Modeling coronal magnetic field using spherical geometry: cases with several active regions

The magnetic fields in the solar atmosphere structure the plasma, store free magnetic energy and produce a wide variety of active solar phenomena, like flare and coronal mass ejections (CMEs). The distribution and strength of magnetic fields are routinely measured in the solar surface (photosphere). Therefore, there is considerable interest in accurately modeling the 3D structure of the coronal magnetic field using photospheric vector magnetograms. Knowledge of the 3D structure of magnetic field lines also help us to interpret other coronal observations, e.g., EUV images of the radiating coronal plasma. Nonlinear force-free field (NLFFF) models are thought to be viable tools for those task. Usually those models use Cartesian geometry. However, the spherical nature of the solar surface cannot be neglected when the field of view is large. In this work, we model the coronal magnetic field above multiple active regions using NLFFF extrapolation code using vector magnetograph data from the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS)/Vector Spectromagnetograph (VSM) as a boundary conditions. We compare projections of the resulting magnetic field lines solutions with their respective coronal EUV-images from the Atmospheric Imaging Assembly (SDO/AIA) observed on October 15, 2011 and November 13, 2012. This study has found that the NLFFF model in spherical geometry reconstructs the magnetic configurations for several active regions which agrees to some extent with observations. During October 15, 2011 observation, there are substantial number of trans-equatorial loops carrying electric current.     

Coronal Mass Ejections as Expanding Force-Free Structures

We model solar coronal mass ejections (CMEs) as expanding force-free magnetic structures and find the self-similar dynamics of configurations with spatially constant ??, where J=?? B, in spherical and cylindrical geometries, expanding spheromaks and Lundquist fields, respectively. The field structures remain force-free, under the conventional non-relativistic assumption that the dynamical effects of the inductive electric fields can be neglected. While keeping the internal magnetic field structure of the stationary solutions, expansion leads to complicated internal velocities and rotation, caused by inductive electric fields. The structure depends only on overall radius R(t) and rate of expansion $\dot{R}(t)$ measured at a given moment, and thus is applicable to arbitrary expansion laws. In case of cylindrical Lundquist fields, magnetic flux conservation requires that both axial and radial expansion proceed with equal rates. In accordance with observations, the model predicts that the maximum magnetic field is reached before the spacecraft reaches the geometric center of a CME.     

Cluster magnetic fields from galactic outflows

We performed cosmological, magnetohydrodynamical simulations to follow the evolution of magnetic fields in galaxy clusters, exploring the possibility that the origin of the magnetic seed fields is galactic outflows during the starburst phase of galactic evolution. To do this, we coupled a semi-analytical model for magnetized galactic winds as suggested by Bertone, Vogt & Enßlin to our cosmological simulation. We find that the strength and structure of magnetic fields observed in galaxy clusters are well reproduced for a wide range of model parameters for the magnetized, galactic winds and do only weakly depend on the exact magnetic structure within the assumed galactic outflows. Although the evolution of a primordial magnetic seed field shows no significant differences to that of galaxy cluster fields from previous studies, we find that the magnetic field pollution in the diffuse medium within filaments is below the level predicted by scenarios with pure primordial magnetic seed field. We therefore conclude that magnetized galactic outflows and their subsequent evolution within the intracluster medium can fully account for the observed magnetic fields in galaxy clusters. Our findings also suggest that measuring cosmological magnetic fields in low-density environments such as filaments is much more useful than observing cluster magnetic fields to infer their possible origin.     

The solar corona as a minimum energy system?

