Solar Force-free Magnetic Fields

The structure and dynamics of the solar corona is dominated by the magnetic field. In most areas in the corona magnetic forces are so dominant that all non-magnetic forces like plasma pressure gradient and gravity can be neglected in the lowest order. This model assumption is called the force-free field assumption, as the Lorentz force vanishes. This can be obtained by either vanishing electric currents (leading to potential fields) or the currents are co-aligned with the magnetic field lines. First we discuss a mathematically simpler approach that the magnetic field and currents are proportional with one global constant, the so-called linear force-free field approximation. In the generic case, however, the relation between magnetic fields and electric currents is nonlinear and analytic solutions have been only found for special cases, like 1D or 2D configurations. For constructing realistic nonlinear force-free coronal magnetic field models in 3D, sophisticated numerical computations are required and boundary conditions must be obtained from measurements of the magnetic field vector in the solar photosphere. This approach is currently of large interests, as accurate measurements of the photospheric field become available from ground-based (for example SOLIS) and space-born (for example Hinode and SDO) instruments. If we can obtain accurate force-free coronal magnetic field models we can calculate the free magnetic energy in the corona, a quantity which is important for the prediction of flares and coronal mass ejections. 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.Comment: 49 pages, 11 figures, Living Reviews in Solar Physics, accepte

On the extrapolation of magneto-hydro-static equilibria on the sun

Modeling the interface region between solar photosphere and corona is challenging, because the relative importance of magnetic and plasma forces change by several orders of magnitude. While the solar corona can be modeled by the force-free assumption, we need to take care about plasma forces (pressure gradient and gravity) in photosphere and chromosphere, here within the magneto-hydro-static (MHS) model. We solve the MHS equations with the help of an optimization principle and use vector magnetogram as boundary condition. Positive pressure and density are ensured by replacing them with two new basic variables. The Lorentz force during optimization is used to update the plasma pressure on the bottom boundary, which makes the new extrapolation works even without pressure measurement on the photosphere. Our code is tested by using a linear MHS model as reference. From the detailed analyses, we find that the newly developed MHS extrapolation recovers the reference model at high accuracy. The MHS extrapolation is, however, numerically more expensive than the nonlinear force-free field (NLFFF) extrapolation and consequently one should limit their application to regions where plasma forces become important, e.g. in a layer of about 2 Mm above the photosphere.Comment: accepted for publication in Ap

Thin current sheets caused by plasma flow gradients in space and astrophysical plasma

Strong gradients in plasma flows play a major role in space and astrophysical plasmas. A typical situation is that a static plasma equilibrium is surrounded by a plasma flow, which can lead to strong plasma flow gradients at the separatrices between field lines with different magnetic topologies, e.g., planetary magnetospheres, helmet streamers in the solar corona, or at the boundary between the heliosphere and interstellar medium. Within this work we make a first step to understand the influence of these flows towards the occurrence of current sheets in a stationary state situation. We concentrate here on incompressible plasma flows and 2D equilibria, which allow us to find analytic solutions of the stationary magnetohydrodynamics equations (SMHD). First we solve the magnetohydrostatic (MHS) equations with the help of a Grad-Shafranov equation and then we transform these static equilibria into a stationary state with plasma flow. We are in particular interested to study SMHD-equilibria with strong plasma flow gradients perpendicular to separatrices. We find that induced thin current sheets occur naturally in such situations. The strength of the induced currents depend on the Alfv\'en Mach number and its gradient, and on the magnetic field.Comment: 10 pages, 6 figures, published in Annales Geophysica

