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4 Solar Modulation Theory

The paradigm of CR transport in the heliosphere has developed soundly over the last ∼ 50 years. The basic processes are considered to be known. However, it is still quite demanding to relate the modulation of galactic CRs and of the ACRs to their true causes and to connect these causes over a period of a solar cycle or more, from both a global and microphysics level. The latter tends to be of a fundamental nature, attempting to understand the physics from first principles (ab initio) whereas global descriptions generally tend to be phenomenological, mostly driven by observations and/or the application of new numerical methods and models. In this context, global numerical models with relevant transport parameters are essential to make progress. Obviously, major attempts are made to have these models based on good assumptions, which then have to agree with all major heliospheric observations.

Understanding the basics of solar modulation of CRs followed only in the 1950s when Parker (1965Jump To The Next Citation Point) formulated a constructive transport theory. At that stage NMs already played a major role in observing solar activity related phenomena in CRs. Although Parker’s equation contained an anti-symmetric term in the embedded tensor accounting for regular gyrating particle motion, it was only until the development of numerical models in the mid-1970s (Fisk, 1971Jump To The Next Citation Point, 1976Jump To The Next Citation Point, 1979Jump To The Next Citation Point) that progress in appreciation the full meaning of transport theory advanced significantly.

4.1 Basic transport equation and theory

A basic transport equation (TPE) was derived by Parker (1965Jump To The Next Citation Point). Gleeson and Axford (1967) came to the same equation more rigorously. They also derived an approximate solution to this TPE, the so called force-field solution, which had been widely used and was surpassed only when numerical models became available (Gleeson and Axford, 1968). For a formal overview of these theoretical aspects and developments, see Schlickeiser (2002Jump To The Next Citation Point). See also Quenby (1984Jump To The Next Citation Point), Fisk (1999), and Moraal (2011) for overviews of the TPEs relevant to CR modulation. The basic TPE follows from the equations of motion of charged particles in fluctuating magnetic fields (on both large and small scales) and averages over the pitch and phase angles of propagation particles. It is based on the reasonable assumption that CRs are approximately isotropic. This equation is remarkably general and is widely used to model CR transport in the heliosphere. The heliospheric TPE according to Parker (1965), but in a rewritten form, is

∂f 1 ∂f --- = − (◟V◝◜◞ + ⟨vd⟩) ⋅ ∇f + ∇ ⋅ (Ks ⋅ ∇f )+--(∇ ⋅ V )------, (5 ) ◟∂◝t◜◞ b ◟◝◜◞ ◟-----◝◜----◞ 3◟------◝◜-∂ ln-P◞ a c d e
where f(r,P, t) is the CR distribution function, P is rigidity, t is time, r is the position in 3D, with the usual three coordinates r, 𝜃, and ϕ specified in a heliocentric spherical coordinate system where the equatorial plane is at a polar angle of 𝜃 = 90°. A steady-state solution has ∂f∕ ∂t = 0 (part a), which means that all short-term modulation effects (such as periods shorter than one solar rotation) are neglected, which is a reasonable assumption for solar minimum conditions. Terms on the right hand side respectively represent convection (part b), with V the solar wind velocity; averaged particle drift velocity ⟨vd⟩ caused by gradients and curvatures in the global HMF (part c); diffusion (part d), with Ks the symmetrical diffusion tensor and then the term describing adiabatic energy changes (part e). It is one of the four major modulation processes and is crucially important for galactic CR modulation in the inner heliosphere. If (∇ ⋅ V ) > 0, adiabatic energy losses are described, which become quite large in the inner heliosphere (see the comprehensive review by Fisk, 1979). If (∇ ⋅ V ) < 0, energy gains are described, which may be the case for ACRs in the heliosheath (illustrated, e.g., by Langner et al., 2006bJump To The Next Citation Point; Strauss et al., 2010bJump To The Next Citation Point). If (∇ ⋅ V ) = 0, no adiabatic energy changes occur for CRs, perhaps the case beyond the TS. This is probably an over simplification but it was shown that the effect is insignificant for galactic CRs but crucially important for ACRs (e.g., Langner et al., 2006aJump To The Next Citation Point). For recent elaborate illustrations of these effects, see Strauss et al. (2010aJump To The Next Citation Point,bJump To The Next Citation Point).

In cases when anisotropies are large, other types of transport equations must be used (see, e.g., Schlickeiser, 2002Jump To The Next Citation Point). Close to the Sun and Jupiter, and even close to the TS, where large observed anisotropies in particle flux sometimes occur (e.g., Kóta, 2012Jump To The Next Citation Point), Equation (5View Equation) thus needs to be modified or even replaced to describe particle propagation based on the Fokker–Planck equation. The ‘standard’ TPE as given by Equation (5View Equation) also needs modification by inserting additional terms relevant to the conditions beyond the TS (e.g., Strauss et al., 2010aJump To The Next Citation Point,bJump To The Next Citation Point).

For clarity on the role of diffusion, particle drifts, convection, and adiabatic energy loss, the TPE can be written in terms of a heliocentric spherical coordinate system as follows:

[ ( ) ] -1 ∂-- r2Krr + ---1-- ∂--(K 𝜃r sin𝜃 ) +--1-- ∂K-ϕr− V ∂f- (6 ) r2 ∂r r sin 𝜃 ∂𝜃 r sin 𝜃 ∂ϕ ∂r [ 1 ∂ 1 ∂ 1 ∂K ϕ𝜃] ∂f + -2 ---(rKr𝜃) + -2--------(K 𝜃𝜃 sin𝜃) +-2----------- --- [r ∂r r sin 𝜃∂𝜃 r sin 𝜃 ∂ ϕ ∂]𝜃 ---1----∂- ---1---∂K-𝜃ϕ- ---1----∂K-ϕϕ- ∂f- + r2 sin 𝜃∂r (rKr ϕ) + r2sin𝜃 ∂ 𝜃 + r2 sin2 𝜃 ∂ ϕ − Ω ∂ϕ 2 2 2 + K ∂-f-+ K-𝜃𝜃 ∂-f-+ --K-ϕϕ--∂-f- rr∂r2 r2 ∂𝜃2 r2 sin2 𝜃∂ ϕ2 2K ∂2f 1 ∂ ( ) ∂f + ----rϕ-------+ ------- r2V -----= 0, r sin 𝜃∂r ∂ϕ 3r2 ∂r ∂ lnp
where it is assumed that the solar wind is axis-symmetrical and directed radially outward, i.e., V = V er. This version of the TPE is rearranged in order to emphasize the various terms that contribute to diffusion, drift, convection, and adiabatic energy losses, so that Equation (6View Equation) becomes
◜---------------------------diffu◞s◟ion---------------------------◝ [ ( ) ] [ ] -1 ∂-- r2Krr + --1---∂K--ϕr ∂f- + ---1--- ∂--(K 𝜃𝜃 sin 𝜃) ∂f (7 ) r2 ∂r r sin 𝜃 ∂ϕ ∂r r2 sin 𝜃 ∂𝜃 ∂ 𝜃 -----------------diffusion------------------ ◜[ ◞◟ ] ◝ + ---1----∂-(rKrϕ) + ----1---∂K--ϕϕ-− Ω ∂f- r2 sin 𝜃∂r r2 sin2 𝜃 ∂ϕ ∂ϕ diffusion ◜----2----------2-----◞◟ -----2-------------2-◝ + K ∂-f-+ K-𝜃𝜃∂--f + --K-ϕϕ--∂-f-+ 2Kr-ϕ--∂-f-- rr ∂r2 r2 ∂ 𝜃2 r2sin2 𝜃∂ϕ2 rsin𝜃 ∂r∂ ϕ drift ◜-------------[--------]◞◟-----[-------------]---◝ + [− ⟨v ⟩ ] ∂f + − 1⟨v ⟩ ∂f- + − --1---⟨v ⟩ ∂f- d r ∂r r d 𝜃 ∂ 𝜃 r sin 𝜃 d ϕ ∂ϕ convection ◜◞◟-◝ ∂f- − V ∂r adiabatic energy losses ◜-------◞◟-------◝ + -1--∂-(r2V ) -∂f---= 0. 3r2 ∂r ∂ ln p

Here Krr, Kr 𝜃, Kr ϕ, K 𝜃r, K𝜃𝜃, K 𝜃ϕ, K ϕr, K ϕ𝜃, and K ϕϕ, are the nine elements of the 3D diffusion tensor, based on a Parkerian type HMF. Note that Krr, Krϕ, K 𝜃𝜃, K ϕr, K ϕϕ describe the diffusion processes and that K r𝜃, K 𝜃r, K 𝜃ϕ, K ϕ𝜃, describe particle drifts, in most cases consider to be gradient, curvature, and current sheet drifts. For this equation it is assumed that the solar wind velocity has only a radial component, which is not the case in the outer parts of the heliosheath, close to the HP.

