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Let us note again that in this Section we have used the model of particle scattering by very small scale inhomogeneities, in which the large scale field has no influence on the nature of the motion of particles at scales of the order of the correlation radius of the very small-scale field. A more general model which takes account of the spiral motion of particles at scales of the order of Lc has been discussed in Toptygin (1973, M1983). It follows from Toptygin (1973, M1983) that all the relationships obtained above keep their form when the influence of the large scale field at scales of the order of Lc is taken into account. It is only necessary to use an expression for A// which takes account of particle motion within the confines of the correlation region of the very small-scale field when doing specific calculations.

2.8.3. Diffusion of CR in a large scale random field

If the scattering frequency of particles by very small scale inhomogeneities significantly exceeds the characteristic frequencies of fluctuations of the large scale field, then the process of propagation of CR is diffusion of CR along the direction of the large-scale field:

Taking account of fluctuations of the large scale field and their influence on the propagation of CR on the basis of Eq. 2.8.52 was discussed by Ptuskin (1985) and by Zybin and Istomin (1985). Another approach to the calculation of the CR diffusion coefficient in a large scale field has been proposed by Berezhko (1985). This problem we discuss here using the functional method of averaging. Expanding the unit vector h(r, t) into a series in powers of the random field H1(r, t) and restricting ourselves to second-order terms, we shall average the expression obtained over the ensemble of realizations of the random field. As a result we obtain

[d + Lo N) — VraTal^H/VriN) + ViYalfr(H/HyVl) , (2.8.53)

where <N> is the average CR density, and

Lo —K- VraKonanlVrl, rcd/3 — naA/ (n) + nlAa/(n) Ya/ — Ko x{Aa//(n)AAY(n) + nanlY + nlnaj3Y } na/Y — 1 na(nfinY-SPy)-nP5aY• (2-8-54)

We shall make use of the Eq. 6.8.13 for averaging the terms on the right hand side of Eq. 2.8.53:

(H/(r,t)VriN[hJ) — |dr^B^r,t;r1,t1 )J^ (H))) ' (2'8'55)

We shall determine the functional derivative, which enters into Eq. 2.8.55 based on perturbation theory:

(^ ) = "Olr {v G(t;r1,t1 )}N(ibt1). (2.8.56) ¿H1r(lb t1)

where G is the Green's function of the operator d/dt + Lo . From Eq. 2.8.55 we have:

(h^ (r, t)VriN[H1 ]) = rgMr J dr\ dpG(p,r)|v (N(r,t)). (2.8.57)

Averaging of the second term on the right hand side of Eq. 2.8.53 leads to the relationship

(h^(i,tHy(i,t)VriN[H1] = B^yir,t;r,t^(N(i,t)). (2.8.58) Substituting Eq. 2.8.57 and Eq. 2.8.58 into Eq. 2.8.53, we obtain the equation:

where

KaX = KonanX + Yaty/Bfy( t;1, t) + raJuj3riAr 1dlG (p,T)VP?VPJB/Y (p'T) . (2.8.60)

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