New Boundary Integral Equation Representation for Finite Energy Force-Free Magnetic Fields in Open Space above the Sun

A boundary integral equation to describe a force-free magnetic field with finite energy content in the open space above the solar surface is found. This is a new representation for a 3-D nonlinear force-free field in terms of the boundary field and its normal gradient at the boundary. Therefore the magnetic field observed on the solar surface can be incorporated into the formulation directly and a standard numerical technique, the boundary element method, can be applied to solve the field. A numerical test case demonstrates the power of the method by recovering the analytical solution to the desired accuracy and its application to practical solar magnetic field problems is straightforward and promising.     

Topology of Magnetic Field and Coronal Heating in Solar Active Regions

Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the transverse components of the observed boundary field are calibrated with a linear force-free field model without changing their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique.     

Magnetic field and current sheets in the corona above active regions

A new method for the calculation of coronal magnetic field is proposed and it is shown to reproduce the EUV features in the corona as observed by Skylab experiments satisfactorily well. One of the remarkable points is that it reproduces the loopy threads in the active region corona and also the large scale field lines connecting active regions. The existence of coronal current is expected wherever the present coronal-current-free model fails to represent the feature. A method of calculating the coronal sheet-current is also developed with the purpose of knowing the shape of the current sheet and the amount of magnetic stress energy stored due the the presence of it by comparing the calculated field configuration with the observed local distortion of the EUV threads. This may be used in pinning down the possible site of the flare and in discussing the flare occurrence in terms of the energy stored there.During the preparation of this work, Poletto et al. (1975) calculated the magnetic field shape in Schmidt's method to compare with the soft X-ray feature obtained by Skylab.     

Numerical Solutions of Three-Dimensional Pressure-Bounded Magnetohydrostatic Flux Tubes

We present a three-dimensional technique for the solution of the magnetohydrostatic equations when we are modeling structures bounded by a current sheet that is free to move to satisfy pressure balance. The magnetic field is expressed in terms of Euler potentials and the equations are transformed to flux coordinates, greatly simplifying the problem of locating the free boundary. Multi-grid techniques are used to rapidly solve the resulting nonlinear elliptic partial differential equations. The method is tested against Low's (1982) exact solution of a bipolar plasma loop. It is shown that fast, accurate solutions can be found.     

Magnetic fields and the structure of the solar corona

Several different mathematical methods are described which use the observed line-of-sight component of the photospheric magnetic field to determine the magnetic field of the solar corona in the current-free (or potential-field) approximation. Discussed are (1) a monopole method, (2) a Legendre polynomial expansion assuming knowledge of the radial photospheric magnetic field, (3) a Legendre polynomial expansion obtained from the line-of-sight photospheric field by a least-meansquare technique, (4) solar wind simulation by zero-potential surfaces in the corona, (5) corrections for the missing flux due to magnetograph saturation. We conclude (1) that the field obtained from the monopole method is not consistent with the given magnetic data because of non-local effects produced by monopoles on a curved surface, (2) that the field given by a Legendre polynomial (which is fitted to the measured line-of-sight magnetic field) is a rigorous and self-consistent solution with respect to the available data, (3) that it is necessary to correct for the saturation of the magnetograph (at about 80 G) because fields exceeding 80 G provide significant flux to the coronal field, and (4) that a zero-potential surface at 2.5 solar radii can simulate the effect of the solar wind on the coronal magnetic field.     

Predicting the evolution of photospheric magnetic field in solar active regions using deep learning

