http://arxiv.org/abs/1612.02331

We consider a random stationary magnetic field with zero average magnetic field, $ <B> = 0 $. In the case when carriers of electric current, that creates a field, are electrons the field is force-free, $\mathrm{curl} {\bf B} = \alpha{\bf B} $. In a small region, in which the coefficient of $ \alpha $ and the strength $ |B| $ can be considered constants, the force-free field is the vector rotating in the direction perpendicular to the plane in which magnetic field lines lie. The motion of a charged particle in such a field is described by a mathematical pendulum and continuously traces the transition from magnetized motion, when the cyclotron radius $ r_c $ less than the characteristic scale of inhomogeneity $L = 2\pi/\alpha$ (trapped particles), to almost free, when $ r_c> L $ (untrapped particles). For the cosmic rays in the Galaxy, in which there are wide ranges of large-scale field strength $B_{LS}$ and the correlation length $L_0 $, averaging over these values gives the dependence of the diffusion coefficient as the fractional power of the Larmor radius $r_m$ in the maximum large-scale field, $ D\propto r_m^{(1- \sigma)/(1+ \beta)}$. The value of $ \beta $ is the index of the spectrum of the large-scale field, $ B_{LS}\propto L_0^{\beta} $, and $ 1- \sigma $ is the index in the distribution function $ f(L_0)$ over scales, $ f(L_0)\propto L_0^{- 1+ \sigma}$. For the Kolmogorov spectrum of the magnetic field, $\beta = 1/3$, and almost flat spectrum over scales, $ \sigma = 1/15 $, the value of the index $(1- \sigma)/(1+ \beta) $ is $ 0.7 $, which corresponds to observed dependence of the diffusion of cosmic rays in the Galaxy over their energy.

Read this paper on arXiv…

Y. Istomin and A. Kiselev
Thu, 8 Dec 16
52/69

Comments: 10 pages, 5 figures