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The phenomenon of quasi-periodic pulsations (QPPs) in solar and stellar flares has been known for over 50 years and significant progress has been made in this research area. It has become clear that QPPs are not rare—they are found in many flares and, therefore, robust flare models should reproduce their properties in a natural way. At least fifteen mechanisms/models have been developed to explain QPPs in solar flares, which mainly assume the presence of magnetohydrodynamic (MHD) oscillations in coronal structures (magnetic loops and current sheets) or quasi-periodic regimes of magnetic reconnection. We review the most important and interesting results on flare QPPs, with an emphasis on the results of recent years, and we present the predicted and prominent observational signatures of each of the fifteen mechanisms. However, it is not yet possible to draw an unambiguous conclusion as to the correct underlying QPP mechanism because of the qualitative, rather than quantitative, nature of most of the models and also due to insufficient observational information on the physical properties of the flare region, in particular the spatial structure of the QPP source. We also review QPPs in stellar flares, where progress is largely based on solar-stellar analogies, suggesting similarities in the physical processes in flare regions on the Sun and magnetoactive stars. The presence of QPPs with similar properties in solar and stellar flares is, in itself, a strong additional argument in favor of the likelihood of solar-stellar analogies. Hence, advancing our understanding of QPPs in solar flares provides an important additional channel of information about stellar flares. However, further work in both theory/simulations and in observations is needed.

The heating of the solar chromosphere and corona to the observed high temperatures, imply the presence of ongoing heating that balances the strong radiative and thermal conduction losses expected in the solar atmosphere. It has been theorized for decades that the required heating mechanisms of the chromospheric and coronal parts of the active regions, quiet-Sun, and coronal holes are associated with the solar magnetic fields. However, the exact physical process that transport and dissipate the magnetic energy which ultimately leads to the solar plasma heating are not yet fully understood. The current understanding of coronal heating relies on two main mechanism: reconnection and MHD waves that may have various degrees of importance in different coronal regions. In this review we focus on recent advances in our understanding of MHD wave heating mechanisms. First, we focus on giving an overview of observational results, where we show that different wave modes have been discovered in the corona in the last decade, many of which are associated with a significant energy flux, either generated in situ or pumped from the lower solar atmosphere. Afterwards, we summarise the recent findings of numerical modelling of waves, motivated by the observational results. Despite the advances, only 3D MHD models with Alfv\'en wave heating in an unstructured corona can explain the observed coronal temperatures compatible with the quiet Sun, while 3D MHD wave heating models including cross-field density structuring are not yet able to account for the heating of coronal loops in active regions to their observed temperature., Comment: accepted in Space Science Reviews

Characterized by cyclic axisymmetric perturbations to both the magnetic and fluid parameters, magnetohydrodynamic fast sausage modes (FSMs) have proven useful for solar coronal seismology given their strong dispersion. This review starts by summarizing the dispersive properties of the FSMs in the canonical configuration where the equilibrium quantities are transversely structured in a step fashion. With this preparation we then review the recent theoretical studies on coronal FSMs, showing that the canonical dispersion features have been better understood physically, and further exploited seismologically. In addition, we show that departures from the canonical equilibrium configuration have led to qualitatively different dispersion features, thereby substantially broadening the range of observations that FSMs can be invoked to account for. We also summarize the advances in forward modeling studies, emphasizing the intricacies in interpreting observed oscillatory signals in terms of FSMs. All these advances notwithstanding, we offer a list of aspects that remain to be better addressed, with the physical connection of coronal FSMs to the quasi-periodic pulsations in solar flares particularly noteworthy., Accepted for publication in Space Science Reviews

Observations have revealed two regimes of kink oscillations of coronal loops. Large amplitude oscillations excited by impulsive energy releases such as coronal mass ejections are characterised by their strong damping by resonant absorption. Lower amplitude oscillations may also be excited and sustained by the ubiquitous motions present in the corona and so are characterised as being decayless. We perform numerical simulations to study the oscillation and evolution of coronal loops in a dynamical environment. We investigate the observational signatures of kink oscillations and the Kelvin-Helmholtz instability in terms of high-resolution seismological and spatial data analysis techniques. We find that low amplitude kink oscillations are capable of generating significant changes in the loop profile which can affect estimates of the transverse loop inhomogeneity based on seismological and forward modelling methods. The disparity between methods may be indicative of nonlinear evolution of coronal loops. The influence on forward modelling estimates could also account for previous observational evidence favouring loops having wider inhomogeneous layers.