The continuous observation of the magnetic field by the Solar Dynamics Observatory(SDO)/Helioseismic and Magnetic Imager(HMI) produces numerous image sequences in time and space.These sequences provide data support for predicting the evolution of photospheric magnetic field. Based on the spatiotemporal long short-term memory(LSTM) network, we use the preprocessed data of photospheric magnetic field in active regions to build a prediction model for magnetic field evolution. Because of the elaborate learning and memory mechanism, the trained model can characterize the inherent relationships contained in spatiotemporal features. The testing results of the prediction model indicate that(1) the prediction pattern learned by the model can be applied to predict the evolution of new magnetic field in the next 6 hours that have not been trained, and predicted results are roughly consistent with real observed magnetic field evolution in terms of large-scale structure and movement speed;(2) the performance of the model is related to the prediction time; the shorter the prediction time, the higher the accuracy of the predicted results;(3) the performance of the model is stable not only for active regions in the north and south but also for data in positive and negative regions. Detailed experimental results and discussions on magnetic flux emergence and magnetic neutral lines finally show that the proposed model could effectively predict the large-scale and short-term evolution of the photospheric magnetic field in active regions. Moreover, our study may provide a reference for the spatiotemporal prediction of other solar activities.     

Magnetic field evolution in simulations with Euler potentials

Using two- and three-dimensional hydromagnetic simulations for a range of different flows, including laminar and turbulent ones, it is shown that solutions expressing the field in terms of Euler potentials (EP) are in general incorrect if the EP are evolved with an artificial diffusion term. In three dimensions, standard methods using the magnetic vector potential are found to permit dynamo action when the EP give decaying solutions. With an imposed field, the EP method yields excessive power at small scales. This effect is more exaggerated in the dynamic case, suggesting an unrealistically reduced feedback from the Lorentz force. The EP approach agrees with standard methods only at early times when magnetic diffusivity did not have time to act. It is demonstrated that the usage of EP with even a small artificial magnetic diffusivity does not converge to a proper solution of hydromagnetic turbulence. The source of this disagreement is not connected with magnetic helicity or the three-dimensionality of the magnetic field, but is simply due to the fact that the non-linear representation of the magnetic field in terms of EP that depend on the same coordinates is incompatible with the linear diffusion operator in the induction equation.     

Unified classical and quantum radiation mechanism for ultrarelativistic electrons in curved and inhomogeneous magnetic fields

We analyse the general radiation emission mechanism from a charged particle moving in a curved inhomogeneous magnetic field. The consideration of the gradient makes the vacuum magnetic field compatible with the Maxwell equations, and adds a non-trivial term to the transverse drift velocity, and, consequently, to the general radiation spectrum. To obtain the radiation spectrum in the classical domain a general expression for the spectral distribution and characteristic frequency of an electron in arbitrary motion is derived, by using Schwinger's method. The radiation patterns of the ultrarelativistic electron are represented in terms of the acceleration of the particle. The same results can be obtained by considering that the motion of the electron can be formally described as an evolution caused by magnetic and electric forces. By defining an effective electromagnetic field, which combines the magnetic field with the fictitious electric field associated to the curvature and drift motion, one can obtain all the physical characteristics of the radiation by replacing the constant magnetic field with the effective field. The power, angular distribution and spectral distribution of all three components (synchrotron, curvature and gradient) of the radiation are considered, in both the classical and the quantum domain, within the framework of this unified formalism. In the quantum domain the proposed approach allows the study of the effects of the inhomogeneities and curvature of the magnetic field on the radiative transition rates of electrons between low-lying Landau levels and the ground state.     

Structure of the magnetic field near the galactic plane

A method is introduced for constructing two-color maps for the in-plane component of the magnetic field of our galaxy in (R, l) and (DM, l) coordinates. It is shown that, in agreement with the standard models of the galactic magnetic field, the magnetic field in neighboring spiral arms reverses direction. However, the magnetic field in the spiral arm of Sagittarius differs significantly from the standard magnetic field model, with the major difference being that the magnetic fields in the southern and northern hemispheres are oppositely directed in the spiral arm of Sagittarius. It is proposed that this distribution of the magnetic field can be explained best by assuming that the spiral arm of Sagittarius, or, at least, a magnetic spiral arm in that region, is not symmetric with respect to the galactic plane and lies mainly in the northern hemisphere.     

