moves out to larger radial distances near the equatorial plane. Therefore, at the TS (atr 1 AU) it follows thatψ → 90◦ forθ ≈ 90◦, so that Eq. 4.2 reduces toκrr ≈ κ⊥r. This makes physical sense, since CRs diffusing along magnetic field lines would have to travel great dis- tances along the Parker spiral to be removed an appreciable radial distance from the TS. Note however that this changes toward the polar regions for a Parker HMF, since Eq. 4.2 reduces to κrr ≈κ||whenψ→0◦. When studying the TS spectrum at low heliographic latitudes though, varyingκ⊥ris sufficient to illustrate the effects of varying the effective radial diffusion. Figure 4.5 demonstrates these effects, showing that smaller values of the scaling factor (κ⊥r,0) shifts the ACR cut-off to higher energies. This follows because scalingκ⊥r downwards necessitates higher energies in order for the relation in Eq. 4.5 to be achieved. Note also that the modelled spectra fall well below the power law in Figure 4.5 for lower values ofκ⊥r,0, though still par- allel to it at high energies. A possible explanation for this is offered in Section 4.7. Another interesting result is that varyingκ||,0, as shown in the left-hand side of Figure 4.6, yields effects that are similar to that in Figure 4.5. One would expect that since the contribution of parallel diffusion in Eq. 4.2 is diminished at the TS, the effects would be smaller. However, recall from Section 3.3 that both κ⊥r andκ⊥θ scale asκ||, so that these effects are essentially a combina- tion of varied radial and polar perpendicular diffusion. It is thus well worth noting that any changes made toκ||in this study are carried through to other modes of diffusion as well.
4.4.2 Polar diffusion
It was already mentioned in the introduction to this section that varying diffusion along the polar direction, along which the TS also extends, would yield marginal effects in terms of par- ticle acceleration. This is supported in the right-hand side of Figure 4.6. Hereκ⊥θ is varied by changing the value ofκ0e in Eq. 3.28. Note that for this section diffusion is not necessarily isotropic, sinceκ0e 6=κ⊥r,0 =⇒ κ⊥r6=κ⊥θ. Qualitatively, varyingκ⊥θ produces similar effects to varyingκ⊥r, with the ACR cut-off shifting to higher energies for smaller diffusion, although marginally. Note also, that by virtue of the effects illustrated above and the enhancement of polar diffusion at high latitudes, the energies attained through acceleration would be generally lower away from the equatorial plane. SeePotgieter[1996, 2000] andHeber and Potgieter[2006]
for reviews on CR modulation at high latitudes.
It appears that with all of the above taken into account,κ⊥rgoverns the most effective transport away from the TS, and that limiting this parameter contributes a great deal toward particle confinement and improving the acceleration efficiency. It also follows from this section as a general consequence that lower diffusion results in acceleration to higher energies.
10−4 10−3 10−2 10−1 10−3
10−2 10−1 100 101 102 103 104
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 100 θ = 90o γ = −1.5
1.0, A < 0 1.0, A > 0 0.55, A < 0 0.55, A > 0 0.0
10−5 10−4 10−3 10−2 10−1 0
0.5 1 1.5 2 2.5 3 3.5 4
Kinetic energy (GeV.nuc−1)
Differential Intensity X E1.5
α = 10o θ = 90o
10−4 10−3 10−2
10−2 10−1 100 101 102 103
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o θ = 10o
γ = −1.5
1.0, A < 0 1.0, A > 0 0.55, A < 0 0.55, A > 0 0.0
10−5 10−4 10−3 10−2 10−1 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Kinetic energy (GeV.nuc−1)
Differential Intensity X E1.5
α = 10o θ = 10o
Figure 4.7:On the left, modelled ACR Oxygen spectra are shown at the TS for the equatorial plane (top panel) and polar regions (bottom panel) atθ = 90◦and10◦ respectively. The solutions are shown for various levels of drift efficiency (κD,0) for both polarities as indicated in the legend. The dashed-dotted line represents the predicted power law with spectral index as labelled. The solutions shown on the right are the same as those on the left, but are multiplied byE1.5and normalised to unity at10−5GeV.
nuc−1. Here the power law is represented as a horizontal line along unity.
seems to be downward and much fainter. Inferring from the fact that the effect varies with magnetic polarity, it is reasonable to expect that drifts are involved.
It is unlikely though that CR drifts are the only cause of this enhancement, because under particular conditions it is also observed in no-drift solutions [see Potgieter and Moraal, 1988;
Florinski and Jokipii, 2003]. It does however seem to be a physical effect rather than a numerical artefact, since this same feature had appeared in previous modulation studies on ACRs using different modelling approaches, e.g. Potgieter and Moraal[1988],Steenkamp[1995],le Roux et al.