We present a new method to track the position and evolution of coronal loops designed for observations such as active regions in which multiple loops appear in close proximity or overlap with each other along the observational line of sight. The method is based on modeling a time–distance map containing one or more loops and fitting the modeled map to observational data, as opposed to the commonly used technique of analyzing each frame independently. This allows us to control the variability of the model, informed by our physical interpretation, and use the trends present to help constrain the model parameters. We apply our method to an observation of a bundle of coronal loops previously investigated using a spatiotemporal autocorrelation method and compare our results. A benefit of our method is that it provides the time series for the position of the loops that may be used for further analysis using established seismological techniques. We demonstrate this by modeling the oscillation of several loops in response to flaring energy releases that occur during the observation, and we find evidence of loop evolution consistent with the Kelvin–Helmholtz instability

We present the Solar Bayesian Analysis Toolkit (SoBAT) which is a new easy to use tool for Bayesian analysis of observational data, including parameter inference and model comparison. SoBAT is aimed (but not limited) to be used for the analysis of solar observational data. We describe a new Interactive Data Language (IDL) code designed to facilitate the comparison of user-supplied model with data. Bayesian inference allows prior information to be taken into account. The use of Markov chain Monte Carlo (MCMC) sampling allows efficient exploration of large parameter spaces and provides reliable estimation of model parameters and their uncertainties. The Bayesian evidence for different models can be used for quantitative comparison. The code is tested to demonstrate its ability to accurately recover a variety of parameter probability distributions. Its application to practical problems is demonstrated using studies of the structure and oscillation of coronal loops.

We present a model for the intensity of optically thin extreme ultraviolet (EUV) emission for a plasma atmosphere. We apply our model to the solar corona as observed using the six optically thin EUV channels of the Solar Dynamics Observatory/Atmospheric Imaging Assembly instrument. The emissivity of the plasma is calculated from the density and temperature using CHIANTI tables and the intensity is then determined by integration along the line of sight. We consider several different profiles for the radial density and temperature profiles, each of which are constrained by the observational data alone with no further physical assumptions. We demonstrate the method first by applying it to a quiet region of the corona, and then use it as the background component of a model including coronal holes, allowing the plasma densities and temperatures inside and outside the hole to be estimated. We compare our results with differential emission measure inversions. More accurate estimates for the coronal density and temperature profiles have the potential to help constrain plasma properties such as the magnetic field strength when used in combination with methods such as seismology.

AH acknowledges funding from the Science and Technology Facilities Council (UK) through the consolidated grant ST/N000609/1. DP and TVD were supported by the GOA-2015-014 (KU Leuven) and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 724326). The transverse structure of coronal loops plays a key role in the physics but the small transverse scales can be difficult to observe directly. For wider loops the density profile may be estimated by forward modelling of the transverse intensity profile. The transverse density profile may also be estimated seismologically using kink oscillations in coronal loops. The strong damping of kink oscillations is attributed to resonant absorption and the damping profile contains information about the transverse structure of the loop. However, the analytical descriptions for damping by resonant absorption presently only describe the behaviour for thin inhomogeneous layers. Previous numerical studies have demonstrated that this thin boundary approximation produces poor estimates of the damping behaviour in loops with wider inhomogeneous layers. Both the seismological and forward modelling approaches suggest loops have a range of layer widths and so there is a need for a description of the damping behaviour that accurately describes such loops. We perform a parametric study of the damping of standing kink oscillations by resonant absorption for a wide range of inhomogeneous layer widths and density contrast ratios, with a focus on the values most relevant to observational cases. We describe the damping profile produced by our numerical simulations without prior assumption of its shape and compile our results into a lookup table which may be used to produce accurate seismological estimates for kink oscillation observations. Publisher PDF

