Dissipation of Standing Slow Magnetoacoustic Waves in Hot Coronal Loops

The damping of standing slow waves in hot (T>6 MK) coronal loops of semicircular shape is revisited in both the linear and nonlinear regimes. Dissipation by thermal conduction, compressive viscosity, radiative cooling, and heating are examined for nonstratified and stratified loops. We find that for typical conditions of hot SUMER loops, thermal conduction increases the period of damped oscillations over the sound-crossing time, whereas the decay times are mostly shaped by compressive viscosity. Damping from optically thin radiation is negligible. We also find that thermal conduction alone results in slower damping of the density and velocity waves compared to the observations. Only when compressive viscosity is added do these waves damp out at the same rate as the observed rapidly decaying modes of hot SUMER loop oscillations, in contrast to most current work, which has pointed to thermal conduction as the dominant mechanism. We compare the linear predictions with numerical hydrodynamic calculations. Under the effects of gravity, nonlinear viscous dissipation leads to a reduction of the decay time compared to the homogeneous case. In contrast, the linear results predict that the damping rates are barely affected by gravity.     

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.     

Propagation and Damping of a Localized Impulsive Longitudinal Perturbation in Coronal Loops

We use linear analysis to simulate the evolution of a coronal loop in response to a localized impulsive event. The disturbance is modeled by injecting a narrow Gaussian velocity pulse near one footpoint of a loop in equilibrium. Three different damping mechanisms, namely viscosity, thermal conduction, and optically thin radiation, are included in the loop calculations. We consider homogeneous and gravitationally stratified, isothermal loops of varying length (50≤L≤400 Mm) and temperature (2≤T≤10 MK). We find that a localized pulse can effectively excite slow magnetoacoustic waves that propagate up along the loop. The amplitudes of the oscillations increase with decreasing loop temperature and increasing loop length and size of the pulse width. At T≥4 MK, the waves are dissipated by the combined effects of viscosity and thermal conduction, whereas at temperatures of 2 MK, or lower, wave dissipation is governed by radiative cooling. We predict periods in the range of 4.6?–?41.6 minutes. The wave periods remain unaltered by variations of the pulse size, decrease with the loop temperature, and increase almost linearly with the loop length. In addition, gravitational stratification results in a small reduction of the periods and amplification of the waves as they propagate up along the loop.     

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.     

Damping of slow magnetoacoustic waves in an inhomogeneous coronal plasma

We study the propagation and dissipation of slow magnetoacoustic waves in an inhomogeneous viscous coronal loop plasma permeated by uniform magnetic field. Only viscosity and thermal conductivity are taken into account as dissipative processes in the coronal loop. The damping length of slow-mode waves exhibit varying behaviour depending upon the physical parameters of the loop in an active region AR8270 observed by TRACE. The wave energy flux associated with slow magnetoacoustic waves turns out to be of the order of 10⁶ erg cm^?2 s^?1 which is high enough to replace the energy lost through optically thin coronal emission and the thermal conduction below to the transition region. It is also found that only those slow-mode waves which have periods more than 240s provide the required heating rate to balance the energy losses in the solar corona. Our calculated wave periods for slow-mode waves nearly match with the oscillation periods of loop observed by TRACE.     

Viscous Damping of Non-Adiabatic MHD Waves in an Unbounded Solar Coronal Plasma

The damping of MHD waves in solar coronal magnetic field is studied taking into account thermal conduction and compressive viscosity as dissipative mechanisms. We consider viscous homogeneous unbounded solar coronal plasma permeated by a uniform magnetic field. A general fifth-order dispersion relation for MHD waves has been derived and solved numerically for different solar coronal regimes. The dispersion relation results three wave modes: slow, fast, and thermal modes. Damping time and damping per periods for slow- and fast-mode waves determined from dispersion relation show that the slow-mode waves are heavily damped in comparison with fast-mode waves in prominences, prominence–corona transition regions (PCTR), and corona. In PCTRs and coronal active regions, wave instabilities appear for considered heating mechanisms. For same heating mechanisms in different prominences the behavior of damping time and damping per period changes significantly from small to large wavenumbers. In all PCTRs and corona, damping time always decreases linearly with increase in wavenumber indicate sharp damping of slow- and fast-mode waves.     

