It is necessary for subsequent analysis of the kinetic equation to specify an explicit form of the tensor Ba^(p,t). The experimental data of Matthaeus and Smith (1981) show that fluctuations of the magnetic field are statistically anisotropic; the spectrum of the fluctuations is axe-symmetric with respect to the direction n of the regular field, so that the spectral tensor

B^M — i(H12){MAa,(k) + B1 (k, r\nan, + x2 xnakl++l )

+ k "1B2 {k,T\na\kn]x + nA[kn]a)+ xk "1B2 (k,r)(([kn]ii + kA[kn]a) + 1B3 (k, T)£aXy (( / k )}, (2.8.29)

where B, B1, and B3 , are even and B2 is an odd fonction of the wave vector k, Aq-ao*) = ^aA - kakAk2 , x = nk/k , and e^iy is a unit vector of the third rank. Using Eq. 2.8.29 we obtain from Eq. 2.8.28:

D = -(H2) JdrJdkdp1Gpp1 (k,r)-^ jB(M) + -2B1 (k,r)l, (2.8.30)

where Gpp1 (k,r) is the Fourier transform of the Green's function, and k// = nk, k= [n[kn]]. It is evident from Eq. 2.8.30 that D = 0 in cases of one-dimensional and two-dimensional turbulence. If the regular field is uniform in space then the Green's function (at x << 1) is

Gpp1 (pt) = -JTS( - P1 )(P - V//t) 5 expi-^tYl + -2>i (ng)Pl (ng1 ), (2.8.31)

where rs = A/4V is the scattering time of particles by very small scale inhomogeneities of the field, ng = np/p, ng1 = np^ p1, and Pl (x ) is the Legendre polynomial. Taking account of Eq. 2.8.31, we obtain from Eq. 2.8.27 and Eq. 2.8.28:

KiaA=nV// 1 dTexP BaÀ(v//T,T), D = Jdrf±paV±pÀBaÀ{pAp=VllT . (2.8.32)

If the spectral tensor B^(k,r) is defined by Eq. 2.8.29, then the coefficient

D = -(flf) J dTj dk exp(zkV//r)-=U^ jB(k,r) + -2 Bi(k,rH, (2.8.33)

and it vanishes in the case in which the tensor Ba^ (k,T) does not explicitly depend on time or in the case of frozen turbulence.

2.8.2. Diffusion approximation

We shall write Eq. 2.8.22, in which we shall set D = 0 , in the form dF/dt + VJ = StF, (2.8.34)

Ja(r,P, t) = V//VaF(i%P, t) - K±aAV±rAF(r,P, t), (2.8.35)

where Ja(r,p,t) is the particle flux in space. It is assumed in writing Eq. 2.8.34 and Eq. 2.8.35 that the distribution function varies weakly at spatial scales of the order of the correlation radius of the large scale field. The particle flux with the specified magnitude of the momentum is

where J// is the particle flux along the direction of the regular field and Jla is the particle flux across the direction of the regular field. The bar in Eq. 2.8.36 and Eq. 2.8.37 denotes averaging over pitch-angle, and x = ng = cos# .

Representing the distribution function in the form of a series of Legandre polynomials and restricting ourselves to two terms of the expansion

where N = F(r,p,t) is the particle density, we obtain from Eq. 2.8.34 a system of equations of the diffusion approximation

^ + Vrnj//(r,t) = VxrOV2 idTexp(- t/ts )x2baä(//Tp^N(r,t -t), (2.8.39) dt 0


1 21

J H (r, t ) = -- V/2 J d/ exp(- t/ts X^Vr )X (r, t - /). (2.8.41)

Substituting Eq. 2.8.41 into Eq. 2.8.39, we obtain the transport equation

^ = Vr J d/exp(- /// NaA(/NrAN (r, t - /), (2.8.42) dt 0

If the scattering time ts is small then small t makes the main contribution to the integrated term in Eq. 2.8.42. In the region of small t the tensor Bai can be replaced by its value at zero. As a result we obtain dN(l% t^ = Vra*aA(0)jdTexp(- t/ts )aN(r, t - t), (2.8.44)

dt which reduces to the telegraph equation (for example, see Earl, 1976; Dorman, Fedorov et al., 1983):

dt S dt2

For large t Eq. 2.8.45 changes into the diffusion equation (Toptygin, 1973, M1983). As follows from Eq. 2.8.45, the diffusion coefficient is kocA(t)= KaA(°)Ts = k//VaVA + KLoA k// = 3VA//> A// = Vts , (2.8.46)

If the tensor Baa is defined by the Eq. 2.8.29, the diffusion coefficient across the lines of force of the regular field is

where the mean free path across the lines of force of the regular field is

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