A source function may also be added to this equation, e.g., if one wants to study the modulation of Jovian electrons (e.g., Ferreira et al., 2001Jump To The Next Citation Point).

The components of the drift velocity are given in the next Section 4.2. It is important to note that the drift velocity in Equation (5View Equation) is multiplied with the gradient in f so when the CR intensity gradients are reduced by changing the contribution from diffusion, drift effects on CR modulation become implicitly reduced. This reduction will unfold differently from changing the drift coefficient explicitly.

It is more useful to calculate the differential intensity of CRs, e.g., as a function of kinetic energy to obtain spectra, which can be compared to observations. In this context, a few useful definitions and relations are as follows:

Particle rigidity is defined as

pc mvc P = ---= ----, (8 ) q Ze
where p is the magnitude of the particle’s relativistic momentum, q = Ze its charge, v its speed, m its relativistic mass, and c the speed of light in outer space.

For a relativistic particle, the total energy in terms of momentum is given by

E2 = (E + E )2 = p2c2 + m2 c4, (9 ) t 0 0
with E the kinetic energy and E0 the rest-mass energy of the particle (e.g., E0 = 938 MeV for protons and E0 = 5.11 × 10− 1 MeV for electrons) and m0 the particle’s rest mass.

The kinetic energy per nucleon in terms of particle rigidity is then

∘ ---------------- ( )2 E = P 2 Ze- + E20 − E0, (10 ) A
so that rigidity can be written in terms of kinetic energy per nucleon as
( ) A ∘ ------------- P = Ze- E (E + 2E0 ), (11 )
with Z the atomic number and A the mass number of the CR particle. The ratio of a particle’s speed to the speed of light, β, in terms of rigidity is given by
β = v-= ∘-------P-------- (12 ) c P2 + (A- )2E2 Ze 0
and in terms of kinetic energy it is
∘ ------------ β = v-= --E-(E-+--2E0). (13 ) c E + E0
The relation between rigidity, kinetic energy, and β is therefore
∘ ------------- ( ) P = A- E (E + 2E0 ) = A- β (E + E0). (14 ) Z Z

Particle density within a region 3 d r, for particles with momenta between p and p + dp, is related to the full CR distribution function (which includes a pitch angle distribution) by

∫ ∫ ⌊ ∫ ⌋ 3 2⌈ ⌉ n = F (r,p,t)d p = p F (r,p,t)dΩ dp, (15 ) p Ω
where 3 2 d p = p dpdΩ. The differential particle density, Up, is related to n by
∫ n = U (r,p,t)dp, (16 ) p
which gives
∫ U (r,p,t) = p2F (r,p,t)dΩ. (17 ) p Ω
The omni-directional (i.e., pitch angle) average of F(r,p, t) is calculated as
∫ F (r,p,t)dΩ ∫ f(r,p,t) = Ω---∫---------= -1- F(r,p, t)dΩ, (18 ) d Ω 4π Ω Ω
which leads to
2 Up (r,p,t) = 4πp f(r,p,t). (19 )

The differential intensity, in units of particles/unit area/unit time/unit solid angle/unit momentum, is

jp = vUp(∫r,p,t)-= vUp(r,p,t)-= vp2f (r,p,t), (20 ) d Ω 4 π Ω
so that
j(r,p,t) = v-U -dp-= -1-U = p2f (r, p,t), (21 ) 4π pdEt 4π p
where j(r,p,t) is the differential intensity in units of particles/area/time/solid angle/energy. The relation between j and f is simply 2 (P ).

4.2 Basic diffusion coefficients

The diffusion coefficients of special interest in a 3D heliocentric spherical coordinate system are

K = K cos2 ψ + K sin2ψ, rr ∥ ⊥r K ⊥𝜃 = K 𝜃𝜃, K = K cos2ψ + K sin2ψ, ϕϕ ( ⊥r ) ∥ Kϕr = K ⊥r − K ∥ cos ψ sin ψ = Kr ϕ, (22 )
where Krr is the effective radial diffusion coefficient, a combination of the parallel diffusion coefficient K || and the radial perpendicular diffusion coefficient K ⊥r, with ψ the spiral angle of the average HMF; K 𝜃𝜃 = K ⊥ 𝜃 is the effective perpendicular diffusion coefficient in the polar direction. Here, K ϕϕ describes the effective diffusion in the azimuthal direction and K ϕr is the diffusion coefficient in the ϕr-plane, etc. Both are determined by the choices for K || and K ⊥. Beyond ∼ 20 AU in the equatorial plane ∘ ψ → 90, so that Krr is dominated by K ⊥r whereas K ϕϕ is dominated by K||, but only if the HMF is Parkerian in its geometry. These differences are important for the modulation of galactic CRs in the inner heliosphere (e.g., Ferreira et al., 2001Jump To The Next Citation Point). The important role of perpendicular diffusion (radial and polar) in the inner heliosphere has become increasingly better understood over the past decade since it was realized that it should be anisotropic (Jokipii, 1973), with reasonable consensus that K ⊥ 𝜃 > K ⊥r away from the equatorial regions. The expressions for these diffusion coefficients become significantly more complicated when advanced geometries are used for the HMF (e.g., Burger et al., 2008Jump To The Next Citation Point; Effenberger et al., 2012).

A typical empirical expression as used in numerical models for the diffusion coefficient parallel to the average background HMF is given by

⌊ ( P )c (P )c ⌋(b−ca) Bn ( P )a P0 + Pk0 K∥ = K ∥,0β ---- --- ⌈ ------(---)c---⌉ , (23 ) Bm P0 1 + PPk0
where (K ||)0 is a constant in units of 1022 cm2 s–1, with the rest of the equation written to be dimensionless with P0 = 1 GV and Bm the modified HMF magnitude with Bn = 1 nT (so that the units remain cm2 s–1). Here, a is a power index that can change with time (e.g., from 2006 to 2009); b = 1.95 and together with a determine the slope of the rigidity dependence respectively above and below a rigidity with the value Pk, whereas c = 3.0 determines the smoothness of the transition. This means that the rigidity dependence of K || is basically a combination of two power laws. The value Pk determines the rigidity where the break in the power law occurs and the value of a specifically determines the slope of the power law at rigidities below Pk (e.g., Potgieter et al., 2013Jump To The Next Citation Point).

Perpendicular diffusion in the radial direction is usually assumed to be given by

K ⊥r = 0.02K ∥. (24 )

This is quite straightforward but still a very reasonable and widely used assumption in numerical models. It is to be expected that this ratio could not just be a constant but should at least be energy-dependent. Advanced and complicated fundamental approaches had been followed, e.g., by Burger et al. (2000Jump To The Next Citation Point) with subsequent CR modulation results that are not qualitatively different from a global point of view. Advances in diffusion theory and subsequent predictions for the heliospheric diffusion coefficients (e.g., Teufel and Schlickeiser, 2002Jump To The Next Citation Point) make it possible to narrow down the parameter space used in typical modulation models, e.g., the rigidity dependence at Earth. This is a work in progress.

