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Quasi-periodic, fast-mode propagating (QFP) wave trains in the corona have been studied intensively over the last decade, thanks to the full-disk, high spatio-temporal resolution, and wide-temperature coverage observations taken by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). In the AIA observations, the QFP wave trains are seen to consist of multiple coherent and concentric wavefronts emanating successively near the epicenter of the accompanying flares. They propagate outwardly either along or across coronal loops at fast-mode magnetosonic speeds from several hundred to more than 2000 km s −1 , and their periods are in the range of tens of seconds to several minutes. Based on the distinctly different properties of QFP wave trains, they might be divided into two distinct categories: narrow and broad ones. For most QFP wave trains, some of their periods are similar to those of the quasi-periodic pulsations (QPPs) in the accompanying flares, indicating that they are probably different manifestations of the same physical process. Currently, candidate generation mechanisms for QFP wave trains include two main categories: the pulsed energy excitation mechanism associated with magnetic reconnection and the dispersion-evolution mechanism related to the dispersive evolution of impulsively generated broadband perturbations. In addition, the generation of some QFP wave trains might be driven by the leakage of three- and five-minute oscillations from the lower atmosphere. As one of the discoveries of SDO, QFP wave trains provide a new tool for coronal seismology to probe the corona parameters, and they are also useful for diagnosing the generation of QPPs, flare processes including energy release, and particle acceleration. This review aims to summarize the main observational and theoretical results of spatially resolved QFP wave trains in extreme-ultraviolet observations and presents briefly a number of questions that deserve further investigation.

We present Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) observation of three types of fast-mode, propagating, magnetosonic waves in a GOES C3.0 flare on 23 April 2013, which was accompanied by a prominence eruption and a broad coronal mass ejection (CME). During the fast-rising phase of the prominence, a large-scale, dome-shaped, extreme-ultraviolet (EUV) wave firstly formed ahead of the CME bubble and propagated at a speed of about 430 km s −1 in the CME’s lateral direction. One can identify the separation process of the EUV wave from the CME bubble. The reflection effect of the on-disk counterpart of this EUV wave was also observed when it interacted with a remote active region. Six minutes after the first appearance of the EUV wave, a large-scale, quasi-periodic EUV train with a period of about 120 seconds, which emanated from the flare epicenter and propagated outward at an average speed up to 1100 km s −1 , appeared inside the CME bubble. In addition, another narrow, quasi-periodic EUV wave train, which also emanated from the flare epicenter, propagated at a speed of about 475 km s −1 and with a period of about 110 seconds, was observed along a closed-loop system connecting two adjacent active regions. We propose that all of the observed waves are fast-mode magnetosonic waves, in which the large-scale, dome-shaped EUV wave ahead of the CME bubble was driven by the expansion of the CME bubble, while the large-scale, quasi-periodic EUV train within the CME bubble and the narrow quasi-periodic EUV wave train along the closed-loop system were excited by the intermittent energy-releasing process in the flare. Coronal seismology application and energy carried by the waves are also estimated based on the measured wave parameters.

We investigate the transverse coronal-loop oscillations induced by the eruption of a prominence-carrying flux rope on 7 December 2012. The flux rope, originating from NOAA Active Region (AR) 11621, was observed in extreme-ultraviolet (EUV) wavelengths by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) spacecraft and in the H α line center by the ground-based telescope at the Big Bear Solar Observatory (BBSO). The early evolution of the flux rope is divided into two steps: a slow-rise phase at a speed of ≈ 230 km s −1 and a fast-rise phase at a speed of ≈ 706 km s −1 . The eruption generates a C5.8 flare and the onset of the fast rise is consistent with the hard X-ray (HXR) peak time of the flare. The embedded prominence has a lower speed of ≈ 452 km s −1 . The eruption is significantly inclined from the local solar normal by ≈ 60 ∘ , suggesting a typical non-radial eruption. During the early eruption of the flux rope, the nearby coronal loops are disturbed and experience independent kink-mode oscillations in the horizontal and vertical directions. The oscillation in the horizontal direction has an initial amplitude of ≈ 3.1 Mm, a period of ≈ 294 seconds, and a damping time of ≈ 645 seconds. It is most striking in 171 Å and lasts for three to four cycles. The oscillations in the vertical directions are observed mainly in 171, 193, and 211 Å. The initial amplitudes are in the range of 3.4 – 5.2 Mm, with an average value of 4.5 Mm. The periods are between 407 seconds and 441 seconds, with an average value of 423 seconds. The oscillations are damping and last for nearly four cycles. The damping times are in the range of 570 – 1012 seconds, with an average value of 741 seconds. Assuming a semi-circular shape of the vertically oscillating loops, we calculate the loop lengths according to their heights. Using the observed periods, we carry out coronal seismology and estimate the internal Alfvén speeds (988 – 1145 km s −1 ) and the magnetic-field strengths (12 – 43 G) of the oscillating loops.

