Comparison of Five Numerical Codes for Automated Tracing of Coronal Loops

The three-dimensional (3D) modeling of coronal loops and filaments requires algorithms that automatically trace curvilinear features in solar EUV or soft X-ray images. We compare five existing algorithms that have been developed and customized to trace curvilinear features in solar images: i) the oriented-connectivity method (OCM), which is an extension of the Strous pixel-labeling algorithm (developed by Lee, Newman, and Gary); ii) the dynamic aperture-based loop-segmentation method (developed by Lee, Newman, and Gary); iii) unbiased detection of curvilinear structures (developed by Steger, Raghupathy, and Smith); iv) the oriented-direction method (developed by Aschwanden); and v) ridge detection by automated scaling (developed by Inhester). We test the five existing numerical codes with a TRACE image that shows a bipolar active region and contains over 100 discernable loops. We evaluate the performance of the five codes by comparing the cumulative distribution of loop lengths, the median and maximum loop length, the completeness or detection efficiency, the accuracy, and flux sensitivity. These algorithms are useful for the reconstruction of the 3D geometry of coronal loops from stereoscopic observations with the STEREO spacecraft, or for quantitative comparisons of observed EUV loop geometries with (nonlinear force-free) magnetic field extrapolation models.     

Slow-Mode Oscillations and Damping of Hot Solar Coronal Loops

The effect of temperature inhomogeneity on the periods, their ratios (fundamental versus first overtone), and the damping times of the standing slow modes in gravitationally stratified solar coronal loops are studied. The effects of optically thin radiation, compressive viscosity, and thermal conduction are considered. The linearized one-dimensional magnetohydrodynamic (MHD) equations (under low-?? condition) were reduced to a fourth-order ordinary differential equation for the perturbed velocity. The numerical results indicate that the periods of nonisothermal loops (i.e., temperature increases from the loop base to apex) are smaller compared to those of isothermal loops. In the presence of radiation, viscosity, and thermal conduction, an increase in the temperature gradient is followed by a monotonic decrease in the periods (compared with the isothermal case), while the period ratio turns out to be a sensitive function of the temperature gradient and the loop lengths. We verify that radiative dissipation is not a main cooling mechanism in both isothermal and nonisothermal hot coronal loops and has a small effect on the periods. Thermal conduction and compressive viscosity are primary mechanisms in the damping of slow modes of the hot coronal loops. The periods and damping times in the presence of compressive viscosity and/or thermal conduction dissipation are consistent with the observed data in specific cases. By tuning the dissipation parameters, the periods and the damping times could be made consistent with the observations in more general cases.     

Automated Temperature and Emission Measure Analysis of Coronal Loops and Active Regions Observed with the Atmospheric Imaging Assembly on the Solar Dynamics Observatory (SDO/AIA)

We developed numerical codes designed for automated analysis of SDO/AIA image datasets in the six coronal filters, including: i) coalignment test between different wavelengths with measurements of the altitude of the EUV-absorbing chromosphere, ii) self-calibration by empirical correction of instrumental response functions, iii) automated generation of differential emission measure [DEM] distributions with peak-temperature maps [T _p(x,y)] and emission measure maps [EM _p(x,y)] of the full Sun or active region areas, iv) composite DEM distributions [dEM(T)/dT] of active regions or subareas, v) automated detection of coronal loops, and vi) automated background subtraction and thermal analysis of coronal loops, which yields statistics of loop temperatures [T _e], temperature widths [σ _T], emission measures [EM], electron densities [n _e], and loop widths [w]. The combination of these numerical codes allows for automated and objective processing of numerous coronal loops. As an example, we present the results of an application to the active region NOAA 11158, observed on 15 February 2011, shortly before it produced the largest (X2.2) flare during the current solar cycle. We detect 570 loop segments at temperatures in the entire range of log(T _e)=5.7?–?7.0 K and corroborate previous TRACE and AIA results on their near-isothermality and the validity of the Rosner–Tucker–Vaiana (RTV) law at soft X-ray temperatures (T?2 MK) and its failure at lower EUV temperatures.     