© 2018. The American Astronomical Society. All rights reserved. The shape of the damping profile of kink oscillations in coronal loops has recently allowed the transverse density profile of the loop to be estimated. This requires accurate measurement of the damping profile that can distinguish the Gaussian and exponential damping regimes, otherwise there are more unknowns than observables. Forward modeling of the transverse intensity profile may also be used to estimate the width of the inhomogeneous layer of a loop, providing an independent estimate of one of these unknowns. We analyze an oscillating loop for which the seismological determination of the transverse structure is inconclusive except when supplemented by additional spatial information from the transverse intensity profile. Our temporal analysis describes the motion of a coronal loop as a kink oscillation damped by resonant absorption, and our spatial analysis is based on forward modeling the transverse EUV intensity profile of the loop under the isothermal and optically thin approximations. We use Bayesian analysis and Markov chain Monte Carlo sampling to apply our spatial and temporal models both individually and simultaneously to our data and compare the results with numerical simulations. Combining the two methods allows both the inhomogeneous layer width and density contrast to be calculated, which is not possible for the same data when each method is applied individually. We demonstrate that the assumption of an exponential damping profile leads to a significantly larger error in the inferred density contrast ratio compared with a Gaussian damping profile. ispartof: ASTROPHYSICAL JOURNAL vol:860 issue:1 status: published

EUV observations of a multi-thermal coronal loop, taken by the Atmospheric Imaging Assembly of the Solar Dynamics Observatory, which exhibits decay-less kink oscillations are presented. The data cube of the quiet-Sun coronal loop was passed through a motion magnification algorithm to accentuate transverse oscillations. Time–distance maps are made from multiple slits evenly spaced along the loop axis and oriented orthogonal to the loop axis. Displacements of the intensity peak are tracked to generate time series of the loop displacement. Fourier analysis on the time series shows the presence of two periods within the loop: ${P}_{1}={10.3}_{-1.7}^{+1.5}$ minutes and ${P}_{2}={7.4}_{-1.3}^{+1.1}$ minutes. The longer period component is greatest in amplitude at the apex and remains in phase throughout the loop length. The shorter period component is strongest further down from the apex on both legs and displays an anti-phase behavior between the two loop legs. We interpret these results as the coexistence of the fundamental and second harmonics of the standing kink mode within the loop in the decay-less oscillation regime. An illustration of seismological application using the ratio P 1/2P 2 ~ 0.7 to estimate the density scale height is presented. The existence of multiple harmonics has implications for understanding the driving and damping mechanisms for decay-less oscillations and adds credence to their interpretation as standing kink mode oscillations.\ud \ud

Context. The strong damping of kink oscillations of coronal loops can be explained by mode coupling. The damping envelope depends on the transverse density profile of the loop. Observational measurements of the damping envelope have been used to determine the transverse loop structure which is important for understanding other physical processes such as heating. Aims. The general damping envelope describing the mode coupling of kink waves consists of a Gaussian damping regime followed by an exponential damping regime. Recent observational detection of these damping regimes has been employed as a seismological tool. We extend the description of the damping behaviour to account for additional physical effects, namely a time-dependent period of oscillation, the presence of additional longitudinal harmonics, and the decayless regime of standing kink oscillations. Methods. We examine four examples of standing kink oscillations observed by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). We use forward modelling of the loop position and investigate the dependence on the model parameters using Bayesian inference and Markov Chain Monte Carlo (MCMC) sampling. Results. Our improvements to the physical model combined with the use of Bayesian inference and MCMC produce improved estimates of model parameters and their uncertainties. Calculation of the Bayes factor also allows us to compare the suitability of different physical models. We also use a new method based on spline interpolation of the zeroes of the oscillation to accurately describe the background trend of the oscillating loop. Conclusions. This powerful and robust method allows for accurate seismology of coronal loops, in particular the transverse density profile, and potentially reveals additional physical effects.

Aims. The transverse density structuring of coronal loops was recently calculated for the first time using the general damping profile for kink oscillations. This seismological method assumes a density profile with a linear transition region. We consider to what extent this density profile accounts for the observed intensity profile of the loop, and how the transverse intensity profile may be used to complement the seismological technique.Methods. We use isothermal and optically transparent approximations for which the intensity of extreme ultraviolet (EUV) emission is proportional to the square of the plasma density integrated along the line of sight. We consider four different models for the transverse density profile; the generalised Epstein profile, the step function, the linear transition region profile, and a Gaussian profile. The effect of the point spread function is included and Bayesian analysis is used for comparison of the models.Results. The two profiles with finite transitions are found to be preferable to the step function profile, which supports the interpretation of kink mode damping as being due to mode coupling. The estimate of the transition layer width using forward modelling is consistent with the seismological estimate.Conclusions. For wide loops, that is those observed with sufficiently high spatial resolution, this method can provide an independent estimate of density profile parameters for comparison with seismological estimates. In the ill-posed case of only one of the Gaussian or exponential damping regimes being observed, it may provide additional information to allow a seismological inversion to be performed. Alternatively, it may be used to obtain structuring information for loops that do not oscillate.

