Force-free Approximation Research Articles

Launching of jets by cold, magnetized disks in Kerr metric

We confirm recent discovery by Cao that in the vicinity of fast rotating black holes jets can be launched centrifugally by cold, magnetized disks even for nearly vertically shaped magnetic flux surfaces. Outflows produced under such extreme conditions are investigated via studying kinematics of test particles in the force-free magnetosphere approximation. Implications of a possibility of magneto-centrifugal launching of very well collimated central outflows around the fast rotating black holes are discussed in the general context of the jet formation scenarios in AGNs.

EFFECT OF POLARIMETRIC NOISE ON THE ESTIMATION OF TWIST AND MAGNETIC ENERGY OF FORCE-FREE FIELDS

The force-free parameter $\alpha$, also known as helicity parameter or twist parameter, bears the same sign as the magnetic helicity under some restrictive conditions. The single global value of $\alpha$ for a whole active region gives the degree of twist per unit axial length. We investigate the effect of polarimetric noise on the calculation of global $\alpha$ value and magnetic energy of an analytical bipole. The analytical bipole has been generated using the force-free field approximation with a known value of constant $\alpha$ and magnetic energy. The magnetic parameters obtained from the analytical bipole are used to generate Stokes profiles from the Unno-Rachkovsky solutions for polarized radiative transfer equations. Then we add random noise of the order of 10$^{-3}$ of the continuum intensity (I$_{c}$) in these profiles to simulate the real profiles obtained by modern spectropolarimeters like Hinode (SOT/SP), SVM (USO), ASP, DLSP, POLIS, SOLIS etc. These noisy profiles are then inverted using a Milne-Eddington inversion code to retrieve the magnetic parameters. Hundred realizations of this process of adding random noise and polarimetric inversion is repeated to study the distribution of error in global $\alpha$ and magnetic energy values. The results show that : (1). the sign of $\alpha$ is not influenced by polarimetric noise and very accurate values of global twist can be calculated, and (2). accurate estimation of magnetic energy with uncertainty as low as 0.5% is possible under the force-free condition.

The X17.2 flare occurred in NOAA 10486: an example of filament destabilization caused by a domino effect

Context. It is now possible to distinguish between two main models describing the mechanisms responsible for eruptive flares : the standard model, which assumes that most of the energy is released, by magnetic reconnection, in the region hosting the core of a sheared magnetic field, and the breakout model, which assumes reconnection occurs at first in a magnetic arcade overlaying the eruptive features. Aims. We analyze the phenomena observed in NOAA 10486 before and during an X17.2 flare that occurred on 2003 October 28, to study the relationship between the pre-flare and flare phases and determine which model is the most suitable for interpreting this event. Methods. We performed an analysis of multiwavelength data set available for the event using radio data (0.8–4.5 GHz), images in the visible range (WL and Hα), EUV images (1600 and 195 A), and X-ray data, as well as MDI longitudinal magnetograms. We determined the temporal sequence of events occurring before and during the X17.2 flare and the magnetic field configuration in the linear force-free field approximation. Results. The active region was characterized by a multiple arcade configuration and the X17.2 flare was preceded, by ∼ 2h , by the partial eruption of one filament. This eruption caused reconnection at null points located in the low atmosphere and a decrease in magnetic tension in the coronal field lines overlaying other filaments present in the active region. As a consequence, these filaments were destabilized and the X17.2 flare occurred. Conclusions. The phenomena observed in NOAA 10486 before and during the X17.2 flare cannot be explained by a simple scenario such as the standard or breakout model, but instead in terms of a so-called domino effect, involving a sequence of destabilizing processes that triggered the flare.

