P

where u is the velocity of matter (|u| << c) and Pg is the gas-dynamical pressure of plasma.

3.3. Effects of CR kinetic stream instability

The importance of CR kinetic stream instability effects for CR acceleration and propagation was first noted by Ginzburg (1965), and then by Wentzel (1974) and Cesarsky (1980). This type of instability is well known in plasma physics (Tsytovich, M1971; Vedenov and Rjutov, 1972; Akhiezer et al., M1974; Artsimovich and Sagdeev, M1979). The CR kinetic stream instability generates a broad spectrum of waves in space plasma, but generation of high-frequency waves (Langmuir and whistler types) is not effective because of a big damping of these waves; moreover, these waves are not effective for CR particle scattering. Another situation is for magneto-hydrodynamic waves that are effective for CR scattering. The growth rate is the largest for waves propagated along magnetic field. The main interaction between CR particles and these waves is based on cyclotron resonant on the first harmonic; the growth rate rc (k) will be determined by equation (Berezinsky et al., M1990):

dp kv

7eHc

7eHc

Here ^ is the cosine of particle's pitch-angle, ma (k) is the frequency of an Alfven wave with wave number k = 2n/A . Let us suppose that the CR distribution function is characterized by isotropic part fo (p) and some small anisotropy with amplitude A which implies:

v dp

is the effective velocity of the observer relative to the system of coordinate with CR isotropic distribution function fo(p) to obtain anisotropy with amplitude A. If we take into account only resonance scattering with

pres

7eH

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