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Fig. 3.4.1. Profiles of CR-plasma systems in the hydrodynamic approach. The parameters are the following: O = 1.0, Fm = 1.6, Ftot = 31.26 and u = 4.0, Pth = 0.4, Pc = 0.8,

P+ = 0.1, P- = 0.25 at x = 0 ; moreover © = 1.842, Ptot = 7.95, WA = 12.82. According to Ko (1999).

Fig. 3.4.2. Profiles of CR-plasma systems in the hydrodynamic approach. The parameters

Fig. 3.4.2. Profiles of CR-plasma systems in the hydrodynamic approach. The parameters are the following: O = 1.0, Fm = 1.6, Ftot = 26.30 and u = 4.0, Pth = 0.4, Pc = 0.8

Ptot

P+ = 10 6, Pw = 0.2 at x = 0; moreover © = 1.842, Ptot = 7.80, WA = 6.25. According to

Ptot

Fig. 3.4.3. Profiles of CR-plasma systems in the hydrodynamic approach. The parameters are the following: O = 1.0, Fm = 4.0, Ftot = 63.53 and u = 4.0, Pth = 1.0, Pc = 0.8,

P+ = 0.4, P- = 0.01 at x = 0; moreover © = 1.0, Ptot = 18.21, WA = 35.65. According to Ko (1999).

Fig. 3.4.4. Profiles of CR-plasma systems in the hydrodynamic approach. The parameters are the following: O = 1.5, Fm = 1.0, Ftot = 70.4 and u = 10.0, Pth = 0.02, Pc = 10-8,

P+ = 0.4, P-= 0.2 at x = 0 ; moreover © = 0.9283, Ptot = 10.62, WA = 34.33. According to Ko (1999).

The most significant features in Fig. 3.4.1-3.4.4 are the flow and pressure profiles which can be non-monotonic, and are in sharp contrast with systems without waves or systems with a unidirectional wave. Fig. 3.4.1 is a reminiscence of the non-linear test particle picture of Jiang, Chan and Ko (1996), where the CR pressure can be increasing non-monotonically. In Fig. 3.4.2 the downstream state closely resembles a system without forward wave (P+ = 0), but the upstream state is totally different. Fig. 3.4.3 shows a prominent peak in velocity and a valley in backward wave pressure, while Fig. 3.4.4 shows the opposite. In these examples the CR pressure far downstream is always larger than the CR pressure far upstream, i.e., CR are always accelerated.

According to Ko (1999) the rich morphology of structures is the result of the interplay between the three basic energy transfer mechanisms (see Eqs. 3.4.3-3.4.6): (i) work done by plasma flow; (ii) CR streaming instability; and (iii) stochastic acceleration. Thus (i) and (ii) are facilitated by pressure gradients, (ii) and (iii) involve energy exchange between CR and waves. Ko (1999) notes that (i) and (ii) can accelerate or decelerate CR, while (iii) can only accelerate. As shown in the non-linear test particle picture, work done by plasma flow is, in general, the major accelerating mechanism for CR. The relative contributions of the three mechanisms along x produce the fine details of the profiles.

Let us note also that according to Ko (1999) the other class of CR-plasma systems with a sub-shock is rather complicated mathematically, but one thing is clear: the profile structure ought to be qualitatively different from the structure of systems without waves. Besides being non-monotonic, the downstream state will not be uniform (recall that the uniform state is the only physically allowable downstream state available to systems without waves). As far as the upstream state has a wave, both forward and backward waves are generated downstream by the shock. When CR and both waves are present, no uniform state is possible because of the stochastic acceleration.

