Comparison of Nonlinear Force-Free Field and Potential Field in the Quiet Sun

In this paper, a potential field extrapolation and three nonlinear force-free (NLFF) field extrapolations (optimization, direct boundary integral (DBIE), and approximate vertical integration (AVI) methods) are used to study the spatial configuration of magnetic field in the quiet Sun. It is found that differences in the computed field strengths among the three NLFF and potential fields exist in the low layers. However, they tend to disappear as the height increases, and the differences are of the order of 0.1 gauss when the height exceeds ≈ 2000 km above the photosphere. The difference in azimuth angles between each NLFF field model and the potential field is as follows: for the optimization field, it decreases evidently as the height increases; for the DBIE field, it almost stays constant and shows no significant change as the height increases; for the AVI field, it increases slowly as the height increases. Our analysis shows that the reconstructed NLFF fields deviate significantly from the potential field in the quiet Sun.     

Topological Differences Between Force-Free Field Models

The potential and linear force-free field models for the magnetic field in the solar corona are often used in the analysis of flares. The field is calculated using boundary values measured in the low solar atmosphere. The topology of the field calculated using these models is then compared to the position of flare emissions. We demonstrate that the topology of the field according to each of these models, with the same boundary conditions in place, is not in general even qualitatively equivalent. An argument is given for a similar discrepancy between a linear force-free field solution and a nonlinear force-free field solution.     

The Energetics of White-light Flares Observed by SDO/HMI and RHESSI

White-light(WL) flares have been observed and studied for more than a century since their first discovery. However, some fundamental physics behind the brilliant emission remains highly controversial.One of the important facts in addressing the flare energetics is the spatio-temporal correlation between the WL emission and the hard X-ray(HXR) radiation, presumably suggesting that energetic electrons are the energy sources. In this study, we present a statistical analysis of 25 strong flares(≥M5) observed simultaneously by the Helioseismic and Magnetic Imager(HMI), on board the Solar Dynamics Observatory(SDO),and the Reuven Ramaty High Energy Solar Spectroscopic Imager(RHESSI). Among these events, WL emission was detected by SDO/HMI in 13 flares, associated with HXR emission. To quantitatively describe the strength of WL emission, equivalent area(EA) is defined as the integrated contrast enhancement over the entire flaring area. Our results show that the EA is inversely proportional to the HXR power-law index,indicating that stronger WL emission tends to be associated with a larger population of high energy electrons. However, no obvious correlation is found between WL emission and flux of non-thermal electrons at50 ke V. For the other group of 13 flares without detectable WL emission, the HXR spectra are softer(larger power-law index) than those flares with WL emission, especially for the X-class flares in this group.     

Multi-height Scattering Phase Shifts from Sunspots and Phase Differences Between Photospheric Heights with SDO/HMI and AIA Data

Following Couvidat (Solar Phys. 282, 15, 2013), we analyze data from the Helioseismic and Magnetic Imager (HMI) and the Atmospheric Imaging Assembly (AIA) instruments onboard the Solar Dynamics Observatory. Doppler velocity and continuum intensity at 6173 Å as well as intensity maps at 1600 Å and 1700 Å are studied on 14 active regions and four quiet-Sun regions at four heights in the solar photosphere. A Hankel–Fourier analysis is performed around these regions of interest. Outgoing–ingoing scattering phase shifts at a given atmospheric height are computed, as well as ingoing–ingoing and outgoing–outgoing phase differences between two atmospheric heights. The outgoing–ingoing phase shifts produced by sunspots show little dependence on measurement height, unlike the acoustic power reduction measured in Couvidat (2013). Phenomena happening above the continuum level, like acoustic glories, appear not to have a significant effect on the phases. However, there is a strong dependence on sunspot size, line-of-sight magnetic flux, and intensity contrast. As previously suggested by other groups, the region of wave scattering appears both horizontally smaller and vertically less extended than the region of acoustic power suppression, and occurs closer to the surface. Results presented here should help refine theoretical models of acoustic wave scattering by magnetic fields. Ingoing–ingoing phase differences between two measurement heights are also computed and show a complex pattern highly dependent on atmospheric height. In particular, a significant sensitivity of AIA signals to lower chromospheric layers is shown. Finally, ingoing–ingoing minus outgoing–outgoing phase differences between various measurement heights are discussed.     

