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GV in the middle interval, and Arr ^

R for R > 3 GV and R << 3 GV in the other two intervals (Fig. 2.34.4). It was proposed and demonstrated first by Moraal et al. (1999), in an empirical approach to the CR diffusion tensor, that a similar three interval R-dependence for Arr , with the weakest R-dependence in the center interval, is necessary for the simulation of both observed galactic and anomalous CR spectra.

Ricter Scale Images
Fig. 2.34.4. The rigidity dependence of the radial mean free path Arr in the ecliptic plane for the NP model of k_l . From le Roux et al. (1999a).

2.34.4. Summarizing and comparison of used three models

Le Roux et al. (1999a) summarized main results as following. The parallel, perpendicular and radial mean free paths for CR were determined theoretically on the basis of three plausible theories for k__ assuming that field line random walk is more important than resonant perpendicular diffusion. A MHD model for field turbulence transport in the solar wind (Zank et al., 1996, 1998) was used to calculate the spatial dependence of the mean free paths. Concerning the MQLT model for A_, A// contributes solely to Atr, and consequently a big drop in Atr across the termination shock implying a strong galactic CR modulation barrier, is predicted. For the MAD model of A__, A// produces a strong contribution to Arr for r > 20 AU from the Sun resulting in a large r-dependence for Arr for r > 30 AU

upstream. For both models, Arr ^ R at large r due to the resonant interaction of CRwith the energy range of the power spectra. Regarding the NP model for A_l, A// contributes significantly to Arr for R < 3 GV at large r but A// dominates in Arr for R > 3 GV. This leads to a complex three interval R-dependence for Arr , with the weakest R-dependence Arr x R given by the middle interval, and A x R in the other two intervals. A similar R-dependence was first proposed empirically by Moraal et al. (1999) as a necessary condition for the simulation of both observed galactic and anomalous CR spectra. The NP model tentatively provides a theoretical basis for the work of Moraal et al. (1999).

2.35. On the role of drifts and perpendicular diffusion in CR propagation

2.35.1. Main equations for CR gradient and curvature drifts in the interplanetary magnetic field

Jokipii and Levy (1977) show that the CR gradient and curvature drifts in an Archimedean-spiral magnetic field produce a significant effect in the galactic CR propagation and modulation in the Heliosphere. The effects of drifts are due to the fact that CR small energy particles for which the drift velocity is comparable to the solar wind velocity have more rapid access (in case when the drift velocity directed to the Sun) to the inner Heliosphere than in the absence of drifts; in the opposite case, when the drift velocity directed from the Sun, the result could be inverse. Although drifts are explicitly contained in standard transport theories (e.g., Parker, 1965; Dorman, 1965; Axford, 1965a,b; Jokipii and Parker, 1970) they have been neglected in all models of galactic CR or SEP propagation in the interplanetary space. Jokipii and Levy (1977) note that Jokipii (1971), Levy (1975, 1976a,b), Barnden and Bercovitch (1975) pointed out some consequences of drifts, but did not construct complete models. Jokipii and Levy (1977) suggest using the term 'drift' to refer to gradient and curvature drifts, and not to the convection with the solar wind. Jokipii and Levy (1977), Jokipii et al. (1977) use the general formulation of CR transport written down by Jokipii and Parker (1970) and start from decompose the CR diffusion tensor Ky into its symmetric and anti-symmetric parts Kjj s and Ky a . Then the average particle drifts may be written as

Noting that according to Levy (1976a)

One may write the equation for the CR density n as a function of position r, time t, and kinetic energy Ek as dn = d dt dxi j dix:: + vdr,i}>

For simplicity Jokipii and Levy (1977), Jokipii et al. (1977) assume that the electromagnetic conditions in the interplanetary space are symmetric about the Sun's rotation axis, and that Krr, kqq are independent of r and 6, and define the new function

The resulting equation for f is df_ 1 d2

dt 2 dr2

-aEkf

Eq. 2.35.6 is a Fokker-plank equation (Chandrasekhar, 1943) with transition moments

^Ar2J = 2KrrAt, {A02J = 2KeeAt, (Ar) = (/r + usw + vdr r

(A6) = (cot)/r2 + vdr,elr) (AEk) = -2utwOEij3r. (2.35.7)

Jokipii and Levy (1977) consider the steady-state solution to Eq. 2.35.7 with outer boundary condition n = no (Ek) at r = ro; it is presumed that the inner boundary at r = ra is an absorber of CR (because the inner absorbs boundary occupies relatively very small region of space, the its nature makes very little impact on the solution). The solution for f is obtained by introducing particles at r = ro, distributed in 6 as sin 6 . Each particle random walks in r and 6 according to the prescription r+1 = r ± [(Ar2]1/2 + (Ar), 6+1 = 6 ± [(A62)^ + (A6),

where the time step At is chosen to be some convenient value, and the plus or minus signs are chosen randomly. Each particle is followed for successive time steps until r+1 is either greater than ro or less than ra , at which point it is regarded as having escaped from the system and a new particle is introduced at r = ro. The space in r,6, Ek is divided into bins, and at each step the bin in which the particle is located is incremented by 1; the resulting r,6, Ek histogram corresponds to the time independent solution for f. The corresponding solution for n is obtained by dividing 2

2.35.2. The using of Archimedean-spiral model of interplanetary magnetic field

The used in Jokipii and Levy (1977) the model of interplanetary magnetic field corresponds to the classical Parker's (M1963) Archimedean-spiral magnetic field for constant radial solar wind velocity, but in which the field changes sign at the solar equator. The field may be written as

r2 uswr

where S()) is the Heaviside step function and A is a constant; the sign of A changes with successive 11-year solar cycles and is positive for positive CR particles between general solar magnetic field reversals from even to odd cycle (as in 19501960, 1970-1980), and negative from odd to even cycle (as in 1960-1970, 1980-

1990, and so on). For negative CR particles the situation is inversely. On the opinion of Jokipii and Levy (1977) this is a very good approximation to that magnetic field which was shown by Levy (1975, 1976a,b) to provide a natural interpretation of interplanetary sector-structure observations. The corresponding particle velocity drift for positive CR particles is given by (Jokipii et al., 1977):

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