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.
0 10 20 30 40 50 60 70 80 90 1.4
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
Compression Ratio
0 10 20 30 40 50 60 70 80 90
−4
−3
−2
−1
Polar angle (degrees)
Spectral Index
s(θ)
V1: Particle data V1: HMF V2: SW (TS−3) V2: HMF (max) V2: SW (TS−2) V2: SW (TS−3)
γ (s(θ))
Figure 4.9:A latitude-dependent compression ratio (top panel) with Voyager measurements during TS encounters (Table 4.1), and the corresponding spectral indices of the power laws expected from DSA theory for a 1-D planar shock (bottom panel).
θc= 45◦andθb= 9◦. The corresponding latitude-dependent expression for the spectral indices of the power laws expected for the compression ratios determined by Eq. 4.6, is given by
γ(θ) = s(θ) + 2
2−2s(θ). (4.7)
Figure 4.9 showss(θ)andγ(θ)as functions of θ, along with the Voyager measurements listed in Table 4.1. Note that the minimum and maximum values ofs(θ)respectively correspond to spectral indices ofγmin=−3.5andγmax=−1.5.
Of course, varying the compression ratio in this manner holds implications for the SW flow speed in the heliosheath. Figure 4.10 shows the SW speed in a meridional cross section of the heliosphere for solar minimum conditions, and compares the effects of SW compression varying with latitude with that of a fixed compression ratio at the TS. While the flow in the heliosheath remains unchanged near the equator, the fast SW streams near the poles decrease by a smaller ratio (i.e. 1.5 as opposed to 2.5) across the TS. This results in a SW speed in the heliosheath of∼500km.s−1. It is not unlikely, depending on the local plasma properties, that this flow is still supersonic, or indeed that the SW speed beyond the TS at high latitudes may be supersonic in general. This follows from Voyager 2 measurements confirming supersonic SW flow in the heliosheath [Li et al., 2008;Chalov and Fahr, 2011]. Contrarily, Voyager 1 reports much slower SW flow along its trajectory [Richardson and Burlaga, 2013]. Of course, if the SW speed in the heliosheath remains fast, it would have run-on effects on all parameters dependent on it as well, e.g. the HMF magnitude.
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o
0 100 200 300 400 500 600 700 800
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o
0 100 200 300 400 500 600 700 800
Figure 4.10: Contour plots illustrating the SW speedVsw(according to the scale in the colour bar with units km. s−1) in the meridional cross section of the heliosphere for solar minimum conditions (α= 10◦).
A fixed compression ratio was assumed in the left panel (i.e. s(∀θ) = 2.5) while it is varied on the right with latitude according to Eq. 4.6. Recall that incompressible flow is assumed for the heliosheath. The TS is shown as a dashed white halfcircle at 90 AU, with the Sun at the origin.
Aside from consequences for heliosheath plasmas however, CR transport is also affected by a varying compression ratio. Plotting the modelled TS spectra for ACR Oxygen at selected polar angles, as illustrated in Figure 4.11, shows harder power laws near the equatorial plane, while the spectra become softer toward the poles. These solutions are shown for both magnetic polar- ities. As expected, the spectrum atθ= 90◦is similar to the spectrum displayed in Figure 4.4 for a fixed, global compression ratio ofs = 2.5. This is however not the case for higher latitudes, e.g.θ= 10◦, where Figure 4.11 shows a pronounced intensity enhancement developing at ener- gies ranging from roughly 0.1 to 10 MeV. nuc−1, which is not visible for the equivalents= 1.5 spectrum in Figure 4.4. This enhancement is therefore a consequence of a latitude-dependent compression ratio, and arises due to the poleward migration of ACRs produced in the equato- rial regions. This migration happens mainly by polar diffusion, especially considering the en- hancement included in Eq. 3.27 for high latitudes, but also by drifts along the TS while A>0.
Indeed, since the intensities at θ = 10◦ in this enhancement region are higher by a factor of
∼10 for A>0 than for A<0, drifts during the former polarity cycle facilitate the poleward migration while drifts during the latter act to inhibit it. These drift patterns are illustrated for both polarities, with s = s(θ), using contour plots in Figure 4.12, and are consistent with the patterns described in Section 4.5.