Study of Differences Between Sunspot and White Light Facular Area Data Determined from SDO/HMI and SOHO/MDI Observations

Sunspot and white light facular areas are important data for solar activity and are used, for example, in the study of the evolution of sunspots and their effect on solar irradiance. Solar Dynamic Observatory??s Helioseismic and Magnetic Imager (SDO/HMI) solar images have much higher resolution (??0.5????pixel^?1) than Solar and Heliospheric Observatory??s Michelson Doppler Imager (SOHO/MDI) solar images (??2????pixel^?1). This difference in image resolution has a significant impact on the sunspot and white light facular areas measured in the two datasets. We compare the area of sunspots and white light faculae derived from SDO/HMI and SOHO/MDI observations. This comparison helps the calibration of the SOHO sunspot and facular area to those in SDO observations. We also find a 0.22 degree difference between the North direction in SDO/HMI and SOHO/MDI images.     

Nonlinear Force-Free Magnetic Field Extrapolations: Comparison of the Grad Rubin and Wheatland Sturrock Roumeliotis Algorithm

We compare the performance of two alternative algorithms which aim to construct a force-free magnetic field given suitable boundary conditions. For this comparison, we have implemented both algorithms on the same finite element grid which uses Whitney forms to describe the fields within the grid cells. The additional use of conjugate gradient and multigrid iterations result in quite effective codes. The Grad Rubin and Wheatland Sturrock Roumeliotis algorithms both perform well for the reconstruction of a known analytic force-free field. For more arbitrary boundary conditions the Wheatland Sturrock Roumeliotis approach has some difficulties because it requires overdetermined boundary information which may include inconsistencies. The Grad Rubin code on the other hand loses convergence for strong current densities. For the example we have investigated, however, the maximum possible current density seems to be not far from the limit beyond which a force-free field cannot exist anymore for a given normal magnetic field intensity on the boundary.     

Magnetic Connectivity Between Active Regions 10987, 10988, and 10989 by Means of Nonlinear Force-Free Field Extrapolation

Extrapolation codes for modelling the magnetic field in the corona in Cartesian geometry do not take the curvature of the Sun’s surface into account and can only be applied to relatively small areas, e.g., a single active region. We apply a method for nonlinear force-free coronal magnetic field modelling of photospheric vector magnetograms in spherical geometry which allows us to study the connectivity between multi-active regions. We use Vector Spectromagnetograph (VSM) data from the Synoptic Optical Long-term Investigations of the Sun (SOLIS) survey to model the coronal magnetic field, where we study three neighbouring magnetically connected active regions (ARs 10987, 10988, 10989) observed on 28, 29, and 30 March 2008, respectively. We compare the magnetic field topologies and the magnetic energy densities and study the connectivities between the active regions. We have studied the time evolution of the magnetic field over the period of three days and found no major changes in topologies, as there was no major eruption event. From this study we have concluded that active regions are much more connected magnetically than the electric current.     

Nonlinear Force-Free Modeling of Coronal Magnetic Fields Part I: A Quantitative Comparison of Methods

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad–Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error E_n = 〈 |B−b|〉/〈 |B|〉 of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while the total energy in the magnetic field is approximated to within 2%. The models converge towards the central, strong input field at speeds that differ by a factor of one million per iteration step. The fastest-converging, best-performing model for these analytical test cases is the Wheatland, Sturrock, and Roumeliotis (2000) optimization algorithm as implemented by Wiegelmann (2004).     