Automated Technique For Comparison Of Magnetic Field Inversion Lines With Filament Skeletons From The Solar Feature Catalogue

We present an automated technique for comparison of magnetic field inversion-line maps from SOHO/MDI magnetograms with solar filament data from the Solar Feature Catalogue created as part of the European Grid of Solar Observations project. The Euclidean distance transform and connected component labelling are used to identify nearest inversion lines to filament skeletons. Several filament inversion-line characteristics are defined and used to automate the decision whether a particular filament/inversion-line pair is suitable for quantitative comparison of orientation and separation. The technique is tested on 551 filaments from 14 Hα images at various dates, and the distributions of angles and distances between filament skeletons and line-of-sight (LOS) magnetic inversion lines are presented for six levels of magnetic field smoothing. The results showed the robustness of the developed technique which can be applied for a statistical analysis of magnetic field in the vicinity of filaments. The method accuracy is limited by the static filament detection which does not distinguish between filaments, fibrils, pre-condensations and filament barbs and this may increase the asymmetries in magnetic distributions and broadening in angular distributions that requires the incorporation of a feature tracking technique.     

Potential field extrapolation using three components of a solar vector magnetogram with a finite field of view

The potential magnetic field from a finite planar boundary is extrapolated into the upper hemisphere using information from all three magnetic field components. The method determines, first, the transverse field associated with the observed normal magnetic intensity. Then by subtraction, the method determines the associated transverse magnetic field observed in the interior (i.e., in the field of view) of the magnetogram which is due to the normal flux exterior to the field of view of the magnetogram. Inverting this information gives an approximation to the exterior normal flux. The combination of the observed normal flux of the interior and the approximation of the exterior normal flux is employed to calculate the potential field. The formulation of the problem results in an ill-posed integral inversion problem in which a regularized solution is obtained using the singular value decomposition (SVD) technique in conjunction with an appropriate Tikhonov-Phillips filter. The technique can be applied to correcting potential field calculations which are influenced by out-of-view fluxes, e.g., for a high spatial resolution vector magnetogram with a small field of view in which there is no supporting exterior data. The problem studied is also important in providing a regularized solution of the Cauchy potential problem. The method provides a much larger range of convergence than the method of Gary and Musielak (1992), and, in fact, is stable in the total upper hemisphere.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Magnetic Field Extrapolations into the Corona: Success and Future Improvements

The solar atmosphere being magnetic in nature, the understanding of the structure and evolution of the magnetic field in different regions of the solar atmosphere has been an important task over the past decades. This task has been made complicated by the difficulties to measure the magnetic field in the corona, while it is currently known with a good accuracy in the photosphere and/or chromosphere. Thus, to determine the coronal magnetic field, a mathematical method has been developed based on the observed magnetic field. This is the so-called magnetic field extrapolation technique. This technique relies on two crucial points: i) the physical assumption leading to the system of differential equations to be solved, ii) the choice and quality of the associated boundary conditions. In this review, I summarise the physical assumptions currently in use and the findings at different scales in the solar atmosphere. I concentrate the discussion on the extrapolation techniques applied to solar magnetic data and the comparison with observations in a broad range of wavelengths (from hard X-rays to radio emission).     

The cooling of a sunspot

A mechanism is proposed to explain the cooling of a sunspot in terms of the detailed interactions between the magnetic field and the convective motions. The mechanism provides that an axially symmetric concentration of magnetic field deforms the normal supergranule cell pattern below the sunspot into a radial outflow of plasma over a region of diameter 60 Mm.The flow occurs at depths where the magnetic and kinetic energy densities are approximately equal ( 5 Mm) and is described in terms of a Carnot refrigeration cycle. Application of the hydromagnetic equations to a very simple model shows that, because the magnetic field concentration causes the outflow, the field will itself decay in a time short compared with the lifetime of a spot. However, a slightly more sophisticated model does suggest conditions under which this decay is considerably reduced.Observations of the outward drift of magnetic knots around sunspots and of supergranule-type surface motions extending radially outwards from the penumbra of a spot to the nearest faculae are discussed in relation to the mechanism.     

