Evolution of Relative Magnetic Helicity: Method of Computation and Its Application to a Simulated Solar Corona above an Active Region

For a better understanding of solar magnetic field evolution it is appropriate to evaluate the magnetic helicity based on observations and to compare it with numerical simulation results. We have developed a method for calculating the vector potential of a magnetic field given in a finite volume; the method requires the magnetic flux to be balanced on all the side boundaries of the considered volume. Our method uses a fast Laplace/Poisson solver to obtain the vector potentials for a given magnetic field and for the corresponding potential (current-free) field. This allows an efficient calculation of the relative magnetic helicity in a finite 3D volume. We tested our approach on a theoretical model (Low and Lou, Astrophys. J. 352, 343, 1990) and also applied our method to the magnetic field above active region NOAA 8210 obtained by a photospheric-data-driven MHD model. We found that the amount of accumulated relative magnetic helicity coincides well with the relative helicity inflow through the boundaries in the ideal and non-ideal cases. The temporal evolution of relative magnetic helicity is consistent with that of magnetic energy. The maximum value of normalized helicity, H _m/Φ²=0.0298, is reached just before a drastic energy release by magnetic reconnection. This value is close to the corresponding value inferred from the formula that connects the magnetic flux and the accumulated magnetic helicity based on the observations of solar active regions.     

Emission and absorption of cyclotron radiation in non-equilibrium plasma

From the kinetic Klimontovich theory we derive the equation of radiative transfer for the case of homogeneous and stationary nonequilibrium plasma in a magnetic fiel. For the sake of simplicity we shall consider only the propagation of cyclotron-radiation in the direction of the magnetic field. Explicit expressions for the coefficients of absorption and spontaneous emission, valid also at the cyclotron frequency, will be obtained. In the final results the plasma is allowed to be weakly inhomogeneous in the z-direction. Application is made of the theory to the simple equilibrium case as a test yields the Rayleigh-Jeans formula of the black-body-radiation.     

Magnetic field configurations associated with polarity intrusion in a solar active region

This paper presents a new class of exact solutions describing the non-linear force-free field above a spatially localized photospheric bipolar magnetic region. An essential feature is the variation in all three Cartesian directions and this could not be modelled adequately with previously known symmetric force-free fields. Sequences of force-free fields are constructed and analyzed to simulate the slow growth of a pair of spots on the photosphere. The axis connecting the spots executes rotational motion, distorting the photospheric neutral line separating fluxes of opposite signs. We show directly from the analytic solutions that the resulting reversal of the positions of the spots relative to the background field is associated with (i) the creation of magnetic free energy, (ii) the severe shearing of localized low-lying loops in the vicinity where the photospheric transverse field aligns with the photospheric neutral line, and (iii) the emergence and disappearance of flux from the photosphere at these highly stressed regions. The model relates theoretically for the first time these different magnetic field features that have been suggested by observation and theoretical considerations to be flare precursors. A general formula, based on the virial theorem, is also given for the free energy of a force-free field, strictly in terms of the field value at the photosphere. This formula has obvious practical application.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Theoretical model of a magnetic white dwarf

In this paper a theoretical model of a magnetic white dwarf is studied. All numerical calculations are performed under the assumption of a spherically symmetric star. The obtained equation of state is stiffer with the increase of value of the magnetic field (B). Numerical values of the maximum mass and radius are presented. The influence of the magnetic field on the results is evident. Finally the departure from the condition of isothermality of a degenerate electron gas in the gravitational field is discussed.     

Comparison of two methods for determining the magnetic fields near black holes

We compare two methods for estimating the magnetic field near a black hole—based on a statistical formula that relates the magnetic field on the horizon of the black hole to its mass and on a direct magnetic field determination from optical polarimetric observations. An optically thick magnetized accretion disk is usually formed around the black hole in active galactic nuclei. The linear polarization of the radiation emerging from the disk is related to the local magnetic field and the polarization in Milne’s classical problem. The magnetic field in the region from where the radiation with the observed wavelength comes can be estimated by measuring this polarization. On the other hand, this field can be estimated from the above statistical formula if the field near the black hole is extrapolated into the accretion disk using a standard power law. Comparison of the magnetic field obtained from the observed polarization with the extrapolated field can serve to estimate the accuracy of the latter and the exponent in the adopted extrapolation law. As an example, we compare these two methods for the source NGC 4258.     