Figure 4.12 also shows that implementing a latitude-dependent compression ratio generally lowers the global ACR Oxygen intensities from what is observed whens(∀θ) = 2.5(compare with e.g. Figure 4.3). Since fewer ACRs are produced near the polar regions, as is also visible from the flattened contours for the A<0 cycle, large latitudinal gradients are observed. What
10−5 10−4 10−3 10−2 10−1 10−3
10−2 10−1 100 101 102 103 104 105
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o A > 0
10o 35o 55o 90o
γ = −3.5
γ = −1.5
10−5 10−4 10−3 10−2 10−1
10−3 10−2 10−1 100 101 102 103 104 105
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o A < 0
10o 35o 55o 90o
γ = −3.5
γ = −1.5
Figure 4.11:Modelled ACR Oxygen TS spectra at varying polar angles (as shown in the figure legend) with a latitudinal-dependent compression ratio for both the A>0 (top panel) and A<0 (bottom panel) magnetic polarities. The expected power laws (dotted lines) and their spectral indices are shown for the maximum and minimum compression ratios atθ= 90◦andθ= 10◦respectively
emerges is a region of preferred acceleration near the equatorial nose region, and from Figure 4.12, at the interface where the TS meets the HCS during the A > 0 cycle. Similar results are reported by Strauss[2010], although with the inclusion of a latitude-dependent injection efficiency as well (see Section 4.6.2). Consensus had however not yet been reached on where the predominant region for acceleration at the TS resides. Early studies [Jokipii, 1986] suggest, in stark contrast to the above, that ACRs are accelerated chiefly at the poles, while the model byMcComas and Schwadron[2006] favours the flanks of the heliosphere.
While this latitude-dependent compression ratio is well-substantiated and was shown to have yielded satisfactory results when modelling ACR intensities [Strauss, 2010], it also makes the model increasingly layered if implemented. To study the effects of individual processes and parameters, a fixed compression ratio is assumed in the following chapters, especially since these chapters are more concerned with the study of electrons at a single latitude along the Voyager 1 trajectory, where estimates for the compression ratio are available.
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A > 0
6 MeV/nuc
−3.5
−3
−2.5
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A < 0
6 MeV/nuc
−3.5
−3
−2.5
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
Figure 4.12:Contour plots illustrating 6 MeV. nuc−1ACR Oxygen intensities in the meridional plane of the heliosphere, for A>0 (left) and A<0 (right) polarities and a latitude-dependent compression ratio.
The colour bar shows intensities as percentages of the maximum on a logarithmic scale. 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.
4.6.2 The pick-up ion source strength
Since the quantitative details of the PUI seed population, from which ACRs arise, are not gen- erally known, the intensity of the PUI source function is chosen arbitrarily and the acceler- ated solutions are normalised to observations (or on some generally approved reference value).
Studies have been conducted from which the spatial distribution of PUIs and their rate of in- jection at the TS are inferred. These include those by e.g. Fahr et al.[2008] andScherer and Fahr [2009], which confirm a longitudinal dependence of the PUI injection efficiency and predict a maximum injection rate at the cross-wind (or flank) regions of the heliosphere. Earlier work by Scherer et al.[2006] had also found a latitudinal dependence for PUI injection. This was conse- quently modelled byNgobeni[2006] by specifying the source strength, denoted asI(θ), in the expression for the PUI source function. That is,
Q(θ) =Q∗I(θ)δ
r−rT S r0
δ
P −Pmin P0
, (4.8)
which is just a modified form of Eq. 3.51. The source strength acts as a scaling constant that varies with polar angle, similar to Eq. 4.6, as follows:
I(θ) =Imin+ Imax−Imin
1 + exp [(θc−θ)/θb], (4.9) withθcandθbas previously defined,Imax=I(90◦) = 1.0, andImin =I(0◦) = 0.1.
10−6 10−5 10−4 10−3 10−2 10−1 10−4
10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o θ = 90o
0.005 0.010 0.020 0.005 0.010 0.020 Pmin = 0.03 GV P
min = 0.3 GV
Figure 4.13: Modelled TS spectra for ACR Oxygen in the equatorial plane for varying values ofκ⊥r,0 and source functions representing the PUI seed population injected atPmin = 0.03GV (in blue) and Pmin = 0.3GV (in red) respectively. Here, drifts are disabled (κD,0= 0), and the expected power laws (withγ=−1.5fors= 2.5) for each set of solutions indicated with dotted lines.
Physically, this means that all the PUIs injected at the TS in the equatorial plane are accelerated to ACRs, while ten times less are injected at the poles.Strauss[2010] reports that this generally results in solutions that are a factor of ∼ I(θ) lower than those shown in Figure 4.11, with slight spectral changes occurring at high energies due to a modified intensity gradient,∇f. For similar reasons to that stated for refraining from the use of the latitude-dependent compression ratio, the injection efficiency is considered fixed for all polar angles in subsequent chapters.