[1996],Steenberg and Moraal[1996] andFlorinski and Jokipii[2003]. Indeed, there are a number of mechanisms capable of influencing the form of the TS spectrum in this fashion. These not only include current sheet and TS drifts [Kota and Jokipii, 1994], but also CR-modified shocks [le Roux and Fichtner, 1997] and a (helio)latitude-dependentκ⊥r [Langner and Potgieter, 2008]. Florinski and Jokipii[2003] attributed this feature to the spherical geometry of the heliosphere, proposing that it limits the amount of phase space available to particles so as to increase the efficiency of
10−4 10−3 10−2 10−1 10−2
10−1 100 101 102 103
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
θ = 90o γ = −1.5
10o, A < 0 10o, A > 0 10o, no drift 70o, A < 0 70o, A > 0 70o, no drift
10−4 10−3 10−2 10−1 0
0.5 1 1.5 2 2.5 3
Kinetic energy (GeV.nuc−1)
Differential Intensity X E1.5
θ = 90o 10o, A < 0 10o, A > 0 10o, no drift 70o, A < 0 70o, A > 0 70o, no drift
Figure 4.8: Similar to Figure 4.7, but for two different HCS tilt angles (α) as shown in the legend, and for both polarities. Recall thatα = 10◦ andα = 70◦ respectively represent low and moderately high solar activity conditions. Here, drift efficiencies are taken at the reference value of 55%, except for the no-drift solutions which correspond toκD,0= 0.
their acceleration in specific energy regions. However the root cause of this enhancement, if assumed not to be a combination of various processes, remains less obvious, the effects of drift on its prominence are evident.
Figure 4.7 demonstrates the deviation of the modelled TS spectra from the predicted power law for different drift efficiencies during each polarity cycle. These effects are accentuated in the right-hand side of the figure, where the scale is chosen such that protrusions above and below unity respectively signify the hardening or softening of the spectrum from the original power-law distribution. Note that in the equatorial plane, intensity enhancements emerge for A<0 and intensity depressions for A>0. Also note the opposite of this effect near the poles.
This is explained as a consequence of global CR drift patterns: For A<0, ACRs produced at the TS near the equatorial plane are unable to escape downstream due to inward drifts along the HCS in the heliosheath. The particles are thus confined and may be injected into the shock repeatedly until they become energetic enough and escape. These intensities are bolstered further, because particles drift along the TS from the polar regions to the equator. For A>0, on the other hand, these drift patterns are reversed: positive particles drift outward along the HCS and upward along the TS to the poles so that intensities are enhanced at high latitudes. See also the interpretation byKota and Jokipii[1994]. From Figure 4.7, the intensity enhancements and depressions at the poles appear much less pronounced than at the equator. This is likely because HCS drifts are absent from the poles, so that fewer particles are carried to and from the TS in the polar regions as compared to near the equator. This may also be, in part, due to the assumption of enhanced polar diffusion at high latitudes, since it is expected that larger scales of diffusion reduce acceleration efficiency (See Section 4.4).
The prominence of drift effects is also coupled to solar activity. Figure 4.7 is shown for solar
minimum conditions, during which the HCS has an inclination from the equator of only 10◦, is only marginally wavy, and drifts along this interface are well-defined. As the cycle progresses toward maximum activity conditions, the tilt angle increases and drift effects diminish near the equatorial plane due to an ill-defined HCS structure. Figure 4.8 demonstrates this, showing that forα = 70◦ the TS spectrum at ∼6 MeV. nuc−1 is only a factor of roughly 1.5 higher for the A <0 cycle than for A>0, in contrast to a factor of at least 10 forα = 10◦. Intriguingly, all of the solutions for solar maximum conditions, including the no-drift solutions, display the intensity enhancement discussed above. This reinforces the notion expressed earlier that CR drifts, subject to magnetic polarity, serve either to obscure or emphasise this enhancement, while it likely arises due to other mechanisms as well. Note that the no-drift solutions for the two tilt angles shown in Figure 4.8 do not coincide. This follows because the scaling factors in the perpendicular diffusion coefficients of Section 3.3.3 are scaled with tilt angle to increase toward maximum solar activity. For recent reviews on drift effects, seePotgieter[2013, 2014b].
While the identification of the mechanism responsible for the intensity enhancement still war- rants further investigation, the visibility of this enhancement is shown to be drift-dependent.
Thereby it is implied that its appearance is also subject to change with solar activity and the magnetic polarity cycle.