Relativistic force-free electrodynamic simulations of neutron star magnetospheres

Abstract The luminosity and structure of neutron star magnetospheres are crucial to our understanding of pulsar and plerion emission. A solution found using the force-free approximation would be an interesting standard with which any model with more physics could be compared. Prior quasi-analytic force-free solutions may not be stable, while prior time-dependent magnetohydrodynamic models used unphysical model parameters. We use a time-dependent relativistic force-free electrodynamics code with no free parameters to find a unique stationary solution for the axisymmetric rotating pulsar magnetosphere in a Minkowski space–time in the case of no surface currents on the star. The solution is similar to the force-free quasi-analytic solution found in 1999 by Contopoulos, Kazanas & Fendt and the numerical magnetohydrodynamic solution found in 2006 by Komissarov. The magnetosphere structure and the usefulness of the classical y-point in the general dissipative regime are discussed. The pulsar luminosity is found to be L≈ 0.99 ± 0.01 μ2Ω4★/c3 for a dipole moment μ and stellar angular frequency Ω★.

On the force-free magnetosphere of an aligned rotator

We investigate in detail the properties of the stationary force-free magnetosphere of an aligned rotator assuming the last closed field line is lying in the equatorial plane at large distances from pulsar. The pulsar equation is solved numerically using a multigrid code with high numerical resolution, and physical properties of the magnetosphere are obtained with high accuracy. We found a set of solutions with different sizes of the closed magnetic field line zone and verify the applicability of the force-free approximation. We discuss the role of electron-positron cascades in supporting the force-free magnetosphere and argue that the closed field line zone should grow with time at a slower rate than the light cylinder. This yields a pulsar breaking index of less than 3. It is shown that models of an aligned rotator magnetosphere with a widely accepted configuration of the magnetic field, such as the one considered in this paper, have serious difficulties. We discuss the solutions of this problem and argue that in any case pulsar energy losses should evolve with time differently than is predicted by the magnetodipolar formula.

Emergence of undulatory magnetic flux tubes by small scale reconnections

With Flare Genesis Experiment (FGE), a balloon borne observatory launched in Antarctica on January 2000, series of high spatial resolution vector magnetograms, Dopplergrams, and Hα filtergrams have been obtained in an emerging active region (AR 8844). Previous analyses of this data revealed the occurence of many short-lived and small-scale H α brightenings called ‘Ellerman bombs’ (EBs) within the AR. We performed an extrapolation of the field above the photosphere using the linear force-free field approximation. The analysis of the magnetic topology reveals a close connexion between the loci of EBs and the existence of “Bald patches” (BP) regions (BPs are regions where the vector magnetic field is tangential to the photosphere). Some of these EBs/BPs are magnetically connected by low-lying field lines, presenting a serpentine shape. This results leads us to conjecture that arch filament systems and active regions coronal loops do not result from the smooth emergence of large scale Ω-loops, but rather from the rise of flat undulatory flux tubes which get released from their photospheric anchorage by reconnection at BPs, which observational signature is Ellerman bombs.

Comparing magnetic field extrapolations with measurements of magnetic loops

We compare magnetic field extrapolations from a photospheric magnetogram with the observationally inferred structure of magnetic loops in a newly developed active region. This is the first time that the reconstructed 3D-topology of the magnetic field is available to test the extrapolations. We compare the observations with potential fields, linear force-free fields and non-linear force-free fields. This comparison reveals that a potential field extrapolation is not suitable for a reconstruction of the magnetic field in this young, developing active region. The inclusion of field-line-parallel electric currents, the so called force-free approach, gives much better results. Furthermore, a non-linear force-free computation reproduces the observations better than the linear force-free approximation, although no free parameters are available in the former case.

Transport of Magnetic Helicity and Dynamics of Solar Active Regions

AbstractWe introduce a method to calculate the magnetic helicity density in a given active-region vector magnetogram, and a lower limit of it, based on a linear force-free (Iff) approximation. Moreover, we provide a lower limit of the total magnetic helicity in the active region (AR). A time series of magnetograms can be used to calculate the rate of helicity transport. The results can be then correlated with manifestations of the dynamical activity in ARs, such as flares and filament eruptions.