The determination of vector magnetic fields in prominences from the observations of the Stokes profiles in the D3 line of helium

A method is presented to measure the magnetic field vector in prominences by means of the polarimetric observations in the D₃ line of He obtained with the High Altitude Observatory Stokes polarimeter. The characteristics of the observed Stokes profiles are discussed. The theory of the Hanle effect is reformulated in the representation of the irreducible tensors of the density matrix, and is generalized to derive the circular polarization profiles across the spectral line in terms of the intensity and direction of the prominence magnetic field. The circular polarization profile so deduced can be employed to obtain useful information which adds to that carried by the linear polarization observations. A non-linear least-squares algorithm is proposed to derive the measurement of the magnetic field from the observations, and a consistency check is suggested to test the adequacy of the theoretical model to describe the physics of the He I atomic excitation in prominences.On leave from: Astrophysical Observatory of Arcetri, Largo E. Fermi, 5, 50125 Firenze, Italy.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Comparison of STEREO/EUVI Loops with Potential Magnetic Field Models

The Solar Terrestrial Relations Observatory (STEREO) provides the first opportunity to triangulate the three-dimensional coordinates of active region loops simultaneously from two different vantage points in space. Three-dimensional coordinates of the coronal magnetic field have been calculated with theoretical magnetic field models for decades, but it is only with the recent availability of STEREO data that a rigorous, quantitative comparison between observed loop geometries and theoretical magnetic field models can be performed. Such a comparison provides a valuable opportunity to assess the validity of theoretical magnetic field models. Here we measure the misalignment angles between model magnetic fields and observed coronal loops in three active regions, as observed with the Extreme Ultraviolet Imager (EUVI) on STEREO on 30 April, 9 May, and 19 May 2007. We perform stereoscopic triangulation of some 100?–?200 EUVI loops in each active region and compute extrapolated magnetic field lines using magnetogram information from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO). We examine two different magnetic extrapolation methods: (1) a potential field and (2) a radially stretched potential field that conserves the magnetic divergence. We find considerable disagreement between each theoretical model and the observed loop geometries, with an average misalignment angle on the order of 20°?–?40°. We conclude that there is a need for either more suitable (coronal rather than photospheric) magnetic field measurements or more realistic field extrapolation models.     

Dynamics of charged particles and perpendicular diffusion in turbulent magnetic field

A test particle code is employed to explore the dynamics of charged particles and perpendicular diffusion in turbulent magnetic field, where a three-dimensional (3D) isotropic turbulence model is used in this paper. The obtained perpendicular diffusion at different particle energies is compared with that of the nonlinear guiding center (NLGC) theory. It is found that the NLGC theory is consistent with test particle simulations when the particle energies are small. However, the difference between the NLGC theory and test particle simulations tends to increase when the particle energy is sufficiently large, and the threshold is related to the turbulence bend-over length. In the NLGC theory, the gyrocenter of a charged particle is assumed to follow the magnetic field line. Therefore, when the particle has sufficiently large energy, its gyroradius will be larger than the turbulence bend-over length. Then the particle can cross the magnetic field lines, and the difference between the test particle simulations and NLGC theory occurs.     

Simulating the generation of the solar toroidal magnetic field by differential rotation

Within the kinematic dynamo theory, we construct a mathematical model for the evolution of the solar toroidal magnetic field, excited by the differential rotation of the convective zone in the presence of a poloidal field of a relic origin. We use a velocity profile obtained by decoding the data of helioseismological experiments. For the model of ideal magnetic hydrodynamics, we calculate the latitudinal profiles of the increasing-with-time toroidal field at different depths in the solar convection zone. It is found that, in the region of differential rotation, the excited toroidal field shows substantial fluctuations in magnitude with depth. Based on the simulations results, we propose an explanation for the “incorrect polarity” of magnetic bipolar sunspot groups in solar cycles.     

An observational examination of models for sunspot magnetic field configurations

The magnetic field configuration for an isolated, nearly circular sunspot has been obtained from measurements at some 100 points in the spot. The observations are compared with theoretical models for the field configuration of an isolated, axially-symmetric sunspot. The results support the Schlüter, Temesváry model with no return flux and a limiting field line inclination of ~70°.     

Time-dependent transport of energetic particles in magnetic turbulence: computer simulations versus analytical theory

We explore numerically the transport of energetic particles in a turbulent magnetic field configuration. A test-particle code is employed to compute running diffusion coefficients as well as particle distribution functions in the different directions of space. Our numerical findings are compared with models commonly used in diffusion theory such as Gaussian distribution functions and solutions of the cosmic ray Fokker–Planck equation. Furthermore, we compare the running diffusion coefficients across the mean magnetic field with solutions obtained from the time-dependent version of the unified non-linear transport theory. In most cases we find that particle distribution functions are indeed of Gaussian form as long as a two-component turbulence model is employed. For turbulence setups with reduced dimensionality, however, the Gaussian distribution can no longer be obtained. It is also shown that the unified non-linear transport theory agrees with simulated perpendicular diffusion coefficients as long as the pure two-dimensional model is excluded.     

