Let us apply these results to a scattering of the low energy solar particles. According to the Parker's model in a region inside the Earth's orbit 2 for a regular field Ho . The spectrum index v = 1.5 ± 0.2 . If magnetic inhomogeneities are generated near the Sun and then transferred into interplanetary space by the solar wind, the intensity H1 should be varied proportionally to Ho . This assumption is founded by measurements for a region outside the Earth's orbit. Thus according to the data of Jokipii and Coleman (1968) the value was decreased by 2.4 times and H2 was decreased by 2.5 times when the distance from the Sun was increased from 1 AU to 1.44 AU Substituting ¡ = 2, v = 1.5 in Eq. 2.10.17 we find that 02 is independent of z. If in this case 02 < 1, the particles generated on the Sun should come to the Earth in the form of a flux with a pronounced anisotropy. These anisotropic fluxes of protons with the energies of 1-10 MeV and a duration of several hours were recurrently registered in the experiments of Vernov et al. (1968a).
Earl (1976) also paid his attention to the effect of adiabatic focusing in the propagation of charged particles of CR in stochastic magnetic fields on the background of a regular field with divergent magnetic lines of force in the direction of the field weakening. In this paper the kinetic equation is obtained for which the proper functions of scattering are found (these functions appeared to be symmetric with respect to cos0 where 0 is a pitch-angle) and the proper functions of focusing (which appeared to be asymmetric with respect to cos0). Numerical calculations which were carried out by means of these functions show that in the case of a weak divergent field there is obtained a diffusion approximation in the pitch-angle space where the effect of adiabatic focusing can be neglected; in the case of a strong field there appeared to be a mode of coherent propagation of particles which is completely determined by the effect of adiabatic focusing.
Morfill et al. (1976) presented the arguments against the model of a superposition of the averaged over extensive time intervals of a regular spiral magnetic field with small scale inhomogeneities causing a resonance scattering of particles with Larmor radius close to inhomogeneity's dimensions, which is generally applied to calculations of a transfer of galactic CR in interplanetary space. There is proposed a model of an irregular spiral field taking into account the middle-scale variations of the interplanetary field owed to the presence of sector structure, of tangential discontinuities, of jet streams in the solar wind etc. which are statistically described by a frequency distribution of the field direction.
The averaged Fokker-Plank equation is obtained for a pitch-angle diffusion of galactic CR (the effects of direct reflection of particles and their large scale drift are not included). Basing on this equation the spatial coefficients of diffusion were derived and the expected radial gradients were estimated. The obtained values appeared to be in good agreement with the results of radial gradient measurements from the data of synchronous observations from various spacecrafts. Calculations have also been carried out of the expected temporal variations of the diffusion coefficient during a solar activity cycle including the data of corresponding variations of the parameters characterizing the spectra of magnetic inhomogeneities in the range of small and moderate scales.
Alpers et al. (1975) carried out a study consisting of charged particles diffusion in the magnetic field which is a superposition of the regular constant magnetic field Ho and a rapidly varying in space stochastic field H1(r). It was shown that a contribution to the diffusion coefficient in the form of a 8-function for particles with the pitch-angles d (with respect to the force lines of the field Ho) close to 90° is not caused neither by pitch-angle d scattering nor peculiarities of particle propagation along the force lines of the average field Ho. Alpers et al. (1975) have drawn the conclusion that abnormal behavior of the coefficient of pitch-angle diffusion at d~ 90° is caused by the fact that generally used to determine coefficient of diffusion theory of a weak interaction of particles with magnetic inhomogeneities in the same point has a singularity. This singularity, however, has no physical sense. A detailed analysis shows that if in the initial state the ensemble of charged particles has d = 90°, its broadening over pitch-angle occurs considerably longer than in the case when d ^ 90° is the initial state. The matter is that at d = 90° a particle is in the situation as if it was frozen in a magnetic inhomogeneity and a variation of its state is owed only to the extremely slow regular acceleration, whereas the particles with d ^ 90° very rapidly change their energy in a stochastic way due to the action of the statistic acceleration mechanism.
Lee and Volk (1975a,b) have solved in a quasi-linear approximation the equations of diffusion of CR particles in the space of pitch-angles and energies. There is considered an interaction of particles with a field of hydromagnetic waves in the presence of a regular field Ho. The commonly used assumption of an isotropic tensor of the spectrum of power of magnetic inhomogeneities require in the given case the equality of spectra of Alfvén and magneto-sonic waves. For the solar wind plasma this assumption is not considerably realistic. it is shown that the coefficient of pitch-angle diffusion in the case under consideration, as well as in the other quasi-linear approximations, vanishes at the pitchangles ~ 90°. Thus in quasilinear theory there is no reflection of CR particles. This difficulty can be overcome by non-linear treatment of particle dynamics.
Basing on the kinetic Vlasov equation Goldstein (1976) has derived a diffusion approximation for the function of distribution over the pitch-angle variable u = cos 6 (where 6 is a pitch-angle) for a propagation of CR in strongly turbulent magneto-active plasma in which the bonds are considered to be weak, according to Kadomtsev's hypothesis. With the assumption that the correlation function of a stochastic magnetic field has the exponential character, the detailed calculations of the diffusion coefficient DU over the pitch-angle variable have been carried out. Special attention is paid to the behavior of DU at 2 (i.e. at
U^ 0). It was shown that Du(j = 0) has a finite value in a good agreement with the results of Monte Carlo numerical calculations, in contrast to the works of the other authors based on the linear theory when it was assumed that Du(j = 0) = 0 or
Du(u = 0)°= S(u) , where 8(j) is the Dirac 8-function.