General relativistic magnetohydrodynamic simulations of monopole magnetospheres of black holes

In this paper, we report on the results of the first ever time-dependent general relativistic magnetohydrodynamic simulations of the magnetically dominated monopole magnetospheres of black holes. It is found that the numerical solution evolves towards a stable steady-state solution which is very close to the corresponding force-free solution found by Blandford & Znajek. Contrary to the recent claims, the particle inertia does not become dynamically important near the event horizon and the force-free approximation provides a proper framework for magnetically dominated magnetospheres of black holes. For the first time, our numerical simulations show the development of an ultrarelativistic particle wind from a rotating black hole. However, the flow remains Poynting-dominated all the way up to the fast critical point. This suggests that the details of the so-called ‘astrophysical load’, where the electromagnetic energy is transferred to particles, may have no effect on the efficiency of the Blandford–Znajek mechanism.

Electrodynamics of black hole magnetospheres

The main goal of this research is to get better insights into the properties of the plasma-filled magnetospheres of black holes by means of direct numerical simulations and, ultimately, to resolve the controversy surrounding the Blandford-Znajek mechanism. Driven by the need to write the equations of black hole electrodynamics in a form convenient for numerical applications, we constructed a new system of 3 + 1 equations, which not only has a more traditional form than the now classic 3 + 1 system of Thorne and Macdonald but also is more general. To deal with the magnetospheric current sheets, we also developed a simple model of radiative resistivity based on the inverse Compton scattering of background photons. The results of numerical simulations combined with simple analytical arguments allow us to make a number of important conclusions on the nature of the Blandford-Znajek mechanism. We show that, just like in the Penrose mechanism and in the magnetohydrodynamic models of Punsly and Coroniti, the key role in this mechanism is played by the black hole ergosphere. The poloidal currents are driven by the gravitationally induced electric field, which cannot be screened within the ergosphere by any static distribution of the electric charge of locally created pair plasma. Contrary to what is expected in the membrane paradigm, the energy and angular momentum are extracted not only along the magnetic field lines penetrating the event horizon but also along all field lines penetrating the ergosphere. In dipolar magnetic configurations symmetric relative to the equatorial plane, the force-free approximation breaks down within the ergosphere, where a strong current sheet develops along the equatorial plane. This current sheet supplies energy and angular momentum at infinity to the surrounding force-free magnetosphere. The Blandford-Znajek monopole solution is found to be asymptotically stable and causal. The so-called horizon boundary condition of Znajek is shown to be a regularity condition at fast critical surface.

An idealized pulsar magnetosphere: the relativistic force-free approximation

The non-dissipative relativistic force-free condition should be a good approximation to describe the electromagnetic field in much of the pulsar magnetosphere, but we may plausibly expect it to break down in singular domains. Self-consistent magnetospheric solutions are found with field lines closing both at and within the light-cylinder. In general, the detailed properties of the solutions may be affected critically by the physics determining the appropriate choice of equatorial boundary condition beyond the light-cylinder.

Force‐free Waves and Black Hole Magnetospheric Causality

The force free approximation is often useful when describing tenuous plasmas in strong cosmic magnetic fields. Time evolution of any such system is governed by the information that can be transported along the characteristics of the plasma modes allowed by the force free constraint. This brief article elucidates, for the first time, the nature of the information that can be transported along the characteristics for each of the two plasma modes, the Alfven and fast modes. It is reassuring that these modes behave as one would expect if the perfect magnetohydrodynamic (MHD) modes in a magnetically dominated plasma were evaluated in the limit of zero plasma inertia (i.e., only allowed to propagate information consistent with the force free constraint). There are no properties of these waves that do not exist within the MHD theory, when this additional constraint on the current is imposed. The results of the characteristic analysis elucidates the nature of the causality violation in force free black hole magnetospheres. The most significant result from the standpoint of global causality is that charge and current perturbations can not be transported along the fast mode short wavelength characteristics in the force free limit.

The minimization of residual stresses in magnetic systems with quasi-force-free current distribution

The feasibility of a magnet experiencing very low mechanical stresses is studied. Megagauss fields can be generated without failure of the solenoid winding and excessively increasing the winding dimensions if the current distribution is force-free in its inner part and the axial current closes in its outer part. The approximation of the quasi-force-free current distribution by discrete conductive layers is considered. Analysis and computations are performed for two force-free winding approximations. In the former case, the conductors are divided into N paired balanced layers where the azimuth and poloidal currents, respectively, pass. The equal and oppositely directed forces arising in the layers of a pair are transferred to insulating spacers between the layers. Stresses in the spacers can be reduced down to values N times as low as the magnetic field pressure B 0 2 /(2μ0) at the solenoid axis. If the current direction in each of the layers meets the balance condition, the residual stresses can be N2 times lower than B 0 2 /(2μ0), because the tensile and compressive forces in the layers partially cancel.

