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.     

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.     

Propagation of the Slow Magnetoacoustic Waves in Coronal Loops Above Sunspots

The observations from the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) have revealed the weak dis- turbances (WDs) propagating in the fan-like coronal loops of the active region (AR 11092) at 171 ?A, 193 ?A, and 211 ?A. These WDs seem to be a common phenomenon in this part of the active region. The disturbances originate from the bright loop foot, and propagate along the loops. The observed propagation speed decreases with the increasing temperature, and varies between 40 km/s and 121 km/s, close to and less than the sound speed in coronal loops. Consid- ering the projection effect and the different angles of the loops with respect to the line of sight, this is exactly what the slow-wave model expects. The wavelet analysis shows that the periods of the WDs observed in different wavebands have no signi?cant difference, the two distinct periods, 3 min and more than 10 min, are all detected in the three EUV wavebands. Not only the coronal loops but also the sunspot region in the chromosphere exhibit intensity oscillations with a period of the order of 3 min. This result suggests that the sunspot oscillations can propagate into the corona through the chromosphere and transition region.     

慢磁声波在黑子上方冕环中的传播

太阳动力学观测站(Solar Dynamics Observatory,SDO)装载的太阳大气成像系统(Atmospheric Imaging Assembly,AIA)通过3个极紫外波段(171 (A)、193 (A)和211 (A))的观测资料展现了太阳活动区(AR11092)扇形冕环微弱的扰动传播,这种波动现象普遍存在于活动区冕环.该扰动传播起源于明亮的环足处,其传播沿冕环方向.冕环中观测到的扰动传播速度随温度升高而减小,速度范围是40～121 km/s,其值接近且小于声速.考虑投影效应以及环与视线方向的倾角大小,这正好是慢波模型所预期的.冕环的扰动周期在不同波段(3个极紫外波段)没有明显的区别,都存在3 min以及10多分钟的周期.此外,不仅仅冕环扰动表现出3 min左右的周期,色球层的黑子区域同样存在相同的振荡周期.这个结果表明黑子振荡是可以穿过色球以及过渡区到达日冕的.     

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.     

Effect of Flow on Resonant Absorption of Slow MHD Waves in Coronal Arcades

Resonant absorption of slow MHD waves is studied numerically by using the SGH method and is applied to a model of a coronal arcade in the presence of equilibrium plasma flows. The arcade is approximated by a 1D horizontal magnetic slab that is non-uniform along the vertical direction and which is surrounded by two homogeneous media. While propagating from the photosphere upwards into the corona, the magneto-acoustic waves can be resonantly absorbed in the inhomogeneous region of the arcade. Computational results show that the resonant absorption of the impinging waves strongly depends on the equilibrium model and on the characteristics of the driving wave. The results also indicate that the presence of an equilibrium plasma flow along the magnetic field of the arcade reduces the resonant absorption for the flow speed parameters considered.     

The Effect of Twisted Magnetic Field on the Resonant Absorption of MHD Waves in Coronal Loops

The standing quasi-modes in a cylindrical incompressible flux tube with magnetic twist that undergoes a radial density structuring is considered in ideal magnetohydrodynamics (MHD). The radial structuring is assumed to be a linearly varying density profile. Using the relevant connection formulae, the dispersion relation for the MHD waves is derived and solved numerically to obtain both the frequencies and damping rates of the fundamental and first-overtone modes of both the kink (m=1) and fluting (m=2,3) waves. It was found that a magnetic twist will increase the frequencies, damping rates and the ratio of the oscillation frequency to the damping rate of these modes. The period ratio P ₁/P ₂ of the fundamental and its first-overtone surface waves for kink (m=1) and fluting (m=2,3) modes is lower than two (the value for an untwisted loop) in the presence of twisted magnetic field. For the kink modes, particularly, the magnetic twists B _φ /B _z =0.0065 and 0.0255 can achieve deviations from two of the same order of magnitude as in the observations. Furthermore, for the fundamental kink body waves, the frequency band width increases with increasing magnetic twist.     