Long-term oscillations of sunspots and a special class of artifacts in SOHO/MDI and SDO/HMI data

A specific type of artifacts (named as “p2p”), that originate due to displacement of the image of a moving object along the digital (pixel) matrix of receiver are analyzed in detail. The criteria of appearance and the influence of these artifacts on the study of long-term oscillations of sunspots are deduced. The obtained criteria suggest us methods for reduction or even elimination of these artifacts. It is shown that the use of integral parameters can be very effective against the “p2p” artifact distortions. The simultaneous observations of sunspot magnetic field and ultraviolet intensity of the umbra have given the same periods for the long-term oscillations. In this way the real physical nature of the oscillatory process, which is independent of the artifacts have been confirmed again. A number of examples considered here confirm the dependence between the periods of main mode of the sunspot magnetic field long-term oscillations and its strength. The dependence was derived earlier from both the observations and the theoretical model of the shallow sunspot. The anti-phase behavior of time variations of sunspot umbra area and magnetic field of the sunspot demonstrates that the umbra of sunspot moves in long-term oscillations as a whole: all its points oscillate with the same phase.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components B _x (x), B _y (x), B _z (x), the force-free parameter α(x), the electric current density j(x), and are accurate to second-order (of the nonlinear force-free α-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.     

Fractal and Multifractal Properties of Active Regions as Flare Precursors: A Case Study Based on SOHO/MDI and SDO/HMI Observations

Several studies indicate that fractal and multifractal parameters inferred from solar photospheric magnetic field measurements may help assessing the eruptive potential of Active Regions (ARs) and also predicting their flare activity. We further investigate this topic, by exploring the sensitivity of some parameters already used in the literature on data and methods employed for their estimation. In particular, we measured the generalized fractal dimensions D ₀ and D ₈, and the multifractal parameters C _div and D _div, on the time series of photospheric magnetograms of the flaring AR NOAA 11158 obtained with the SOHO/MDI and SDO/HMI. The observations by the latter instrument are characterized by a higher spatial and temporal resolution, as well as higher flux sensitivity, than the ones obtained from SOHO/MDI, which were widely employed in earlier studies. We found that the average and peak values of complexity parameters measured on the two data sets agree within measurement uncertainties. The temporal evolution of the parameters measured on the two data sets show rather similar trends, but the ones derived from the SOHO/MDI observations show larger and spurious variations over time than those deduced from analysis of the corresponding SDO/HMI data. We also found a larger sensitivity of these measurements to characteristics of the data analyzed than reported by earlier studies. In particular, analysis of the higher resolution and higher cadence SDO/HMI data allows us also to detect slight variations of the complexity indicators that cannot be derived from the analysis of the SOHO/MDI data. These variations occur right after the major events in the analyzed AR. They may be the signature of photospheric effects of coronal magnetic field re-arrangement.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. II. Numeric Code and Tests

Based on a second-order approximation of nonlinear force-free magnetic field solutions in terms of uniformly twisted field lines derived in Paper I, we develop here a numeric code that is capable to forward-fit such analytical solutions to arbitrary magnetogram (or vector magnetograph) data combined with (stereoscopically triangulated) coronal loop 3D coordinates. We test the code here by forward-fitting to six potential field and six nonpotential field cases simulated with our analytical model, as well as by forward-fitting to an exactly force-free solution of the Low and Lou (Astrophys. J. 352, 343, 1990) model. The forward-fitting tests demonstrate: i) a satisfactory convergence behavior (with typical misalignment angles of μ≈1^°?–?10^°), ii) relatively fast computation times (from seconds to a few minutes), and iii) the high fidelity of retrieved force-free α-parameters (α _fit/α _model≈0.9?–?1.0 for simulations and α _fit/α _model≈0.7±0.3 for the Low and Lou model). The salient feature of this numeric code is the relatively fast computation of a quasi-force-free magnetic field, which closely matches the geometry of coronal loops in active regions, and complements the existing nonlinear force-free field (NLFFF) codes based on photospheric magnetograms without coronal constraints.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. III. The Free Energy

Stereoscopic white-light imaging of a large portion of the inner heliosphere has been used to track interplanetary coronal mass ejections. At large elongations from the Sun, the white-light brightness depends on both the local electron density and the efficiency of the Thomson-scattering process. To quantify the effects of the Thomson-scattering geometry, we study an interplanetary shock using forward magnetohydrodynamic simulation and synthetic white-light imaging. Identifiable as an inclined streak of enhanced brightness in a time–elongation map, the travelling shock can be readily imaged by an observer located within a wide range of longitudes in the ecliptic. Different parts of the shock front contribute to the imaged brightness pattern viewed by observers at different longitudes. Moreover, even for an observer located at a fixed longitude, a different part of the shock front will contribute to the imaged brightness at any given time. The observed brightness within each imaging pixel results from a weighted integral along its corresponding ray-path. It is possible to infer the longitudinal location of the shock from the brightness pattern in an optical sky map, based on the east–west asymmetry in its brightness and degree of polarisation. Therefore, measurement of the interplanetary polarised brightness could significantly reduce the ambiguity in performing three-dimensional reconstruction of local electron density from white-light imaging.     

