We review novel data analysis techniques developed or adapted for the field of coronal seismology. We focus on methods from the last ten years that were developed for extreme ultraviolet (EUV) imaging observations of the solar corona, as well as for light curves from radio and X-ray. The review covers methods for the analysis of transverse and longitudinal waves; spectral analysis of oscillatory signals in time series; automated detection and processing of large data sets; empirical mode decomposition; motion magnification; and reliable detection, including the most common pitfalls causing artefacts and false detections. We also consider techniques for the detailed investigation of MHD waves and seismological inference of physical parameters of the coronal plasma, including restoration of the three-dimensional geometry of oscillating coronal loops, forward modelling and Bayesian parameter inference.

We consider the use of propagating kink waves, such as those observed by the Coronal Multi-channel Polarimeter, as a diagnostic technique. The transverse structuring of the plasma may be inferred by the frequency-dependent wave damping, which is attributed to resonant absorption. We include the effect of reflection of waves at the loop footpoints, which leads to the asymmetry parameter, describing the ratio of driven wave power at the footpoints becoming weakly constrained. The classical model of resonant absorption based on an exponential damping profile significantly overestimates the damping rate in coronal loops with low density contrast ratios. The use of the exponential profile in an analysis of observations therefore leads to underestimates for the density contrast ratio and associated parameters such as the heating rate following phase mixing.

ABSTRACT We consider the behaviour of Alfvén waves propagating in a medium with random density perturbations. The imposed density perturbations have a broad-band spectrum and their characteristic spatial scale may be defined according to the peak in the spectrum. The interaction of the boundary driven Alfvén waves with the medium generates reflections most efficiently when their wavelength is comparable to the spatial scale of the density perturbations. For our monotonic driver, this leads to the generation of quasi-periodic oscillations. The periods of oscillation of the propagating Alfvén waves is no longer only associated with the driver. Additional periodicities may be associated with one or more characteristic spatial scales in the density profile, or with beating between other spectral components. Multiple wave reflections cause oscillatory power to be retained at low altitudes, increasing opportunities to contribute to heating at those locations.

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 IDL code designed to facilitate the comparison of a user-supplied model with data. Bayesian inference allows prior information to be taken into account. The use of Markov Chain Monte Carlo 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.

Kink oscillations of coronal loops, i.e., standing kink waves, is one of the most studied dynamic phenomena in the solar corona. The oscillations are excited by impulsive energy releases, such as low coronal eruptions. Typical periods of the oscillations are from a few to several minutes, and are found to increase linearly with the increase in the major radius of the oscillating loops. It clearly demonstrates that kink oscillations are natural modes of the loops, and can be described as standing fast magnetoacoustic waves with the wavelength determined by the length of the loop. Kink oscillations are observed in two different regimes. In the rapidly decaying regime, the apparent displacement amplitude reaches several minor radii of the loop. The damping time which is about several oscillation periods decreases with the increase in the oscillation amplitude, suggesting a nonlinear nature of the damping. In the decayless regime, the amplitudes are smaller than a minor radius, and the driver is still debated. The review summarises major findings obtained during the last decade, and covers both observational and theoretical results. Observational results include creation and analysis of comprehensive catalogues of the oscillation events, and detection of kink oscillations with imaging and spectral instruments in the EUV and microwave bands. Theoretical results include various approaches to modelling in terms of the magnetohydrodynamic wave theory. Properties of kink oscillations are found to depend on parameters of the oscillating loop, such as the magnetic twist, stratification, steady flows, temperature variations and so on, which make kink oscillations a natural probe of these parameters by the method of magnetohydrodynamic seismology.

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 Alfvén 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.

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 characterized 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 characterized 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 modeling methods. The disparity between methods may be indicative of non-linear evolution of coronal loops. The influence on forward modeling estimates could also account for previous observational evidence favoring loops having wider inhomogeneous layers.

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.