The role of polar perpendicular diffusion, K ⊥ 𝜃, in the inner heliosphere has become increasingly better understood over the past decade since it was realized that perpendicular diffusion should be anisotropic, with reasonable consensus that K ⊥𝜃 > K ⊥r away from the equatorial regions (Potgieter, 2000Jump To The Next Citation Point). Numerical modeling shows explicitly that in order to explain the small latitudinal gradients observed for protons by Ulysses during solar minimum modulation in 1994, an enhancement of latitudinal transport with respect to radial transport is required (e.g., Heber and Potgieter, 2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point, and reference therein). The perpendicular diffusion coefficient in the polar direction is thus assumed to be given by

K = 0.02K f , (25 ) ⊥𝜃 ∥ ⊥𝜃
with
f = A+ ∓ +A − tanh [8 (𝜃 − 90∘ ± 𝜃 )]. (26 ) ⊥𝜃 A F

Here A ± = (d ± 1)∕2, 𝜃F = 35∘, 𝜃A = 𝜃 for 𝜃 ≤ 90∘ but 𝜃A = 180 ∘ − 𝜃 with 𝜃 ≥ 90∘, and d = 3.0. This means that K𝜃𝜃 = K ⊥𝜃 is enhanced towards the poles by a factor d with respect to the value of K|| in the equatorial regions of the heliosphere. This enhancement is an implicit way of reducing drift effects by changing the CR intensity gradients significantly. For motivations, applications and discussions, see Potgieter (2000Jump To The Next Citation Point), Ferreira et al. (2003aJump To The Next Citation Point,bJump To The Next Citation Point), Moeketsi et al. (2005Jump To The Next Citation Point), and Ngobeni and Potgieter (2008, 2011Jump To The Next Citation Point). The procedure usually followed to solve Equation (6View Equation) is described by, e.g., Ferreira et al. (2001Jump To The Next Citation Point) and Nkosi et al. (2008Jump To The Next Citation Point).

4.3 The drift coefficient

The pitch angle averaged guiding center drift velocity for a near isotropic CR distribution is given by ⟨vd⟩ = ∇ × (KdeB ), with eB = B ∕B where B is the magnitude of the background HMF usually assumed to have a basic Parkerian geometry in the equatorial plane. This geometry gives very large drifts over the polar regions of the heliosphere so that it is standard practice to modify it in the polar regions, e.g., Smith and Bieber (1991) and Potgieter (1996, 2000).

Under the assumption of weak scattering, the drift coefficient is straightforwardly given as

βP (Kd )ws = (Kd )0----, (27 ) 3Bm
where (Kd)0 is dimensionless; if (Kd )0 = 1.0, it describes what Potgieter et al. (1989Jump To The Next Citation Point) called 100% drifts (i.e., full ‘weak scattering’ gradient and curvature drifts). Drift velocity components in terms of K r𝜃, K 𝜃r, K ϕ𝜃, K 𝜃ϕ are
A ∂ ⟨vd⟩r = − ---------(sin 𝜃K 𝜃r), r si[n 𝜃∂ 𝜃 ] ⟨v ⟩ = − A- -1---∂-(K ) + ∂--(rK ) , d 𝜃 r sin 𝜃 ∂ϕ ϕ𝜃 ∂r r𝜃 A ∂ ⟨vd⟩ϕ = − -----(K 𝜃ϕ ), (28 ) r ∂ 𝜃
with A = ±1; when this value is positive (negative) an A > 0 (A < 0) polarity cycle is described. The polarity cycle around 2009 is indicated by A < 0, as was also the case for the years around 1965 and 1987. During such a cycle, positively charged CRs are drifting into the inner heliosphere mostly through the equatorial regions, thus having a high probability of encountering the wavy HCS.

Idealistic global drift patterns of galactic CRs in the heliosphere are illustrated in Figure 7View Image for positively charged particles in an A > 0 and A < 0 magnetic polarity cycle respectively, together with a wavy HCS as expected during solar minimum conditions. Different elements of the diffusion tensor are also shown in the left panel with respect to the HMF spiral for illustrative purposes.

View Image

Figure 7: The parallel and two perpendicular diffusion orientations, indicated by the corresponding elements of the diffusion tensor, are shown with respect to the HMF spiral direction (left) for illustrative purposes. The arrows with V indicate the radially expanding solar wind (convection). Idealistic global drift patterns of positively charged particles in an A > 0 and A < 0 magnetic polarity cycle are schematically shown in the right panel, together with a wavy HCS as expected during solar minimum conditions. Image reproduced by permission from Heber and Potgieter (2006Jump To The Next Citation Point), copyright by Springer.

A formal and fundamental description of global curvature, gradient and current sheet drifts in the heliosphere is still unsettled. The spatial and rigidity dependence of Kd is entirely based on the assumption of weak-scattering. A deviation from this weak scattering form is given by

( ) -P- 2 K = (K ) ---Pd0----. (29 ) d dws (-P-)2 1 + Pd0
This means that below Pd0 (in GV) particle drifts are progressively reduced with respect to the weak scattering case. This is required to explain the small latitudinal gradients at low rigidities observed by Ulysses (Heber and Potgieter, 2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point; De Simone et al., 2011Jump To The Next Citation Point). This reduction is in line with what Potgieter et al. (1989), Webber et al. (1990), Ferreira and Potgieter (2004Jump To The Next Citation Point), and Ndiitwani et al. (2005Jump To The Next Citation Point) found when describing modulation in terms of drifts with a HCS tilt angle dependence in numerical models. Theoretical arguments and numerical simulations have been presented requiring the reduction of particle drifts, in particular with increasing solar activity. For a summary of the essence of the problem, see Tautz and Shalchi (2012). This aspect must fit into the picture where Ulysses observations of CR latitudinal gradients especially at lower energies, require particle drifts to be reduced. It thus remains a theoretical challenge to explain why reduced drifts in the heliosphere is needed to explain some of these major CR observations, and what type of particle drifts apart from gradient and curvature drifts may occur in the heliosphere.

4.4 Gradient, curvature, and current sheet drifts

The realization that particle drifts could not be neglected in the solar modulation of CRs was elevated by the development of numerical models, which reached sophisticated levels already in the late 1980s and early 1990s including a full tensor. The importance of particle drifts in the heliosphere was at first discussed only theoretically but it was soon discovered compellingly in various existing CR observations. It should be noted that it was rather fortuitous that the sharp peak in the 1965 intensity-time profiles of CRs was followed by a really flat intensity-time profile in the 1970s, not to repeat again so evidently. The mini-modulation-cycle around 1974 had little to do with drifts (e.g., Wibberenz et al., 2001). The recent A < 0 polarity cycle also did not produce such a sharp peak as during previous A < 0 cycles as shown in Figure 6View Image. See also the review by Potgieter (2013Jump To The Next Citation Point).

Convincing theoretical arguments for the importance of particle drifts were presented by Jokipii et al. (1977) and later followed by persuasive numerical modeling (e.g., Jokipii and Kopriva, 1979Jump To The Next Citation Point; Jokipii and Thomas, 1981Jump To The Next Citation Point; Kóta and Jokipii, 1983Jump To The Next Citation Point; Potgieter and Moraal, 1985Jump To The Next Citation Point), which illustrated that gradient and curvature drifts could cause charge-sign dependent modulation and a 22-year cycle. The main reason for this to occur is that the solar magnetic field reverses polarity every ∼ 11 years so that galactic CRs of opposite charge will reach Earth from different heliospheric directions. This also makes the wavy HCS of the HMF a very important modulation feature with its tilt angle (Hoeksema, 1992) being a very useful modulation parameter. When protons drift inwards mainly through the equatorial regions of the heliosphere (A < 0 polarity cycles) they encounter the dynamic HCS and get progressively reduced by its increasing waviness as solar activity surges. This produces the sharp peaks in the galactic CR intensity-time profiles whereas during the A > 0 cycles the profiles are generally flatter (for recent updates see, e.g., Potgieter, 2011Jump To The Next Citation Point; Krymsky et al., 2012Jump To The Next Citation Point; Potgieter, 2013Jump To The Next Citation Point). The effect reverses for negatively charged galactic CRs causing charge-sign dependent effects as reviewed by, e.g., Heber and Potgieter (2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point) and Kóta (2012Jump To The Next Citation Point).

Figure 8View Image is an illustration of the computed intensity distribution caused by drifts for the two HMF polarity cycles, in this case for 1 MeV/nuc anomalous oxygen in the meridional plane of the heliosphere. The position of the rTS = 90 AU is indicated by the white dashed line. The contours are significantly different in the inner heliosphere, with the intensity reducing rapidly towards Earth. The distribution is quite different beyond the TS, where the region of preferred acceleration of these ACRs is assumed to be away from the TS, closer to the HP near the equatorial plane, positioned at 140 AU, for illustrative purposes (Strauss et al., 2010aJump To The Next Citation Point,b).