Context. Kink oscillations of solar coronal loops are frequently observed to be strongly damped. The damping can be explained by mode coupling on the condition that loops have a finite inhomogeneous layer between the higher density core and lower density background. The damping rate depends on the loop density contrast ratio and inhomogeneous layer width.\ud Aims. The theoretical description for mode coupling of kink waves has been extended to include the initial Gaussian damping regime in addition to the exponential asymptotic state. Observation of these damping regimes would provide information about the structuring of the coronal loop and so provide a seismological tool.\ud Methods. We consider three examples of standing kink oscillations observed by the Atmospheric Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO) for which the general damping profile (Gaussian and exponential regimes) can be fitted. Determining the Gaussian and exponential damping times allows us to perform seismological inversions for the loop density contrast ratio and the inhomogeneous layer width normalised to the loop radius. The layer width and loop minor radius are found separately by comparing the observed loop intensity profile with forward modelling based on our seismological results.\ud Results. The seismological method which allows the density contrast ratio and inhomogeneous layer width to be simultaneously determined from the kink mode damping profile has been applied to observational data for the first time. This allows the internal and external Alfvén speeds to be calculated, and estimates for the magnetic field strength can be dramatically improved using the given plasma density.\ud Conclusions. The kink mode damping rate can be used as a powerful diagnostic tool to determine the coronal loop density profile. This information can be used for further calculations such as the magnetic field strength or phase mixing rate.

Aims: \ud \ud Strongly damped standing and propagating kink oscillations are observed in the solar corona. This can be understood in terms of mode coupling, which causes the wave energy to be converted from the bulk transverse oscillation to localised, unresolved azimuthal motions. The damping rate can provide information about the loop structure, and theory predicts two possible damping profiles. \ud \ud Methods: \ud We used the recently compiled catalogue of decaying standing kink oscillations of coronal loops to search for examples with high spatial and temporal resolution and sufficient signal quality to allow the damping profile to be examined. The location of the loop axis was tracked, detrended, and fitted with sinusoidal oscillations with Gaussian and exponential damping profiles. \ud \ud Results: \ud Using the highest quality data currently available, we find that for the majority of our cases a Gaussian profile describes the damping behaviour at least as well as an exponential profile, which is consistent with the recently developed theory for the damping profile due to mode coupling.

Context. Rapidly propagating coronal EUV disturbances recently discovered in the solar corona are interpreted in terms of guided fast magnetoacoustic waves. Fast magnetoacoustic waves experience geometric dispersion in waveguides, which causes localised, impulsive perturbations to develop into quasi-periodic wave trains. Aims. We consider the formation of fast wave trains in a super-radially expanding coronal hole modelled by a magnetic funnel with a field-aligned density profile that is rarefied in comparison to the surrounding plasma. This kind of structure is typical of coronal holes, and it forms a fast magnetoacoustic anti-waveguide as a local maximum in the Alfven speed. Methods. We performed 2D MHD numerical simulations for impulsively generated perturbations to the system. Both sausage and kink perturbations are considered and the role of the density contrast ratio investigated. Results. The anti-waveguide funnel geometry refracts wave energy away from the structure. However, in this geometry the quasi-periodic fast wave trains are found to appear, too, and so can be associated with the observed rapidly propagating coronal EUV disturbances. The wave trains propagate along the external edge of the coronal hole. The fast wave trains generated in coronal holes exhibit less dispersive evolution than in the case of a dense waveguide. Conclusions. We conclude that an impulsive energy release localised in a coronal plasma inhomogeneity develops into a fast wave train for both kink and sausage disturbances and for both waveguide and anti-waveguide transverse plasma profiles.

Magnetohydrodynamic (MHD) processes are important for the transfer of energy over large scales in plasmas and so are essential to understanding most forms of dynamical activity in the solar atmosphere. The introduction of transverse structuring into models for the corona modifies the behavior of MHD waves through processes such as dispersion and mode coupling. Exploiting our understanding of MHD waves with the diagnostic tool of coronal seismology relies upon the development of sufficiently detailed models to account for all the features in observations. The development of realistic models appropriate for highly structured and dynamical plasmas is often beyond the domain of simple mathematical analysis and so numerical methods are employed. This paper reviews recent numerical results for seismology of the solar corona using MHD.

