r case that 9 = -0.5 and in the three-dimensional case a = 3p (for example, a = 0.3 and 9 = 0.1). It has been shown that this mechanism is reasonable to consider in interplanetary space only for those particles with energies not above 1012 eV and in interstellar space for those particles with energies not above 1015 eV.

4.11.6. The magnetic pumping mechanism in the case of field variations according to the power law

The mechanism of particle acceleration by magnetic pumping was initially proposed by Alfven (1959) for the case of slow periodic change of a homogeneous magnetic field:

(here 3 < 1 is the so called pumping parameter; o is the frequency of the field variation) in turbulent plasma accompanied by conservation of the adiabatic invariants p// = const, p!/h = const. In this case an exponential increase of the total momentum on hydro-magnetic turbulence in time and the betatron acceleration are possible. Because of the particle scattering by hydromagnetic turbulence, the portion of the momentum accumulated during the magnetic field enhancement (Eq. 4.11.37) owed to betatron acceleration (p^^ H ) is transferred to the parallel component of the momentum. As a result, if the scattering time is small (much smaller than the period of field variation), the particle momentum loss proves to be smaller than the momentum increase during the field enhancement; it is this circumstance that results in the acceleration. The mechanism for this case of magnetic pumping was further developed by Schluter (1957) who has shown that the mode of periodic variations of the field is of no importance in principle and that in the particular case described by Eq. 4.11.36 the acceleration effect for an ensemble of particles of the same energy is a maximum at vejj ~ o, where vejp is the effective frequency of particle scattering.

4.11.7. Kinetic theory of particle acceleration by magnetic pumping

The most consistent theory of particle acceleration in variable magnetic fields was developed in the works by Bakhareva et al. (1970a,b). It is in these works that the particle scattering by hydromagnetic turbulence was proposed as the scattering mechanism and the problem of particle acceleration was formulated on the basis of the equations of quasi-linear kinetic which permitted both the accelerated particle spectrum and the spectrum of turbulent pulsations causing the particle scattering to be determined.

It may be expected that in the presence of a variable magnetic field the cyclotron instability associated with the anisotropy of the angular distribution of charged particle velocities is the source of intense turbulence. In the initial isotropic plasma such anisotropy appears because of conservation of the adiabatic invariant p2/H = const and is owed to two-dimensional compression (expansion) of the Larmor orbits of particles owing to periodic variations of magnetic field. The above mentioned instability is owed to the cyclotron resonance between waves and particles on Larmor frequency including the Doppler effect and appears at very small anisotropy for sufficiently high particle velocities.

The equation set of quasi-linear approximation for the examined case may be written in the form df p sin 9 dH dt 2H dt

dp p d9

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