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Figure 8: An illustration of the computed intensity distribution caused by drifts for the two HMF polarity cycles, in this case for 1 MeV/nuc ACR oxygen in the meridional plane of the heliosphere. The position of the TS at 90 AU is indicated by the white dashed line. Note how the coloured contours differ in the inner heliosphere for the two cycles and how the intensity decreases towards Earth, and how the distribution is quite different beyond the TS. In this case, the region of preferred acceleration for these ACRs is assumed near the equatorial plane and close to the HP at 140 AU. Image reproduced by permission from Strauss et al. (2011b), copyright by COSPAR.

Another indication of the role of gradient and curvature drifts came in the form of a 22-year variation in the direction of the daily anisotropy vector in the galactic CR intensity as measured by NMs from one polarity cycle to another (Levy, 1976; Potgieter and Moraal, 1985Jump To The Next Citation Point). Potgieter et al. (1980) also discovered a 22-year cycle in the differential response function of NMs used for geomagnetic latitude surveys at sea-level in 1965 and 1976 (see also Moraal et al., 1989).

Some of the key outcomes of gradient, curvature and current sheet drifts as applied to the solar modulation of CRs, are:

  1. Particles of opposite charge will experience solar modulation differently because they sample different regions of the heliosphere during the same polarity epoch before arriving at Earth or at another observation point. Particle drift effects inside the heliosheath, on the other hand, may be different from upstream (towards the Sun) of the TS.
  2. A well-established 22-year cycle occurs in the solar modulation of galactic CRs, which is not evident in other standard proxies for solar activity (see Figure 6View Image). This is also evident in the directional changes of the diurnal anisotropy vector with every HMF polarity reversal (e.g., Potgieter and Moraal, 1985Jump To The Next Citation Point; Nkosi et al., 2008; Ngobeni and Potgieter, 2010Jump To The Next Citation Point) and from the changes in differential response functions of NMs obtained during geomagnetic latitude surveys.
  3. The wavy HCS plays a significant role in establishing the features of this 22-year cycle in the solar modulation of CRs.
  4. Cosmic ray latitudinal and radial intensity gradients in the heliosphere are significantly different during the two HMF polarity cycles and repeated ideal modulation conditions will display a 22-year cycle (e.g., Heber and Potgieter, 2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point; Potgieter et al., 2001Jump To The Next Citation Point; De Simone et al., 2011Jump To The Next Citation Point). An illustration is shown in Figure 9View Image of how the computed radial gradients change with kinetic energy for the two polarity cycles at different positions in the heliosphere. In Figure 10View Image the difference caused by drifts in the computed latitudinal proton gradients between the two polarity cycles is shown as a function of rigidity in comparison with the observed gradient from PAMELA and Ulysses for the period 2007 (De Simone et al., 2011Jump To The Next Citation Point).
  5. Cosmic ray proton spectra are softer during A > 0 cycles so that below 500 MeV the A > 0 solar minima spectra are always higher than the corresponding A < 0 spectra (Beatty et al., 1985; Potgieter and Moraal, 1985Jump To The Next Citation Point). This means that the adiabatic energy losses that CRs experience in A < 0 cycles are somewhat different than during the A > 0 cycles (Strauss et al., 2011aJump To The Next Citation Point,cJump To The Next Citation Point). This also causes the proton spectra for two consecutive solar minima to cross at a few GeV (Reinecke and Potgieter, 1994).
  6. Drift effects are not necessarily the same during every 11-year cycle, not even at solar minimum because the recent solar minimum was different than other A < 0 cycles. The intriguing interplay among the major modulation mechanisms changes with the solar cycles (Potgieter et al., 2013Jump To The Next Citation Point).
  7. Drifts also influence the effectiveness by which galactic CRs are re-accelerated at the solar wind TS (e.g., Jokipii, 1986Jump To The Next Citation Point; Potgieter and Langner, 2004aJump To The Next Citation Point). However, it is unclear whether particle drifts play a significant role in the heliosheath and to what extent drift patterns are different than in the inner heliosphere.

Particle drifts, as a modulation process, has made a major impact on solar modulation theory and was eclipsed only in the early 2000s when major efforts were made to understand diffusion theory better together with the underlying heliospheric turbulence theory (as reviewed by Bieber, 2003Jump To The Next Citation Point and McKibben, 2005). It was also then finally realized that particle drifts do not dominate solar modulation over a complete solar cycle but that it is part of an intriguing interplay among basically four mechanisms and that this play-off changes over the solar cycle and from one cycle to another. The latest prolonged solar minimum brought additional insight in how this interplay can change as the Sun keeps on surprising us (Potgieter et al., 2013Jump To The Next Citation Point).

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Figure 9: Left panels: Computed radial gradients for galactic protons, in % AU–1, as a function of kinetic energy for both polarity cycles and for solar minimum conditions in the equatorial plane at 1, 50, and 91 AU, respectively (top to bottom panels). Right panels: Similar but at a polar angle 𝜃 = 55°. Two sets of solutions are shown in all panels, first without a latitude dependence (black lines) and second with a latitude-dependent compression ratio for the TS (red lines). In this case, the TS is at 90 AU and the HP is at 120 AU. Image reproduced by permission from Ngobeni and Potgieter (2010), copyright by COSPAR.
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Figure 10: The difference caused by drifts in the computed latitudinal gradients (% degree–1) for protons in the inner heliosphere for the two HMF polarity cycles as a function of rigidity Potgieter et al. (2001Jump To The Next Citation Point). Marked by the black line-point is the latitudinal gradient calculated from a comparison between Ulysses and PAMELA observations for 2007. Image reproduced by permission from De Simone et al. (2011).

4.5 Aspects of diffusion and turbulence theory relevant to solar modulation

Progress has been made over the last decade to improve the understanding of heliospheric turbulence, in particular to overcome some of the deficiencies of standard scattering theories such as quasi-linear theory (QLT). New approaches have taken into account the dynamical character and the three-dimensional geometry of magnetic field fluctuations. Historically, it is important to note that in order to reconcile observations with the theoretical mean free paths, Bieber et al. (1994) advocated a composite model for the turbulence, which consists of ∼ 20% slab and ∼ 80% 2D fluctuations. At high rigidities the simpler theory can be used to describe particle transport parallel to the mean field, but at low rigidities the dynamical theory predicts a much more efficient scattering, which reduces the parallel mean path compared to standard QLT for ions, but gives large values for electrons. Dröge (2005) showed that standard QLT underestimated mean free paths of low energy electrons by almost two orders of magnitude even with a corrected slab fraction for magnetic turbulence. A general conclusion seems to be that the parallel mean free path of CRs is a key input parameter for CR transport but not nearly the only one. Teufel and Schlickeiser (2002), amongst others, produced analytical formulae for the parallel mean free path as a function of rigidity at Earth for CR protons and electrons, which differ significantly at lower energies for these two species.

The diffusion of particles across the mean HMF is an important area of study. It is very nearly radially directed towards the Sun for distances beyond 10 AU. For CRs to reach Earth, they must cross the mean HMF, otherwise it would require large parallel mean-free paths and subsequently very large anisotropies, which are not observed. Cross-field diffusion remains puzzling, with the dimensionality of the underlying turbulence of critical importance. Conceptually, it is understood that the local transport across individual magnetic lines and the motion of particles along spatially meandering magnetic field lines can both occur. Simulations of particle transport in irregular magnetic fields were performed by, e.g., Giacalone and Jokipii (1994, 1999) and Qin and Shalchi (2012) with insightful results. Later, a new theory with different assumption, was developed by Matthaeus et al. (2003) and Bieber et al. (2004). This nonlinear guiding-center (NLGC) theory is promising for understanding perpendicular transport. This topic is evidently a very specialized field of theoretical research and for these details the reader is referred to the reviews by Bieber (2003) and Giacalone (2011). Developing a full ab initio theory of CR transport and modulation, especially for perpendicular diffusion, by integrating turbulence quantities with diffusion coefficients throughout the heliosphere is a work in progress (e.g., Burger et al., 2000Jump To The Next Citation Point; Pei et al., 2010). Unfortunately, the ab initio approach cannot as yet produce elements of the diffusion tensor than can explain all observations consistently when used in global modulation models so that phenomenological approaches remain very useful. For comprehensive monographs on transport theory, see Schlickeiser (2002) and Shalchi (2009).

