The last term in the right hand part of Eq. 2.11.11 describes the adiabatic cooling of charged particles associated with radial divergence of solar wind plasma with the frozen in magnetic field inhomogeneities. A consistent derivation of the Eq. 2.11.11, on the basis of the kinetic equation was first considered by Dolginov and Toptygin (1966a,b).
2.11.2. Including of magnetic inhomogeneities velocity fluctuations
The Eq. 2.11.11 is obtained with the assumption that a proper motion of magnetic inhomogeneities is neglected, i.e. u1 = 0 . Including a stochastic velocity of magnetic inhomogeneities is equivalent to appearance of stochastic electric fields resulting in acceleration of particles (Fermi mechanism of acceleration). Owing to general properties of the Fermi acceleration mechanism (see Chapter 4) the necessary condition for the efficiency of this mechanism is a high degree of isotropy of particle distribution in the momentum space. Therefore the acceleration of particles can be considered in a diffusion approximation. The procedure similar to that used in deducing equation Eq. 2.11.11 results in the equation of anisotropic diffusion including the effect of particle acceleration (Dolginov and Toptygin, 1967):
= -r— Kap(x, p)--uoa — + + ——p2£(r, p)—, (2.11.12)
is the coefficient of particle diffusion in the momentum space; ^u2^ is the mean-square velocity fluctuation.
2.11.3. Diffusion approximation including the second spherical harmonic
Dorman, Katz and Fedorov (1977, 1978a,b), basing on the kinetic equation which includes an interaction of charged particles with stochastic magnetic fields in space, have obtained a set of equations for the diffusion approximation taking into account the second spherical harmonic. Let us start from the kinetic equation describing a propagation of CR in magnetized moving plasma dt+ Lo )F( P, t) = Da IdTexp(- Lgt)
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