Goldstein (1977) has presented a critical analysis of theoretical models of pitch-angle scattering and spatial diffusion of particles based on a quasi-linear approximation of the kinetic theory which were developed in the literature. Goldstein (1977) gives also a generalization of the resonance theory of disturbed trajectories of particle pitch-angle diffusion in a model of magneto-static turbulence; the results obtained are used for numerical calculations of the spatial coefficient of the field-aligned diffusion. in this case it is possible to eliminate the all divergences which are proper to a quasi-linear formalism for a spectrum of magnetic field fluctuations of the type k-v at v > 2 (here k is the wave number). It was found that different methods give, in the first approximation, for 1 < v < 2 close values for the spatial diffusion coefficient; the method of disturbed trajectories which is used in the paper only gives systematically slightly lower values of the diffusion coefficients than the models of quasi-linear theory.
In the works of Jones et al. (1973), Jones (1975) the expression for the coefficient of a diffusion over pitch-angles has been obtained on the basis of a kinetic equation for a distribution function averaged over fluctuations. It was shown that the developed theory gives the values of the diffusion coefficient coincident with those which are expected in quasi-linear theory at 0.6<j< 1 (where u = cos6 and 6 is a pitch-angle); at lower values of u the new theory gives the values for the diffusion coefficient which are considerably higher than the values expected in the usual theory.
Jones et al. (1978) obtained the kinetic equation describing particle interaction with turbulent fluctuations of a magnetic field, using the non-linear theory which has been developed in Jones et al. (1973), Jones (1975), and made it possible to overcome correctly the difficulties proper to quasi-linear theory. The effect of fluctuations in the method developed by Jones et al. (1978) is determined from particle orbits which, in their turn, include a statistical averaging over a series of possible configuration of turbulence. In the method of a partially averaged field the averaging procedure is made from a sample of all realizations for which the field intensity takes a fixed value in a given point. Using the new method, the calculations of the coefficient DUjU of a diffusion over pitch-angles for particles interacting with a 'stratified' model of magnetic turbulence in which the fluctuations of magnetic field are linearly polarized transverse to the direction of the average magnetic field (h) . The results obtained are compared with the data of quasi-linear theory and of Monte Carlo a numerical model experiment. The conclusion was drawn that the main result of quasi-linear theory consists in determining Dju in the pitch-angle range near 90° where quasi-linear pproximation is violated. Using Dju value, the coefficient k// of the spatial diffusion along the direction of magnetic field (h) has been estimated. It was noted that the method of partially averaged field is not restricted by a criterion of smallness of the amplitudes of fluctuating fields, and therefore it is not a perturbation theory.
Developing this study, Kaiser (1975), Kaiser et al. (1978) presented the results of numerical Monte Carlo simulating the process of charged particles diffusion over the velocities in a stochastic turbulent magnetic field. The coefficient of diffusion over pitch-angles was determined by means of exact calculation of the orbits of particles moving in a great ensemble of realizations of a stochastic magnetic field with the statistic properties which are selected in a certain way. The calculations were carried out for a wide range of particle rigidities and of mean square intensities of a magnetic field. A comparison has been made of the results given by standard quasi-linear theory with the conclusions of non-linear theory which uses partially averaged fields.
Moussas et al. (1975), Moussas and Quenby (1977) have carried out numerical calculations of the diffusion coefficient d(ju) of CR in the pitch-angle space ( u = vH/vH , where v is the particle velocity and H is the interplanetary magnetic field) basing on the data of the three-dimensional structure of interplanetary magnetic field. A comparison was made with d(jJ which is expected according to the usual quasi-linear kinetic theory of CR propagation. It was found that at j ^ 0 the numerical calculations basing on the interplanetary magnetic field data of HEOS-2 give the values =-1.5 for lg d(j), whereas quasi-linear theory gives lgd(u)< -3, i.e. there is a discrepancy of almost two orders. At j from 0.3 to 0.8 a discrepancy is also about 3-4 times. Analytical corrections for quasi-linear theory providing determination of correct results at u^ 0 in the case of very weak stochastic perturbations of a regular component of interplanetary field have been obtained. The problem of a rotational discontinuity effect on the pitch-angle distribution of solar CR has been solved numerically assuming in this case that the distribution in a stream of CR before passing through a discontinuity was axially-symmetric. The results were presented for numerical calculations of the expected particle distribution over the phases p and over j values depending on a distance from the discontinuity. it was predicted that there arises a two-
directed pitch-angle distribution after passing through a rotational discontinuity. It was observed that arising of this peculiarity in the pitch-angle distribution of solar CR is not connected with acceleration of particles.
2.11. Fokker-Planck CR transport equation for diffusion approximation
2.11.1. Diffusion approximation including the first spherical mode
At distances exceeding the free large angle scattering path of particles the distribution function is close to the isotropic distribution. In this case the CR propagation may be described by using the diffusion approximation equation (Dolginov and Toptygin, 1966a,b). After series expanding the function in spherical harmonics we obtain
where n(r, p, t) is the particle concentration; j(r, p, t) is the density of the particle flux. Substituting Eq. 2.11.1 in Eq. 2.3.7 and multiplying Eq. 2.3.7 by 1 and by p. we shall integrate the obtained expressions over vector angles p taking account of the first non-vanishing terms in powers uo/v. The resultant set of equations for n(r, p, t) and j(r, p, t) is dn T U2 — + divJ = -2-dt 9k2
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