For elaborate reviews on turbulence effects in the heliosphere and on the fundamental process of reconnection and acceleration of particles, see Fisk and Gloeckler (2009Jump To The Next Citation Point, 2012), Lazarian et al. (2012), Lee et al. (2012), Matthaeus and Velli (2011), and several other contributions in the same issue of Space Science Reviews.

4.6 Development of numerical modulation models

The wide-ranging availability of fast computers has brought significant advances in numerical modeling of solar modulation. In-situ observations have always been limited so that numerical modeling plays an important role to broaden our understanding of solar modulation. For comprehensive global modeling it is essential to have a sound transport theory, reliable numerical schemes with appropriate boundary conditions, local interstellar spectra as initial input spectra and properly considered transport parameters. Furthermore, a basic knowledge of the solar wind and HMF and how they change throughout the whole heliosphere is required. This is quite a task and takes major efforts to accomplish.

Fisk (1971) developed the first numerical solution of the TPE by assuming a steady-state and spherical symmetry, i.e., a one-dimensional (1D) model with radial distance as the only spatial variable, and of course an energy dependence. Later, a polar angle dependence was included to form an axisymmetric (2D) steady-state model without drifts (Fisk, 1976). In 1979, Jokipii and Kopriva (1979Jump To The Next Citation Point) and Moraal et al. (1979) presented their separately developed 2D steady-state models including gradient and curvature drifts for a flat HCS. The first 2D models to emulate the waviness of the HCS were developed by Potgieter and Moraal (1985Jump To The Next Citation Point) and Burger and Potgieter (1989). Three-dimensional (3D) steady-state models including drifts with a 3D wavy HCS were developed by Jokipii and Thomas (1981) and Kóta and Jokipii (1983) with similar models developed later by Hattingh et al. (1997) and Gil et al. (2005). Haasbroek and Potgieter (1998Jump To The Next Citation Point), Fichtner et al. (2000Jump To The Next Citation Point) and Ferreira et al. (2001Jump To The Next Citation Point) independently developed steady-state models including the Jovian magnetosphere as a source of low-energy electrons.

The first 1D time-dependent model (numerically thus three dimensions: radial distance, energy, and time) was developed by Perko and Fisk (1983). Extension to two spatial dimensions was done by Le Roux and Potgieter (1990) including drifts and the effect of outwards propagating GMIRs at large radial distance thus enabling the study of long-term CR modulation effects (Potgieter et al., 1993Jump To The Next Citation Point). Fichtner et al. (2000Jump To The Next Citation Point) developed a 3D time-dependent model for electrons, but approximated adiabatic cooling of electrons at lower energies by doing a momentum averaging of the Parker TPE.

The inclusion of the effects of a heliospheric TS was done by Jokipii (1986) who developed the first 2D time-dependent, diffusion shock acceleration model. Potgieter and Moraal (1988) demonstrated that it was possible to include shock acceleration in a steady-state spherically symmetric model by specifying the appropriate boundary conditions with regard to the CR streaming and spectra at the TS. This model was expanded to 2D by Potgieter (1989). Later on 2D shock acceleration models with discontinuous and continuous transitions of the solar wind velocity across the TS were developed to study ACRs (e.g., Steenberg and Moraal, 1996; Le Roux et al., 1996; Langner and Potgieter, 2004aJump To The Next Citation Point,bJump To The Next Citation Point). Haasbroek and Potgieter (1998) developed a model that could handle all possible geometrical elongation of the heliosphere by assuming a non-spherical heliospheric boundary geometry. All the above mentioned numerical models were developed using the Crank–Nicholson and Alternating Direction Implicit (ADI) schemes, with some deviations depending on the complexity of the studied physics (Fichtner, 2005).

The intricacy of the TPEs applicable to CR modulation may cause numerical models to have notorious problems with instability when solving in higher (five numerical) dimensions. Solving the relevant TPEs by means of stochastic differential equations (SDEs) has become therefore quite popular after earlier noteworthy attempts were not truly appreciated (e.g., Fichtner et al., 1996; Gervasi et al., 1999; Yamada et al., 1999; Zhang, 1999). This method has several advantages, most notably unconditional numerical stability and an independence of a spatial grid size. The method is highly suitable for parallel processing. Recently, models developed around SDEs have become rather sophisticated (e.g., Kopp et al., 2012) but not always focussed on additional insight into CR modulation. It has been illustrated that additional physical insights can be extracted from this approach, e.g., Florinski and Pogorelov (2009), Strauss et al. (2011aJump To The Next Citation Point,cJump To The Next Citation Point, 2012cJump To The Next Citation Point, 2013aJump To The Next Citation Point), and Bobik et al. (2012Jump To The Next Citation Point). Some aspects are discussed next.

4.6.1 Illustrations of SDE based modeling

The following figures are illustrations of CR modulation based on the SDE approach to solar modulation as examples of what can be done additionally to the ‘standard’ numerical approaches. Figure 11View Image shows pseudo-particle traces for 100 MeV protons in the A < 0 cycle and varying values of the tilt angle α, projected onto the meridional plane in comparison with a projection of the wavy HCS onto the same plane. These particles propagate mainly in latitudes covered by the HCS, and their transport is greatly affected by the waviness of the HCS but mainly for small values of α, as diffusion disrupts the drift pattern. Diffusion can in principle almost wipe out these drift patterns as illustrated by Strauss et al. (2012cJump To The Next Citation Point). This realistic picture of the combination of global and HCS drifts is in sharp contrast to the idealist picture shown in Figure 7View Image. Studies that are focusing on drifts as the sole modulation process give scenarios that are unrealistic (e.g., Roberts, 2011). Perfect drift dominated CR transport does not exist because CR modulation essentially is a convection-diffusion process. As mentioned, inspection of Equation (5View Equation) shows that when changing diffusion the CR intensity gradients are changed directly, which subsequently changes the effects of drifts implicitly (Potgieter et al., 2013Jump To The Next Citation Point) while explicit changes in drifts and consequently of drift effects are done by changing the drift coefficient directly (Equation (27View Equation)).

In Figure 12View Image binned propagation times for 100 MeV galactic electrons between Earth and the HP (at 140 AU) for three scenarios: the A < 0 polarity cycle (left panel), the A > 0 cycle (middle panel), and the non-drift case (right panel). For the latter, the propagation time follows a normal distribution, peaking at ∼ 400 days, while for the different drift cycles, the distribution tends to be more Poisson like (CR cannot reach Earth infinitely fast) with lower propagation times. The reason for the shorter propagation times, ∼ 240 days for the A > 0 cycle and ∼ 110 days for the A < 0 cycle, is that drifts cause a preferred direction of transport for these CR electrons, thereby allowing them to propagate faster to Earth. The propagation times for the A < 0 cycle is shorter than for the A > 0 cycle because these electrons can easily escape through the heliospheric poles than drifting along the HCS in the A > 0 cycle (Strauss et al., 2011aJump To The Next Citation Point,cJump To The Next Citation Point).

The propagation times and energy loss of 100 MeV protons propagating from the HP (at 100 AU) to Earth as a function of the HCS tilt angle (α) for A < 0 polarity cycles of the HMF are shown in Figure 13View Image. Note that the increase in propagation time significantly slows down above α = 40°. The energy loss levels off above α = 40°. See Strauss et al. (2011aJump To The Next Citation Point,c) for additional illustrations.

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Figure 11: Pseudo-particle traces (trajectories) for galactic protons in the A < 0 HMF cycle projected onto the meridional plane for four values of the HCS tilt angle, shown as red lines. The HP position is indicated by the dashed lines, while the dotted lines show a projection of the waviness of HCS onto the same plane. The simulation is done for 100 MeV protons. Image reproduced by permission from Strauss et al. (2012cJump To The Next Citation Point), copyright by Springer.
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Figure 12: Binned propagation times for galactic electrons released at Earth at 100 MeV for the A < 0 (left panel), A > 0 (middle panel), and for the no-drift scenarios (right panel). For each computation 10000 particle trajectories were integrated using the SDE approach to modulation modeling. Image reproduced by permission from Strauss et al. (2011a), copyright AAS.
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Figure 13: The propagation times and energy loss of 100 MeV protons propagation from the HP to Earth as a function of the HCS tilt angle (α) for A < 0 polarity cycles of the HMF. Note the change in propagation time at α = 40° and how the energy loss levels off above α = 50°. Image reproduced by permission from Strauss et al. (2012c), copyright by Springer.