Non-reflective Propagation of Kink Waves in Coronal Magnetic Loops

Propagating kink waves are ubiquitously observed in solar magnetic wave guides. We consider the possibility that these waves propagate without reflection although there is some inhomogeneity. We briefly describe the general theory of non-reflective, one-dimensional wave propagation in inhomogeneous media. This theory is then applied to kink-wave propagation in coronal loops. We consider a coronal loop of half-circle shape embedded in an isothermal atmosphere, and assume that the plasma temperature is the same inside and outside the loop. We show that non-reflective kink-wave propagation is possible for a particular dependence of the loop radius on the distance along the loop. A viable assumption that the loop radius increases from the loop footpoint to the apex imposes a lower limit on the loop expansion factor, which is the ratio of the loop radii at the apex and footpoints. This lower limit increases with the loop height; however, even for a loop that is twice as high as the atmospheric scale height, it is small enough to satisfy observational constraints. Hence, we conclude that non-reflective propagation of kink waves is possible in a fairly realistic model of 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.     

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.     

Nonlinear Resonant Excitation of Fast Sausage Waves in Current-Carrying Coronal Loops

We consider a model of a coronal loop that is a cylindrical magnetic tube with two surface electric currents. Its principal sausage mode has no cut-off in the long-wavelength limit. For typical coronal conditions, the period of the mode is between one and a few minutes. The sausage mode of flaring loops could cause long-period pulsations observed in microwave and hard X-ray ranges. There are other examples of coronal oscillations: long-period pulsations of active-region quiet loops in the soft X-ray emission are observed. We assume that these can also be caused by sausage waves. The question arises of how the sausage waves are generated in quiet loops. We assume that they can be generated by torsional oscillations. This process can be described in the framework of the nonlinear three-wave interaction formalism. The periods of interacting torsional waves are similar to the periods of torsional oscillations observed in the solar atmosphere. The timescale of the sausage-wave excitation is not much longer than the periods of interacting waves, so that the sausage wave is excited before torsional waves are damped.     

MHD Modeling of Coronal Large-Amplitude Waves Related to CME Lift-off

We have employed a two-dimensional magnetohydrodynamic simulation code to study mass motions and large-amplitude coronal waves related to the lift-off of a coronal mass ejection (CME). The eruption of the filament is achieved by an artificial force acting on the plasma inside the flux rope. By varying the magnitude of this force, the reaction of the ambient corona to CMEs with different acceleration profiles can be studied. Our model of the ambient corona is gravitationally stratified with a quadrupolar magnetic field, resulting in an ambient Alfvén speed that increases as a function of height, as typically deduced for the low corona. The results of the simulations show that the erupting flux rope is surrounded by a shock front, which is strongest near the leading edge of the erupting mass, but also shows compression near the solar surface. For rapidly accelerating filaments, the shock front forms already in the low corona. Although the speed of the driver is less than the Alfvén speed near the top of the atmosphere, the shock survives in this region as well, but as a freely propagating wave. The leading edge of the shock becomes strong early enough to drive a metric type II burst in the corona. The speed of the weaker part of the shock front near the surface is lower, corresponding to the magnetosonic speed there. We analyze the (line-of-sight) emission measure of the corona during the simulation and recognize a wave receding from the eruption site, which strongly resembles EIT waves in the low corona. Behind the EIT wave, we clearly recognize a coronal dimming, also observed during CME lift-off. We point out that the morphology of the hot downstream region of the shock would be that of a hot erupting loop, so care has to be taken not to misinterpret soft X-ray imaging observations in this respect. Finally, the geometry of the magnetic field around the erupting mass is analyzed in terms of precipitation of particles accelerated in the eruption complex. Field lines connected to the shock are further away from the photospheric neutral line below the filament than the field lines connected to the current sheet below the flux rope. Thus, if the DC fields in the current sheet accelerate predominantly electrons and the shock accelerates ions, the geometry is consistent with recent observations of gamma rays being emitted further out from the neutral line than hard X-rays.     

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.     

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.     

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.     

Resonantly Damped Surface and Body MHD Waves in a Solar Coronal Slab with Oblique Propagation

The theory of magnetohydrodynamic (MHD) waves in solar coronal slabs in a zero-β configuration and for parallel propagation of waves does not allow the existence of surface waves. When oblique propagation of perturbations is considered, both surface and body waves are able to propagate. When the perpendicular wavenumber is larger than a certain value, the body kink mode becomes a surface wave. In addition, a sausage surface mode is found below the internal cutoff frequency. When nonuniformity in the equilibrium is included, surface and body modes are damped by resonant absorption. In this paper, first, a normal-mode analysis is performed and the period, the damping rate, and the spatial structure of the eigenfunctions are obtained. Then, the time-dependent problem is solved, and the conditions under which one or the other type of mode is excited are investigated.     

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).     

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.