asymptotic behavior of the Bessel function was used and cos x replaced by its mean value cos2 x = 1/2. The item 3 describes the particles which are in

Cherenkov resonance with the waves (Galperin et al., 1971).

Generally speaking, as we consider the particles with a velocity far more than the Alfven velocity (va ~ 60 km/sec in the solar wind plasma), it could appear that the Cherenkov resonance is of no importance but the interaction is caused by cyclotron resonances of all orders. However, the Cherenkov resonance must be included when evaluating the time of isotropization and acceleration of particles.

The limiting transition y^ 0 was carried out in all of the items' cyclotron summands (in all items except for ¡3) when calculating D1,D2,D3 (Galperin et al., 1971). This approximation is not applicable for calculation of 3 as the limit condition y ^ 0 means that the effective time particle interaction with the waves appears to be infinitely large. In fact, the presence of an imaginary part of a frequency gives a finite width to the region of interaction of a separate Fourier harmonic of the wave with moving particles. This property results in the occurrence of a finite interaction time between a wave and moving particles. As is known (Braginsky, 1963), the Alfven waves fade out with the decrement of fading where cr± is the coefficient of the transverse conductivity of plasma. Substituting the Eq. 2.6.25 for y(k ) in Eq. 2.6.18 we obtain the following expression for ¡3 (at r(k) = c2k2j4na1_ ,

The importance of the Cherenkov resonance was emphasized by Galperin et al., 1971 (see also Vedenov et al., 1962). The Eq. 2.6.17-2.6.27 together with Eq. 2.6.12 describe completely a particle motion in low-turbulent magnetized plasma. The Eq. 2.6.12 at D2 = D3 = 0 represents the process of particle diffusion in the angular space with the energy conservation. This case was studied in detail by Tverskoy (1967b). The equation of the type of Eq. 2.6.12 at ¡ = 0 was minutely investigated also in the works of Tverskoy (1967a,b). Another possible cause of broadening of the Cherenkov resonance is the scattering of particles which was studied in details by Galperin et al. (1971).

2.7. Green's function of the kinetic equation and the features of propagation of low energy particles

Let us write Eq. 2.6.12 in the spherical coordinates in the momentum space (pz = pcos9, p± = psin9):

dt dz 2V ' d9 p2sin9d9 99 d9

psin9d9 dp p2 dp d9 p2 dp dp where

Dee = p2D1 + cos 0(D3 cos G - 2pD2 ), Dpp = D3 sin G,

The coefficients D1, D2, D3 are determined by Eq. 2.6.13-2.6.24 with the corresponding substitution of variables. At Dgp = DpG = Dpp = 0 the Eq. 2.7.1

describes a diffusion process in the angular space taking place with energy conservation. Let us consider the Eq. 2.7.1 for this case. In the general case with arbitrary values G of a particle's pitch-angle it was not possible to obtain a solution of the Eq. 2.7.1; however for the angles 6<< 1 there is the analytical solution of the Eq. 2.7.1. At 6<< 1 the diffusion coefficient in the angular space is determined by the Eq. 2.6.22-2.6.24 where vz = v cosG. Using Eq. 2.6.22-2.6.24

and Eq. 2.7.2 we write the Eq. 2.7.1 for the stationary case including in the right-hand part of the equation a point source with the coordinates z0 and 0o, i.e. consider the equation for the Green's function G// (z) of the kinetic equation:

Was this article helpful?

0 0

Post a comment