4.7 Charge-sign dependent modulation

It was not until around 1976 – 1978 that particle drifts were considered seriously as a competitive modulation process, in addition to the conventional convective, diffusive, and adiabatic energy loss processes. It was experienced as controversial when introduced and it took 10 years to become widely accepted. Even today the full extent of its relevance and importance over a complete solar activity cycle is debated. See, as an example, the critical review by Cliver et al. (2011Jump To The Next Citation Point). The early stages of this theoretical development, and the status of the research round that time, were reviewed by Quenby (1984). The realization also came that the only way to understand the full scope of particle drifts on galactic CR modulation was with numerical modeling. Therefore, since the early 1980s increasingly more sophisticated numerical models had been introduced that kept on improving as discussed above. See Potgieter (2013) for a full review on this topic, parts of which is also repeated here.

Simultaneous measurements of CR electrons and positrons (protons and anti-protons) serve as a crucial test of our present understanding of how large charge-sign dependent modulation in the heliosphere is, as a function of energy and position over a complete solar activity cycle. It is expected that the effects of drifts on CRs should become more evident closer to minimum solar activity. Observations of CR particles and their anti-particles have been done over the years and are presently been made simultaneously by PAMELA (e.g., Boezio et al., 2009Jump To The Next Citation Point; Sparvoli, 2012Jump To The Next Citation Point; Boezio and Mocchiutti, 2012Jump To The Next Citation Point) and the AMS-02 mission (e.g., Battiston, 2010). PAMELA is a satellite-borne experiment designed for cosmic-ray antimatter studies. The instrument is flying on board the Russian Resurs-DK1 satellite since June 2006, following a semi-polar near-Earth orbit. For an overview of balloon-based observations, see Seo (2012).

Before these simultaneous measurements, charge-sign dependent solar modulation was mostly studied using ‘electrons’, which was actually the sum of electrons and positrons, together with CR protons and helium of the same rigidity. The first sturdy observational evidence of charge-sign dependent solar modulation was reported by Webber et al. (1983Jump To The Next Citation Point) and modelled by Potgieter and Moraal (1985Jump To The Next Citation Point) using a first generation drift model. This is shown in Figure 14View Image as electron spectra during two consecutive solar minimum modulation periods in 1965 and 1977. The corresponding charge-sign dependent effect is illustrated in Figure 15View Image, comparing proton and electron measurements made during two consecutive solar minimum periods (1965 as A < 0 and 1977 as A > 0). The ratio of differential intensities, DI(1977)/DI(1965 – 66), is shown for both electrons and protons as a function of E. Evidently, electrons behaved different from protons during these consecutive solar minimum epochs, again convincingly revealing a 22-year modulation cycle and charge-sign dependence.

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Figure 14: Galactic CR electron observations for two consecutive solar minimum modulation periods in 1965 (open circles) and 1977 (filled circles) compared to the predictions of a first generation drift-modulation model (band between solid lines) containing gradient, curvature, and current sheet drifts. Clearly, a 22-year modulation cycle is portrayed (Potgieter and Moraal, 1985Jump To The Next Citation Point, and references therein).
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Figure 15: Ratios of proton and electron measurements for 1977 (A > 0 polarity cycle) to 1965 – 66 (A < 0 polarity cycle) as a function of kinetic energy compared to the predictions made with a drift-modulation model illustrating how differently protons behave to electrons during two solar minimum periods with opposite solar magnetic field polarity (Webber et al., 1983; Potgieter and Moraal, 1985, and references therein).

A newer generation drift-modulation model was used by Ferreira (2005Jump To The Next Citation Point) to illustrate how the modulation of galactic electrons differ form one polarity cycle to another as shown in Figure 16View Image (right panel). For electrons the influence of particle drifts is evident over an energy range from 50 MeV to 5 GeV with a maximum effect around 200 MeV to 500 MeV as shown in the left panel. It also illustrates how the drift effect changes with increasing distance from the Sun.

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Figure 16: Panel (a): Computed differential intensities of galactic electron at 1, 5, 60 and 90 AU (from bottom to top) in the heliospheric equatorial plane for the A > 0 and A < 0 polarity cycles. Panel (b): Ratio of the computed intensities for the A > 0 and A < 0 cycles as a function of kinetic energy and for the radial distance as in (a). Image reproduced by permission from Ferreira (2005Jump To The Next Citation Point), copyright by COSPAR.

Drift-modulation models predicted that during A > 0 polarity cycles the ratio of electron to proton (e− ∕p) intensities as a function of time (with solar activity as described by the wavy HCS) should exhibit an inverted V shape while during A < 0 cycles it should exhibit an upright V around minimum modulation periods (see the reviews by Potgieter et al., 2001; Heber and Potgieter, 2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point; Strauss et al., 2012b). This means that as a function of time electrons would exhibit a sharper intensity time profile than protons or helium during A > 0 solar epochs. This was displayed eloquently by Ulysses observations of electrons, helium and protons at 1.3 GV and 2.5 GV for the period 1990 to 2004 as reproduced in Figure 17View Image. The e− ∕p ratio indeed formed an inverted V around the 1997 solar minimum shown in the bottom panel. This effect was also shown by Ferreira et al. (2003aJump To The Next Citation Point) and Ferreira et al. (2003bJump To The Next Citation Point) for the 1987 solar minimum (A < 0 cycle) when a V-shape was displayed in the − e ∕He ratio.

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Figure 17: Observed % changes respectively of helium (1.2 GV), electrons (1.2 GV and 2.5 GV), and protons (2.5 GV), as a function of time (solar activity) for the Ulysses mission from 1990 to 2005. The period from 1990 to 2000 was an A > 0 polarity epoch but changed to an A < 0 epoch around 2000 – 2001. Clearly the electrons exhibited a sharper profile over this A > 0 cycle than protons and helium in accord with predictions of drift-modulation models. Adapted by Heber from Heber et al. (2002Jump To The Next Citation Point, 2003Jump To The Next Citation Point, 2009Jump To The Next Citation Point).
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Figure 18: A comparison of the observed anti-proton to proton ratio (below 10 GeV with first generation drift-model computations for solar minimum conditions with the HCS tilt angle α = 10° and for solar maximum conditions with α = 70°. The – and + signs indicate A < 0 and A > 0 polarity cycles, respectively. The corresponding ratio for the galactic spectra is indicated as IS. Image reproduced by permission from Webber and Potgieter (1989Jump To The Next Citation Point), copyright by AAS.

A first illustration of the charge-sign dependent effect for the modulation of protons and anti-protons was made by Webber and Potgieter (1989). This is shown in Figure 18View Image. These first generation drift models predicted a change of a factor of 10 in the ¯p∕p ratio at 200 MeV from A > 0 to A < 0 solar minima but less for increased solar activity periods. This was done in an attempt to reproduce the data displayed in this figure with the GS as assumed in those days. Comprehensive modeling has later been done for electron-positron (− + e ∕e) and proton-antiproton modulation (Langner and Potgieter, 2004aJump To The Next Citation Point,b; Potgieter and Langner, 2004a) and heavier CR species such as boron and carbon (Potgieter and Langner, 2004b). They relied heavily on the computations of galactic spectra with the GALPROP code (e.g., Moskalenko et al., 2002; Strong et al., 2007). Apart from drifts, their model also included the effects of the solar wind TS. These results are shown in Figure 19View Image respectively for − + e ∕e and ¯p∕p as a function of E for the two solar polarity cycles during typical solar minimum modulation conditions. This is compared to the corresponding ratio of the local interstellar spectra (LIS). See also Della Torre et al. (2012).

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Figure 19: Left: Computed electron to positron ratios at Earth for two polarity cycles (A > 0, e.g., 1997, 2020 and A < 0, e.g., 1985, 2009) of the HMF compared to the ratio of the LIS at the heliopause (120 AU). Differences above about 80 MeV are caused by gradient, curvature, and current sheet drifts in the heliosphere during solar minimum activity. Right: Similar to left panel, but for computed ratios of galactic protons to anti-protons (Langner and Potgieter, 2004aJump To The Next Citation Point).