Quantitative Magnetohydrodynamic Modeling of the Solar Transition Region

The transition region (TR) is assumed to be a collision-dominated plasma. The dissipation and transport of energy in such a plasma is accurately described by the classical transport coefficients, which include the electrical and thermal conductivity, viscosity, and thermoelectric tensors. These tensors are anisotropic and are functions of local values of temperature, density, and magnetic field strength. The transport coefficients are valid for all magnetic field strengths and so may be used to study the physics of weakly as well as strongly magnetized regions of the TR. They may be used in an MHD model to obtain a self-consistent, realistic description of the TR. The physics of kinetic processes is included in the MHD model through the transport coefficients. As a first step in studying heating and cooling processes in the TR in a realistic, quantitative manner, a 1.5 dimensional, steady state MHD model with a specified temperature profile is developed. The momentum equation includes the inertial, pressure, magnetic, and gravitational forces. Ohm's law includes the exact expressions for the electrical conductivity and thermoelectric tensors. It is found that the contribution of the dissipation of large-scale electric currents to in situ heating of the TR is negligible, but that thermal energy flowing into the TR from the corona can provide the energy required to heat the TR. The possibility that significant in situ heating of the TR takes place through viscous dissipation or small-scale electric current dissipation such as may occur in current sheets or filaments is discussed, although these processes are not described by the model. The importance of thermoelectric and electron pressure gradient effects in Ohm's law, and in determining the electron heat flux, is demonstrated. Results of the model suggest that the force-free approximation is not valid over most of the TR. Justification for assuming that the TR is collision dominated is presented. In particular, a self-consistent calculation of the ratio of the electric field parallel to the magnetic field to the Dreicer electric field yields a value 10-3, which suggests that anomalous transport processes are not important. The necessity of using a realistic description of transport processes in modeling heating mechanisms in the solar atmosphere is stressed.

The force-free magnetosphere around an oblique rotator

The force-free magnetosphere around an oblique rotator

9.15. Theoretical method for treating force-free black hole magnetosphere in non-stationary and non-axisymmetric state

The force-free approximation is a very useful method in studies of the global electromagnetic structure of the magnetospheres around the relativistic astronomical bodies. However, its application has almost been restricted to the stationary and axisymmetric configuration. In this work, we remove this restriction and construct a general method for treating the force-free electromagnetic field.

Theory of force-free electromagnetic fields. I. General theory

A general method to deal with the (relativistic) force-free electromagnetic field is developed. We formulate the theory without assuming symmetry of the electromagnetic field configuration. Thus we can apply it to any object where the force-free approximation is justified, e.g., the pulsar magnetosphere, the black-hole magnetosphere, and the magnetosphere around the accretion disk. We describe the force-free electromagnetic field by a classical field theory. The basic variables are the Euler potentials extended to the relativistic degenerate electromagnetic field. The basic equation is given by a two-component nonlinear equation for two Euler potentials. The theory has a close connection with geometry. It is based on the concept of the flux surface. The flux surface is a geometric entity corresponding to the world sheet of the magnetic field line. We give both the covariant and the 3+1 expression of the basic equation. By the latter form, the causal development of the force-free electromagnetic field is discussed. It is shown that the theory describes the causal development of the force-free electromagnetic field self-consistently as far as $F\ensuremath{\cdot}F>0$. The basic equation contains arbitrariness. It does not determine the solution uniquely. Although this arbitrariness originates from the gauge freedom of the electromagnetic field, it differs from the arbitrariness in the ordinary gauge field theories. Namely, the dynamics of the Euler potentials itself does not contain arbitrariness. It appears from nonuniqueness in correspondence between the Euler potentials and the electromagnetic field. Further, we discuss the breakdown of the force-free approximation.