This paper is an exploration of the possibility that the large-scale equilibrium of plasma and magnetic fields in the solar corona is a minimum energy state. Support for this conjecture is sought by considering the simplest form of that equilibrium in a dipole solar field, as suggested by the observed structure of the corona at times of minimum solar activity. Approximate, axisymmetric solutions to the MHD equations are constructed to include both a magnetically closed, hydrostatic region and a magnetically open region where plasma flows along field lines in the form of a transonic, thermally-driven wind. Sequences of such solutions are obtained for various degrees of magnetic field opening, and the total energy of each solution is computed, including contributions from both the plasma and magnetic field. It is shown that along a sequence of increasingly closed coronal magnetic field, the total energy curve is a non-monotonic function of the parameter measuring the degree of magnetic field opening, with a minimum occurring at moderate field opening.For reasonable choices of model parameters (coronal temperature, base density, base magnetic field strength, etc.), the morphology of the minimum energy solution resembles the observed quiet, solar minimum corona. The exact location energy minimum along a given sequence depends rather sensitively on some of the adopted parameter values. It is nevertheless argued that the existence of an energy minimum along the sequences of solutions should remain a robust property of more realistic coronal wind models that incorporate the basic characteristics of the equilibrium corona- the presence of both open and closed magnetic regions.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Equilibrium of isolated magnetostatic flux structure

Two-dimensional magnetostatic models of flux structure confined in stratified atmosphere are discussed in the present paper. The magnetic field in the flux structure is assumed to be force-free at the first step. Numerical solutions for this nonlinear free boundary problem are obtained by finite element method. Results show clearly the relation between the inside fields and outside pressure, especially the influence of atmospheric pressure distribution on the flux structure.     

On magnetostatic equilibrium in a stratified atmosphere

This is a study of the relationship between a magnetic field and its embedding plasma in static equilibrium in a uniform gravity. The ideal gas law is assumed. A system invariant in a given direction is treated first. We show that an exact integral of the equation for force balance across field lines can be derived in a closed form. Using this integral, exact solutions can be generated freely by integrating directly for the distributions of pressure, density and temperature necessary to keep a given magnetic field in equilibrium. Particular solutions are presented for illustration with the solar atmosphere in mind. Extending the treatment to the general system depending on all three spatial coordinates, we arrive at the general form of a theorem of Parker that a magnetic field in static equilibrium must possess certain symmetries. We derive an equation involving the Euler potentials of the magnetic field stipulating these necessary symmetries. Only those magnetic fields satisfying this equation can be in static equilibrium and for these fields, the endowed symmetries make the construction of exact solutions an essentially two dimensional problem as exemplified by the special case of invariance in a given direction.     

Magnetic field in a cavity of a perfect diamagnetic material

A new method using an integral equation has been proposed in this paper for calculating the magnetic field enclosed within a perfect diamagnetic material. A magnetic field is assumed to exist inside an arbitrary cavity in a perfect diamagnetic material. Self-consistent magnetic fields inside the cavity and currents on its surface are calculated, resulting as it should in zero magnetic field outside the cavity. This method has been tested in a special case and the results have been compared to the analytical solutions. This method should be also applicable to any case concerning the interaction between magnetic fields and perfect diamagnetic materials, e.g., to cases where a perfect diamagnetic material is surrounded by a magnetic field.     

Reconstructing the IRAS Point Source Catalog Redshift Survey with a generalized PIZA

We study the influence of different magnetic boundary conditions on the generation of magnetic fields by turbulent convection. It is found that the structure and strength of the generated field in the vicinity of the boundary is strongly dependent on the choice of boundary conditions. In the convective interior, however, the solutions remain largely insensitive to the boundary conditions. In all cases the overall efficiency of the dynamo process remains high with a steady state magnetic energy density between 12 and 25 per cent of the turbulent kinetic energy, and peak field values exceeding the equipartition level. These results support the idea that the solar granulation may constitute a dynamo source for magnetic fields in the quiet photosphere.     

