Damped Oscillations of Coronal Loops

A mechanism of damped oscillations of a coronal loop is investigated. The loop is treated as a thin toroidal flux rope with two stationary photospheric footpoints, carrying both toroidal and poloidal currents. The forces and the flux-rope dynamics are described within the framework of ideal magnetohydrodynamics (MHD). The main features of the theory are the following: i) Oscillatory motions are determined by the Lorentz force that acts on curved current-carrying plasma structures and ii) damping is caused by drag that provides the momentum coupling between the flux rope and the ambient coronal plasma. The oscillation is restricted to the vertical plane of the flux rope. The initial equilibrium flux rope is set into oscillation by a pulse of upflow of the ambient plasma. The theory is applied to two events of oscillating loops observed by the Transition Region and Coronal Explorer (TRACE). It is shown that the Lorentz force and drag with a reasonable value of the coupling coefficient (c _d ) and without anomalous dissipation are able to accurately account for the observed damped oscillations. The analysis shows that the variations in the observed intensity can be explained by the minor radial expansion and contraction. For the two events, the values of the drag coefficient consistent with the observed damping times are in the range c _d ≈2 – 5, with specific values being dependent on parameters such as the loop density, ambient magnetic field, and the loop geometry. This range is consistent with a previous MHD simulation study and with values used to reproduce the observed trajectories of coronal mass ejections (CMEs).     

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.     

Nonlinear Kink Oscillations of Coronal Magnetic Loops

We studied nonlinear kink oscillations of a thin magnetic tube using the cold-plasma approximation. We assumed that the plasma density varies along the tube but does not vary in the radial direction. Using the regular perturbation method, we show that the nonlinearity does not affect the oscillation amplitude. We also calculated the nonlinear correction to the oscillation frequency, which is proportional to the oscillation amplitude squared. As an example, we considered nonlinear oscillations of a coronal magnetic loop of half-circle shape in an isothermal atmosphere with equal plasma temperatures inside and outside the loop.     

The Oscillations of Coronal Loops Including the Shell

We investigate the MHD waves in a double magnetic flux tube embedded in a uniform external magnetic field. The tube consists of a dense hot cylindrical cord surrounded by a co-axial shell. The plasma and the magnetic field are taken to be uniform inside the cord and also inside the shell. Two slow and two fast magnetosonic modes can exist in the thin double tube. The first slow mode is trapped by the cord, the other is trapped by the shell. The oscillations of the second mode have opposite phases inside the cord and shell. The speeds of the slow modes propagating along the tube are close to the tube speeds inside the cord and the shell. The behavior of the fast modes depends on the magnitude of Alfvén speed inside the shell. If it is less than the Alfvén speed inside the cord and in the environment, then the fast mode is trapped by the shell and the other may be trapped under the certain conditions. In the opposite case when the Alfvén speed in the shell is greater than those inside the cord and in the environment, then the fast mode is radiated by the tube and the other may also be radiated under certain conditions. The oscillation of the cord and the shell with opposite phases is the distinctive feature of the process. The proposed model allows to explain the basic phenomena connected to the coronal oscillations: i) the damping of oscillations stipulated in the double tube model by the radiative loss, ii) the presence of two different modes of perturbations propagating along the loop with close speeds, iii) the opposite phases of oscillations of modulated radio emission, coming from the near coronal sources having sharply different densities.     

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.     

Transverse Oscillations of Coronal Loops with Slowly Changing Density

In this paper we study kink oscillations of coronal loops with the density varying along the loop and also slowly changing with time. Using the Wentzel?CKramers?CBrillouin (WKB) method we obtain the adiabatic invariant that determines the time dependence of the oscillation amplitude. The obtained general results are applied to kink oscillations of cooling loops. The main conclusion of this study is that cooling causes the amplification of kink oscillations.     

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.     

Observations of Excitation and Damping of Transversal Oscillations in Coronal Loops by AIA/SDO

The excitation and damping of the transversal coronal loop oscillations and quantitative relation between damping time, damping property (damping time per period), oscillation amplitude, dissipation mechanism and the wake phenomena are investigated. The observed time series data with the Atmospheric Imaging Assembly (AIA) telescope on NASA’s Solar Dynamics Observatory (SDO) satellite on 2015 March 2, consisting of 400 consecutive images with 12 s cadence in the 171 \(\mathring{\mathrm{A}}\) pass band is analyzed for evidence of transversal oscillations along the coronal loops by the Lomb–Scargle periodgram. In this analysis signatures of transversal coronal loop oscillations that are damped rapidly were found with dominant oscillation periods in the range of \(\mathrm{P}=12.25\,\text{--}\,15.80\) min. Also, damping times and damping properties of the transversal coronal loop oscillations at dominant oscillation periods are estimated in the range of \({\tau_{\mathrm{d}}=11.76}\,\text{--}\,{21.46}\) min and \({\tau_{\mathrm{d}}/\mathrm{P}=0.86}\,\text{--}\,{1.49}\), respectively. The observational results of this analysis show that damping properties decrease slowly with increasing amplitude of the oscillation, but the periods of the oscillations are not sensitive functions of the amplitude of the oscillations. The order of magnitude of the damping properties and damping times are in good agreement with previous findings and the theoretical prediction for damping of kink mode oscillations by the dissipation mechanism. Furthermore, oscillations of the loop segments attenuate with time roughly as \(t^{-\alpha}\) and the magnitude values of \(\alpha\) for 30 different segments change from 0.51 to 0.75.     