A Quantitative Method to Optimise Magnetic Field Line Fitting of Observed Coronal Loops

Many authors use magnetic-field models to extrapolate the field in the solar corona from magnetic data in the photosphere. The accuracy of such extrapolations is usually judged qualitatively by eye, where a less judgemental quantitative approach would be more desirable. In this paper, a robust method for obtaining the best fit between a theoretical magnetic field and intensity observations of coronal loops on the solar disk will be presented. The method will be applied to Yohkoh data using a linear force-free field as an illustration. Any other theoretical model for the magnetic field can be used, provided there is enough freedom in the model to optimize the fit.     

Magnetic field strength estimation in synchrotron radiation sources

In this paper a method of estimating the magnetic field strength,B, in a homogeneous microwave burst source with simplified expressions for the synchrotron radiation is presented. An approximate formula of the magnetic field is obtained using the method. Once the magnetic field is estimated the total number of energetic electrons along the line of sightN L can be estimated also. The errors ofB andN L have been given. It is found that this method is useful for semiquantitative investigations of models of radio burst sources.     

Comparisons of measured and calculated potential magnetic fields

Photospheric line-of-sight and transverse magnetic field data obtained with the Marshall Space Flight Center vector magnetograph system for an isolated sunspot are described. A study of the linear polarization patterns and of the calculated transverse field lines indicates that the magnetic field of the region is very nearly potential. The H fibril structures of this region as seen in high resolution photographs corroborate this conclusion. Consequently, a potential field calculation is described using the measured line-of-sight fields together with assumed Neumann boundary conditions; both are necessary and sufficient for a unique solution. The computed transverse fields are then compared with the measured transverse fields to verify the potential field model and assumed boundary values. The implications of these comparisons on the validity of magnetic field extrapolations using potential theory are discussed.     

Buried Magnetic Field in Accreting Neutron Star

The evolution of the neutron star magnetic field is correlated to the accretion physical process in the binary X-ray phase. It is assumed that the original strong magnetic field is buried by the accreted materials, which is on account of the ferromagnetic physical property of the accreted matter. The obtained theoretical conclusions of our model are that: (1) the magnetic field decays in the X-ray binary phase, and relates inversely to the accreted mass as M^-1, (2) The internal magnetic field strength still remains large although the surface field strength decays.     

An equatorially symmetric rotator model for magnetic stars

Suggested by theoretical and empirical investigations, a very simple magnetic field distribution at the surface of magnetic stars is proposed. This model corresponds to the superposition of a dipole and a quadrupole, both lying in the equatorial plane with their axes in the same direction. The observed variations of the effective magnetic field strength of the sixteen stars investigated can be represented surprisingly well with this model. When the quadrupole contribution is large, the variation of the magnetic field vector over the surface of the star differs characteristically from that of the decentred dipole model; thus the difficulty with the latter, regarding the representation of both the effective magnetic field and the mean surface field with the same parameters may perhaps be avoided.     

Aspects of Three-Dimensional Magnetic Reconnection – (Invited Review)

In this review paper we discuss several aspects of magnetic reconnection theory, focusing on the field-line motions that are associated with reconnection. A new exact solution of the nonlinear MHD equations for reconnective annihilation is presented which represents a two-fold generalization of the previous solutions. Magnetic reconnection at null points by several mechanisms is summarized, including spine reconnection, fan reconnection and separator reconnection, where it is pointed out that two common features of separator reconnection are the rapid flipping of magnetic field lines and the collapse of the separator to a current sheet. In addition, a formula for the rate of reconnection between two flux tubes is derived. The magnetic field of the corona is highly complex, since the magnetic carpet consists of a multitude of sources in the photosphere. Progress in understanding this complexity may, however, be made by constructing the skeleton of the field and developing a theory for the local and global bifurcations between the different topologies. The eruption of flux from the Sun may even sometimes be due to a change of topology caused by emerging flux break-out. A CD-ROM attached to this paper presents the results of a toy model of vacuum reconnection, which suggests that rapid flipping of field lines in fan and separator reconnection is an essential ingredient also in real non-vacuum conditions. In addition, it gives an example of binary reconnection between a pair of unbalanced sources as they move around, which may contribute significantly to coronal heating. Finally, we present examples in TRACE movies of geometrical changes of the coronal magnetic field that are a likely result of large-scale magnetic reconnection. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005248007615     

Magnetic-field accumulation in supergranules

Some theoretical models illustrating the growth of magnetic fields around the boundaries of supergranulation cells are given. The evolution of a magnetic field in a prescribed velocity field is calculated for the idealized case of hexagonal cells. The calculations are carried out both for a perfect conductor and a fluid of finite conductivity. The theory is developed in such a way as to be virtually free of specific assumptions about the depth dependence of the supergranulation flow. The results demonstrate the tendency of the field to accumulate at those boundary points where the flow is most strongly converging.