V drv V
The Eq. 2.11.26, or its extreme cases Eq. 2.11.39 and Eq. 2.11.40, together with the equation of anisotropic diffusion Eq. 2.11.32 and the expression for the vector of particle flux density Eq. 2.11.30 solve the problem of deducing the set of equations in the diffusion approximation including the second spherical harmonic. Observe that this equation set has a remarkable feature: the tensor f,v does not give a contribution either in the equation of anisotropic diffusion Eq. 2.11.32 or in the expression for the particle flux density Eq. 2.11.30. The circumstance mentioned simplifies substantially the analysis of propagation of CR particles when it is necessary to take into account the third spherical harmonic.
2.11.4. Drift effects in a diffusion propagation of CR
With the assumption that the regular magnetic field component, with lines of force in the form of Archimedean spirals, consists of several sectors with alternating field direction (anti-Sunward and Sunward), Barnden and Bercovitch (1975) have carried out Monte Carlo calculations of trajectories of test particles of cosmic radiation in their stochastic wandering in the solar system. The calculations were carried out including parallel with diffusion, a convective transfer of particles by the solar wind, as well as their energy losses owing to adiabatic deceleration caused by radially divergent inhomogeneities. It was shown that for a transfer of CR in interplanetary space the latitude drift of a particle is of substantial significance, which arises owing to a curvature of lines of force of the regular component of the magnetic field and to the presence of the field gradients in a vicinity of the sector boundaries. It was found that particles coming to the Earth with energy lower than 100 GeV have a wide distribution of the energy losses and of the duration of their wandering in the solar system, and that they come from the Galaxy in a wide ranges of helio-latitudes.
Forman (1975) considered the general expression, which describes a formation of CR anisotropy in interplanetary space and includes four terms. The first term represents a convective transfer of CR in a radial direction from the Sun by the solar wind (the Compton-Getting effect). The second term reflects a diffusion along magnetic lines of force (inverse to the gradient of CR density). The third term is owed to diffusion across the field lines of force, and the fourth term is caused by a transverse gradient drift that is directed normally to the field lines of force and normally to the gradient of CR density. To account for the experimental data obtained by means of the neutron super-monitors, according to which the anisotropy vector of CR in interplanetary space in 20-35% of observations occurs to be directed at the angle above 30° to the direction of the magnetic field projection onto the ecliptic plane (averaged over 24 hours), the following two possibilities are analyzed: either the expression stated for CR anisotropy in interplanetary space is not valid, or CR diffusion across magnetic lines of force and (or) a transverse gradient drift are of substantially greater importance sometimes than is usually considered. Forman (1975) draws the conclusions that the second possibility is, rather, realized. in this case the important role must be played by the transverse gradient drift whereas the case k±/k// = 1 (where k± and k// are the components of the coefficient of diffusion across and along the field) is realized in rare days.
Jokipii and Levy (1977), Jokipii et al. (1977) have shown that a drift of CR particles in twisted in the Archimedean spirals interplanetary magnetic field which is related to the gradient of magnetic field and the curvature of the lines of force, affects considerably diffusion propagation and the effect of modulation of galactic CR by the solar wind. The case is that the drift velocities of CR particles (the rigidities R > 0.3 GV) appear to be higher than the solar wind velocity, and the value of the radial component of drift velocity is comparable to or higher than the wind's velocity. Preliminary results of Monte Carlo calculations are presented for CR modulation in a spherically symmetric solar wind carrying magnetic field in the form of Archimedean spiral. The calculations show that including particle drift can result in a considerable decrease of modulation, heliocentric gradient, and energy variation (for particles with R ~ 1 GV) inside the solar system. It was observed that, though the calculations have been carried out for a magnetic field of certain configuration, the drift effect should act as well in a more general case.
Therefore, always when the drift velocity is comparable to or higher than the velocity of the solar wind, it easier for galactic CR to penetrate inside the solar system, and this results in a decrease of energy variation and in a decrease of the radial gradient of CR.
In the paper of Dorman, Dremukhina and Okulov (1977a,b) the component of CR current was considered which is caused to gradient drift of high energy charge particles of CR in a stationary non-uniform interplanetary magnetic field. As charge particles with the energy E in the field H have the magnetic moment
where 6 is a particle pitch angle, there will arise in a non-uniform field the force
under the action of which the particles will drift with the velocity vdr - F XH . (2.11.44)
To determine from Liouville's theorem the CR current arising in the stationary case, the equation was obtained for the distribution function of particles in the six-dimensional phase space r, p (where r are the spatial coordinates, p is the particle momentum) through Poissonian brackets with the Hamiltonian of the system in which the gradient drift is included. Parker's model of interplanetary field in the form of Archimedean spirals was considered as an example. It was shown that if the transverse coefficient of diffusion is = 5 x1020 cm2/sec, the radial gradient of CR density is = 10%/AU , then with a solar wind velocity ~ 400 km/sec the anisotropy of CR owing to gradient drift will have the value of about 0.2% which is comparable with the components of solar anisotropy owing to the other causes. it is shown that in the presence of sector structure of the interplanetary field under the action of gradient drift there will also arise abruptly changing on the sector boundaries North-South asymmetry of CR which is comparable in amplitude with that observed by means of the global net of neutron monitors and muon telescopes.
it is necessary to take into account the gradient and centrifugal drifts when considering a process of CR transfer in interplanetary space was also proved by Isenberg and Jokipii (1978). If the radial diffusion coefficient Krr is independent of particle energy and is proportional to the radial distance from the sun, it will be possible to obtain a solution of the Fokker-Plank equation including convective and drift transfer, anisotropic diffusion and adiabatic cooling. For this purpose a transformation was made to the pitch-angle variable u = cos 6, where 6 is the pitch-angle of CR particles relative to a line of force of the regular component of interplanetary magnetic field (which is chosen, according to Parker's model, in the form of spiral field, with a neutral sheet in the equatorial plane). The authors presented graphs of the expected modulation depth, radial gradient and radial flux of CR including the reversal of interplanetary field direction in a 22-year cycle of the solar activity as the functions of r and 6 with the solar wind velocity 4 x107 cm/sec, Krr = 1.68 x1021 and 1.25 x 1020 cm2/sec (corresponding to protons with a rigidity of 1 GV and an energy 10 MeV). It was shown that including the drift affects substantially the results of calculations of the expected modulation of galactic CR (for example, for particles with a rigidity ~ 1 GV, including the drift results in a decrease of the relative radial gradient almost by 5 times).
2.11.5. General poloidal magnetic field effects in a diffusion propagation of CR
Gall et al. (1977) considered Stormer's theory for a model of interplanetary magnetic field in the form of an Archimedean spiral:
where r is the distance from the solar center, ro = 2.5 x rs (rs is the solar radius), A is the helio-latitude, Q is the angular velocity of solar rotation, u is the solar wind velocity. The field values were given in this case, according to Altschuler et al. (1974). Stormer's constant of motion ySt is derived from a Lagrangian of the equation set of motion of a charged particle in a similar way as it is made for a dipole magnetic field (see Dorman et al., M1971). As a result, Stormer's cut-off rigidity of interplanetary CR was determined as
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