Comparative Analyses of the CSSS Calculation in the UCSD Tomographic Solar Observations

We describe a new method to derive the interplanetary magnetic field (IMF) out to 1 AU from photospheric magnetic field measurements. The method uses photospheric magnetograms to calculate a source surface magnetic field at 15R_⊙. Specifically, we use Wilcox Solar Observatory (WSO) magnetograms as input for the Stanford Current-Sheet Source-Surface (CSSS) model. Beyond the source surface the magnetic field is convected along velocity flow lines derived by a tomographic technique developed at UCSD and applied to interplanetary scintillation (IPS) observations. We compare the results with in situ data smoothed by an 18-h running mean. Radial and tangential magnetic field amplitudes fit well for the 20 Carrington rotations studied, which are largely from the active phase of the solar cycle. We show exemplary results for Carrington rotation 1965, which includes the Bastille Day event.     

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.     

Dayside induced magnetic field in the ionosphere of Mars

The Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS) onboard the Mars Express spacecraft has occasionally displayed surprising features. One such feature is the occurrence of a series of broadband, low-frequency echoes at equally spaced delay times after the sounder transmitter pulse. The interval between the echoes has been shown to be at the cyclotron period of electrons orbiting in the local magnetic field. The electrons are believed to be accelerated by the large voltages applied to the antenna by the sounder transmitter. Measurements of the period of these “electron cyclotron echoes” provide a simple technique for determining the magnitude of the magnetic field near the spacecraft. These measurements are particularly useful because Mars Express carries no magnetometer, so this is the only method available for measuring the magnetic field magnitude. Using this technique, results are presented showing the large scale structure of the draped field inside the magnetic pile-up boundary. The magnitude of the draped field is shown to vary from about 40 nT at a solar zenith angle of about 25°, to about 25 nT at a solar zenith angle of 90°. The results compare favorably with similar results from the Mars Global Surveyor spacecraft. A fitting technique is developed to derive the vector direction and magnitude of the draped magnetic field in cases where the spacecraft passes through regions with significant variation in the crustal field. The magnetic field directions are consistent with current knowledge of the draping geometry of the magnetic field around Mars.     

Steady state charge neutral models of the magnetopause

Plane models of the magnetopause are investigated under the assumption that ionospheric electrons are able to short-circuit electric fields (exact charge neutrality). Using the Vlasov theory a general method is presented for constructing distribution functions that lead to given magnetic field and tangential bulk velocity profiles. As an example we describe the magnetic field transition in terms of error functions and obtain particle distributions in explicit form, including bulk velocities.It is thus shown that bulk velocities in the direction of the magnetic field do not necessarily lead to a non-equilibrium magnetopause which investigations by Parker and Lerche seem to suggest.Of the European Space Research Organisation (ESRO).     

Study of interplanetary magnetic field with Ground State Alignment

We demonstrate a new way of studying interplanetary magnetic field—Ground State Alignment (GSA). Instead of sending thousands of space probes, GSA allows magnetic mapping with any ground telescope facilities equipped with spectropolarimeter. The polarization of spectral lines that are pumped by the anisotropic radiation from the Sun is influenced by the magnetic realignment, which happens for magnetic field (<1 G). As a result, the linear polarization becomes an excellent tracer of the embedded magnetic field. The method is illustrated by our synthetic observations of the Jupiter’s Io and comet Halley. Polarization at each point was constructed according to the local magnetic field detected by spacecrafts. Both spatial and temporal variations of turbulent magnetic field can be traced with this technique as well. The influence of magnetic field on the polarization of scattered light is discussed in detail. For remote regions like the IBEX ribbons discovered at the boundary of interstellar medium, GSA provides a unique diagnostics of magnetic field.     

Equilibrium structure of stars with radiation pressure in the presence of a magnetic field

Equilibrium configuration of the upper Main-Sequence stars, with significant radiation pressure and having an interior magnetic field (matching with an external dipole field) has been cosidered. The structural parameters have been calculated for low and high magnetic fields by using a first-order perturbation method and a modified perturbation technique respectively. With the increase of radiation pressure, the star is seen to become more centrally condensed.