2.1. The problem of CR propagation and a short review of a development of the basically concepts

Propagation of charged particles of CR in space plasmas (interstellar and interplanetary media, intergalactic space, various types of CR sources) is one of the actual problems of CR astrophysics and geophysics. The foundations of the theory of CR interaction with magnetized space plasma where developed more than fifty years ago: Fermi (1949) showed that charged particles of CR in a 'collision' with 'clouds' of magnetized plasma moving in the inverse direction should be accelerated, and in the opposite case - decelerated. Detailed studies of the processes of CR propagation in space began, however, after giant solar flare event at 23 February 1956 and International Geophysical Year (IGY) in 1957-1958. These studies were stimulated by extensive development of experimental methods connected with the formation of the world-wide network of ground based neutron monitors (NM) and muon telescopes (MT), and with direct observations in the magnetosphere and in space by satellites and spacecrafts.

The initial attempts at forming a statistical theory of CR propagation were based on a simple model of isotropic diffusion (Meyer et al., 1956; Dorman, M1957, 1958, M1963a,b; Parker, M1963; Ginzburg and Syrovatsky, M1963; Dorman and Miroshnichenko, M1968; Dorman, M1975a,b; Berezinsky et al., M1990). This theory was complicated later on the basis of phenomenological considerations and experimental data. A detail description of the theory of propagation of CR and a detailed analysis of various effects in the framework of the isotropic diffusion model has been presented in a series of monographs (Parker, M1963; Ginzburg and Syrovatsky, M1963; Dorman, M1963a,b). The theory of anisotropic diffusion, including the case of expanding space plasma, was developed for the first time in papers of Parker (1965), Dorman (1965, 1967). A treatment and development of the theory of CR propagation on the basis of kinetic approach were presented in (Shishov, 1966; Dolginov and Toptygin, 1966a,b, l967, 1968a,b; Tverskoy, 1967a,b, 1969; Galperin et al., 1971; Toptygin, 1972, 1973a,b; Dorman and Katz, 1972a,b, 1973, 1974a,b, 1977a,b,c; Klimas and Sandri, 1973, 1975; Scudder and Klimas, 1975; Jokipii et al., 1995; Earl et al., 1995; Kota, 1995, 2000; Otsuka and Hada, 2003). The corresponding reviews of the kinetic theory of CR propagation were presented in (Jokipii, 1971; Volk, 1975;

Dorman and Katz, 1977d; DrĂ¶ge, 2000). In De Koning and Bieber (2001) there are analyzed the particle-field correlation in a flowing plasma. The case of small pitch-angle scattering was considered by Shakhov and Stehlik (2001), and in Dorman, Shakhov, and Stehlik (2003) the second order pitch-angle approximation for the CR Fokker-Planck kinetic equation was considered. Burgoa (2003) proposed a Lagrangian density for obtaining the Fokker-Planck CR transport equation and determining the energy-momentum tensor and CR currents.

In the work of Dolginov and Toptygin (1966a) a consistent theory of CR propagation in an inhomogeneous medium (where on a background of a regular magnetic field the stochastic inhomogeneities of magnetic field are present which are transferred by the cosmic plasma with a certain velocity) was developed basing on the collisionless Boltzman equation. The kinetic equation obtained, Dolginov and Toptygin (1966a,b) made a transformation to the diffusion approximation, and found the expression for the flux density of particles including the variations of particle energy, and considered some special problems of the theory of CR propagation.

A substation progress in a study of the processes of CR propagation is connected with the exploration of propagation of the solar CR. The analysis of the data on the tremendous flare of CR on February 23, 1956 has already shown that propagation of high energy particles from the Sun is well described by the diffusion theory (Meyer et al., 1956; Dorman, M1957, l958). The interaction of galactic CR with the front of solar corpuscular streams (or as it is now called, with the front of the 'coronal mass ejection') leads to CR particles acceleration and increasing of CR intensity on the Earth (Dorman, 1959b). The analysis of the observational data of the flare on May 1959 has shown that a transfer of low energy solar CR in a trap formed in the frontal part of the solar corpuscular stream is possible (Dorman, 1959a).

Tverskoy (1967a,b, 1969) has formed the hypothesis of a transfer of fast particles behind the front of a shock wave where a developed hydromagnetic turbulence arises. It was assumed there that the Larmor radius of particles is far less than the main scale of turbulence. In this case the kinetic coefficients describing the process of particle propagation are determined by a detailed form of the spectrum function of a stochastic magnetic field. Note that a developed hydromagnetic turbulence (Alfven waves) was recognized as a result of numerous direct measurements in interplanetary space (see, for example, Belcher and Davis, 1969, 1971).