Slow magnetoacoustic waves observed in the solar corona are used as seismological probes of plasma parameters. It has been shown that the dispersion properties of such waves can vary significantly under the influence of the wave-induced thermal misbalance. In the current research, we study the effect of misbalance on waves inside the magnetic-flux tube under the second-order thin-flux-tube approximation. Using the parameters of active-region-fan coronal loops, we calculated wave properties such as the phase speed and decrement. It is shown that neglecting thermal misbalance may be the reason for the substantial divergence between seismological and spectrometric estimations of plasma parameters. We also show that the frequency dependence of the phase speed is affected by two features, namely the geometric dispersion and the dispersion caused by the thermal misbalance. In contrast to the phase speed, the wave decrement primarily is affected by the thermal misbalance only. The dependencies of the phase speed and decrement of the slow wave on the magnetic field and tube cross-section are also analysed.

We analyse the phase shifts of standing, slow magnetohydrodynamic (MHD) waves in solar coronal loops using a linear MHD model taking into account the role of thermal conductivity, compressive viscosity, radiative losses, and heating–cooling misbalance. We estimate the phase shifts in time and space of density and temperature perturbations with respect to velocity perturbations and also calculate the phase difference between density and temperature perturbations. The overall significance of compressive viscosity is found to be negligible for most of the loops considered in the study. For loops with high background density and/or low background temperature, the role of radiative losses (with heating–cooling misbalance) is found to be more significant. Also, the effect of heating–cooling misbalance with a temperature- and density-dependent heating function is found to be more significant in the case of longer loop lengths ( L = 500  Mm). We derived a general expression for the polytropic index [ γ eff ] and found that under linear MHD the effect of compressive viscosity on the polytropic index is negligible. The radiative losses with constant heating lead to a monotonic increase of γ eff with increasing density, whereas the consideration of an assumed heating function [ H ( ρ , T ) ∝ ρ a T b , where a = − 0.5 and b = − 3 ] makes the γ eff peak at a certain loop density. We also explored the role of different heating functions by varying the free parameters a and b for a fixed loop of ρ 0 = 10 − 11  kg m −3 , T 0 = 6.3  MK, and loop length L = 180  Mm. We find that the consideration of different heating functions [ H ( ρ , T ) ] leads to a significant variation of the phase difference between density and temperature perturbations; however, the polytropic index remains close to a value of 1.66.

The capability of the motion-magnification technique for the detection of transverse oscillations, such as kink oscillations of solar coronal loops observed with an imaging telescope, in the sub-pixel regime is investigated. The technique is applied to artificial-image sequences imitating harmonic transverse displacements of the loop, observed in the optically thin regime. Motion magnification is found to work well on the analysis of sub-pixel, ≥ 0.01  pixel oscillations, and it is characterised by linear scaling between the magnified amplitude and input amplitude. Oscillations of loops with transverse density profiles of different steepness are considered. After magnification, the original transverse profiles are preserved sufficiently well. The motion-magnification performance is found to be robust in noisy data, for coloured noise with spectral indices ranging from 0 to 3, and additional Poisson noise with a signal-to-background-noise ratio down to unity. Our findings confirm the reliability of the motion-magnification technique for applications in magnetohydrodynamic seismology of the solar corona.

We examine resonantly damped kink modes in straight coronal slabs, paying special attention to the effects of the formulation for the transverse density distribution (“profile”). We work in the framework of pressure-less, gravity-free, resistive magnetohydrodynamics, and we adopt the dissipative-eigenmode perspective. The density profile is restricted to be one-dimensional, but nonetheless allowed to take a generic form characterized by a continuous transition layer connecting a uniform interior to a uniform exterior. A dispersion relation (DR) is derived in the thin-boundary limit, yielding analytical expressions for the eigenfrequencies that generalize known results in various aspects. We find that the analytical rather than the numerical solutions to the thin-boundary DR serve better the purpose for validating our self-consistent resistive solutions. More importantly, the eigenfrequencies are found to be sensitive to profile specifications, the ratio of the imaginary to the real part readily varying by a factor of two when one profile is used in place of another. Our eigenmode computations are also examined in the context of impulsively excited kink waves, suggesting the importance of resonant absorption for sufficiently oblique components when the spatial scale of the exciter is comparable to the slab half-width.