Characteristics of transverse oscillations in a coronal loop arcade

TRACE observations from 15 April 2001 of transverse oscillations in coronal loops of a post-flare loop arcade are investigated. They are considered to be standing fast kink oscillations. Oscillation signatures such as displacement amplitude, period, phase and damping time are deduced from 9 loops as a function of distance along the loop length. Multiple oscillation modes are found with different amplitude profile along the loop length, suggesting the presence of a second harmonic. The damping times are consistent with the hypothesis of phase mixing and resonant absorption, although there is a clear bias towards longer damping times compared with previous studies. The coronal magnetic field strength and coronal shear viscosity in the loop arcade are derived.     

Characteristics of transverse oscillations in a coronal loop arcade

TRACE observations from 15 April 2001 of transverse oscillations in coronal loops of a post-flare loop arcade are investigated. They are considered to be standing fast kink oscillations. Oscillation signatures such as displacement amplitude, period, phase and damping time are deduced from 9 loops as a function of distance along the loop length. Multiple oscillation modes are found with different amplitude profile along the loop length, suggesting the presence of a second harmonic. The damping times are consistent with the hypothesis of phase mixing and resonant absorption, although there is a clear bias towards longer damping times compared with previous studies. The coronal magnetic field strength and coronal shear viscosity in the loop arcade are derived.     

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.     

Attenuation of small-amplitude oscillations in a prominence–corona model with a transverse magnetic field

Observations show that small-amplitude prominence oscillations are usually damped after a few periods. This phenomenon has been theoretically investigated in terms of non-ideal magnetoacoustic waves, non-adiabatic effects being the best candidates to explain the damping in the case of slow modes. We study the attenuation of non-adiabatic magnetoacoustic waves in a slab prominence embedded in the coronal medium. We assume an equilibrium configuration with a transverse magnetic field to the slab axis and investigate wave damping by thermal conduction and radiative losses. The magnetohydrodynamic equations are considered in their linearised form and terms representing thermal conduction, radiation and heating are included in the energy equation. The differential equations that govern linear slow and fast modes are numerically solved to obtain the complex oscillatory frequency and the corresponding eigenfunctions. We find that coronal thermal conduction and radiative losses from the prominence plasma reveal as the most relevant damping mechanisms. Both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In addition, the energy transfer between the prominence and the corona caused by thermal conduction has a noticeable effect on the wave stability, radiative losses from the prominence plasma being of paramount importance for the thermal stability of fast modes. We conclude that slow modes are efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The presence of the corona causes a decrease of the damping times with respect to those of an isolated prominence slab, but its effect is still insufficient to obtain damping times of the order of the period in the case of fast modes.     

The formation of flare loops by magnetic reconnection and chromospheric ablation

Slow-mode shocks produced by reconnection in the corona can provide the thermal energy necessary to sustain flare loops for many hours. These slow shocks have a complex structure because strong thermal conduction along field lines dissociates the shocks into conduction fronts and isothermal subshocks. Heat conducted along field lines mapping from the subshocks to the chromosphere ablates chromospheric plasma and thereby creates the hot flare loops and associated flare ribbons. Here we combine a non-coplanar compressible reconnection theory with simple scaling arguments for ablation and radiative cooling, and predict average properties of hot and cool flare loops as a function of the coronal vector magnetic field. For a coronal field strength of 100 G the temperature of the hot flare loops decreases from 1.2 × 10⁷ K to 4.0 × 10⁶ K as the component of the coronal magnetic field perpendicular to the plane of the loops increases from 0% to 86% of the total field. When the perpendicular component exceeds 86% of the total field or when the altitude of the reconnection site exceeds 10⁶km, flare loops no longer occur. Shock enhanced radiative cooling triggers the formation of cool H flare loops with predicted densities of 10¹³ cm^–3, and a small gap of 10³ km is predicted to exist between the footpoints of the cool flare loops and the inner edges of the flare ribbons.     

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.     