Self-similar solution of hot accretion flows with ordered magnetic field and outflow

Observations and numerical magnetohydrodynamic (MHD) simulations indicate the existence of outflows and ordered large-scale magnetic fields in the inner region of hot accretion flows. In this paper, we present the self-similar solutions for advection-dominated accretion flows (ADAFs) with outflows and ordered magnetic fields. Stimulated by numerical simulations, we assume that the magnetic field has a strong toroidal component and a vertical component in addition to a stochastic component. We obtain the self-similar solutions to the equations describing the magnetized ADAFs, taking into account the dynamical effects of the outflow. We compare the results with the canonical ADAFs and find that the dynamical properties of ADAFs such as radial velocity, angular velocity and temperature can be significantly changed in the presence of ordered magnetic fields and outflows. The stronger the magnetic field is, the lower the temperature of the accretion flow will be and the faster the flow rotates. The relevance to observations is briefly discussed.     

The double inverse polarity paradigm

A model is presented for the origin of inverse polarity magnetic fields in the perpendicular as well as in the axial direction of quiescent prominences. The model is based on the presence of a discrete coronal arcade structure where magnetic separating surfaces can be identified. On the crossing of these separating surfaces magnetic reconnection driven by photospheric shear and converging motions can create the observed field direction in quiescent prominences.Dedicated to Cornelis de Jager     

Magnetic Null Points due to Multiple Sources of Solar Photospheric Flux

How common are magnetic null points in the highly complex magnetic field of the solar atmosphere? In this work we seek to model the magnetic structure of quiet regions by placing magnetic sources and sinks on a hexagonal network of supergranule cells to represent the intense magnetic fields that occur at the boundaries of these cells. The resulting potential coronal magnetic field is then computed analytically and searched numerically for magnetic null points, which are classified according to their types and spine directions. Two relations from the theory of vector fields relate the numbers of null points to the numbers of sources and sinks and these are used to check the numerical results. Previous results relating these quantities for monopolar and dipolar magnetic fields are described and a new one for a particular class of quadrupolar fields arising in this study is derived. We model a three-cell configuration and study the effects of increasing the strength of a central sink and of moving the central sink. A twelve-cell configuration is studied in lesser detail.

     

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.     

Gravitational collapse of polytropic, magnetized, filamentary clouds

For the case in which the gas of a magnetized filamentary cloud obeys a polytropic equation of state, gravitational collapse of the cloud is studied using a simplified model. We concentrate on the radial distribution and restrict ourselves to a purely toroidal magnetic field. If the axial motions and poloidal magnetic fields are sufficiently weak, we could reasonably expect our solutions to be a good approximation. We show that while the filament experiences gravitational condensation and the density at the centre increases, the toroidal flux-to-mass ratio remains constant. A series of spatial profiles of density, velocity and magnetic field for several values of the toroidal flux-to-mass ratio and the polytropic index, is obtained numerically and discussed.     

Theoretical interpretation of cosmic magnetic fields

The paper discusses the possibilty of interpreting the magnetic fields of astronomical bodies in the framework of a unified field theory.Using one of the solutions of the generalized field theory, a direct relation between the polar magnetic field, the angular velocity and the gravitational potential of the body considered, is obtained. The geometric model used for applications has spherical symmetry and is of the type (FIGI).The predictions of the theoretical formula, obtained from the model, are compared with available observational data, and with the empirical formula of Blackett. The theoretical formula gives a possible interpretation of a seed magnetic field which will develop and produce the large-scale magnetic field observed for celestial objects. The formula shows that the field is generated as a result of rotation of a massive object.     

Milestones in the Observations of Cosmic Magnetic Fields

Magnetic fields are observed everywhere in the universe. In this review, we concentrate on the observational aspects of the magnetic fields of Galactic and extragalactic objects. Readers can follow the milestones in the observations of cosmic magnetic fields obtained from the most important tracers of magnetic fields, namely, the star-light polarization, the Zeeman effect, the rotation measures (RMs, hereafter) of extragalactic radio sources, the pulsar RMs, radio polarization observations, as well as the newly implemented sub-mm and mm polarization capabilities. The magnetic field of the Galaxy was first discovered in 1949 by optical polarization observations. The local magnetic fields within one or two kpc have been well delineated by starlight polarization data. The polarization observations of diffuse Galactic radio background emission in 1962 confirmed unequivocally the existence of a Galactic magnetic field. The bulk of the present information about the magnetic fields in the Galaxy comes from anal     