Failed Eruption of a Filament as a Driver for Vertical Oscillations of Coronal Loops

We present observations of a failed eruption of a magnetic flux rope recorded during the M6.2 flare of 14 July 2004. The observations were mainly made with TRACE 171 Å and 1600 Å filters. The flare was accompanied by a destabilization of a magnetic structure observed as a filament eruption. After an initial acceleration, the eruption slowed down and finally was stopped by the overlying coronal loops. The observations suggest that the whole event is well described by the quadrupole model of a solar flare. The failed eruption stretched the overlying loops, and they were then observed to be oscillating. We were able to observe clear vertical polarization of the oscillatory motion in the TRACE images. The derived parameters of the oscillatory motion are an initial amplitude of 9520 km, a period of 377 s, and an exponential damping time of 500 s. Differences between the existing models and the observations have been found. The analyzed event is the second sample for global vertical kink waves found besides the first by Wang and Solanki (Astrophys. J. Lett. 421, 33, 2004).     

Effects of Field-Aligned Flows on Standing Kink and Sausage Modes Supported by Coronal Loops

Fundamental standing modes and their overtones play an important role in coronal seismology. We examine the effects of a significant field-aligned flow on standing modes that are supported by coronal loops, which are modeled here as cold magnetic slabs. Of particular interest are the period ratios of the fundamental to its (n?1)th overtone [P ₁/nP _n ] for kink and sausage modes, and the threshold half-width-to-length ratio for sausage modes. For standing kink modes, the flow significantly reduces P ₁/nP _n in general, the effect being particularly strong for higher n and weaker density contrast [ $\rho_{0}/\rho_{\rm e}$ ] between loops and their surroundings. That said, even when $\rho_{0}/\rho_{\rm e}$ approaches infinity, this effect is still substantial, reducing the minimal P ₁/nP _n by up to 13.7?% (24.5?%) for n=2 (n=4) relative to the static case, when the Alfvén Mach number [M _A] reaches 0.8, where M _A measures the loop flow speed in units of the internal Alfvén speed. Although it is not negligible for standing sausage modes, the flow effect in reducing P ₁/nP _n is not as strong. However, the threshold half-width-to-length ratio is considerably higher in the flowing case than in its static counterpart. For $\rho_{0}/\rho_{\rm e}$ in the range [9,1024] and M _A in the range [0,0.5], an exhaustive parameter study yields that this threshold is well fitted by $(d/L)_{\rm cutoff, fit} = \frac{1}{2}\sqrt{\frac{1}{\rho_{0}/\rho_{\rm e}-1}} \exp (3.7 M_{\mathrm{A}}^{2} )$ , which involves the two parameters in a simple way. This allows one to analytically constrain the combination $(\rho_{0}/\rho_{\rm e}, M_{\mathrm {A}})$ for a loop with a known width-to-length ratio when a standing sausage oscillation is identified. It also allows one to examine the idea of partial sausage modes in more detail, and the flow is found to significantly reduce the spatial extent where partial modes are allowed.     

The Effect of Magnetic and Density Differences on the Fast Kink Oscillations of Neighboring Coronal Loops

It is reasonable that neighboring coronal loops may obtain similar momentum during a flare. The fast kink oscillations (FKOs) between them are thus mainly influenced by their physical differences. We discuss the dependencies of FKO on the physical properties of coronal loops in a low-\(\beta \) thin-tube approximation. From the analysis, we obtain the analytic relationship between the density [\(\rho _{\mathrm{i}}\)] and magnetic field [\(B\)] of loops and the corresponding period [\(\tau \)] and amplitude [\(A\)] of FKO, which may provide us with a powerful tool to diagnose the physical differences between neighboring loops.     

Transverse Oscillations in Coronal Loops Observed with TRACE – II. Measurements of Geometric and Physical Parameters

Solar Physics - We measure geometric and physical parameters oftransverse oscillations in 26 coronal loops, out of the 17 events described in Paper I by Schrijver, Aschwanden, and Title (2002)....     

Cross-Sectional Properties of Coronal Loops

Careful examination of 43 soft X-ray loops observed by Yohkoh has revealed a number of interesting properties of the loop cross section. First, the loops tend to be only slightly ( 30%) wider at their midpoints than at their footpoints, implying less-than-expected expansion of the magnetic field. Second, the variation of width along each loop tends to be modest, implying that the cross section has an approximately circular shape. And third, cross-axis intensity profiles tend to be singly-peaked and simple, implying that the cross section is approximately uniformly filled on resolvable scales. We conclude that the energy which heats the plasma is either dissipated axially symmetrically on a scale equal to a loop diameter ( 11000 km) or else is dissipated with any spatial structure, but on a scale much smaller than a loop diameter, and then transported laterally in an axisymmetric fashion (perhaps via conduction along chaotic field lines). In their present form, none of the theoretical ideas concerning the magnetic structure and heating of loops are obviously capable of explaining all of the observed properties.     