Quasi-periodic pulsations (QPPs) are routinely observed in a range of wavelengths during flares, but in most cases the mechanism responsible is unknown. We present a method to detect and characterize QPPs in time series such as light curves for solar or stellar flares based on forward modeling and Bayesian analysis. We include models for QPPs as oscillations with finite lifetimes and nonmonotonic amplitude modulation, such as wave trains formed by dispersive evolution in structured plasmas. By quantitatively comparing different models using Bayes factors, we characterize the QPPs according to five properties: sinusoidal or nonsinusoidal, finite or indefinite duration, symmetric or asymmetric perturbations, monotonic or nonmonotonic amplitude modulation, and constant or varying period of oscillation. We demonstrate our method and show examples of these five characteristics by analyzing QPPs in white-light stellar flares observed by the Kepler space telescope. Different combinations of properties may be able to identify particular physical mechanisms and so improve our understanding of QPPs and allow their use as seismological diagnostics. We propose that three observational classes of QPPs can be distinguished: decaying harmonic oscillations, finite wave trains, and nonsinusoidal pulsations.

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.

Aims. An observation of a coronal loop standing kink mode is analysed to search for higher harmonics, aiming to reveal the relation between different harmonics’ quality factors. Methods. Observations of a coronal loop were taken by the Atmospheric Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO). The loop’s axis was tracked at many spatial positions along the loop to generate time series data. Results. The distribution of spectral power of the oscillatory transverse displacements throughout the loop reveals the presence of two harmonics, a fundamental at a period of ∼8 min and its third harmonic at ∼2.6 min. The node of the third harmonic is seen at approximately a third of the way along the length of the loop, and cross correlations between the oscillatory motion on opposing sides of the node show a change in phase behaviour. The ratio of periods P 1 /3 P 3 was found to be ∼0.87, indicating a non-uniform distribution of kink speed through the loop. The quality factor for the fundamental mode of oscillation was measured to be ∼3.4. The quality factor of the third harmonic was measured for each spatial location and, where data was reliable, yielded a value of ∼3.6. For all locations, the quality factors for the two harmonics were found to agree within error as expected from 1d resonant absorption theory. This is the first time a measurement of the signal quality for a higher harmonic of a kink oscillation has been reported with spatially resolved data.

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 modeling 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 behavior for thin inhomogeneous layers. Previous numerical studies have demonstrated that this thin boundary approximation produces poor estimates of the damping behavior in loops with wider inhomogeneous layers. Both the seismological and forward modeling approaches suggest loops have a range of layer widths and so there is a need for a description of the damping behavior 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.

Context . Frequent observations of quasi-periodic rapidly-propagating wave trains in coronal structures have been made in the last decade. The dispersive evolution of fast magnetohydrodynamic waves propagating in coronal waveguides can provide a physical interpretation for many of these observations. Aims . Previous studies have considered the generation of fast wave trains by impulsive drivers which deposit energy instantaneously. The signatures of dispersively formed wave trains must depend on the temporal nature of the driver. We investigate the effect of varying the temporal width of the driving perturbation. Methods . 2D magnetohydrodynamic numerical simulations of impulsively generated wave trains in a guiding field-aligned density enhancement were performed with the novel addition of a time-dependant driver. Results . The final spatial and spectral signatures of the guided wave trains are found to depend strongly on the temporal duration of the initial perturbation. In particular, the wavelength (or frequency) of highest spectral amplitude is found to increase (decrease) with increasing temporal duration, whereas the spectral width decreases. Additionally, the efficiency of generation of fast wave trains is found to decrease strongly with increasing temporal width of the driver, with a cut-off at twice the internal Alfvén crossing time.

We present a model for the intensity of optically thin 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 SDO/AIA 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 (DEM) 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.

Quasi-periodic pulsations (QPPs) appear to be a common feature observed in the light curves of both solar and stellar flares. However, their quasi-periodic nature, along with the fact that they can be small in amplitude and short-lived, makes QPPs difficult to unequivocally detect. In this paper, we test the strengths and limitations of state-of-the-art methods for detecting QPPs using a series of hare-and-hounds exercises. The hare simulated a set of flares, both with and without QPPs of a variety of forms, while the hounds attempted to detect QPPs in blind tests. We use the results of these exercises to create a blueprint for anyone who wishes to detect QPPs in real solar and stellar data. We present eight clear recommendations to be kept in mind for future QPP detections, with the plethora of solar and stellar flare data from new and future satellites. These recommendations address the key pitfalls in QPP detection, including detrending, trimming data, accounting for colored noise, detecting stationary-period QPPs, detecting QPPs with nonstationary periods, and ensuring that detections are robust and false detections are minimized. We find that QPPs can be detected reliably and robustly by a variety of methods, which are clearly identified and described, if the appropriate care and due diligence are taken.

