Being charged particles, CRs are expected to undergo drifts in response to the large-scale HMF structure as illustrated in Figure 3.8. Specifically, drift motions associated with gradients in the HMF magnitude, the curvature of the field, and with changes in the field direction across the HCS [e.g.Burger and Potgieter, 1989], are induced. It is known that drift effects are reduced in the presence of turbulence, although the appropriate characterisation of this turbulence and of the ensuant drift effects is an ongoing endeavour [Burger and Visser, 2010;Engelbrecht and Burger, 2015]. The significance of drifts in CR modulation was first remarked byJokipii et al.[1977], and indeed, they were later revealed to be necessary to explain dependences of CR observations on the HMF polarity [e.g.Potgieter and Moraal, 1985]. See also the review byPotgieter[2014b].
In this section, a general discussion on the drift concepts used in this study is provided. Note
Figure 3.8: Meridional projection of the drift trajectories of positively charged CRs in the inner helio- sphere during the A>0 polarity cycle, with arrows indicating drift directions. The bold line near the equator represents the wavy HCS. See Strauss[2013] for a discussion on possible drift patterns in the outer heliosphere. Figure obtained fromJokipii and Thomas[1981].
that HCS drifts are emulated in this study using the wavy current sheet model by Hattingh and Burger [1995], while its modification and implementation in a 2-D modulation model is discussed thoroughly byLangner[2004].
Drifts are represented in the TPE by the pitch-angle-averaged guiding centre drift velocity, which in a very general case is given by
h~vDi= pv 3Q
(ωτd)2
1 + (ωτd)2∇ × B~
B2, (3.29)
where the suppression of drifts through scattering is represented by(ωτd)2/(1 + (ωτd)2) [see alsoMinnie et al., 2007]. Here,ω is the particle’s gyro-frequency andτda time scale between scattering events. Eq. 3.29 is written in terms of the maximal Larmor radius (rL=P/Bc) as
h~vDi = ∇ ×v 3rL
(ωτd)2
1 + (ωτd)2eˆB (3.30)
≡ ∇ ×κD ˆeB,
witheˆB = B/B~ a unit vector directed alongB, and where the drift coefficient,~ κD, is prelimi- narily defined as
κD = v
3rL (ωτd)2
1 + (ωτd)2. (3.31)
The drift coefficient, and hence also the drift velocity, assume maximum values under the as- sumption of weak scattering, that is, assuming a particle undergoes several gyrations before being scattered. This implies thatωτd 1. On the other hand, the drift coefficient diminishes
10−5 10−4 10−3 10−2 10−1 100 101 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100
Rigidity (GV)
Drift Scale (AU)
10−6 10−5 10−4 10−3 10−2 10−1 100 101 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100 101
Kinetic Energy (GeV)
Drift Scale (AU)
rL λD electrons, rL
protons, rL ACR Oxygen, rL electrons, λD protons, λD ACR Oxygen, λD
Figure 3.9: CR drift scales (λD) as functions of kinetic energy (left panel) and rigidity (right panel) at Earth (1 AU,θ = 90◦). Solid lines represent the weak-scattering limit, where drift scales are equal to the Larmor radius,rL, while dashed lines represent the drift scales following from Eq. 3.33. In the left panel, lines in red, black and blue respectively representλDfor electrons, protons and ACR Oxygen.
whenωτd1. Thus, in these limits it follows that κD u
( v
3rL ωτd1 weak scattering
0 ωτd1 strong scattering (3.32)
An intermediate value is likely the more realistic though, and is attained through incorporat- ing a realistic suppression factor to represent the effects of turbulence. FollowingBurger et al.
