downstream ___^ upstream downstream ___^ upstream

Fig. 4.16.6. Contour plot of the distribution function at a Q-Perp compression region with b A// = 0.2 (darker regions have a higher particle density). According to Malakit et al. (2003).

Malakit et al. (2003) conclude that the mirroring effect leads to more effective acceleration, especially at low energy, evidenced by the hardened spectrum. Another result is that spectra of particles accelerated by compression regions are generally not power laws but rather are hardened at high energy (see Fig. 4.16.7).

Fig. 4.16.7. Particle spectrum for a compression region with b/ A// = 2.0. According to Malakit et al. (2003).

Fig. 4.16.7. Particle spectrum for a compression region with b/ A// = 2.0. According to Malakit et al. (2003).

Furthermore, Malakit et al. (2003) also found that the spectral index at a given particle energy increases approximately linearly with the compression width for wide compressions.

4.17. The cumulative acceleration mechanism near the zero lines of magnetic field

4.17.1. Injection-less acceleration of particles and the mechanism of magnetic field annihilation

It was shown above (see Sections 4.2-4.12) that the statistical acceleration mechanisms exhibited a high sensitivity to the type of the accelerated particles. At the same time the above mentioned features of the statistical acceleration are not observed for quite a number of events. Namely, the composition of nuclei accelerated in solar flares repeats the composition of the Sun's atmosphere (Dorman and Miroshnichenko, M1968; Dorman, M1978; Dorman and Venkatesan, 1993; Miroshnichenko, M2001) and the acceleration of the electrons observed directly and on the basis of their radio-emission and X-rays (Korchak, 1967) proves to be highly effective. A great number of the accelerated electrons are also present in the Earth's magnetosphere and magnetospheric tail, in the magnetospheres of Jupiter and other planets of the Solar system, in the galactic CR, in the supernova shells, in quasars and radio galaxies. The excess of heavy nuclei in the galactic CR, though observable, is not so high as may have been expected in the case of only statistical acceleration.

The above-mentioned data have to be explained based on some other mechanism of acceleration, or at least injection, which would not display the high sensitivity to the type of the accelerated particles inherent to the statistical mechanisms. It is of great interest from this viewpoint to consider the mechanism of annihilation of oppositely directed magnetic fields, because in this case the electric field induced in the vicinities of the zero line will accelerate all the particles of the medium in a region of a relatively small volume. It should be noted that amongst numerous theoretical mechanisms involving the magnetic field's annihilation; when interpreting the flare processes on the Sun and the particle acceleration, the theories considering the rapid rearrangement and dissipation of magnetic field in the class of two-dimensional streams seem to be most promising at present.

The first step in this direction was made by Sweet (1958), who examined the one-dimensional stationary compression of the plasma between two anti-parallel layers with due account of plasma streaming in the transverse direction along the layer. After qualitatively estimating Sweet's model, however, Parker (1963) showed that such model gave excessively long times to be necessary for the energy to be released. In order to obtain smaller times of magnetic field dissipation as compared with the values obtained by Sweet's model, Sweet's mechanism was examined by including the ambipolar diffusion. The essence of ambipolar diffusion is that the magnetic field moves with the electrically conducting ionized plasma component whose motion relative to the neutral component is retarded by the friction owed to ion collisions with neutral atoms. Owing to this phenomenon, a decrease of the effective conductivity may be included in the estimate of the magnetic field's dissipation rate. Petchek (1964) reconsidered the Sweet-Parker mechanism and showed that even if the mechanism of ambipolar diffusion is included the time necessary for the magnetic energy to be converted into thermal energy still remains too great. In Petchek (1964) the Sweet-Parker model is also supplemented with magneto-hydrodynamic waves whose propagation will be of great importance to the stable plasma streams of high conductivity. According to Petchek's estimate for a compressible flux corresponding to solar flares, the energy necessary to a flare may be released within ~102 sec. Thus the model of Petchek (1964) can satisfactorily explain the rapid heating and ejections of plasma in the region of solar flares. In that theory, however, the problem of generation of acceleration particles remains to be solved.

4.17.2. Current sheets and rapid rearrangement of magnetic fields

The mechanism of particle acceleration in the case of magnetic field dissipation was further developed in the works of Syrovatsky (1966, 1967, 1968, 1969, 1971) which deal with the cumulative mechanism of acceleration near the zero lines of magnetic field, the mechanism that ensures acceleration in individual small regions of plasma for all charged particles irrespective of their properties. According to Syrovatsky (1966, 1967, 1968, 1969, 1971) such acceleration must take place in the vicinities of current sheets in the case of rapid rearrangement of magnetic field. According to the above-mentioned works, the magnetic field in the space is rapidly rearranged under the conditions of seeming ideal frozenness of the field. Such rearrangement is accompanied by particle acceleration, for example in chromospheric flares. Syrovatsky (1971) examined the rearrangement mechanism in detail. It is assumed in Syrovatsky (1971) for the sake of simplicity that the magnetic fields are plane and the plasma moves in a strong magnetic field, i.e., p/puA << 1; u2/U2 << 1, (4.17.1)

where P and p is the pressure and density of the plasma; ua = H is the

Alfvén velocity; u is the velocity of plasma motion. In this case the equations of magneto-hydrodynamics for the perfectly conducting plasma will be written in the form dA dA . A. „ du . _ dP

-=-+ uVA = 0; AA = 0; — xVA = 0; — = —odivu . (4.17.2)

dt dt dt dt

Here ^(x, y, t) is the unique non-zero component of the vector-potential A, so

It has been shown in Syrovatsky (1971) that if the field of the external field sources in the volume studied has singular zero points then there exist regions of non-analytical solution of the equation system described by Eq. 4.17.2. By virtue of the assumption Eq. 4.17.1 the non-analyticity region may consist of only isolated points (linear currents) and cuts (plane currents). If the intensity of the linear current was zero at t = 0 (the simplest zero point of the type X exists at that moment, see Fig. 2.17.1a), and then increased gradually with time, the zero point should be 'doubled' and two points of the type X with a region of closed force lines between them should, appear (see Fig. 2.17.1b).

Fig. 4.17.1. When linear current I appears at the singular zero point (panel a), the zero point is 'doubled' (panel b). According to Syrovatsky (1971).

Under the condition of frozenness, however, the closed force lines cannot be obtained as a result of permanent deformation of the initial field which did not comprise such lines. Thus the isolated singular points cannot be used to construct a solution for Eq. 4.17.2 that will be continuous in the rest of the space. As a result only the solutions with cuts have to be accepted, and Syrovatsky (1971) asserts the following: if the external field comprises singular zero points the plane currents or (which is the same) the current sheets are formed in the plasma near such points. The location of the cuts corresponding to current sheets on the complex plane should be such that the boundary problem for Eq. 4.17.2 would have an infinite solution everywhere beyond the cuts. The following rule may be formulated to determine the location of the cuts. A cut must include the initial zero point and all zero points appearing if the initial zero point contain a linear current varying from zero to some finite value. In this case, the linear current direction must coincide with the electric

Fig. 4.17.1. When linear current I appears at the singular zero point (panel a), the zero point is 'doubled' (panel b). According to Syrovatsky (1971).

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