Calculations using Eq. 2.33.25 for a ~ 0.7Q yield A// «1 AU for galactic CR with an energy of 10 GeV. The numerical value and rigidity dependence of A// ^ R is in agreement with experimental data on galactic CR modulation in the outer Heliosphere (Fujii and McDonald, 1995).
For galactic CR with energies above 4 GeV scattered in the interstellar medium, when the random magnetic field has a Kolmogorov spectrum with v ~ 1.7, we obtain the following order-of-magnitude estimate from Eq. 2.33.23:
Assuming that a -1.8Q, L// ~ 100 pc, and H1 ~ 0.3 nT, we obtain for relativistic protons
where E is the particle energy in GeV. Calculated value of A// is close to the experimental mean free path (Ptuskin, 2001).
2.34. CR perpendicular diffusion calculations on the basis of MHD transport models
As it is mention in le Roux et al. (1999a), quasi-linear theory (QLT) for the parallel diffusion (diffusion coefficient K//) of CR appears to be understood reasonably well, unlike perpendicular diffusion (k__ ). This hampers our understanding of CR modulation in the context of well-established CR transport theory. le Roux et al. (1999a) present calculations of the radial cosmic ray diffusion coefficient in the ecliptic plane on the basis of three different theories for perpendicular diffusion assuming that large-scale field line random walk dominates resonant perpendicular diffusion. The radial dependence of Krr is determined completely theoretically using a promising recent model for the combined transport of a predominantly 2D component (80%), and a minor slab component (20%) of MHD turbulence in the solar wind.
2.34.2 Three models for perpendicular diffusion coefficient
On the basis of standard QLT for the cyclotron resonant interaction of CR with random Heliospheric magnetic field (HMF) slab fluctuations le Roux et al. (1999a) derived the CR parallel mean free path
where rg is the particle gyro-radius, lb is the wavelength for slab turbulence at the break point in the power spectrum of HMF fluctuations, A is the normalized amplitude of the x-component of the slab fluctuations (A = SBx/B, where B is the magnitude of the mean HMF; (SBX/B) = 0.05 and B = 5 nT at 1 AU). The first model for k__ is given by
where v is CR particle speed, lc is the correlation length of slab turbulence, and the amplitude (lc = 0.79256 lb where lb = 0.03 AU at 1 AU), A is the sum of slab and 2D turbulence amplitudes. Eq. 2.34.2 corresponds to the QLT of Jokipii (1971) for slab turbulence and implies that CR are tied to and moving along a large-scale random-walking field line without experiencing resonant spatial diffusion. The only modification is that A denotes the sum of slab and 2D turbulence instead of just the slab component. This theory is tied to the condition A//>> lc indicating applicability for rigidities R >> 2X10-4 GV at 1 AU. Thereby, all R-values of relevance for CR modulation are covered. This model is referred to as the modified QLT (MQLT) model (see also Zank et al., 1998).
In the limit A// << lc or R << 2x 10-4 GV at 1 AU, k_ is given by k_= 0.5 A2k// , (2.34.3)
where A is the sum of the amplitudes of slab and 2D turbulence. This expression is an outflow of the QLT by Chuvilgin and Ptuskin (1993) on anomalous perpendicular diffusion. It means that CRs are resonantly diffusing primarily along and weakly across large-scale random walking field lines, so allowing CR to change field lines. It implies that the rate at which large-scale neighboring field lines separate then plays a major role in determining the effective k__ of CR across B (see large ratio of k__/k// in Eq. 2.34.3). Unfortunately, this model applies at R below that of interest for CR modulation. However, test particle simulations by Giacalone (1998) suggest that large-scale field line separation effects also occur at energies relevant for CR modulation, but at a reduced level. Using their work as a guide, le Roux et al. (1999) assume for the 2nd model of k__ that k_
where A is the amplitude of the turbulence at 1 AU. This model is referred to as the modified anomalous diffusion (MAD) model.
The 3rd model makes use of the well-known basic expression for k__ given by
3 1 + m t where a is the particle gyro-frequency, and t is the scattering or relaxation time. Bieber and Matthaeus (1997) suggest that aTcan be expressed as
where Db _ describes large-scale field line wandering across B. The expression for Db _ is given by
In Eq. 2.34.8, Dsl describes the magnetic field wandering for slab turbulence, Asl is the amplitude of this turbulence, and lsi is its correlation length along B, while D2d describes the magnetic field wandering for 2D turbulence, A2D is the amplitude of this turbulence, and l2D is its correlation length across B (Matthaeus et al., 1995). The advantage of this approach is that there is a clear distinction between lc along and across B tied to the 2 components of solar wind turbulence. The MQLT model allows just for lc parallel to B. In addition, the approach is not limited to small amplitudes as QLT. Assuming cat >> 1, and D2D = 0, in Eq. 2.34.5 is the same as in Eq. 2.34.2 for slab turbulence. Although not well known, the expectation is that 12d >> lsl so that k_ is larger compared to the MQLT model in the limit cot >> 1 and smaller when cot << 1. Le Roux et al.
(1999a) use ¡2d = 100 lsi and for reference they call this model the nonperturbative (NP) model (see also Zank et al., 1998).
The dependence of the diffusion coefficients on radial distance r is theoretically determined with the MHD model for HMF turbulence transport in the solar wind according to Zank et al. (1996). The model gives a good reproduction of the observed r-dependence in the energy density of HMF fluctuations and also specifies the r-dependence of lc . Key elements in its success are the generation of turbulence by corotating interaction regions close to the Sun, and by isotropizing pickup ion (PI) ring distributions beyond the ionization cavity (r > 6 AU). In an extended version of the model (le Roux et al., 1999b), it is shown that near isotropic PI distributions can also damp turbulence for r > 30 AU, but that turbulence generation by PI still dominates. The increase in the energy density of the turbulence and the decrease in lc across the termination shock is estimated simply with the extended model.
In Fig. 2.34.1 are shown theoretically calculated mean free paths for 930 MV CR He+ with the MQLT model for k__ in the ecliptic plane as a function of increasing r from the Sun.
The radial mean free path Arr in Fig. 2.34.1 is calculated according to
where yis the Parker field spiral angle, and A__ = 3k_j_ / v is the perpendicular mean free path. The three important results to emerge from the MQLT model are as follows (see Fig. 2.34.1):
(1.1) Arr is determined solely by A// without any contribution from A__ so that large negative radial gradients in Arr exist for R << 1 GV.
(1.2) There is a big decrease in the magnitude of A// across the termination shock at
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