Leske et al.[2013] and the references therein for ACR Oxygen intensities during recent solar cycles. Although the reproduction of ACR observations is not an objective of this study, the solutions displayed throughout this chapter are nonetheless scaled at the TS to measurements from Voyager 1 and 2 [Webber et al., 2007]. The above model configuration consequently yields the solutions shown in Figures 4.1, 4.2 and 4.3. These may be used as references in subsequent sections, and any departures from this configuration for further solutions in this chapter will be stated explicitly.
The reference solutions already reveal a number of intriguing features with regards to CR ac- celeration. Figure 4.1 illustrates how the accelerated TS spectrum is modulated from the TS position to Earth for both magnetic polarities in the equatorial plane (θ=90◦). The first notable feature is of course the power-law structure of the TS spectrum – a characteristic of DSA. At energies higher than about10MeV. nuc−1, however, this power-law structure diminishes and transitions into an exponential decay. This is referred to as the ACR cut-off or roll-over energy, above which anomalous intensities fall below those of GCRs [Webber et al., 2007]. In the energy region immediately preceding this cut-off, the TS spectrum bulges slightly upward for the A
< 0 polarity cycle, while deviating downward for A> 0. This polarity dependence hints at the involvement of CR drifts. The three spectral features identified above form the basis for discussion in following sections. Furthermore, note from Figure 4.2 that the modelled radial intensity profiles attain a maximum at the TS in the equatorial plane (also visible in Figure 4.3). This is however in the absence of any heating mechanisms other than acceleration at the TS, and in apparent contradiction with Voyager observations that show a definitive increase in ACRs from the TS into the heliosheath [McDonald et al., 2007;Webber et al., 2007]. These obser- vations have given rise to some debate on the location of the chief source of ACRs, whether it is indeed at the TS [as argued byJokipii and Kota, 2011] or further into the heliosheath as obser- vations suggest [Stone et al., 2009]. The latter notion was corroborated byStrauss et al.[2010b]
who modelled ACR intensities in the heliosheath with the inclusion of adiabatic heating and stochastic (second-order Fermi) acceleration. These mechanisms, though summarised briefly in Section 3.6, are not explored further in this study. See Strauss[2010] for a comprehensive review.
The omission of alternative acceleration sources does however prove useful to explore some of the more general modulation features in the heliosheath that would otherwise have been ob- scured. For instance, in the absence of adiabatic energy changes (by virtue of the assumption of incompressible SW flow), Figure 4.2 shows constant intensities throughout the heliosheath for particles with energies below 1.0 MeV. This ensues because ACRs produced at the TS are merely convected outward into the heliosheath with little to no modulation. On the other hand, the intensities of these low-energy particles plummet quickly into the heliospheric interior due
1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 10−7
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Radial distance (AU) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
A > 0 α = 10o θ = 90o 0.1 MeV
1.0 MeV 6.0 MeV 22 MeV r TS
1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 10−7
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Radial distance (AU) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
A < 0 α = 10o θ = 90o 0.1 MeV
1.0 MeV 6.0 MeV 22 MeV r TS
Figure 4.2:Radial profile of modelled ACR Oxygen intensities in the equatorial plane (θ= 90◦) for the A>0 (top panel) and A<0 (bottom panel) magnetic polarities during solar minimum (α= 10◦). The intensities are shown for CR energies of 0.1, 1.0, 6.0 and 22 MeV, with the TS at 90 AU (indicated by solid vertical line) and the HP assumed at 140 AU.
to small MFPs (see Figure 3.6), but also due to energy losses. Indeed, Figure 4.1 shows that toward lower energies modulated spectra are forced into thej ∝E+1 relation at Earth that is associated with adiabatic cooling [see e.g.Moraal and Potgieter, 1982]. For higher-energy par- ticles, however, CR drifts are more pronounced. Figure 4.2 conveys that during the A < 0 polarity, while raising intensities upstream of the TS (see also Figure 4.1), the inward-directed drifts along the HCS serve to diminish intensities of ACRs further into the heliosheath. On the other hand, during the A>0 cycle, outward drifts and impaired diffusion in the heliosheath collectively yield intensities that are nearly constant across this region. This holds for the equa- torial plane as demonstrated in Figure 4.2, while the global drift patterns explained in Section 3.4 are illustrated more clearly with contours in Figure 4.3. Nearer to Earth, however, both dif- fusion and drift scales become smaller as shown in Figures 3.6 and 3.10, reducing the amount of particles transported here and resulting in similar intensities at Earth for both polarities.
The prevalent qualities of the solutions presented here are consistent with that illustrated by Strauss[2010] under the same assumptions. It is however those features discussed above that
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α = 10o A > 0
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α = 10o A < 0
6 MeV/nuc
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Figure 4.3: Contour plots illustrating 6 MeV. nuc−1 ACR Oxygen intensities in the meridional plane of the heliosphere, for A >0 (left) and A <0 (right) polarities. The colour bar shows intensities as percentages of the maximum on a logarithmic scale (e.g. 2 ≡ 102 %). The dashed white halfcircle indicates the TS position (at 90 AU), while the dash-dotted white lines indicate the polar extent of the HCS forα= 10◦. The Sun is located at the origin.
pertain to the acceleration of ACR Oxygen at the TS that are of interest in this study. As men- tioned before, these features form the basis for discussion in subsequent sections where they are investigated in greater detail.