[2000, 2008], it is approximated that(ωτd)∼P, so that κD =κD,0v
3rL (P/PD)2
1 + (P/PD)2, (3.33)
withPD = 1/√
10[see alsoLangner et al., 2003;Ngobeni and Potgieter, 2015], and where κD,0 ∈ [0,1]is introduced as a scaling parameter to indicate the drift efficiency. During solar maximum conditions, for instance, when more turbulent conditions are expected, Ferreira and Potgieter [2004] employedκD,0 ∈ [0,0.1]to reproduce observations. This quantity is scaled as needed in the current study, althoughκD,0 = 0.55 may be considered a default value [Langner et al., 2003; Ngobeni, 2015]. To aid in visualisation, κD is represented more intuitively as a length scale, λD, similar to how MFPs represent diffusion coefficients (Eq. 3.9). Note that by using this representation, it follows from Eq. 3.32 thatλD =rLunder the weak scattering limit.
The rigidity profile ofλDis illustrated in the right-hand side panel of Figure 3.9 for full drift ef- ficiency (κD,0= 1). The scattering-induced suppression of drifts significantly reduceλDat low
10 20 30 40 50 60 70 80 90 100 110 120 10−3
10−2 10−1 100
Radial distance (AU)
Drift Scale (AU)
Figure 3.10: Electron drift scale (λD) as function of radial distance at P = 1 GV, where the weak- scattering limit has already been attained. Note thatλD ∝ 1/B. The TS position is indicated with a dash-dotted line at 94 AU.
rigidities and become altogether negligible atP PD. The weak scattering limit is attained atP > PD, where the drift scales predicted by Eq. 3.33 converge with the Larmor radius,rL, and continue increasing proportional toP. In further chapters, drift effects on CR intensities are presented in terms of kinetic energy, in terms of which the drift scales vary appreciably depending on the CR species considered. This is shown in the left panel of Figure 3.9. The drift scales of the heavier species remain significant down to lower energies, whereas those of electrons become negligible quickly below ∼ 100 MeV. Indeed, drift effects for electrons are shown in Chapter 5 to be most pronounced in the approximate energy interval of 100 MeV to 1 GeV. See also the comprehensive study on electron modulation byNndanganeni[2015]. Note also that if the rigidity parameters in Eq. 3.33 are expressed in terms of energy according to Eq. 3.3, the dependence ofλD onE also changes relative to the relevant rest-mass energy; this is reflected in Figure 3.9, where theλD-curves of electrons and the heavier species steepen at
∼0.5MeV and 1 GeV respectively. The radial dependence ofλD, as a result of its dependence onrL, follows as1/B, which is evident when comparing Figures 3.10 and 2.7.
WithκD specified,h~vDiis also known. The drift directions of CRs can consequently be solved fromd~l× h~vDi = 0, which would yield streamlines similar to those shown in Figure 3.8. The figure demonstrates that while A > 0 positively charged particles drift downward from the poles to the equator and outward along the HCS. Since the divergence of a curl of a vector is zero, these streamlines are expected to be directed back to the poles at some point in the outer heliosphere, e.g. along the TS. Note that these patterns are shown for qA >0, which is the notation used in modulation models when solving the TPE. For positively charged particles, qA>0 =⇒ A>0, sinceq >0, whereas for negatively charged particlesqA>0 =⇒ A<0, becauseq <0. The drift directions for electrons are hence reversed from those shown in Figure 3.8. Of course, when the magnetic polarity reverses, and magnetic field lines are directed in the opposite direction, these drift directions are also reversed.
Drifts can be moderated both directly and indirectly in the TPE and the modulation model that
solves it. Firstly, the drift coefficient can be scaled explicitly as done in Eq. 3.33 through e.g.
κD,0 or the rigidity-dependent suppression factor. Implicitly though, any process that acts to decrease intensity gradients and hence the gradient of the CR distribution function,∇f, in Eq.
3.1, causes the producth~vDi · ∇f to be reduced and thereby inhibits drift effects. Note finally that the drift velocity as described by Eq. 3.30 can be rewritten in terms of the asymmetrical drift tensor,KD. This follows because the curl of a vector may be rewritten as the divergence of a second rank tensor, leading to
h~vDi=∇ ×κD ˆeB =−∇ ·KD, (3.34) with
KD ≡
0 0 0
0 0 κD
0 −κD 0
. (3.35)
This identity allows the TPE to be rewritten in a more compact form and facilitates the trans- formation to spherical coordinates. This is discussed in the next section.