Radial oscillations of coronal loops and microwave radiation from solar flares

We consider the damping mechanisms for the radial oscillations of solar coronal loops in the approximation of a thin magnetic flux tube. We show that the free tube oscillations can have a high Q if the plasma density inside the magnetic flux tube is much higher than the density outside. We analyze the effect of radial coronal-loop magnetic-field oscillations on the modulation of the microwave radiation from solar flares. In the case of a nonthermal gyrosynchrotron mechanism, the fluxes from optically thin and optically thick sources are modulated in antiphase. Based on our model, we diagnose the flare plasma. For the event of May 23, 1990, we estimate the spectral index for accelerated electrons, α≈4.4, and the magnetic-field strength in the region of energy release, B≈190 G.     

Slow-mode oscillations of large-scale coronal loops

On several occasions, repetitive X-ray brightenings, sometimes accompanied by mass injections into adjacent loops, appeared quasi-periodically with mean periods close to 20 minutes. In all cases when X-ray images were available, the sites of these brightenings were in active regions which were associated with large-scale coronat loops of length (2 – 3) × 10⁵ km. Therefore, the primary source of these long-periodic pulsations might be slow-mode oscillations in these large-scale loops. Free MHD oscillations, proposed earlier by Roberts, Edwin, and Benz (1984), may fit the observed data.     

Kink oscillations of asymmetric coronal loops

The free oscillations of coronal loops with a constant density and a variable magnetic field changing according to parabolic laws are investigated. Using our developed method, we derive the wave equations with constant coefficients that describe the kink oscillations of symmetric and asymmetric magnetic flux tubes. For such models, we obtain analytical expressions for the oscillation spectra and amplitudes as well as the magnitudes and directions of the displacements of the extrema of the fundamental and first modes relative to their values for homogeneous tubes. For the first mode of an asymmetric loop, we have determined the dependence of the coordinate displacement for the internal node on the ratios of the magnetic field strengths in its asymmetric parts and the ratio of the amplitudes at the extremum points.     

Acoustic damping of fast kink oscillations of coronal loops

The damping of fast kink oscillations of solar coronal loops attributable to the radiation of MHD waves into the surroundings is considered in the thin-tube approximation. The oscillation damping decrement is calculated both by using a new energy method and by solving the dispersion equation for magnetic-tube eigenmodes. The two approaches are in good agreement under appropriate assumptions. The damping is negligible if MHD waves are radiated perpendicular to the magnetic field. The low Q factor of the loop oscillations in active regions found with the TRACE space telescope is associated with the generation of running waves that propagate along magnetic field lines.     

Electron acceleration by electric fields near the footpoints of current-carrying coronal magnetic loops

We analyze the electric fields that arise at the footpoints of a coronal magnetic loop from the interaction between a convective flow of partially ionized plasma and the magnetic field of the loop. Such a situation can take place when the loop footpoints are at the nodes of several supergranulation cells. In this case, the neutral component of the converging convective flows entrain electrons and ions in different ways, because these are magnetized differently. As a result, a charge-separating electric field emerges at the loop footpoints, which can efficiently accelerate particles inside the magnetic loop under appropriate conditions. We consider two acceleration regimes: impulsive (as applied to simple loop flares) and pulsating (as applied to solar and stellar radio pulsations). We have calculated the fluxes of accelerated electrons and their characteristic energies. We discuss the role of the return current when dense beams of accelerated particles are injected into the corona. The results obtained are considered in light of the currently available data on the corpuscular radiation from solar flares.     

The resonant excitation of transverse oscillations in coronal loops

We suggest a way of self-consistently solving the problem of the excitation and rapid damping of coronal loop oscillations observed from the TRACE (Transition Region and Coronal Explorer) satellite. Oscillations are excited on the dispersion branch of fast magnetoacoustic waves, which propagate mainly across the magnetic field. The rapid damping of the observed oscillations is governed by the dispersion spreading of the pulse of these waves that was produced, for example, by a solar flare. The fundamental oscillation period is close to the period of the fundamental mode. Dissipative processes attributable to the nonideality of the plasma and the coronal-loop footpoints play no fundamental role.     

Resonantly damped oscillations of elliptically shaped stratified emerging coronal loops

The effects of both elliptical shape and stage of emergence of the coronal loop on the resonant absorption of standing kink oscillations are studied. To do so, a typical coronal loop is modeled as a zero-beta longitudinally stratified cylindrical magnetic flux tube. We developed the connection formulae for the resonant absorption of standing transversal oscillations of a coronal loop with an elliptical shape, at various stages of its emergence. Using the connection formulae, the dispersion relation is derived and solved numerically to obtain the frequencies and damping rates of the fundamental and first-overtone kink modes. Our numerical results show that both the elliptical shape and stage of emergence of the loop alter the frequencies and damping rates of the tube as well as the ratio of frequencies of the fundamental and its first-overtone modes. However, the ratio of the oscillation frequency to the damping rate is not affected by the tube shape and stage of its emergence and also is independent of the density stratification parameter.     

