Substituting the solution of Eq. 3.5.16 into Eq. 3.5.12 and expanding the exponent one can obtain the saturated non-linear damping rate:

where

~T T(v4)0 WFVTexP

3.5.3. On the possible role of nonlinear damping saturation in the CR-plasma systems

According to Zirakashvili et al. (1999), the trapping of thermal particles is essential for the damping of Alfven waves if the frequency of collisions is small enough. For trapped particles |u| < ¡u*, where ¡u* ~ SB/B for Alfven waves. Hence the escape time is tesc /v. (3.5.20)

It should be compared with the period of particle oscillations inside the trap,

This gives the condition for saturation of nonlinear damping:

The saturated damping rate can be estimated as the unsaturated damping rate multiplied by the ratio T/tesc . It is easy to see that such an estimate is in accordance with Eq. 3.5.18.

In the self-consistent model of galactic wind flow developed by Zirakashvili et al. (1996), Ptuskin et al. (1997) where the unsaturated damping rate was used,

(SB/B) 10-2 and is determined by the power of CR sources in the Galactic disk. For this case the critical value for the collision frequency is 10-12 sec-1 for a

wavenumber k ~ 10 cm that is in resonance with 1 GeV CR protons. This value is close to the value of the collision frequency of a hot rarefied plasma with number density 10- cm~ and temperature 106 K. Therefore in the absence of other scattering processes, trapping effects might be relevant for Alfven wave damping in our Galaxy (see in more details below Sections 3.13-3.14).

Another important feature of saturated damping according to Zirakashvili et al. (1999) is the possibility of not only damping but also energy transfer to smaller wavenumbers. This property is absent for unsaturated damping of unpolarized (((k) = I(- k)) waves. Such energy transfer can be important for diffusive shock acceleration because it permits small energy particles to generate Alfven waves that are in resonance with particles of greater energies and, hence determines the rate of acceleration.

3.6. Interplanetary CR modulation, possible structure of the Heliosphere and expected CR nonlinear effects

3.6.1. CR hysteresis effects and dimension of the modulation region; importance of CR nonlinear effects in the outer Heliosphere

The studies of the neutron component data have made it possible to find the hysteresis character of the relationships between the variations in solar activity (SA) and in CR intensity (Simpson, 1963; Dorman and Dorman, 1967a,b,c,d; Dorman. M1974, M1975a). This effect arises from the delay of the interplanetary processes (responsible for CR modulation) with respect to the initiating solar processes which correspond to some effective velocity of solar wind propagation (see Fig. 3.6.1 for ro/u = 10 months and Fig. 3.6.2 for ro/u = 20 months, where ro is an effective radius of modulation region and u is an effective radial velocity of solar wind).

= 1° months (which corresponds to ro = 67 AU at u = 4 x 1° cm/s) as function of sunspot number W flattened over the 12 month period and comparison with observation data of the neutron component in Chicago, also flattened over the 12 month period (circles connected with the corresponding theoretical values by the dashed vertical lines, the right hand scale).

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