is integral CR momentum spectrum and ni is the plasma density. Eq. 3.3.5 shows that rc (k)> 0 for Uo > Va . This condition is necessary for the development of CR stream instability and the generation of Alfven turbulence.

For the spectrum described by Eq. 3.3.6 the increment rc (k)«: kY-1, but if the spectrum has a maximum at p = p* then for p < p* (i.e. for k > k*) the increment rc (k)^ k-1. Here it is assumed that rc (k)<<m(k), i.e. that AWc << H2/4n. Usually this condition is valid because the amplitude of the anisotropy A < 0.1 and CR energy density Wc < H2/4n (in the Galaxy, in the main part of Heliosphere, in the processes of particle acceleration by shock waves, and in regions of magnetic field reconnection). Perhaps only in very powerful compact sources of CR can the accelerated particle anisotropy and energy density be so high that it becomes necessary to consider a stronger approximation.

3.4. On the structure and evolution of nonlinear CR-space plasma systems

3.4.1. Principles of hydrodynamic approach to the CR-space plasma nonlinear system

As was considered in Chapters 1 and 2, and in the previous Sections 3.2 and 3.3, CR interact with thermal plasma via magnetic clouds, hydromagnetic irregularities, and hydromagnetic waves in the plasma. Scattered by the magnetic irregularities (mostly by gyro-resonant scattering), CR propagate and diffuse through the plasma. CR acquire energy from the plasma if the plasma flow is systematically converging. This process is called the first order Fermi acceleration. Because CR are anisotropic their interaction with the plasma excite hydromagnetic waves via streaming instability. When waves of different phase velocities are present, CR diffuse in the momentum space also. This is called the second order Fermi acceleration, or stochastic acceleration. The system is self-consistent and is called a CR-plasma system (Ko, 1999).

As one can imagine solving the system in distribution function approach is very difficult (see e.g., Malkov, 1997a,b). On other hand, basing on the papers of Drury and Volk (1981), Axford, Leer, and McKenzie (1982), McKenzie and Volk (1982), Ko (1992), authors Jiang et al. (1996), Ko et al. (1997), Ko (1998, 1999) came to conclusion that the hydrodynamic approach is a fairly good approximation for studying the structure and evolution of any CR-plasma system. In this approach every component is considered as a fluid. For instance, CR and waves are treated as massless fluids but with significant energy density and pressure. For example, Ko (1999) consider a four-fluid model which comprises the thermal plasma, CR, and two oppositely propagating Alfven waves. It was shown that in general there are three energy exchange mechanisms: 1) work done by plasma flow, 2) CR streaming instability, and 3) stochastic acceleration. In Ko

(1999) are presented several steady state profiles of the CR-plasma system which demonstrate the interplay between these three energy exchange mechanisms.

3.4.2. Four-fluid model for description CR-plasma system

Ko (1992) proposed a fairly comprehensive version of the hydrodynamic approach. That is a four-fluid model which comprises thermal plasma, CR and two oppositely propagating Alfven waves. The governing equations are the total mass and momentum equations, and energy equations of various components (i.e., kinetic energy and thermal energy of plasma, CR energy and wave energies). In the one dimensional approximation with magnetic field parallel to the plasma flow, these equations will be

dt dx

dEth dEth dPth .

dt dx dx T

dt dx dx dx 2t where p and u are the density and velocity of the plasma; the indexes k, th, c and w denote the kinetic part of the plasma, the thermal part of the plasma, CR and wave parts, respectively; and ± denote forward and backward propagating waves. The energy densities and energy fluxes are given by:

Ek =(1/2)Pk = (1/2)pu2, El = Pth/(g -1), Ec = Pj(c -1), E± = 2PW,

Fk = uEk, Fth = u(Eth + Pth I Fc =(u + Va (e+- e-)))Ec + Pc )-K(Ec/dx I


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