Evidently, the charge-sign effect is significant but seems less than what was predicted by the first generation models. Strauss et al. (2012a) applied a newly developed 3D modulation model based on the SDE approach to study the modulation of galactic protons and anti-protons inside the heliosphere. They also found a maximum drift effect of a factor of 10 below ∼ 100 MeV for the ¯p∕p, which gradually subsides with increasing E to become negligible above 10 GeV. They predict that for the next solar minimum period (A > 0 cycle) this ratio will be lower than in the preceding A < 0 cycle and noted that if modulation conditions during the next solar minimum would be the same as in the previous A < 0 minimum of 2009, CR intensities will be even higher at Earth because of drifts. They illustrated convincingly with pseudo-particle traces how protons drift differently than anti-protons through the heliosphere to Earth during the same modulation cycle. See also Bobik et al. (2011, 2012). For an illustration of charge-sign-dependent effects in the outer heliosphere and heliosheath see, e.g., Langner and Potgieter (2004aJump To The Next Citation Point).

In order to establish how large the effect is over an energy range considered relevant to solar modulation, precise and simultaneous measurements of CRs and their antiparticles by the same instrumentation are necessary. The PAMELA mission has introduced an era of such precise measurements of protons (anti-protons) and electrons (positron) done simultaneously to energies down to ∼ 100 MeV so that solar modulation can also be studied and particle drifts thoroughly tested. The preliminary proton and electron data as monthly averages, from mid-2006 to the end of December 2009, were reported in the PhD theses of Di Felice (2010Jump To The Next Citation Point) and De Simone (2011Jump To The Next Citation Point), and the Master’s theses of Vos (2011Jump To The Next Citation Point) and Munini (2011). The finalized proton spectra for this period were reported by Adriani et al. (2013Jump To The Next Citation Point) with the electron and positron data for this period to be finalized in 2013. Yearly averages were reported by Sparvoli (2012Jump To The Next Citation Point) and Boezio and Mocchiutti (2012Jump To The Next Citation Point). The PAMELA data shown here as Figure 20View Image illustrate how protons between 0.5 – 1.0 GV responded differently over the mentioned period than electrons at the same rigidity. Protons had increased by an average of a factor of ∼ 2.5 over 4.5 years whereas electrons of the same rigidity had increased by only a factor of ∼ 1.4 over the same period. This trend once again reveals the inverted V shape as discussed above.

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Figure 20: Normalized proton (light-red lines) and electron (light-blue lines) differential intensities at (0.75 ± 0.2) GV as a function of time, from July 2006 to December 2009. The red and blue symbols represent the average intensities of protons and electrons, respectively. These intensity-time profiles are normalized to the intensity measured in July 2006. Image reproduced by permission from Vos (2011). See also Di Felice (2010) and De Simone (2011Jump To The Next Citation Point).

De Simone (2011) reported for the year 2009 that − + e ∕e = 6 at 200 MeV changed to − + e ∕e = 15 at 8 GeV, which demonstrates a similar trend as the prediction in Figure 6View Image but seemly larger. This provides ample motivation to apply a comprehensive drift model to the PAMELA data for CRs with opposite charge to establish the exact extent of drifts during the recent prolonged solar minimum.

What happens to the − e ∕p or − e ∕He ratios at times of solar maximum activity was demonstratively discussed by, e.g., Ferreira (2005Jump To The Next Citation Point), Heber and Potgieter (2006Jump To The Next Citation Point, 2008Jump To The Next Citation Point), and Potgieter (2011). PAMELA observations after 2009 could be most helpful in this regard.

It follows from Figure 19View Image that the optimum energy range for observations to test the extent of drift effects for electrons and positrons is between 50 MeV and 5 GeV. Below 50 MeV, electrons observed at Earth get ‘contaminated’ by Jovian electrons (Potgieter and Nndanganeni, 2013aJump To The Next Citation Point).

4.8 Main causes of the complete 11-year and 22-year solar modulation cycles

Apart from the 11-year and 22-year cycles, regular steps are superposed on the intensity-time profiles of CR modulation. A departure point for these time-dependent steps (both increases and decreases) from a global point of view is that ‘propagating barriers’ are formed and later dissipate in the outer heliosphere during the 11-year activity cycle. These ‘barriers’ are basically formed by solar wind and magnetic field co-rotating structures which are inhibiting the easy access of CRs to a relative degree. This is especially applicable to the phase of the solar activity cycle before and after solar maximum conditions when large steps in the particle intensities had been observed. A wide range of interaction regions occur in the heliosphere, with GMIRs the largest, as introduced by Burlaga et al. (1993). They observed that a clear relation exists between CR decreases (recoveries) and the time-dependent decrease (recovery) of the HMF magnitude and extent local to the observation point. The paradigm on which these modulation barriers is based is that interaction and rarefaction regions form with increasing radial distance from the Sun (see also Potgieter and le Roux, 1989, and references therein). These relatively narrow interaction regions can grow latitudinally and especially azimuthally and as they propagate outwards they spread, merge and interact to form eventually GMIRs that can become large in extent and capable of causing the large step-like changes in CRs. Potgieter et al. (1993) illustrated that their affects on long-term modulation depend on their rate of occurrence, the radius of the heliosphere (i.e., how long they stay inside the modulation volume), the speed with which they propagate, their spatial extent and amplitude, especially their latitudinal extent (to disturb drifts), and the background modulation conditions they encounter. Drifts on the other hand, normally dominate periods of solar minimum modulation up to four years so that during an 11-year cycle a transition must occur (depending how solar activity develops) from a period dominated by drifts to a period dominated by these propagating structures. The largest of the step decreases and recoveries shown in Figure 6View Image are caused by these GMIRs.

The 11-year and 22-year cycles together with the step-like modulation are good examples of the interplay among the main modulation mechanisms as illustrated by Le Roux and Potgieter (1995). They showed that it was possible to simulate, to the first order, a complete 22-year modulation cycle by including a combination of drifts with time-dependent tilt angles and GMIRs in a time-dependent modulation model. See also Potgieter (1995, 1997) and references therein. Le Roux and Fichtner (1999) confirmed that a series of GMIRs cannot on their own reproduce the fully observed 11-year modulation cycle. This is achieved by adding a well-defined time variation in the propagation process such as for the diffusion coefficients, or using the time-dependent wavy HCS. A major issue with time-dependent modeling, apart from the global dynamic features such as the wavy HCS, is what to assume for the time dependence of the diffusion coefficients mentioned above.

A subsequent step in understanding long-term modulation came when Cane et al. (1999) pointed out that the step decreases observed at Earth could not be primarily caused by GMIRs because they occurred well before any GMIRs could form beyond 10 – 20 AU. Instead, they suggested that time-dependent global changes in the HMF over an 11-year cycle are responsible for long-term modulation. Potgieter and Ferreira (2001) and Ferreira and Potgieter (2004Jump To The Next Citation Point) combined these changes with time-dependent drifts, naming it the compound modeling approach. It was assumed that all the diffusion coefficients change with time proportional to B (t)− n(P,t), with B (t) the observed, time-dependent HMF magnitude close to Earth, and n(P, t) a function of rigidity and the HCS tilt angle, which introduces an additional time dependence related to solar activity. These changes are then propagated outwards at the solar wind speed to form propagating modulation barriers throughout the heliosphere, changing with the solar cycle. With simply n = 1, and with B (t) changing by an observed factor of 2 over a solar cycle, this approach resulted in a variation of the diffusion coefficients by a factor of 2 only, which is perfect for simulating the 11-year modulation at NM energies at Earth, as seen in Figure 6View Image, but not at all for lower rigidities. In order to reproduce spacecraft observations at energies below a few GeV, n (P,t) must depend on time (solar activity) and on rigidity. For a more advanced treatment and recent application of this approach, see Manuel et al. (2011aJump To The Next Citation Point,bJump To The Next Citation Point).

Ferreira and Potgieter (2004) confirmed that using the HCS tilt angles as the only time-dependent modulation parameter resulted in compatibility only with solar minimum observations. Using the compound approach resolved this problem. Applied at Earth and along the Ulysses and Voyager 1 and 2 trajectories, this approach is remarkably successful over a period of 22 years, e.g., when compared with 1.2 GV electron and helium observations at Earth, it produces the correct modulation amplitude and most of the modulation steps. Some of the simulated steps did not have the correct magnitude and phase, indicating that refinement of this approach is needed by allowing for merging of the propagating structures. However, solar maximum modulation could be largely reproduced for different CR species using this relatively simple concept, while maintaining all the other major modulation features during solar minimum, such as charge-sign dependence discussed above.