Nonlinear force-free coronal magnetic stereoscopy

Getting insights into the 3D structure of the solar coronal magnetic field have been done in the past by two completely different approaches: (1.) Nonlinear force-free field (NLFFF) extrapolations, which use photospheric vector magnetograms as boundary condition. (2.) Stereoscopy of coronal magnetic loops observed in EUV coronal images from different vantage points. Both approaches have their strength and weaknesses. Extrapolation methods are sensitive to noise and inconsistencies in the boundary data and the accuracy of stereoscopy is affected by the ability of identifying the same structure in different images and by the separation angle between the view directions. As a consequence, for the same observational data, the computed 3D coronal magnetic field with the two methods do not necessarily coincide. In an earlier work (Paper I) we extended our NLFFF optimization code by the inclusion of stereoscopic constrains. The method was successfully tested with synthetic data and within this work we apply the newly developed code to a combined data-set from SDO/HMI, SDO/AIA and the two STEREO spacecraft. The extended method (called S-NLFFF) contains an additional term that monitors and minimizes the angle between the local magnetic field direction and the orientation of the 3D coronal loops reconstructed by stereoscopy. We find that prescribing the shape of the 3D stereoscopically reconstructed loops the S-NLFFF method leads to a much better agreement between the modeled field and the stereoscopically reconstructed loops. We also find an appreciable decrease by a factor of two in the angle between the current and the magnetic field which indicates the improved quality of the force-free solution obtained by S-NLFFF.Comment: 9 pages, 7 figure

The Magnetic Field in the Solar Atmosphere

This publication provides an overview of magnetic fields in the solar atmosphere with the focus lying on the corona. The solar magnetic field couples the solar interior with the visible surface of the Sun and with its atmosphere. It is also responsible for all solar activity in its numerous manifestations. Thus, dynamic phenomena such as coronal mass ejections and flares are magnetically driven. In addition, the field also plays a crucial role in heating the solar chromosphere and corona as well as in accelerating the solar wind. Our main emphasis is the magnetic field in the upper solar atmosphere so that photospheric and chromospheric magnetic structures are mainly discussed where relevant for higher solar layers. Also, the discussion of the solar atmosphere and activity is limited to those topics of direct relevance to the magnetic field. After giving a brief overview about the solar magnetic field in general and its global structure, we discuss in more detail the magnetic field in active regions, the quiet Sun and coronal holes.Comment: 109 pages, 30 Figures, to be published in A&AR

Fragmentation of electric currents in the solar corona by plasma flows

We consider a magnetic configuration consisting of an arcade structure and a detached plasmoid, resulting from a magnetic reconnection process, as is typically found in connection with solar flares. We study spontaneous current fragmentation caused by shear and vortex plasma flows. An exact analytical transformation method was applied to calculate self-consistent solutions of the nonlinear stationary MHD equations. The assumption of incompressible field-aligned flows implies that both the Alfven Mach number and the mass density are constant on field lines. We first calculated nonlinear MHS equilibria with the help of the Liouville method, emulating the scenario of a solar eruptive flare configuration with plasmoids and flare arcade. Then a Mach number profile was constructed that describes the upflow along the open magnetic field lines and implements a vortex flow inside the plasmoid. This Mach number profile was used to map the MHS equilibrium to the stationary one. We find that current fragmentation takes place at different locations within our configuration. Steep gradients of the Alfven Mach number are required, implying the strong influence of shear flows on current amplification and filamentation of the MHS current sheets. Crescent- or ring-like structures appear along the outer separatrix, butterfly structures between the upper and lower plasmoids, and strong current peaks close the lower boundary. Impressing an intrinsic small-scale structure on the upper plasmoid results in strong fragmentation of the plasmoid. Hence fragmentation of current sheets and plasmoids is an inherent property of MHD theory. Transformations from MHS into MHD steady-states deliver fine-structures needed for plasma heating and acceleration of particles and bulk plasma flows in dissipative events that are typically connected to magnetic reconnection processes in flares and coronal mass ejections.Comment: 12 pages, 7 figures, accepted for publication in Astronomy and Astrophysic

MHD flows at astropauses and in astrotails

The geometrical shapes and the physical properties of stellar wind -- interstellar medium interaction regions form an important stage for studying stellar winds and their embedded magnetic fields as well as cosmic ray modulation. Our goal is to provide a proper representation and classification of counter-flow configurations and counter-flow interfaces in the frame of fluid theory. In addition we calculate flows and large-scale electromagnetic fields based on which the large-scale dynamics and its role as possible background for particle acceleration, e.g. in the form of anomalous cosmic rays, can be studied. We find that for the definition of the boundaries, which are determining the astropause shape, the number and location of magnetic null points and stagnation points is essential. Multiple separatrices can exist, forming a highly complex environment for the interstellar and stellar plasma. Furthermore, the formation of extended tail structures occur naturally, and their stretched field and streamlines provide surroundings and mechanisms for the acceleration of particles by field-aligned electric fields.Comment: 10 pages, 4 Figure