Latent Heating of Coronal Loops

This paper is aimed at establishing the relationship between the large-scale magnetic fields (LSMF), coronal holes (CH), and active regions (AR) in the Sun. The LSMF structure was analyzed by calculating the vector photospheric magnetic field under a potential approximation. Synoptic maps were drawn to study the distribution of the B field component and to isolate regions where the open field lines of the unipolar magnetic field are most radial. These are the sites of occurrence of X-ray and Hei 10830 Å coronal holes detected from the SXT/Yohkoh images. It is shown that coronal holes are usually located in LSMF regions with a typical pattern of divergentB vectors and a so-called saddle configuration.B vectors from the conjugate (spaced by 90°) coronal holes converge towards the active regions between CH. Variations in AR distort coronal holes and change their boundaries. This implies that the energy regime in CH depends on the energy supply from the active region. The LSMF structure is more stable than coronal holes, remaining practically unchanged during tens of rotations of the Sun. Thus, a peculiar magnetically coupled system of LSMF/CH/AR has been revealed. A model has been suggested to describe the interaction of the emerging toroids in the convection zone and in the photosphere. The cellular convection, that develops at the center of the toroids, is responsible for the occurrence of active regions. The model qualitatively describes the observed particularities of the LSMF/CH/AR system.     

Ballooning Instability in Coronal Flare Loops

Within the framework of ideal magnetohydrodynamics the excitation of the ballooning instability in a toroidal coronal loop with a radius of cross section a and a radius of curvature R is analyzed by using the energy method. Kink oscillations are able to excite the ballooning instability when the plasma beta parameter β>2a/R. It has been suggested that this can result in the formation of cusp-shaped coronal loops. Modulation of gyrosynchrotron emission caused by kink oscillations is considered. The intensity of gyrosynchrotron emission for optically thin sources is the most sensitive to Alfvén disturbances. The obtained theoretical results are discussed in the light of Yohkoh, SOHO, TRACE, RHESSI, and Nobeyama observations.     

Magnetic Stereoscopy of Coronal Loops in NOAA 8891

The Solar TErrestrial RElations Observatory (STEREO) requires powerful tools for the three-dimensional (3D) reconstruction of the solar corona. Here we test such a program with data from SOHO and TRACE. By taking advantage of solar rotation, a newly developed stereoscopy tool for the reconstruction of coronal loops is applied to the solar active region NOAA 8891 observed from 1 March to 2 March 2000. The stereoscopic reconstruction is composed of three steps. First, we identify loop structures in two TRACE images observed from two vantage viewpoints approximately 17 degrees apart, which corresponds to observations made about 30 hours apart. In the second step, we extrapolate the magnetic field in the corona with the linear force-free field model from the photospheric line-of-sight SOHO/MDI data. Finally, combining the extrapolated field lines and one-dimensional loop curves from two different viewpoints, we obtain the 3D loop structures with the magnetic stereoscopy tool. We demonstrate that by including the magnetic modeling this tool is more powerful than pure geometrical stereoscopy, especially in resolving the ambiguities generated by classical stereoscopy. This work will be applied to the STEREO mission in the near future.     

Segmentation of Loops from Coronal EUV Images

We present a procedure to extract bright loop features from solar EUV images. In terms of image intensities, these features are elongated ridge-like intensity maxima. To discriminate the maxima, we need information about the spatial derivatives of the image intensity. Commonly, the derivative estimates are strongly affected by image noise. We therefore use a regularized estimation of the derivative, which is then used to interpolate a discrete vector field of ridge points; these “ridgels” are positioned on the ridge center and have the intrinsic orientation of the local ridge direction. A scheme is proposed to connect ridgels to smooth, spline-represented curves that fit the observed loops. Finally, a half-automated user interface allows one to merge or split curves or eliminate or select loop fits obtained from this procedure. In this paper we apply our tool to one of the first EUV images observed by the SECCHI instrument onboard the recently launched STEREO spacecraft. We compare the extracted loops with projected field lines computed from near-simultaneous magnetograms measured by the SOHO/MDI Doppler imager. The field lines were calculated by using a linear force-free field model. This comparison allows one to verify faint and spurious loop connections produced by our segmentation tool and it also helps to prove the quality of the magnetic-field model where well-identified loop structures comply with field-line projections. We also discuss further potential applications of our tool such as loop oscillations and stereoscopy.     

Short-Wave Oscillations of Coronal Magnetic Arcades

We investigate MHD waves in potential and force-free magnetic arcades describing bipolar active regions. The eikonal method allows us to study analytically the short waves, which are divided into Alfvén and magnetosonic waves. The eigen-modes of magnetic arcades are formed as a result of their reflection at the photosphere. The Alfvén mode oscillations of a certain frequency take place on magnetic surfaces. The fast-mode oscillations also take place on some surfaces but they are not magnetic surfaces. Both the Alfvén and fast-mode eigen-frequencies change continuously from one such surface to another. Each oscillation surface has a discrete set of eigen-frequencies.