On the Lifetime of Hot Coronal Plasmas Arising from Nanoflares

The cooling of plasmas in closed coronal loops by thermal conduction is important when considering their detectability at X-ray and EUV wavelengths. A non-local formalism of thermal conduction originating in laboratory plasmas is used and it is shown that while the effect is unlikely to be important for loops that are in a steady state, it does play a significant role in loops that are impulsively heated (e.g., by nanoflares). Such loops are “under-dense”, and so hot electrons have a relatively long mean free path. Analytic and numerical models are presented, and it is shown that conduction cooling times are lengthened quite considerably. A comparison of various cooling times with ionisation times is also presented, and it is noted that this conductive physics may enhance the chances of observing hot nanoflare-heated plasma.     

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.     

Dynamical Features and Evolutional Characteristics of Brightening Coronal Loops

We present a detailed study of coronal loop brightenings observed in an active region on the solar limb. These brightening loops show expanding and shrinking motions in EUV coronal line images and also show downflow along the loops in Lα and Hα images. By means of time-slice analysis of the images, we found that both the expanding and shrinking motions of the loops are not real motions of plasma but apparent motions like post-flare loops, where the loops at the different height are successively heated and cooled. From a temperature analysis, the time delay between the brightenings of hot 195 Å and cool Lα loops is found to be nearly equal to the time-scale of the conduction cooling. We conclude that these loop brightenings are sources of so called Hα coronal rains.     

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.     

Dissipating waves in a hot,compressible, magnetic,structured atmosphere

The damping of ducted, fast, magnetohydrodynamic (MHD) waves by ion viscosity and electron heat conduction in a radiating, optically thin, warm, structured atmosphere has been evaluated. Dissipation is more effective in a warm plasma than in a cold one but, for waves ducted by solar coronal loops, dissipation is only efficient if the periods of the waves are shorter than a few tens of seconds and only if the background magnetic field is less than about 15 G. It appears that MHD waves of longer periods and in stronger magnetic fields will survive the dissipative mechanisms considered here and may be manifest as observable coronal oscillations.     

The thermal stability of coronal loops numerical simulations

We have studied the radiative stability of thermally isolated coronal loops with free-flow boundary conditions by nonlinear numerical simulation. We first establish a chromosphere-to-corona loop equilibrium (including the option of a deep chromosphere) by following the nonlinear evolution from an initial isothermal state with rigid boundaries. We then change the end conditions, to allow free flow and to fix the temperature, and investigate the response to non-isobaric perturbations. Within a family of loops of the same pressure, we find long hot loops to be stable and short cool loops to be unstable to the thermal chromosphericexpansion mode. The stable cases remain so, even when long chromospheric ends and/or gravity are added. In those cases which are unstable, we follow the subsequent nonlinear evolution which exhibits swelling of the chromosphere until the entire loop becomes cool and dense.     

A model of hot loops associated with solar flares

A dynamical model is proposed for the formation of soft X-ray emitting hot loops in solar flares. It is examined by numerical simulations how a solar model atmosphere in a magnetic loop changes its state and forms a hot loop when the flare energy is released in the form of heat liberation either at the top part or around the transition region in the loop.When the heat liberation takes place at the top part of the loop which arches in the corona, the plasma temperature around the loop apex rises rapidly and, as the result, the downward thermal conductive flux is increased along the magnetic tube of force. Soon after the thermal conduction front rushes into the upper chromosphere, a local peak of pressure is produced near the conduction front and the chromospheric material begins to expand into the corona to form a high-temperature (10⁷ K-3 × 10⁷ K at the loop apex) and high-density (10¹⁰ cm^–3-10¹¹ cm^–3 at the loop apex) loop. The velocity of the expanding material can reach a few hundred kilometres per second in the coronal part. The thermal conduction front also plays a role of piston pushing the chromospheric material downward and gives birth to a shock wave which propagates through the minimum temperature region into the photosphere. If, on the other hand, the heat source is placed around the transition region in the loop, the expansion of the material into the corona occurs from the beginning of the flare and the formation process of the hot loop differs somewhat from the case with the heat source at the top part of the loop.Thermal components of radiations emitted from flare regions, ranging from soft X-rays to radio wavelengths, are interpreted in a unified way by using physical quantities obtained as functions of time and position in our flare loop model as will be discussed in detail in a following paper.     

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.