Dissipation of Longitudinal Oscillations in Stratified Nonisothermal Hot Coronal Loops

We investigate the damping of longitudinal (i.e., slow or acoustic) waves in nonisothermal, hot (T≥ 5.0 MK), gravitationally stratified coronal loops. Motivated by SOHO/SUMER and Yohkoh/SXT observations, and by taking into account a range of dissipative mechanisms such as thermal conduction, compressive viscosity, radiative cooling, and heating, the nonlinear governing equations of one-dimensional hydrodynamics are solved numerically for standing-wave oscillations along a magnetic field line. A semicircular shape is chosen to represent the geometry of the coronal loop. It was found that the decay time of standing waves decreases with the increase of the initial temperature, and the periods of oscillations are affected by the different initial footpoint temperatures and loop lengths studied by the numerical experiments. In general, the period of oscillation of standing waves increases and the damping time decreases when the parameter that characterises the temperature at the apex of the loop increases for a fixed footpoint temperature and loop length. A relatively simple second-order scaling polynomial between the damping time and the parameter determining the apex temperature is found. This scaling relation is proposed to be tested observationally. Because of the lack of a larger, statistically relevant number of observational studies of the damping of longitudinal (slow) standing oscillations, it can only be concluded that the numerically predicted decay times are well within the range of values inferred from Doppler shifts observed by SUMER in hot coronal loops.     

Automated Detection of Coronal Loops Using a Wavelet Transform Modulus Maxima Method

We propose and test a wavelet transform modulus maxima method for the automated detection and extraction of coronal loops in extreme ultraviolet images of the solar corona. This method decomposes an image into a number of size scales and tracks enhanced power along each ridge corresponding to a coronal loop at each scale. We compare the results across scales and suggest the optimum set of parameters to maximize completeness, while minimizing detection of noise. For a test coronal image, we compare the global statistics (e.g. number of loops at each length) to previous automated coronal-loop detection algorithms.     

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.     

Automated Detection of Low-Contrast Solar Features Using the Phase-Congruency Algorithm

We propose a new feature-detection technique based on phase-congruency (PC) measurements to automatically recognize or enhance faint features in solar observations, such as off-limb coronal loops and umbral dots. Compared with other feature-detection methods that are based on gradient illuminance and imaging filtering, PC-based measurements are particular efficient for recognizing faint features, which generally have a low-intensity contrast to their surroundings. In the present article, we carry out a PC-based measurement of the local weighted mean phase angle (LWMPA) at each point in an image to indicate or highlight low-contrast features. We first used artificial images to check the detection accuracy and sensitivity to the noise of this approach. Subsequently, we applied this approach to an EUV observation obtained by the Solar Dynamics Observatory/Atmospheric Imaging Assembly to highlight off-limb coronal loops, and a photospheric observation obtained by the Hinode/Solar Optical Telescope to recognize faint dots within the cores of sunspots and pores. The results illustrate that this PC-based measurement of the LWMPA is a robust detection method for faint structures in solar observations.     

