and x = r,p,t; r is the coordinate, p is the momentum, v is the particle velocity; t is the time; Ho is the strength of the regular magnetic field, e is the particle charge, c is the speed of light, u is the velocity of the magnetic field, and G^x, x1) is the one-particle Green function that is the solution of the linear kinetic equation. Mel'nikov (2005a) choused the correlation tensor of the random anisotropic magnetic field H1 for a power-law spectrum in the form (Matthaeus et al. 1990; Toptygin 1985; Chuvilgin and Ptuskin 1993):
and k is the wave vector, v is the spectral index, ho = Ho/Ho^ k// = (kho)ho^ k± = k-k//^ q// = (qho)ho^ q± = q-q^ (2.33.5)
q± = 2nL—, q// = 2nL-K Av = 2^2 ♦ l^qf (h?^^j]"1 ,(2.33.6)
r is the Gamma function. Passing to the drift approximation, we obtain:
where the coordinate z is along the vector ho, O = (f) , 0 is the angle between p and ho , u = cos 0, y is the azimuthally angle between p and ho . The nonlinear average collision integral is
where the kinetic coefficient is b(u) = —er~2 JdT\dkP(k)cos(pk -p)cos(pk -p-Q.T)T0(m)exp(/'kAr(r)),(2.33.9)
m is the particle mass, pk is the azimuthally angle of the vector k, Q is the gyro-frequency in the regular magnetic field, a is the gyro-frequency in the random magnetic field, Ar(r) is the change in the radius vector of the particle in the regular magnetic field, r0 (m) is a factor that is related to the additional Green function of the particle in the nonlinear collision integral and that yields the damping of the resonant wave-particle interaction:
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