Rapid damping of the oscillations of coronal loops with an azimuthal magnetic field

We consider the MHD oscillations of an inhomogeneous coronal loop that consists of a dense cord surrounded by a shell. The magnetic field is longitudinal in the cord and has only an azimuthal component in the shell. The parameters of the loop are chosen to be such that there are no resonances; i.e., the resonance points are cut off. This choice is dictated by the formulated problem of considering the influence of the radiation of MHD waves into the surrounding space on the loop oscillations, thereby ruling out the possibility of resonant energy absorption. The wave radiation efficiency is high and allows low oscillation Q-factors, which are equal in order of magnitude to their observed values, to be obtained.     

Damped large amplitude oscillations in a solar prominence and a bundle of coronal loops

We investigate the evolutions of two prominences(P1, P2) and two bundles of coronal loops(L1, L2), observed with SDO/AIA near the east solar limb on 2012 September 22. It is found that there were large-amplitude oscillations in P1 and L1 but no detectable motions in P2 and L2. These transverse oscillations were triggered by a large-scale coronal wave, originating from a large flare in a remote active region behind the solar limb. By carefully comparing the locations and heights of these oscillating and non-oscillating structures, we conclude that the propagating height of the wave is between 50 Mm and130 Mm. The wave energy deposited in the oscillating prominence and coronal loops is at least of the order of 10~(28) erg. Furthermore, local magnetic field strength and Alfv ′en speeds are derived from the oscillating periods and damping time scales, which are extracted from the time series of the oscillations. It is demonstrated that oscillations can be used in not only coronal seismology, but also to reveal the properties of the wave.     

High-frequency coronal oscillations and coronal heating

At the 1980 total solar eclipse, we searched for high-frequency (0.1–2 Hz) oscillations in the intensity of the 5303-Å coronal green line, as a test of predictions of theories of coronal heating via magnetohydrodynamic waves. Portions of the image 2.5- or 5-arc sec across were fed to cooled photomultipliers using fiber-optic probes. We detected excess power in Fourier transforms of the data for the region between 0.5 and 2 Hz at the level of 1% or 2% of the incident power. Such oscillations could be associated with Alfvén waves that are trapped on loops a few thousand kilometers long or with fast waves that are trapped on loops a few thousand kilometers in diameter. Additional observations at the 1983 eclipse are planned to resolve atmospheric and instrumental contributions.     

Models of dynamic coronal loops

This paper reviews the basic ideas underlying one-dimensional fluid dynamic models of coronal loops and presents some of their most recent applications. These models are an important theoretical support to explore the new scenario provided by the data of Yohkoh, SOHO, and TRACE, and are useful to interpret observations, when supplemented by appropriate spectral synthesis codes. Possible developments are also discussed.     

The structure of coronal loops

With the advent of space telescopes, coronal magnetic loops, both within and outside active regions, are being observed with renewed interest. This paper is an attempt to outline some general physical considerations pertinent to such loops, as a prelude to more sophisticated modelling. For example, a loop that is stretched (or possibly twisted) too much may be subject to a thermal instability that cools its core to a new equilibrium below 10⁵ K. Also a simple consequence of hydrostatic balance along an equilibrium loop is that, under some circumstances, the density inside a cool loop can be comparable with that outside, despite the much smaller scale height. Finally, when the equilibrium loop density is less than the ambient density, several small scale magnetohydrodynamic instabilities are sometimes efficient enough to generate a circulation that tends to equalize the densities.     

The effects of density stratification on standing fast body oscillations in coronal loops and dissipation

The standing magnetohydrodynamic (MHD) quasi-linear modes in a zero-β cylindrical magnetic flux tube that undergoes a longitudinal density stratification and radial density structuring are considered. The radial structuring is assumed to have a step-like density profile. The dispersion relation for the fast MHD body waves is derived and solved numerically to obtain the frequencies of the fundamental, first-overtone and second-overtone   k = 1, 2, 3  modes of both kink  ( m = 1)  and fluting  ( m = 2)  waves, where k and m are the longitudinal and azimuthal mode numbers, respectively. Damping rates due to both viscous and resistive dissipations in the presence of the density stratification are derived and solved numerically for the first three modes of both kink and fluting waves.     

Plasma flow around coronal loops

Current dissipation models of coronal loop heating are studied. Turbulent current dissipation is shown to lead to a time dependent process because of an enormous mass motion induced in the current layer. A stationary heating process involves only ohmic heating, which requires a large current layer. To insure MHD stability, the loop must be composed of many elements with the oppositely directed currents. A stationary current dissipation process induces the plasma motion across the magnetic field into the loop and down the loop with the speeds v 10⁴ cm s^–1 and v 10⁴ cm s^–1, respectively. The pressure of the loop is also estimated to be proportional to the current density: p/J=6.3 × 10^-8dyn/statamp.     