Recent observations with the Atmospheric Imaging Assembly (AIA) instrument on the SDO spacecraft have revealed the existence of decayless coronal kink oscillations. These transverse oscillations are not connected to any external phenomena like flares or coronal mass ejections, and show significantly lower amplitudes than the externally excited decaying oscillations. Numerical studies have managed to reproduce such decayless oscillations in the form of footpoint driven standing waves in coronal loops, and to treat them as a possible mechanism for wave heating of the solar corona. Our aim is to investigate the correlation between the observed amplitudes of the oscillations and input the energy flux from different drivers. We perform 3D MHD simulations in single, straight, density-enhanced coronal flux tubes for different drivers, in the presence of gravity. Synthetic images at different spectral lines are constructed with the use of the FoMo code. The development of the Kelvin-Helmholtz instability leads to mixing of plasma between the flux tube and the hot corona. Once the KHI is fully developed, the amplitudes of the decayless oscillations show only a weak correlation with the driver strength. We find that low amplitude decayless kink oscillations may correspond to significant energy fluxes of the order of the radiative losses for the Quiet Sun. A clear correlation between the input energy flux and the observed amplitudes from our synthetic imaging data cannot be established. Stronger drivers lead to higher vales of the line width estimated energy fluxes. Finally, estimations of the energy fluxes by spectroscopic data are affected by the LOS angle, favoring combined analysis of imaging and spectroscopic data for single oscillating loops.

Context . Kink oscillations of a coronal loop are observed and studied in detail because they provide a unique probe into the structure of coronal loops through magnetohydrodynamics (MHD) seismology and a potential test of coronal heating through the phase mixing of Alfvén waves. In particular, recent observations show that standing oscillations of loops often involve higher harmonics in addition to the fundamental mode. The damping of these kink oscillations is explained by mode coupling with Alfvén waves. Aims . We investigate the consequences for wave-based coronal heating of higher harmonics and which coronal heating observational signatures we may use to infer the presence of higher harmonic kink oscillations. Methods . We performed a set of non-ideal MHD simulations in which we modelled the damping of the kink oscillation of a flux tube via mode coupling. We based our MHD simulation parameters on the seismological inversion of an observation for which the first three harmonics are detected. We studied the phase mixing of Alfvén waves, which leads to the deposition of heat in the system, and we applied seismological inversion techniques to the MHD simulation output. Results . We find that the heating due to phase mixing of Alfvén waves triggered by the damping of kink oscillation is relatively small. We can however illustrate how the heating location drifts from subsequent damping of lower order harmonics. We also address the role of higher order harmonics and the width of the boundary shell in the energy deposition. Conclusions . We conclude that the coronal heating due to phase mixing does not seem to provide enough energy to maintain the thermal structure of the solar corona even when multi-harmonic oscillations are included; these oscillations play an inhibiting role in the development of smaller scale structures.

Recent developments in the observation and modelling of kink oscillations of coronal loops have led to heightened interest over the last few years. The modification of the Transverse Density Profile (TDP) of oscillating coronal loops by non-linear effects, in particular the Kelvin-Helmholtz Instability (KHI), is investigated. How this evolution may be detected is established, in particular, when the KHI vortices may not be observed directly. A model for the loop's TDP is used which includes a finite inhomogeneous layer and homogeneous core, with a linear transition between them. The evolution of the loop's transverse intensity profile from numerical simulations of kink oscillations is analysed. Bayesian inference and forward modelling techniques are applied to infer the evolution of the TDP from the intensity profiles, in a manner which may be applied to observations. The strongest observational evidence for the development of the KHI is found to be a widening of the loop's inhomogeneous layer, which may be inferred for sufficiently well resolved loops, i.e $>$ 15 data points across the loop. The main signatures when observing the core of the loop (for this specific loop model) during the oscillation are: a widening inhomogeneous layer, decreasing intensity, an unchanged radius, and visible fine transverse structuring when the resolution is sufficient. The appearance of these signatures are delayed for loops with wider inhomogeneous layers, and quicker for loops oscillating at higher amplitudes. These cases should also result in stronger observational signatures, with visible transverse structuring appearing for wide loops observed at SDO/AIA resolution.

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) similar to 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.