A Unified and Very Fast Way for Computing the Global Potential and Linear Force-Free Fields

We present a fast solver for computing potential and linear force-free fields (LFFF) above the full solar disk with a synoptic magnetic map as input. The global potential field and the LFFF are dealt with in a unified way by solving a three-dimensional Helmholtz equation in a spherical shell and a two-dimensional Poisson equation on the solar surface. The solver is based on a combination of the spectral method and the finite-difference scheme. In the longitudinal direction the equation is transformed into the Fourier spectral space, and the resulting two-dimensional equations in the r?C?? plane for the Fourier coefficients are solved by finite differencing. The solver shows an extremely fast computing speed, e.g., the computation for a magnetogram with a resolution of 180(??)×360(?) is completed in less than 2 s. Even on a high-resolution 600×1200 grid, the solution can be obtained within only about one minute on a single CPU. The solver can potentially be applied directly to the original resolution of observed magnetograms from SDO/HMI for routinely analyzing daily full-disk data.     

Sounding Rocket Observations of Active Region Soft X-Ray Spectra Between 0.5 and 2.5 nm Using a Modified SDO/EVE Instrument

Spectrally resolved measurements of individual solar active regions (ARs) in the soft X-ray (SXR) range are important for studying dynamic processes in the solar corona and their associated effects on the Earth’s upper atmosphere. They are also a means of evaluating atomic data and elemental abundances used in physics-based solar spectral models. However, very few such measurements are available. We present spectral measurements of two individual ARs in the 0.5 to 2.5 nm range obtained on the NASA 36.290 sounding rocket flight of 21 October 2013 (at about 18:30 UT) using the Solar Aspect Monitor (SAM), a channel of the Extreme Ultaviolet Variability Experiment (EVE) payload designed for underflight calibrations of the orbital EVE on the Solar Dynamics Observatory (SDO). The EVE rocket instrument is a duplicate of the EVE on SDO, except the SAM channel on the rocket version was modified in 2012 to include a freestanding transmission grating to provide spectrally resolved images of the solar disk with the best signal to noise ratio for the brightest features, such as ARs. Calibrations of the EVE sounding rocket instrument at the National Institute of Standards and Technology Synchrotron Ultraviolet Radiation Facility (NIST/SURF) have provided a measurement of the SAM absolute spectral response function and a mapping of wavelength separation in the grating diffraction pattern. We discuss techniques (incorporating the NIST/SURF data) for determining SXR spectra from the dispersed AR images as well as the resulting spectra for NOAA ARs 11877 and 11875 observed on the 2013 rocket flight. In comparisons with physics-based spectral models using the CHIANTI v8 atomic database we find that both AR spectra are in good agreement with isothermal spectra (4 MK), as well as spectra based on an AR differential emission measure (DEM) included with the CHIANTI distribution, with the exception of the relative intensities of strong Fe?xvii lines associated with \(2p^{6}\)–\(2p^{5}3{s}\) and \(2p^{6}\)–\(2p^{5}3{d}\) transitions at about 1.7 nm and 1.5 nm, respectively. The ratio of the Fe?xvii lines suggests that the AR 11877 is hotter than the AR 11875. This result is confirmed with analysis of the active regions imaged by X-ray Telescope (XRT) onboard Hinode.     