The Magnetic Properties of Heating Events on High-Temperature Active Region Loops

Understanding the relationship between the magnetic field and coronal heating is one of the central problems of solar physics. However, studies of the magnetic properties of impulsively heated loops have been rare. We present results from a study of 34 evolving coronal loops observed in the Fe XVIII line component of AIA/SDO 94 A filter images from three active regions with different magnetic conditions. We show that the peak intensity per unit cross-section of the loops depends on their individual magnetic and geometric properties. The intensity scales proportionally to the average field strength along the loop ($B_{avg}$) and inversely with the loop length ($L$) for a combined dependence of $(B_{avg}/L)^{0.52\pm0.13}$. These loop properties are inferred from magnetic extrapolations of the photospheric HMI/SDO line-of-sight and vector magnetic field in three approximations: potential and two Non Linear Force-Free (NLFF) methods. Through hydrodynamic modeling (EBTEL model) we show that this behavior is compatible with impulsively heated loops with a volumetric heating rate that scales as $\epsilon_H\sim B_{avg}^{0.3\pm0.2}/L^{0.2\pm^{0.2}_{0.1}}$.Comment: Astrophysical Journal, in pres

Coronal magnetic field extrapolation using a specific family of analytical 3D MHS equilibria

Funding: LN acknowledges financial support by the University of St Andrews, and TN acknowledges support by the United Kingdom’s Science and Research Council (STFC) via Consolidated Grant ST/W001195/1.With current observational methods it is not possible to determine the magnetic field in the solar corona accurately. Therefore, coronal magnetic field models have to rely on extrapolation methods using photospheric magnetograms as boundary conditions. In recent years, due to the increased resolution of observations and the need to resolve non-force-free lower regions of the solar atmosphere, there have been increased efforts to use magnetohydrostatic (MHS) field models instead of force-free extrapolation methods. Although numerical methods to calculate MHS solutions can deal with non-linear problems and hence provide more accurate models, analytical three-dimensional MHS equilibria can also be used as a numerically relatively “cheap” complementary method. We discuss a family of analytical MHS equilibria that allows for a transition from a non-force-free region to a force-free region. The solution involves hypergeometric functions and while routines for the calculation of these are available, this can affect both the speed and the numerical accuracy of the calculations. Therefore, we look into the asymptotic behaviour of this solution in order to numerically approximate it through exponential functions aiming to improve the numerical efficiency. We present an illustrative example by comparing field line profiles, density and pressure differences between the exact solutions, the asymptotic solution and a hybrid model where the use of the hypergeometric function is restricted to an area around the transitional region between the non-force-free and the force-free domain.Publisher PDFNon peer reviewe

Doppler shift of hot coronal lines in a moss area of an active region

The moss is the area at the footpoint of the hot (3 to 5 MK) loops forming the core of the active region where emission is believed to result from the heat flux conducted down to the transition region from the hot loops. Studying the variation of Doppler shift as a function of line formation temperatures over the moss area can give clues on the heating mechanism in the hot loops in the core of the active regions. We investigate the absolute Doppler shift of lines formed at temperatures between 1 MK and 2 MK in a moss area within active region NOAA 11243 using a novel technique that allows determining the absolute Doppler shift of EUV lines by combining observations from the SUMER and EIS spectrometers. The inner (brighter and denser) part of the moss area shows roughly constant blue shift (upward motions) of 5 km/s in the temperature range of 1 MK to 1.6 MK. For hotter lines the blue shift decreases and reaches 1 km/s for Fe xv 284 {\AA} (~2 MK). The measurements are discussed in relation to models of the heating of hot loops. The results for the hot coronal lines seem to support the quasi-steady heating models for non-symmetric hot loops in the core of active regions.Comment: 11 pages, 15 Figures, Astronomy and Astrophysics (in press