Non-equilibrium ionization in coronal loops

The ionization conditions in coronal loops are investigated in the temperature range 2 × 10⁵–2 × 10⁶K, assuming velocity, density and temperature distributions computed for a siphon model of a pure hydrogen plasma. Use is made of the set of the carbon ions as an example of the general behaviour of the ions characteristic of that temperature range. It is found that the deviation from equilibrium ionization is large for subsonic-supersonic flow if the density is less than 5 × 10⁹cm^–-3, with the exception of the lower part of the first leg of very cool loops (T 2 × 10 K). With this exception cooler loops, given their larger density drop along the axis, show deviations from ionization equilibrium more easily than hotter ones, in spite of their lower flow velocity. We conclude that the possibility of a non-equilibrium state must be taken into account when deducing from measurements of line intensities the temperature of loops in which a flow may occur.Now at Institute for Plasma Research, Stanford University, as an E.S.A. Fellow.     

Longitudinal intensity oscillations in coronal loops observed with TRACE – II. Discussion of Measured Parameters

In this paper, we give a detailed discussion of the parameters of longitudinal oscillations in coronal loops, described in Paper I. We found a surprising absence of correlations between the measured variables, with the exception of a relation between the estimated damping length and the period of the intensity variations. Only for 2 out of the 38 cases presented in Paper I did we find a significant perturbation in the 195 Å TRACE data. The loops supporting the propagating disturbances were typically stable, quiescent loops and the total luminosity of the analyzed structures generally varied by no more than 10%. The observed density oscillations are unlikely to be flare-driven and are probably caused by an underlying driver exciting the loop footpoints. It was demonstrated that the rapid damping of the perturbations could not simply be explained as a consequence of the decreasing intensity along the loops. However, we found that (slightly enhanced) thermal conduction alone could account for the observed damping lengths and wavelengths, and, additionally, explain the correlation between propagation period and damping length.     

Thermal equilibria of coronal magnetic loops

Equations of thermal equilibrium along coronal loops with footpoint temperatures of 2 × 10⁴ K are solved. Three fundamentally different categories of solution are found, namely hot loops with summit temperatures above about 4 × 10⁵ K, cool loops which are cooler than 8 × 10⁴ K along their whole length and hot-cool loops which have summit temperatures around 2 × 10⁴ K but much hotter parts at intermediate points between the summit and the footpoints. Hot loops correspond to the hot corona of the Sun. The cool loops are of relevance for fibrils, for the cool cores observed by Foukal and also for active-region prominences where the magnetic field is directed mainly along the prominence. Quiescent prominences consist of many cool threads inclined to the prominence axis, and each thread may be modelled as a hot-cool loop. In addition, it is possible for warm loops at intermediate summit temperatures (8 × 10⁴K to 4 × 10⁵ K) to exist, but the observed differential emission measure suggests that most of the plasma in the solar atmosphere is in either the hot phase or the cool phase. Thermal catastrophe may occur when the length or pressure of a loop is so small that the hot solution ceases to exist and there are only cool loop solutions. Many loops can be superimposed to form a coronal arcade which contains loops of several different types.     

The thermal statics of coronal loops

The thermal statics of constant pressure coronal loops is discussed, with particular emphasis on non-equilibrium and scaling relations. An analytical solution showing explicitly the occurrence of non-equilibrium in radiation dominated loops is presented. In addition, the general scaling law for hot loops is given. However, in view of the uncertainties in the coronal heating function and the observational determined loop parameters, it is suggested that scaling laws are currently of limited value.     

Loss of equilibrium in coronal loops

The loss of equilibrium in coronal magnetic field structures is a possible source of energy for coronal heating and solar flares. We investigate whether such a loss of equilibrium occurs when a coronal loop is progressively twisted by photospheric motions. In studies of 2-D cylindrical equilibria, long loops have been found to be of constant cross-sectional area along most of their length, with axial variations being confined to narrow boundary layers. We use this information to develop a 1-D line-tied model, for a 2-D coronal loop. We specify the twist in terms of the azimuthal field and more physically, in terms of the photospheric footpoint displacement. In the former case we find a loss of equilibrium, but not in the latter. We also examine a twisted loop with a non-zero plasma pressure. The loss of equilibrium is only found at high-plasma . It is conjectured that such high- can occur in flare loops and prior to a prominence eruption. However, when the plasma evolves adiabatically, there is no loss of equilibrium.     

The structure of coronal magnetic loops

We present here a model, based on observations, for the magnetic-field equilibrium of a cool coronal loop. The pressure structure, taken from the Harvard/Skylab EUV data, is used to modify the usual force-free-field form in quasi-cylindrical symmetry. The resulting field, which has the same direction but different strength, is calculated and its variation displayed. Finally, localized interchange stability is evaluated and discussed, as the first step in a subsequent complete magnetohydrodynamic-stability analysis.