Resolving Differences in Absolute Irradiance Measurements Between the SOHO/CELIAS/SEM and the SDO/EVE

The Solar EUV Monitor (SEM) onboard SOHO has measured absolute extreme ultraviolet (EUV) and soft X-ray solar irradiance nearly continuously since January 1996. The EUV Variability Experiment (EVE) on SDO, in operation since April of 2010, measures solar irradiance in a wide spectral range that encompasses the band passes (26?–?34 nm and 0.1?–?50 nm) measured by SOHO/SEM. However, throughout the mission overlap, irradiance values from these two instruments have differed by more than the combined stated uncertainties of the measurements. In an effort to identify the sources of these differences and eliminate them, we investigate in this work the effect of reprocessing the SEM data using a more accurate SEM response function (obtained from synchrotron measurements with a SEM sounding-rocket clone instrument taken after SOHO was already in orbit) and time-dependent, measured solar spectral distributions – i.e., solar reference spectra that were unavailable prior to the launch of the SDO. We find that recalculating the SEM data with these improved parameters reduces mean differences with the EVE measurements from about 20 % to less than 5 % in the 26?–?34 nm band, and from about 35 % to about 15 % for irradiances in the 0.1?–?7 nm band extracted from the SEM 0.1?–?50 nm channel.     

Analysis of Supergranule Sizes and Velocities Using Solar Dynamics Observatory (SDO)/Helioseismic Magnetic Imager (HMI) and Solar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI) Dopplergrams

Co-temporal Doppler images from Solar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI) and Solar Dynamics Observatory (SDO)/Helioseismic Magnetic Imager (HMI) have been analyzed to extract quantitative information about global properties of the spatial and temporal characteristics of solar supergranulation. Preliminary comparisons show that supergranules appear to be smaller and have stronger horizontal velocity flows within HMI data than was measured with MDI. There appears to be no difference in their evolutionary timescales. Supergranule sizes and velocities were analyzed over a ten-day time period at a 15-minute cadence. While the averages of the time-series retain the aforementioned differences, fluctuations of these parameters first observed in MDI data were seen in both MDI and HMI time-series, exhibiting a strong cross-correlation. This verifies that these fluctuations are not instrumental, but are solar in origin. The observed discrepancies between the averaged values from the two sets of data are a consequence of instrument resolution. The lower spatial resolution of MDI results in larger observed structures with lower velocities than is seen in HMI. While these results offer a further constraint on the physical nature of supergranules, they also provide a level of calibration between the two instruments.     

Using Statistical Multivariable Models to Understand the Relationship Between Interplanetary Coronal Mass Ejecta and Magnetic Flux Ropes

In-situ measurements of interplanetary coronal mass ejections (ICMEs) display a wide range of properties. A distinct subset, “magnetic clouds” (MCs), are readily identifiable by a smooth rotation in an enhanced magnetic field, together with an unusually low solar wind proton temperature. In this study, we analyze Ulysses spacecraft measurements to systematically investigate five possible explanations for why some ICMEs are observed to be MCs and others are not: i) An observational selection effect; that is, all ICMEs do in fact contain MCs, but the trajectory of the spacecraft through the ICME determines whether the MC is actually encountered; ii) interactions of an erupting flux rope (FR) with itself or between neighboring FRs, which produce complex structures in which the coherent magnetic structure has been destroyed; iii) an evolutionary process, such as relaxation to a low plasma-β state that leads to the formation of an MC; iv) the existence of two (or more) intrinsic initiation mechanisms, some of which produce MCs and some that do not; or v) MCs are just an easily identifiable limit in an otherwise continuous spectrum of structures. We apply quantitative statistical models to assess these ideas. In particular, we use the Akaike information criterion (AIC) to rank the candidate models and a Gaussian mixture model (GMM) to uncover any intrinsic clustering of the data. Using a logistic regression, we find that plasma-β, CME width, and the ratio O ⁷/O ⁶ are the most significant predictor variables for the presence of an MC. Moreover, the propensity for an event to be identified as an MC decreases with heliocentric distance. These results tend to refute ideas ii) and iii). GMM clustering analysis further identifies three distinct groups of ICMEs; two of which match (at the 86 % level) with events independently identified as MCs, and a third that matches with non-MCs (68 % overlap). Thus, idea v) is not supported. Choosing between ideas i) and iv) is more challenging, since they may effectively be indistinguishable from one another by a single in-situ spacecraft. We offer some suggestions on how future studies may address this.