Context: Recent observations of the corona reveal ubiquitous transverse velocity perturbations that undergo strong damping as they propagate. These can be understood in terms of propagating kink waves that undergo mode coupling in inhomogeneous regions. Aims: The use of these propagating waves as a seismological tool for the investigation of the solar corona depends upon an accurate understanding of how the mode coupling behaviour is determined by local plasma parameters. Our previous work suggests the exponential spatial damping profile provides a poor description of the behaviour of strongly damped kink waves. We aim to investigate the spatial damping profile in detail and provide a guide to the approximations most suitable for performing seismological inversions. Methods: We propose a general spatial damping profile based on analytical results that accounts for the initial Gaussian stage of damped kink waves as well as the asymptotic exponential stage considered by previous authors. The applicability of this profile is demonstrated by a full parametric study of the relevant physical parameters. The implication of this profile for seismological inversions is investigated. Results: The Gaussian damping profile is found to be most suitable for application as a seismological tool for observations of oscillations in loops with a low density contrast. This profile also provides accurate estimates for data in which only a few wavelengths or periods are observed. Publisher PDF

We use images of high spatial and temporal resolution, obtained with the Rapid Oscillations in the Solar Atmosphere instrument at the Dunn Solar Telescope, to reveal how the generation of transverse waves in Type I spicules is a direct result of longitudinal oscillations occurring in the photosphere. Here we show how pressure oscillations, with periodicities in the range 130 - 440 s, manifest in small-scale photospheric magnetic bright points, and generate kink waves in the Sun's outer atmosphere with transverse velocities approaching the local sound speed. Through comparison of our observations with advanced two-dimensional magneto-hydrodynamic simulations, we provide evidence for how magnetoacoustic oscillations, generated at the solar surface, funnel upwards along Type I spicule structures, before undergoing longitudinal-to-transverse mode conversion into waves at twice the initial driving frequency. The resulting kink modes are visible in chromospheric plasma, with periodicities of 65 -220 s, and amplitudes often exceeding 400 km. A sausage mode oscillation also arises as a consequence of the photospheric driver, which is visible in both simulated and observational time series. We conclude that the mode conversion and period modification is a direct consequence of the 90 degree phase shift encompassing opposite sides of the photospheric driver. The chromospheric energy flux of these waves are estimated to be approximately 300,000 W/m^2, which indicates that they are sufficiently energetic to accelerate the solar wind and heat the localized corona to its multi-million degree temperatures., Accepted as an ApJ Letter, 4 figures

Aims: The effect of changing the attack angle for the interaction of a fast MHD wave with a 3D coronal loop is studied, to investigate to what extent the properties of the excited transverse kink mode oscillations of the loop depend on this angle. \ud \ud Methods: 3D numerical simulations are performed of the interaction of a fast MHD wave, generated by a pressure pulse, with a 3D coronal loop. The loop itself is modelled as a density enhancement (with a finite plasma beta) within a magnetic arcade. The initial pressure pulse has a width comparable to the loop diameter and is situated outside of the loop, at the same height as the loop apex. This height is kept fixed but the (horizontal) angle between the pressure pulse and the loop is varied. \ud \ud Results: We find that the global, transverse kink mode is efficiently excited for a range of attack angles and qualitatively in agreement with theoretical expectations. The period and damping time are found to be independent of the attack angle. For larger values of the attack angle, the global (longitudinal) slow wave is excited, whereas for intermediate values the second harmonic kink mode is also present.

Context. Flaring coronal loops are often observed in microwave and sometimes in soft X-rays, as extended structures of variable cross-section or width. The quasi-periodic pulsations associated with flaring coronal loops have been interpreted in terms of standing, fast sausage magnetoacoustic modes and suggested for the diagnostics of the magnetic field outside the oscillating loop.\ud Aims. We investigate the effects of a non-uniform cross-section on a coronal loop with respect to the standing sausage modes it supports.\ud Methods. Numerical simulations of standing sausage modes are performed for a straight, zero-β coronal loop with a varying crosssection. The global sausage mode and higher harmonics are considered, and simulations were performed for a range of density contrast ratios and loop divergence parameter. We consider modifications of the period of oscillation and the spatial profile of the standing\ud modes.\ud Results. As the loop divergence parameter increases, the period of the standing modes decreases. The fractional change period is independent of the density contrast ratio. The spatial profiles of the standing modes are modified by the cross-section inhomogeneity,but this effect is too weak to be observable by modern instruments. For the global sausage mode, the cross-section radius divergence\ud by a factor of 2 is found to cause the decrease in the period of about 5%.