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Register a new userDiffusion of charged particles in a stochastic force-free magnetic field
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We study diffusion of charged particles in stationary stochastic magnetic field ${bf B}$ with zero mean, $langle {bf B} rangle = 0 $. In the case when electric current is carried by electrons, the field is force-free, $mathrm{curl} ,{bf B} = alpha{bf B} $, where $alpha({bf r})$ is an arbitrary scalar function. In a small region where the function $alpha $ and the field magnitude $|{bf B}|$ are approximately constant, the equations of motion of charged particles are integrated and reduced to the equation of mathematical pendulum. The transition from trapped to untrapped particles is continuously traced. Averaging over the magnetic field spectrum gives the spatial diffusion coefficient $D$ of particles as a function of the Larmor radius $r_L$ in the large-scale magnetic fields ($B_{LS}$) and magnetic field correlation length $L_0$. The diffusion coefficient turns out to be proportional to the Larmor radius, $Dpropto r_L $, for $r_L <L_0 / 2pi $, and to the Larmor radius squared, $ D propto r_L^2 $, for $ r_L> L_0 /2pi $. We apply obtained results to the diffusion of cosmic rays in the Galaxy, which contains a large number of independent regions with parameters $L_0$ and $B_{LS}$ varying in wide range. We average over $B_{LS}$ with the Kolmogorov spectrum and over $L_0$ with the distribution function $f(L_0)propto L_0^{- 1+ sigma}$. For the practically flat spectrum $sigma = 1/15$, we have $ Dpropto r_m^{0.7}$, which is consistent with observations.
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3D picture of the coronal magnetic field remains an outstanding problem in solar physics, particularly, in active regions. Nonlinear force-free field reconstructions that employ routinely available full-disk photospheric vector magnetograms represent state-of-the-art coronal magnetic field modeling. Such reconstructions, however, suffer from an inconsistency between a force-free coronal magnetic field and non-force-free photospheric boundary condition, from which the coronal reconstruction is performed. In this study we focus on integrating the additional chromospheric and / or coronal magnetic field data with the vector photospheric magnetograms with the goal of improving the reliability of the magnetic field reconstructions. We develop a corresponding modification of the available optimization codes described in Fleishman et al. (2017) and test their performance using a full-fledged MHD model obtained from the Bifrost code by performing a `voxel-by-voxel comparison between the reconstructed and the model magnetic fields. We demonstrate that adding even an incomplete set of chromospheric magnetic field data can measurably improve the reconstruction of the coronal magnetic field, greatly improve reconstructions of the magnetic connectivity and of the coronal electric current.
Extrapolations of solar photospheric vector magnetograms into three-dimensional magnetic fields in the chromosphere and corona are usually done under the assumption that the fields are force-free. The field calculations can be improved by preprocessing the photospheric magnetograms. We compare two preprocessing methods presently in use, namely the methods of Wiegelmann et al. (2006) and Fuhrmann et al. (2007). The two preprocessing methods were applied to a recently observed vector magnetogram. We examine the changes in the magnetogram effected by the two preprocessing algorithms. Furthermore, the original magnetogram and the two preprocessed magnetograms were each used as input data for nonlinear force-free field extrapolations by means of two different methods, and we analyze the resulting fields. Both preprocessing methods managed to significantly decrease the magnetic forces and magnetic torques that act through the magnetogram area and that can cause incompatibilities with the assumption of force-freeness in the solution domain. Both methods also reduced the amount of small-scale irregularities in the observed photospheric field, which can sharply worsen the quality of the solutions. For the chosen parameter set, the Wiegelmann et al. method led to greater changes in strong-field areas, leaving weak-field areas mostly unchanged, and thus providing an approximation of the magnetic field vector in the chromosphere, while the Fuhrmann et al. method weakly changed the whole magnetogram, thereby better preserving patterns present in the original magnetogram. Both preprocessing methods raised the magnetic energy content of the extrapolated fields to values above the minimum energy, corresponding to the potential field. Also, the fields calculated from the preprocessed magnetograms fulfill the solenoidal condition better than those calculated without preprocessing.
Force-free extrapolations are widely used to study the magnetic field in the solar corona based on surface measurements. The extrapolations assume that the ratio of internal energy of the plasma to magnetic energy, the plasma-beta is negligible. Despite the widespread use of this assumption observations, models, and theoretical considerations show that beta is of the order of a few percent to more than 10%, and thus not small. We investigate what consequences this has for the reliability of extrapolation results. We use basic concepts starting with the force and the energy balance to infer relations between plasma-beta and free magnetic energy, to study the direction of currents in the corona with respect to the magnetic field, and to estimate the errors in the free magnetic energy by neglecting effects of the plasma (beta<<1). A comparison with a 3D MHD model supports our basic considerations. If plasma-beta is of the order of the relative free energy (the ratio of the free magnetic energy to the total magnetic energy) then the pressure gradient can balance the Lorentz force. This is the case in the solar corona, and therefore the currents are not properly described. In particular the error in terms of magnetic energy by neglecting the plasma is of the order of the free magnetic energy, so that the latter can not be reliably determined by an extrapolation. While a force-free extrapolation might capture the magnetic structure and connectivity of the coronal magnetic field, the derived currents and free magnetic energy are not reliable. Thus quantitative results of extrapolations on the location and amount of heating in the corona (through current dissipation) and on the energy storage of the magnetic field (e.g. for eruptive events) are limited.
A formalism for describing charged particles interaction in both a finite volume and a uniform magnetic field is presented. In the case of short-range interaction between charged particles, we show that the factorization between short-range physics and finite volume long-range correlation effect is possible, a Luscher formula-like quantization condition is thus obtained.
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