Assuming it to be isotropic one finds Bzz ~ 3.5 ^Gs. The corresponding component can be smaller in the galactic halo, say =1.0 ^Gs. In that case the results of the calculations of Zirakashvili et al. (1996) with 1.0 ^G regular magnetic field can be used. On the other hand, larger values of magnetic field

strength are also possible. Zirakashvili et al. (2001) take the value Bzz ~ 3.0 ^Gs for the calculations described here. In addition it was used a smaller value of CR

pressure Pco = 1.0 x 10 erg/cm at the base level 3 kpc over the disk in order to maintain approximately the same cosmic ray energy flux in comparison with

calculations of Zirakashvili et al. (1996). A gas number density no = 10 cm at this base level was assumed. Radiative cooling losses are relevant for denser gas at smaller heights above the disk. Numerical results for the flux tube originating at galacto-centric distance ro = 8.5 kpc (Sun's position) are shown in Fig. 3.15.2 and Fig. 3.15.3.

Fig. 3.15.2. Variation of the meridional and azimuthal velocities us and u^, azimuthal and meridional magnetic field strength and BlJs2, and gas temperature T, with distance from the disk. The resulting initial velocity is uo = 31.5 km/sec, and the critical points positions are zs = 7.1 kpc, za = 21.6 kpc, and Zf = 84.8 kpc. The terminal velocity is uf = 698 km/sec. According to Zirakashvili et al. (2001).

Fig. 3.15.2. Variation of the meridional and azimuthal velocities us and u^, azimuthal and meridional magnetic field strength and BlJs2, and gas temperature T, with distance from the disk. The resulting initial velocity is uo = 31.5 km/sec, and the critical points positions are zs = 7.1 kpc, za = 21.6 kpc, and Zf = 84.8 kpc. The terminal velocity is uf = 698 km/sec. According to Zirakashvili et al. (2001).

Fig. 3.15.3. Variation of dynamic pressure pu2s , CR pressure Pc , gas pressure Pg , and magnetic pressure B2 ¡8n with distance z from the disk. According to Zirakashvili et al. (2001).

The height of the slow magneto-sonic point is practically the same zs = 7.1 kpc above the disk. The Alfven point za = 21.6 kpc and the fast magneto-sonic point z f = 84.8 kpc move further out into the flow. The initial wind velocity at the base level is uo = 31.5 km/sec, and the terminal velocity is Uf = 698 km/sec. The magnetic pressure dominates gas and CR pressures practically everywhere. Nevertheless, this is a CR driven wind because CR givs approximately half of the kinetic energy flux, the second half given by rotational effects (see Eq. 3.15.15).

On the basis of the results described, Zirakashvili et al. (2001) conclude that the inclusion of the random field component in the galactic wind model results in the possibility that our Galaxy is surrounded by a large wind halo with a rather strong magnetic field even though this field strength is probably an upper limit. The field geometry is rather simple. The magnetic field disturbances are nearly isotropic in the galactic disk and have a size of about 100 pc. They are strongly elongated in the galactic halo. The elongation estimated is 1:10-1:100. This magnetic field leads to an effective angular momentum transport and a correspondent additional centrifugal acceleration of the flow, which then results in larger terminal velocity of the wind. At large distances the field is practically azimuthal and sign dependent with fluctuating direction. one can expect that CR diffusion in such a field is highly anisotropic, enhanced diffusion being in direction of the elongation. It can be also expected that high energy protons with energies larger than 3 x1017eV are hardly held by such a sign dependent field. The gas heating owed to damping of Alfven waves generated by the CR streaming instability is rather effective, the wind halo being filled by a hot rarefied gas with a temperature of about one million degrees. The angular momentum loss rate of the Galaxy is mainly owing to magnetic torque and is about 50% in 1010 years.

3.16. Nonlinear Alfven waves generated by CR streaming instability and their influence on CR propagation in the Galaxy

3.16.1. On the balance of Alfven wave generation by CR streaming instability with damping mechanisms

Zirakashvili et al. (1999) consider Alfven wave generation by CR streaming instability and nonlinear damping of parallel propagating Alfven waves in high-P plasma. There was also taken into account trapping of thermal ions and Coulomb collisions, saturated damping rate be calculated, and applications was made for CR propagation in the Galaxy. As it was considered above, the CR streaming instability can play an important role in processes of diffusive shock acceleration and CR propagation in the Heliosphere and in Galaxy since it can supply Alfven waves that scatter the particles on pitch angle (Lerche, 1967; Kulsrud and Pearce, 1969; Wentzel, 1969). In order to balance 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. Such waves are not subject to any damping in incompressible magneto-hydrodynamics. The use of compressibility results in a pondermotive force gives a second order plasma velocity 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 saturation of wave damping was also understood many years ago (Kulsrud, 1978; Volk and Cesarsky, 1982). Corresponding saturated damping rates that 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) derive the nonlinear Alfven wave damping rate in the presence of thermal collisions.

3.16.2. Basic equations and their solutions

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

Here F is the velocity distribution of thermal particles that is averaged over the gyro-period, v is the particle velocity, b=B/B is the unit vector along the magnetic field B, ^ = pB/Bp is the cosine of the pitch angle of the particle. The right hand side of Eq. 3.16.1 describes collisions of particles. For Maxwell's equations it is necessary to know the flux of particles. It is given by drift theory (Chandrasekhar, M1960):

where Q is the particle gyro-frequency in the local field. The last term on the left hand side of Eq. 3.16.1 describes 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 we shall deal only with the time asymptotic state in the following. We shall use 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 s along the magnetic field, and the distribution function f = F - Fm one obtains the following equation for f:

ds du ds

du ds

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. 3.16.4. 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 the standard quasi-linear theory according to Galeev and Sagdeev (1979). The function f can be written in the form f = fo + Sf, where fo = (f) is the ensemble averaged distribution function f. We 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 ^ << 1 particles we leave in the collision operator the second derivative on ^ only and come to the equation:

4 du ds du

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

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