Effect of Variable Background on an Oscillating Hot Coronal Loop

We investigate the effect of a variable, i.e. time-dependent, background on the standing acoustic (i.e. longitudinal) modes generated in a hot coronal loop. A theoretical model of 1D geometry describing the coronal loop is applied. The background temperature is allowed to change as a function of time and undergoes an exponential decay with characteristic cooling times typical for coronal loops. The magnetic field is assumed to be uniform. Thermal conduction is assumed to be the dominant mechanism for damping hot coronal oscillations in the presence of a physically unspecified thermodynamic source that maintains the initial equilibrium. The influence of the rapidly cooling background plasma on the behaviour of standing acoustic (longitudinal) waves is investigated analytically. The temporally evolving dispersion relation and wave amplitude are derived by using the Wenzel–Kramers–Brillouin theory. An analytic solution for the time-dependent amplitude that describes the influence of thermal conduction on the standing longitudinal (acoustic) wave is obtained by exploiting the properties of Sturm–Liouville problems. Next, numerical evaluations further illustrate the behaviour of the standing acoustic waves in a system with a variable, time-dependent background. The results are applied to a number of detected loop oscillations. We find a remarkable agreement between the theoretical predictions and the observations. Despite the emergence of the cooling background plasma in the medium, thermal conduction is found to cause a strong damping for the slow standing magneto–acoustic waves in hot coronal loops in general. In addition to this, the increase in the value of thermal conductivity leads to a strong decay in the amplitude of the longitudinal standing slow MHD waves.     

Thermal equilibria of coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. By assuming a form for the plasma pressure in the arcade, the possible thermal structures can be shown to depend on three parameters. Arcades can contain hot loops with summits hotter than 400 000 K, cool loops at temperatures less than 80 000 K along their lengths, hot-cool loops with cool summits and cool footpoints but hotter intermediate portions, and warm loops, cooler than 80 000 K along most of their lengths but with summits as hot as 400 000 K. For certain arcades, there exist regions where more than one kind of loop is possible. If the parameters describing the arcade are varied, it is possible for non-equilibrium to occur when a type of solution ceases to exist. For example, hot or warm loops can cease to exist so that only cool solutions are possible when the arcade size or pressure is decreased, while warm or cool loops may give way to hot-cool loops when the heating is reduced or the pressure is increased.     

A new family of solutions of the force-free field equation

In this paper, we determine the temperature profile along the footpoints of large coronal loops observed by TRACE in both the 171 Å and 195 Å passbands. The temperature along the lower part of these coronal loops only shows small variations and can probably be considered to be isothermal. Using the obtained temperature profile T(s) and an estimate of the column depth along the loop, we then determine the pressure along the lower part of the observed coronal loops and hence the value of the pressure scale length. The obtained scale lengths correspond in order-of-magnitude with the theoretically predicted gravitational scale height. We show that the differences between the observed and predicted scale heights are unlikely to be caused by (significant) flows along the loops but could possibly be a consequence of the inclination of the loops. This implies that the quasi-periodic intensity oscillations observed in the loops are most probably caused by compressive waves propagating upward at the coronal sound speed.     

Spectroscopic studies of solar corona VI: Trend in line-width variation of coronal emission lines with height independent of the structure of coronal loops

We have obtained spectroscopic observations in coronal emission lines by choosing two lines simultaneously, one [Fe x] 6374 Å and the other [Fe xi] 7892 Å or [Fe xiii] 10747 Å or [Fe xiv] 5303 Å. We found that in 95 per cent of the coronal loops observed in 6374 Å, the FWHM of the emission line increases with height above the limb irrespective of the size, shape and orientation of the loop and that in case of 5303 Å line decreases with height in about 89 per cent of the coronal loops. The FWHM of 7892 Å and 10747 Å emission lines show intermediate behavior. The increase in the FWHM of 6374 Å line with height is the steepest among these four lines. We have also studied the intensity ratio and ratio of FWHM of these lines with respect to those of 6374 Å as a function height above the limb. We found that the intensity ratio of 7892 Å and 10747 Å lines with respect to 6374 Å line increases with height and that of 5303 Å to 6374 Å decreases with height above the limb. This implies that temperature in coronal loops will appear to increase with height in the intensity ratio plots of 7892 Å and 6374 Å; and 10747 Å and 6374 Å whereas it will appear to decrease with height in intensity ratio of 5303 Å to 6374 Å lineversus height plot. These findings are up to a height of about 200 arcsec above the limb. The varying ratios with height indicate that relatively hotter and colder plasma in coronal loops interact with each other. Therefore, the observed increase in FWHM with height above the limb of coronal emission lines associated with plasma at about 1 MK may not be due to increase in non-thermal motions caused by coronal waves but due to interaction with the relatively hotter plasma. These findings also do not support the existing coronal loop models, which predict an increase in temperature of the loop with height above the limb.     

