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If the regular field is uniform in space, then having made use of the well known Green's function of the operator d/dt + Lo , we obtain from Eq. 2.8.60:

Consequently small fluctuations of a large scale field do not lead to diffusion of CR across the direction of the regular component of the large scale magnetic field, in accordance with Ptuskin (1985). As is evident from the results of Dorman, Katz, and Stehlik (1988) described in Sections 2.8.1-2.8.3, if the characteristic time of fluctuations of the large scale magnetic field significantly exceeds the scattering time of particles by very small scale inhomogeneities of the magnetic field, diffusion of the CR is described by the telegraph equation, in accordance with Earl (1976) and Dorman et al. (1983), which takes account of the presence of diffusion of CR across the lines of force of the regular field. The mean free path of particles across lines of force is related by the Eq. 2.8.49 to their mean free path with respect to scattering by very small scale inhomogeneities and is determined in the case of anisotropic turbulence of the large scale field (see Eq. 2.8.29) by the two spectral functions B and B1.

2.8.4. CR transport in the random girotropic magnetic field

The problem of propagation of CR in the random girotropic magnetic field has been discussed by Fedorov et al. (1992) and Dolginov and Katz (1994). These examinations are a major preoccupation to the investigation of particles motion in the small-scale random girotropic magnetic field. Katz and Yacobi (1997) considered the effects are owed to existence of the large scale magnetic field. The influence of small-scale magnetic field provides the effective particles scattering whereas the nonzero helicity of the turbulence leads to the particles acceleration. Katz and Yacobi (1997) obtained the drift kinetic equation including these effects and derive the kinetic coefficients describing the particles propagation at these conditions.

The distribution function f (r,p,t) of the ensemble of non-interacting particles moving in the small scale random girotropic magnetic field obeys the equation (Fedorov, et al., 1992; Dolginov and Katz, 1994):

A+ Wr F ' dt r 9p where e f (p,r, t ) = Stf (r, p, t ), (2.8.62)

c is the force acting on the particle with charge e and momentum p corresponding to the velocity V = cp/E and energy E; u is plasma velocity which transfers the large scale magnetic field H frozen into it. The second term in Eq. 2.8.63 is owed to particles acceleration in the small scale random girotropic magnetic field. The coefficient a is expressed in helicity terms of the turbulence. The collision integral Stf (r,p,t) in Eq. 2.8.62 has Fokker-Planck form:

s PaPA saA

where A is particle transport path respectively its scattering on the random inhomogeneities of the small scale magnetic field. If the magnetic field H is sufficiently strong, it is conveniently to use the drift approximation (Toptygin, M1983). In the drift approximation Eq. 2.8.62 reads f + vrR/ + JLf + A_dpIL f = f , (2.8.65)

R = Fllh + u±, h = H/H, ddpj = -Vllp|vrh, dplL =1 Vj2p±Vrh-a(hH), (2.8.66)

dt dt 2

and where /(r, p//, pj, t) is the drift distribution function and indices j and ll mark particle momentum components across and along the direction of the magnetic field H. The symbol (...)h denotes averaging over directions of the momentum vector in the plane normal to vector h. If the large scale magnetic field H is random function of the coordinates and time we have to average the Eq. 2.8.65 over the fluctuations of the large-scale magnetic field. This may be performed if the random component of the large scale magnetic field is small: H = Ho + Hx, where Ho is the regular component and Hx is the random component. Along with this we have to change the variables in Eq. 2.8.62 on the other that are related to the direction of the large scale regular magnetic field Ho. In this case the Eq. 2.8.62 will have the following form

/ + Vrdf/ + ~~TÍ1 / + IP"d-/ = <S/)n • (2.8.67)

dt dt dpi dt dpn dt x /n where

^T = V// {(1 - Hi )n + (1 - H1// )u }+ u± - (uHu )n - (un)Hu, (2.8.68)

In Eq. 2.8.67-2.8.70 the indices j and ll mark the components associated with the direction n = Ho/Ho of the regular magnetic field Ho. The Eq. 2.8.67 is described with accuracy of terms second order relative to the random magnetic field Hl . The last is measured in the units of Ho. The next step according to Katz and Yacobi (1997) is averaging Eq. 2.8.67 over the ensemble of realization of the random magnetic field, and velocity u: f — F . Assuming the Gaussian distribution for random fields we obtain dF d

oo r

= I dTI dxF//Gp (x,t)b«1(x,t) + V// rp (x,t)o4(x,t)

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