In recent years, as a result of the progress in the experimental technology, the new class of CR variation, CR pulsations, i.e. more or less regular variations of CR intensity with the period of several minutes or less is studied. When studying CR pulsations the most informative is a comparison of the theory with experimental data on the variation of correlation function of fluctuations or the function of particle distribution. The theory of pulsation effects in CR was developed in (Shishov, 1968; Dorman and Katz, 1973, 1974a,b, 1977d; Jokipii and Owens, 1976; Vasilyev and Toptygin, 1976a,b; Dorman, Katz, Stehlik, 1976, 1977; Dorman, Katz, Yukhimuk, 1977).

A substational progress in the development of the theory of CR propagation in interplanetary space is undoubtedly connected with wide application of computers for a solution of problems of CR propagation in the conditions close to the actual ones (Urch and Gleeson, 1972; Dorman and Kobylinski, 1968, 1972a,b,c, 1973; Dorman, Kobylinski, and Khadakhanova, 1973; Fisk et al., 1973; Barnden, 1973 a,b; Cecchini and Quenby, 1975; Dorman and Milovidova, 1973, 1974, 1975a,b, 1976a,b,c,d, 1977, 1978; Alaniya and Dorman, 1977, 1978, 1979, M1981; Alania et al., 1976, 1977a,b, 1978, 1979, M1987). A sufficient success was achieved in the solution of the problems of CR propagation at the presence of moving hydromagnetic discontinuities in interplanetary space (Dorman, 1959c, M1963a,b, 1969, 1973a,b, 1975; Parker, M1963; Blokh et al., 1964; Bagdasariyan et al., 1971; Belov et al., 1973, 1975, 1976; Dorman and Shogenov, 1973a,b, 1974a,b, 1975 a,b, 1977, 1979; Dorman, Babayan et al., 1978a), in the problems of distortion of the external anisotropy during propagation of particles in interplanetary space (Parker, 1967; Belov and Dorman, 1969, 1972, 1977), in the self-consistent problems of propagation of CR including their non-linear interaction with the solar wind (Dorman and Dorman, 1968a,b, 1969; Babayan and Dorman, 1976; 1977a,b, 1979a,b,d; see below, Chapter 3).

2.2. The method of the characteristic functional and a deduction of kinetic equation for CR propagation in space in the presence of magnetic field fluctuations

CR moving in interplanetary space can be considered as a flow of non-interacting charged particles in a magnetic field

which has a regular H0 (r, t) and a random H^r, t) components; in this case (h) = Ho, (H^ = 0. The oblique brackets denote an averaging over random fields.

Since a magnetic field in interplanetary space is transferred by the magnetized plasma of the solar wind, we should take into account the field motion with respect to an observer. The velocity which should be assigned to the field depends on a degree of freezing of the magnetic field in plasma. We intend to consider the case when the field is completely frozen in plasma and is involved by plasma into a motion with the velocity uo (r). In the general case this velocity has various values in different points of a space and uo << c. The most complete description of the propagation of CR in the interplanetary space with the magnetic field, which is determined by Eq. 2.2.1, is given by the collisionless kinetic equation for a CR distribution function f (r, p, t) (Dolginov and Toptygin, 1966a,b; Dorman and Katz, I972a):

where

is the operator of particle momentum, p is the momentum, v = cp/E is the velocity and E is the total energy of a particle. The distribution function f (r,p,t) varies fast following a random field variation. Averaged over a stochastic field the distribution function F(r,p, t) = (f (r,p, t)) is of interest. To obtain the equation for

F we shall apply the method which has been developed in the quantum theory of fields (Schwinger, 1951; Fradkin, 1965) and statistical fluid mechanics (Hopf, 1952; see also Monin and Yaglom, M1965; M1967). This method was intensively developed in the problems of wave propagation in the medium with stochastic inhomogeneities (Tatarsky, M1967; Klyatskin and Tatarsky, 1973).

As is known (Hopf, 1952; Tatarsky, Ml967), the statistical properties of a stochastic function F1(r,p, t ) = F1(x,t) (in further consideration we shall frequently denote a set of variables {r,p} by a single letter {x}) are completely determined if its characteristic functional is settled:

where

is the 'scalar product' in the functional space and summation over the repeated indices is assumed here and below. All moments of a stochastic field can be derived from Eq. 2.2.4 as the functional derivatives at zero-valued functional argument n(x, t):

n=o and so on. In particular, the statistical properties of a magnetic field are completely determined if its characteristic functional is set:

According to the above considerations, all the moments of a stochastic magnetic field can be obtained from Eq. 2.2.8 as variation derivatives at the functional argument equal to zero. For instance, the value

n=o is the correlation tensor of a random magnetic field of the second rank which, in general, can be determined from experimental data.

Multiplying Eq. 2.2.2 by expiJ(n(r, t)H(r,t))drdt and averaging the equation obtained over possible realizations of a random magnetic field, we have the equation in the variance derivatives (Dorman and Katz, 1972a)

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