Three-Dimensional Nonlinear Force-Free Field Reconstruction of Solar Active Region 11158 by Direct Boundary Integral Equation

A three-dimensional coronal magnetic field is reconstructed for the NOAA active region 11158 on 14 February 2011. A GPU-accelerated direct boundary integral equation (DBIE) method is implemented which is approximately 1000 times faster than the original DBIE used on solar non-linear force-free field modeling. Using the SDO/HMI vector magnetogram as the bottom boundary condition, the reconstructed magnetic field lines are compared with the projected EUV loop structures as observed in the front-view (SDO/AIA) and the side-view (STEREO-A/B) images for the first time; they show very good agreement three-dimensionally. A quantitative comparison with some stereoscopically reconstructed coronal loops shows that the average misalignment angles in our model are at the same order as the state-of-the-art results obtained from reconstructed coronal loops. It is found that the observed coronal loop structures can be grouped into a number of closed and open field structures with some central bright coronal loop features around the polarity inversion line. The reconstructed highly sheared magnetic field lines agree very well with the low-lying sigmoidal filament along the polarity inversion line. This central low-lying magnetic field loop system must have played a key role in powering the flare. It should be noted that while a strand-like coronal feature along the polarity inversion line may be related to the filament, one cannot simply interpret all the coronal bright features along the polarity inversion line as manifestation of the filament without any stereoscopic information.     

A Research on the Physical Features of Coronal Loop Oscillations

Observations of the solar full-disk were carried out by the Atmo- spheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) with the Fe IX 171 Å line on 16th October 2010. The obtained high-quality data permit us to elaborate on the coronal loop oscillations. It is found that a major flare of GOES (Geostationary Operational Environmental Satellite) class M_2.9 occurred in the active region NOAA 1112 during this period, which triggered a number of coronal loops on the solar surface to oscillate. Among them, there are two coronal loops exhibiting oscillations with different physical features. The oscillation of the coronal loop located at W₄₉₂/S₁₇₀ is a simple harmonic oscillation with a period of 385_s, which abides by the oscillating equation of x = 2.2 sin[2π/385(t–768)], while the other located at W559/S142 is a damping oscillation with a period of 449s, and the oscillating equation is expressed by x = 24.8e - 2π/343 t sin[2π/449(t–1128)], where t is the observational time in units of second.     

Thermal equilibria of isobaric coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of isobaric thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. The possible thermal equilibria can be shown to depend on two parameters L ^* p ^* and h ^*/p ^* representing the ratios of cooling (radiation) to condu and heating to cooling, respectively. Arcades can contain four types of loops: hot loops with summits hotter than 400000 K; cool loops at temperatures less than 80000 K along their lengths; hot-cool loops with cool summits and cool footpoints but hotter intermediate portions; and warm loops, cooler than 80000 K along most of their lengths but with summits as hot as 400000 K. Two possibilities for coronal heating are considered, namely a heating that is independent of magnetic field and a heating that is proportional to the square of the local magnetic field. When the arcade is sheared the thermal structure of the arcade may change, leading in some cases to non-equilibrium or in other cases to the formation of a cool core.     

Coronal Seismology Using Transverse Oscillations of Non-planar Coronal Loops

We continue studying the robustness of coronal seismology. We concentrate on two seismological applications: the estimate of coronal scale height using the ratio of periods of the fundamental harmonic and first overtone of kink oscillations, and the estimate of magnetic-field magnitude using the fundamental harmonic. Our analysis is based on the model of non-planar coronal loops suggested by Ruderman and Scott (Astron. Astrophys. 529, A33, 2011), which was formulated using the linearized MHD equations. We show that the loop non-planarity does not affect the ratio of periods of the fundamental harmonic and first overtone, and thus it is unimportant for the estimates of the coronal scale height. We also show that the density variation along the loop and the loop non-planarity only weakly affect the estimates of the magnetic-field magnitude. Hence, using the simplest model of coronal loops, which is a straight homogeneous magnetic cylinder, provides sufficiently accurate estimates for the magnetic-field magnitude.     