We study the phase shifts of propagating slow magnetoacoustic waves in solar coronal loops invoking the effects of thermal conductivity, compressive viscosity, radiative losses, and heating–cooling imbalance. We derive the general dispersion relation and solve it to determine the phase shifts of density and temperature perturbations relative to the velocity and their dependence on the equilibrium parameters of the plasma such as the background density [ ρ 0 ] and temperature [ T 0 ]. We estimate the phase difference [ Δ ϕ ] between density and temperature perturbations and its dependence on ρ 0 and T 0 . The role of radiative losses, along with the heating–cooling imbalance for an assumed specific heating function [ H ( ρ , T ) ∝ ρ − 0.5 T − 3 ], in the estimation of the phase shifts is found to be significant for the high-density and low-temperature loops. Heating–cooling imbalance can significantly increase the phase difference ( Δ ϕ ≈ 140 ∘ ) for the low-temperature loops compared to the constant-heating case ( Δ ϕ ≈ 30 ∘ ). We derive a general expression for the polytropic index [ ] using the linear MHD model. We find that in the presence of thermal conduction alone, remains close to its classical value 5 / 3 for all the considered ρ 0 and T 0 observed in typical coronal loops. We find that the inclusion of radiative losses (with or without heating–cooling imbalance) cannot explain the observed polytropic index under the considered heating and cooling models. To make the expected match the observed value of 1.1 ± 0.02 in typical coronal loops, the thermal conductivity needs to be enhanced by an order of magnitude compared to the classical value. However, this conclusion is based on the presented model and needs to be confirmed further by considering more realistic radiative functions. We also explore the role of different heating functions for typical coronal parameters and find that although the remains close to 5 / 3 , but the phase difference is highly dependent on the form of the heating function.

We present the analysis of three kinds of oscillating behavior using multi-wavelength observations of the 10 November 2013 (SOL2013-11-10T05:14) circular-ribbon flare. This event is a typical circular-ribbon flare with an outer spine structure and homologous jets. We found three kinds of oscillations (or perturbations): i) flux oscillation (or QPP) with a dominant period of about 20 seconds in X-ray, EUV, and microwave emissions, ii) periodic jets with an intermittent cadence of around 72 seconds, iii) an outer loop perturbing half a cycle with a duration of about 168 seconds. Similar to the periodic jets that could be produced by a nonthermal process, like repeated magnetic reconnection, the flare QPP detected in the thermal emissions could have the same origin as the oscillation seen in the nonthermal emissions. The outer-loop perturbation is possibly triggered by a blast wave driven by the circular-ribbon flare, or it might be modulated by the sausage wave or the slow magnetoacoustic wave. The results obtained provide data for further numerical studies on the physical origin of the flare oscillations.

The processes of coronal plasma heating and cooling were previously shown to significantly affect the dynamics of slow magnetoacoustic (MA) waves, causing amplification or attenuation, and also dispersion. However, the entropy mode is also excited in such a thermodynamically active plasma and is affected by the heating/cooling misbalance too. This mode is usually associated with the phenomenon of coronal rain and formation of prominences. Unlike adiabatic plasmas, the properties and evolution of slow MA and entropy waves in continuously heated and cooling plasmas get mixed. Different regimes of the misbalance lead to a variety of scenarios for the initial perturbation to evolve. In order to describe properties and evolution of slow MA and entropy waves in various regimes of the misbalance, we obtained an exact analytical solution of the linear evolutionary equation. Using the characteristic timescales and the obtained exact solution, we identified regimes with qualitatively different behaviour of slow MA and entropy modes. For some of those regimes, the spatio-temporal evolution of the initial Gaussian pulse is shown. In particular, it is shown that slow MA modes may have a range of non-propagating harmonics. In this regime, perturbations caused by slow MA and entropy modes in a low- β plasma would look identical in observations, as non-propagating disturbances of the plasma density (and temperature) either growing or decaying with time. We also showed that the partition of the initial energy between slow MA and entropy modes depends on the properties of the heating and cooling processes involved. The exact analytical solution obtained could be further applied to the interpretation of observations and results of numerical modelling of slow MA waves in the corona and the formation and evolution of coronal rain.