CAN WE EXTRAPOLATE A MAGNETIC FIELD WHEN ITS TOPOLOGY IS COMPLEX

In order to understand various solar phenomena controlled by the magnetic field, such as X-ray bright points, flares and prominence eruptions, the structure of the coronal magnetic field must be known. This requires a precise extrapolation of the photospheric magnetic field. Presently, only potential or linear force-free field approximations can be used easily. A more realistic modelling of the field is still an active research area because of well-known difficulties related to the nonlinear mixed elliptic-hyperbolic nature of the equations. An additional difficulty arises due to the complexity of the magnetic field structure which is caused by a discrete partition of the photospheric magnetic field. This complexity is not limited to magnetic regions having magnetic nulls (and so separatrices) but also occurs in those containing thin elongated volumes (called Quasi-Separatrix Layers) where the photospheric field-line linkage changes rapidly. There is a wide range for the thickness of such layers, which is determined by the character (bipolar or quadrupolar) of the magnetic region, by the sizes of the photospheric field concentrations and by the intensity of the electric currents. The aim of this paper is to analyse the recent nonlinear force-free field extrapolation techniques for complex coronal magnetic fields.

3D magnetic reconnection at an X-ray bright point

On May 1, 1993, a flaring X-ray bright point (XBP) was observed for about 16 hours in the old, disintegrating, bipolar active region (AR) NOAA 7493. During this period, a minor magnetic bipole (1020 Mx) emerged in the region. We have found observational evidence showing that the XBP brightenings were due to magnetic reconnection between the new bipole and pre-existing plage fields. The aim of the present work is to substantiate with magnetic modelling what has been shown by the observations. For this purpose we extrapolate the observed photospheric magnetic fields in the linear force-free approximation and follow its evolution during the lifetime of the XBP. From the computed coronal field lines we determine the location of regions of drastic change in field-line linkage, called ‘quasi-separatrix layers’ or QSLs. QSLs are open layers that behave physically like separatrices: the break down of ideal magnetohydrodynamics and the release of free magnetic energy may occur at these locations when their thickness is small enough. The extrapolated field lines, with photospheric footpoints on both sides of QSLs, match the observed chromospheric and coronal structures (arch filament system, XBP and faint X-ray loops (FXL)). We study also the evolution of the width of the QSL located over the new negative polarity pore: the calculated QSL is very thin (typically less than 100 m) during the lifetime of the XBP, but becomes much thicker (≥ 104 m) after the XBP has faded. Furthermore we show that peaks in X-ray brightness propagate along the FXL with a velocity of ≈ 670 km s-1, starting from the XBP location, implying that the energy is released where the emerging bipole impacts against pre-existing coronal loops. We discuss the possible mechanism of energy transport and conclude that the energy is conducted to the remote footpoints of the FXL by a thermal front. These results strongly support the supposition that the XBP brightness and flaring are due to the interaction of different flux systems, through 3D magnetic reconnection, at QSLs.

Coronal magnetic fields from microwave polarization observations

The solar active region (AR) 7530 was observed at 6 cm on July 3 and 4, 1993 with the Westerbork Synthesis Radio Telescope, using a multi-channel receiver with very narrow bandwidth. We compare the radio data with Yohkoh SXT observations and with the magnetic field extrapolated from the Marshall vector magnetograms in the force-free and current-free approximations. The comparison with soft X-rays shows that, although a general agreement exists between the shape of the radio intensity map and the X-ray loops, the brightness temperature, Tb, obtained using the parameters derived from the SXT is much lower than that observed. The comparison with the extrapolated photospheric fields shows instead that they account very well for the observed Tb above the main sunspots, if gyroresonance emission is assumed. In the observation of July 4 an inversion and strong suppression of the circular polarization was clearly present above different portions of the AR, which indicates that particular relationships exist between the electron density and the magnetic field in the region where the corresponding lines of sight cross the field quasi-perpendicularly. The extrapolated magnetic field at a much higher level (∼ 1010 cm), satisfies the constraints required by the wave propagation theory all over the AR. However, a rather low electron density is derived.