3.5. Nonlinear Alfven waves generated by CR streaming instability

3.5.1. Possible damping mechanisms for Alfven turbulence generated by CR streaming instability

In Sections 3.1 and 3.3 there was mention that the CR streaming instability can play an important role in processes of CR particles' diffusive propagation through space plasma and in diffusive shock acceleration since it can supply Alfven waves which scatter the particles on different pitch angles (see also Lerche,1967; Kulsrud and pearce,1969; Wentzel, 1969). In order to balance Alfven wave generation some damping mechanism is usually considered. As Alfven waves are weakly linearly damped, various nonlinear effects are currently used. CR streaming generates waves in one hemisphere of wave vectors. It is well known that such waves are not subject to any damping in incompressible magneto-hydrodynamics. Using compressibility results in a ponder-motive force that gives a second order plasma velocities and electric field perturbations along the mean magnetic field. These perturbations can yield wave steepening as well as nonlinear damping, if kinetic effects of thermal particles are included. Those effects were taken into account in order to obtain nonlinear damping rates of parallel propagating Alfven waves (Lee and Volk, 1973; Kulsrud, 1978; Achterberg, 1981). The importance of trapping of thermal particles for nonlinear dissipation of sufficiently strong waves that results in the saturation of wave damping was also understood many years ago (Kulsrud, 1978; Volk and Cesarsky, 1982). Corresponding saturated damping rates which take into account dispersive effects were calculated. Nevertheless dispersive effects can be rather small for Alfven waves that are in resonance with galactic CR nuclei. Hence the effect of Coulomb collisions can be important. Zirakashvili et al. (1999) derived the nonlinear Alfven wave damping rate in presence of thermal collisions.

3.5.2 Basic equations described the nonlinear Alfven wave damping rate in presence of thermal collisions

Zirakashvili et al. (1999) consider Alfven waves propagating in one direction along the ambient magnetic field. It is convenient to write the equations in the frame of coordinates moving with the waves. In such a frame there are only quasi-static magnetic and electric fields slowly varying in time owed to wave dispersion and nonlinear effects. The case of a high-P Maxwellian plasma is considered. Electric fields are negligible for nonlinear damping in such a plasma. Zirakashvili et al. (1999) investigated waves with wavelengths much greater thermal particles gyro-radii and used drift equations for distribution function of those particles (Chandrasekhar, M1960):

were F is the velocity distribution of thermal particles that is averaged on gyro-period, v is particle velocity, b = B/B is the unit vector along the magnetic field B, p = pB/B is the cosine of the pitch angle of the particle. The right hand side of Eq. 3.5.1 describes collisions of particles. For the Maxwell equations it is necessary to know the flux of particles. It is given by drift theory (Chandrasekhar, M1960):

were Q is particle gyro-frequency in local field. The last term on the left hand side of Eq. 3.5.1 describes the mirroring of particles. Because the field is static in this frame, the particle energy is constant, and in a time asymptotic state wave dissipation is absent without collisions. In the presence of wave excitation it will only deal with the time asymptotic state in the following. It will be used for the collision operator a simplified form

where Fm is the Maxwellian distribution function shifted by the Alfven velocity Va ; Av is the Laplace operator in velocity space and v is the collision frequency. This operator tends to make the particle distribution function Maxwellian. Introducing the coordinate 5 along the magnetic field, and the distribution function f = F - Fm one obtains the following equation for f df 1 - u2 df d ln B(s) A 2r 1 -j2 dFM d In B(s)

VjU—---— v—--^-Avvv2f = —— v—M--^ . (3.5.4)

For sufficiently small magnetic field perturbations (conditions for that case will be derived later) one can neglect the mirroring term on the left hand side of Eq. 3.5.4. Without collisions this leads to the well known nonlinear damping mentioned above. Zirakashvili et al. (1999) take into account the mirroring term here and use standard quasilinear theory (Galeev and Sagdeev, 1979). The function f can be written in the form f = fo + ff, where fo = (f) is the ensemble averaged distribution function f. They are interested in the case of a small magnetic field amplitude A << 1, where A = (B - Bo )/Bo. Taking also into account that mirroring is sufficient for small p << 1 of particles they leave in the collision operator the second derivative on u only and come to the equation:

Taking into account that average distribution function is s independent one can obtain equation for Fourier transform ffk = Jdsff (s)exp(- isk):

dp2 4'"'"* dp ikvfk - v^ =1 ikv4^(o + Fm). (3.5.6)

The functions fo and ffk are peaked near u = °. It is convenient to introduce the Fourier transform on

UUfo= Jdjfo(U)exp(- iUj and fffk= Jdjffk(U)exp(- ijj. (3.5.6a) Then Eq. 3.5.6 will transmit into kv f + ve$k = -M f 2nf (^)dFM| u=° + ifo . (3.5.7) d£ 4 I dj )

This equation has a solution

After ensemble averaging of Eq. 3.5.5 and using Eq. 3.5.8 one obtains an equation for f (F):

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