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Figure 21: Computed 1.2 GV e− ∕He ratio at Earth for 1976 – 2000 in comparison with the observed e− ∕He obtained from electron measurements of ISEE3/ICE, He measurements from IMP and electron measurements from KET (Heber et al., 2003Jump To The Next Citation Point). The shaded areas correspond to the period with no well-defined HMF polarity. Two periods indicated by labels A and B with relatively large differences between the computed ratios and the observations require further investigation. Image reproduced by permission from Ferreira et al. (2003aJump To The Next Citation Point), copyright by COSPAR.

An important accomplishment of this compound approach is that it also produces the observed charge-sign dependent CR modulation from solar minimum to maximum activity. Figure 21View Image depicts the computed 1.2 GV electron (e−) to helium ratio at Earth for 1976 – 2000 in comparison with the observed e− ∕He obtained from electron measurements of ISEE3/ICE, helium measurements from IMP and electron measurements from KET (Heber et al., 2002Jump To The Next Citation Point, 2003Jump To The Next Citation Point). The shaded areas correspond to the period when there was not a well defined HMF polarity. Two periods, labelled A and B, were found with relatively large differences between the computed ratios and the observations that require further refinement.

The compound approach also involves two other important modifications. First, a significant increased polar perpendicular diffusion is required to account for the observed latitude dependence of CR protons and the lack thereof for electrons along the Ulysses trajectory over the 22 year cycle. This is mainly to reduce the large latitudinal gradients caused by unmodified drifts and in addition to the time dependence of the diffusion coefficients. This effect is illustrated in Figure 22View Image using the computed 2.5 GV electron to proton ratio (e− ∕p) along the Ulysses trajectory and at Earth in comparison with the 2.5 GV e− ∕p observations from KET (Heber et al., 2002Jump To The Next Citation Point, 2003). The top panels show the position of Ulysses during this period. In order to model the observed e-/p as a function of time, the latitude dependence of both electrons and protons must be correctly modeled (Ferreira et al., 2003bJump To The Next Citation Point). Second, during periods of large solar activity, drifts must be reduced additionally to improve explaining the observed electron to He intensity ratio at Earth and the electron to proton ratio along the Ulysses trajectory during the period when the HMF polarity reverses. For example, drifts had to be reduced from a 50% level at the beginning of 1999 to a 10% level by the end of 1999, to vanish during 2000, but to quickly recover after the polarity reversal during 2001 to levels above 10%. For the period after 2001, the model predicts a steady increase in drifts from 10% to 20% by the end of 2002. This indicates that in order to produce realistic charge-sign dependent modulation during extreme solar maximum conditions, the heliosphere must become diffusion (non-drift) dominated. Ndiitwani et al. (2005Jump To The Next Citation Point) calculated the percentage drifts required over a full modulation cycle, especially during extreme solar maximum, to find compatibility between the compound model and the observed e− ∕p. This is shown in Figure 23View Image in comparison to the observed tilt angles as proxy for solar activity. Obviously, little drifts are required during solar maximum in contrast to ∼ 90% at solar minimum activity. For charge-sign dependent effects in the outer heliosphere and heliosheath, see Langner et al. (2003Jump To The Next Citation Point) and Langner and Potgieter (2004a).

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Figure 22: Computed 2.5 GV electron to proton ratio (e− ∕p) as a function of time along the Ulysses trajectory (solid line) and at Earth (dotted line) in comparison with the 2.5 GV ratio observed with KET (Heber et al., 2002). Top panels show the position of Ulysses. In order to simulate this observed ratio as a function of time, the latitude dependence of both electrons and protons must be correctly modeled (Ferreira et al., 2003a; Ferreira, 2005Jump To The Next Citation Point). Image reproduced by permission from Ferreira et al. (2003b), copyright by EGU.
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Figure 23: Percentage of drifts (continuous line) in the compound model that gives realistic modulation for various stages of the solar cycle for both the 2.5 GV electron and protons. As a proxy for solar activity the tilt angles, as used in the model are shown for illustrative purposes. Image reproduced by permission from Ndiitwani et al. (2005), copyright by EGU.

A question that is relevant within the context of Voyager 1 and 2 observations is how much modulation occurs inside the heliosheath. The process is of course highly energy-dependent. An illustrative example of the amount of modulation that CR protons may experience in the heliosheath in the nose direction is shown in Figure 24View Image. The percentage of modulation in the equatorial plane in the heliosheath is given with respect to the total modulation (between 120 AU and 1 AU) as a function of kinetic energy for both polarity cycles, for solar minimum and moderate maximum conditions (Langner et al., 2003Jump To The Next Citation Point). See also Strauss et al. (2013b) for a recent illustration of modulation in the outer heliosphere.

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Figure 24: Computed percentage of galactic CR modulation in the heliosheath with respect to the total modulation (between 120 AU and 1 AU) for the two magnetic polarity cycles (A > 0 and A < 0), for solar minimum (α = 10°) and for moderate maximum (α = 75°) conditions, in the equatorial plane in the nose direction of the heliosphere. Negative percentages mean that the galactic CRs are reaccelerated at the TS under these assumed conditions (Langner et al., 2003Jump To The Next Citation Point).

Evidently, at E < ∼ 0.02 GeV modulation > 80% may occur in the heliosheath for both polarity cycles. For all four the conditions, the heliosheath modulation will eventually reach 0% (not shown) but at different energies, indicating that it differs significantly with energy as well as with drift cycles. How much gradient and curvature drifts actually occur in the heliosheath is still unanswered. The negative percentages indicate that the intensity is actually increasing in the heliosheath as one moves inward from the outer boundary toward the TS because of the re-acceleration of CRs at the TS. This depends on many aspects, in particular the TS compression ratio as discussed by Langner et al. (2003, 2006a,b).

In the study of the long-term CR modulation in the heliosphere, several issues need further investigation and research. Despite the apparent success of the compound numerical model described above the amount of merging taking place beyond 20 AU needs to be studied with MHD models, especially the relation between CMEs and GMIRs, and how these large barriers will modify the TS and the heliosheath. The full rigidity dependence of the compound model is as yet not well described. A major issue with time-dependent modeling, apart from global dynamic features such as the wavy HCS, is what to use for the time dependence of the diffusion coefficients in Equation (6View Equation), on top of the already complex issue of what their steady-state energy (rigidity) and spatial dependence are in the inner heliosphere. It has now also become pressing to understand the diffusion tensor in the outer heliosphere and beyond the TS. Equation (5View Equation) is probably more complex and has been extended to include mechanisms otherwise considered to be negligible. Fundamentally, from first principles, it is not yet well understood how gradient and curvature drifts reduce with solar activity. This aspect needs now also to be investigated for the region beyond the TS. For example, what happens to the wavy HCS in the heliosheath and how will the strong non-radial components of the solar wind and the associated HMF affect drifts and CR modulation in the heliosheath? To answer this, more study is needed to gain a better understanding of the finer details of these complex magnetic fields and the relation to long-term CR modulation, especially how this field changes with solar activity. An aspect that should be kept in mind is that the study of more realistic fields requires an ever more complex diffusion tensor and description of drifts. Other interesting aspects of CRs beyond the TS are discussed by Potgieter (2008Jump To The Next Citation Point). Another important question is how the heliospheric modulation volume varies with time over scales of hundreds to thousands of years, which are most relevant for the study of space climate. The Sun encounters different interstellar environments during its passage through the galaxy, and hence the outer heliospheric structure should change (e.g., Scherer et al., 2008b).

The paleo-cosmic ray records can be used to study the properties of the heliosphere and solar processes over 10000 and more years. McCracken et al. (2011) reported that both varied greatly over such as period, ranging from ∼ 26 Grand Minima of duration 50 – 100 years when the Sun was inactive, to periods similar to the recent 50 years of strong solar activity. The 11 and 22 year cycles of solar activity continued through the Spörer and Maunder Grand Minima. They speculated that the solar dynamo exhibits a 2300 year periodicity, wherein it alternates between two different states of activity and argued that paleo-cosmic ray evidence suggests that the Sun has now entered a more uniform period of activity, following the sequence of Grand Minima (Wolf, Spörer, Maunder, and Dalton) that occurred between 1000 and 1800 AD. See also the discussion by Usoskin et al. (2001) and the review by Usoskin (2013).


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