2D Feature Recognition And 3d Reconstruction In Solar Euv Images

EUV images show the solar corona in a typical temperature range of T >rsim 1 MK, which encompasses the most common coronal structures: loops, filaments, and other magnetic structures in active regions, the quiet Sun, and coronal holes. Quantitative analysis increasingly demands automated 2D feature recognition and 3D reconstruction, in order to localize, track, and monitor the evolution of such coronal structures. We discuss numerical tools that “fingerprint” curvi-linear 1D features (e.g., loops and filaments). We discuss existing finger-printing algorithms, such as the brightness-gradient method, the oriented-connectivity method, stereoscopic methods, time-differencing, and space–time feature recognition. We discuss improved 2D feature recognition and 3D reconstruction techniques that make use of additional a priori constraints, using guidance from magnetic field extrapolations, curvature radii constraints, and acceleration and velocity constraints in time-dependent image sequences. Applications of these algorithms aid the analysis of SOHO/EIT, TRACE, and STEREO/SECCHI data, such as disentangling, 3D reconstruction, and hydrodynamic modeling of coronal loops, postflare loops, filaments, prominences, and 3D reconstruction of the coronal magnetic field in general.     

Thermal equilibria of coronal magnetic loops with non-constant cross-sectional area

Equations of thermal equilibrium along coronal loops are solved in the absence of gravity but where the cross-sectional area changes along the loop. The footpoint temperature is assumed to be 2 × 10⁴ K. Several fundamental types of solution are found, namely hot loops, cool loops, hot-cool loops (where the footpoints and summits are cool but the intermediate parts are hotter) and warm loops (cool along most of their lengths except the summits). On increasing the cross-sectional area the summit temperature generally increases slightly except for warm loops where no increase in temperature is recorded and hot-cool loops where a dramatic increase in summit temperature may occur. The cool and hot-cool loops may model elementary fibril structures within prominences.     

The XUV structure of solar active regions

XUV spectroheliograms of 2 active regions are studied. The images are due to lines emitted at temperatures between 8 x 10⁴ K and 2 x 10⁶ K and thus are indicative of transition region and coronal structures. The hot coronal lines are formed solely in loop structures which connect regions of opposite photospheric magnetic polarity but are not observed over sunspots. Transition region lines are emitted in plages overlying regions of intense photospheric magnetic field and in loops or loop-segments connecting such regions. The hot coronal loops are supported hydrostatically while only some of the transition zone loops are. The coronal and transition zone loops are distinctly separated and are not coaxial. A comparison of direct measurements of electron densities using density sensitive line ratios with indirect measurements using emission measures and path lengths shows the existence of fine structures of less than a second of arc in transition region loops. From a similar analysis, hot coronal loops do not have any fine structure below about 2 seconds of arc.     

Numerical Simulations of the Flux Tube Tectonics Model for Coronal Heating

In this paper we present results from 3D MHD numerical simulations based on the flux tube tectonics method of coronal heating proposed by Priest, Heyvaerts, and Title (2002). They suggested that individual coronal loops connect to the photosphere in many different magnetic flux fragments and that separatrix surfaces exist between the fingers connecting a loop to the photosphere and between individual loops. Simple lateral motions of the flux fragments could then cause currents to concentrate along the separatrices which may then drive reconnection contributing to coronal heating. Here we have taken a simple configuration with four flux patches on the top and bottom of the numerical domain and a small background axial field. Then we move two of the flux patches on the base between the other two using periodic boundary conditions such that when they leave the box they re-enter it at the other end. This simple motion soon causes current sheets to build up along the quasi-separatrix layers and subsequently magnetic diffusion/reconnection occurs.