The chromospheric magnetograph

By comparison of photoelectric magnetograms with high resolution Hα pictures it is possible to formulate a set of rules by which the magnetic field may be derived directly from the filtergrams. This is possible because of the regularities of magnetic field configurations on the sun and because chromospheric morphology is determined by the magnetic field. Off-band pictures (preferably 0.5 Å red) show a well-defined enhanced chromospheric network, the boundaries of which coincide with the 5 G contour of longitudinal field on the Mt. Wilson magnetograms. The actual fields are presumably more concentrated along the dark structure of the network. Higher fields are marked by filled-in cells. Regions of predominantly transverse fields may be inferred from the absence of normal network structure and the presence of chromospheric fibrils. The quiet chromosphere is recognized by the presence of oscillatory motion and the absence of fibrils or strong network structure. Thus, the chromosphere may be divided into three types of regions: enhanced network, horizontal field, and quiet network. The polarity of the magnetic field may be recognized by plage-antiplage asymmetry; that is, the fact that only following magnetic fields show bright plage in the center of Hα.     

Axisymmetric stars with both poloidal and toroidal magnetic fields

Chandrasekhar and Prendergast have established a result which has been assumed to imply that axisymmetric stars with an internal toroidal magnetic field should have zero external poloidal field. By considering mildly singular functions, the range of solutions is increased, and models can then be constructed which have toroidal and poloidal fields in the interior and a non-zero, external, poloidal field. Both the magnetic field and its associated current are continuous everywhere.     

Probing magnetic turbulence by synchrotron polarimetry: statistics and structure of magnetic fields from Stokes correlators

We describe a novel technique for probing the statistical properties of cosmic magnetic fields based on radio polarimetry data. Second-order magnetic field statistics like the power spectrum cannot always distinguish between magnetic fields with essentially different spatial structure. Synchrotron polarimetry naturally allows certain fourth-order magnetic field statistics to be inferred from observational data, which lifts this degeneracy and can thereby help us gain a better picture of the structure of the cosmic fields and test theoretical scenarios describing magnetic turbulence. In this work we show that a fourth-order correlator of specific physical interest, the tension force spectrum, can be recovered from the polarized synchrotron emission data. We develop an estimator for this quantity based on polarized emission observations in the Faraday rotation free frequency regime. We consider two cases: a statistically isotropic field distribution, and a statistically isotropic field superimposed on a weak mean field. In both cases the tension force power spectrum is measurable; in the latter case, the magnetic power spectrum may also be obtainable. The method is exact in the idealized case of a homogeneous relativistic electron distribution that has a power-law energy spectrum with a spectral index of   p = 3  , and assumes statistical isotropy of the turbulent field. We carry out numerical tests of our method using synthetic polarized emission data generated from numerically simulated magnetic fields. We show that the method is valid, that it is not prohibitively sensitive to the value of the electron spectral index p , and that the observed tension force spectrum allows one to distinguish between e.g. a randomly tangled magnetic field (a default assumption in many studies) and a field organized in folded flux sheets or filaments.     

Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields

Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive general magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric model with a ring current in the equatorial region and a time-dependent model of the Earth's magnetosphere. Our results are compared with the Olson-Pfitzer model.     

Dynamo hysteresis and grand minima of solar activity

A nonlinear dynamo model that allows for the dependence of the turbulent diffusivity on the magnetic field shows the phenomenon of a hysteresis. In a certain range of dynamo numbers, two types of solutions are possible: decaying oscillations of weak fields and magnetic cycles with a constant and large amplitude, which are settled depending on the initial conditions. Fluctuations in α-parameter cause transitions between these two regimes and calculations show the intermittency of magnetic cycles with a relatively large amplitude and epochs of weak magnetic fields. This behavior can serve as a model of grand minima of solar activity like the well-known Maunder minimum.