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Funding: The research leading to these results has received funding from the UK Science and Technology Facilities Council (consolidated grant ST/N000609/1), the European Union Horizon 2020 research and innovation programme (grant agreement No. 647214). IDM received funding from the Research Council of Norway through its Centres of Excellence scheme, project number 262622. We consider the behaviour of Alfvén waves propagating in a medium with random density perturbations. The imposed density perturbations have a broadband 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. Publisher PDF

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 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

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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.

© 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

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., Accepted by ApJ

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.

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 P1/3P3 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.

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., Accepted for publication in ApJSS, 46 pages, 29 figures, 7 tables

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 perform numerical simulations of impulsively generated magnetic swirls in an isolated flux tube which is rooted in the solar photosphere. These swirls are triggered by an initial pulse in a horizontal component of the velocity. The initial pulse is launched either: (a) centrally, within the localized magnetic flux tube; or (b) off-central, in the ambient medium. The evolution and dynamics of the flux tube is described by three-dimensional, ideal magnetohydrodynamic equations. These equations are numerically solved to reveal that in case (a) dipole-like swirls associated with the fast magnetoacoustic kink and $m=1$ Alfv\'en waves are generated. In case (b), the fast magnetoacoustic kink and $m=0$ Alfv\'en modes are excited. In both these cases, the excited fast magnetoacoustic kink and Alfv\'en waves consist of similar flow pattern and magnetic shells are also generated with clockwise and counter-clockwise rotating plasma within them, which can be the proxy of dipole-shaped chromospheric swirls. The complex dynamics of vortices and wave perturbations reveals the channelling of sufficient amount of energy to fulfill energy losses in the chromosphere ($\sim$ 10$^{4}$ W m$^{-1}$) and in the corona ($\sim$ 10$^{2}$ W m$^{-1}$). Some of these numerical findings are reminiscent of signatures in recent observational data.

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.

We perform 3D numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. The presence of inhomogeneities in the density profile leads to the coupling of the driven kink mode to Alfvén modes by resonant absorption. The decay of the propagating kink wave as energy is transferred to the local Alfvén mode is in good agreement with a modified interpretation of the analytical expression derived for standing kink modes. This coupling may account for the damping of transverse velocity perturbation waves which have recently been observed to be ubiquitous in the solar corona.

This research has received funding from the Science and Technology Facilities Council (UK) through the consolidated grant ST/N000609/1 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program ( grant agreement No. 647214). This work is supported by the European Research Council under the SeismoSun Research Project No. 321141 (DJP). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 724326). This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by a BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/K00087X/1, DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. 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 MHD seismology and a potential test of coronal heating through the phase-mixing of Alfvén waves . In particular, recent observations show that standing oscill ations of loops often involve also higher harmonics, beside 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 hea ting of higher harmonics and what coronal heating observational signatures we may use to infer the presence of higher harmonic kink oscillations. Methods. We perform a set of non-ideal MHD simulations where the damping of the kink oscillation of a flux tube via mode coupling is modelled. Our MHD simulation parameters are based on the seismological inversion of an observation for which the first three harmonics are detected. We study the phase-mixing of Alfvén waves that leads to the deposition of heat in the system, and we apply the seismological inversion techniques to the MHD simulation output. Results. We find that the heating due to phase-mixing of the Alfvén wave s triggered by the damping of the kink oscillation is relatively small, however we can illustrate i) how the heating location drifts due to the subsequent damping of lower order harmonics. We also address the role of the 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 see ms not to provide enough energy to maintain the thermal structure of the solar corona even when multi-harmonics oscillations are included, and these oscillations play an inhibiting role in the development of smaller scale structures. Postprint

Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. However, there is ongoing discussion regarding their interpretation as kink or Alfven waves. To investigate the nature of transverse waves propagating in the solar corona and their potential for use as a coronal diagnostic in MHD seismology, we perform three-dimensional numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. We consider the cases of both a uniform medium and one with loop-like density structure and perform a parametric study for our structuring parameters. When density structuring is present, resonant absorption in inhomogeneous layers leads to the coupling of the kink mode to the Alfven mode. The decay of the propagating kink wave as energy is transferred to the local Alfven mode is in good agreement with a modified interpretation of the analysis of Ruderman & Roberts for standing kink modes. Numerical simulations support the most general interpretation of the observed loop oscillations as a coupling of the kink and Alfven modes. This coupling may account for the observed predominance of outward wave power in longer coronal loops since the observed damping length is comparable to our estimate based on an assumption of resonant absorption as the damping mechanism.

The magnetic field strength inside a model coronal loop is estimated using coronal seismology, to examine the reliability of magnetic field strengths derived from observed, transverse coronal loop oscillations. Three-dimensional numerical simulations of the interaction of an external pressure pulse with a coronal loop (modeled as a three-dimensional density enhancement inside a two-dimensional magnetic arcade) are analyzed and the observed properties of the excited transverse loop oscillations are used to derive the value of the local magnetic field strength, following the method of Nakariakov & Ofman. Due to the (unexpected) change in periodicity, the magnetic field derived from our observed oscillation is substantially different from the actual (input) magnetic field value (approximately 50%). Coronal seismology can derive useful information about the local magnetic field, but the combined effect of the loop curvature, the density ratio, and aspect ratio of the loop appears to be more important than previously expected.

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.

The effect of fine multilayered structuring on the resonant periods of global sausage (m=0), fast magnetoacoustic oscillations of coronal loops is studied for the straight magnetic slab model filled with a zero-β plasma. The slab is considered to have a one-dimensional inhomogeneity of the Alfven speed in the direction perpendicular to the axis of the slab, which coincides with the equilibrium, straight, magnetic field. This magnetic configuration models an oscillating coronal loop that is formed by a bundle of field-aligned, smooth, concentric, circular shells of variable density and width. The number of the layers (or shells), along with their density, width, and radial steepness, are randomly distributed. It is shown that the resonant properties of long-wavelength sausage standing modes of the slab are not sensitive to the details of fine structuring. It is concluded that the coronal seismological technique for the determination of the magnetic field strength by the period of global sausage oscillations is not sensitive to the fine multilayered or multishell structuring.

Aims. Analytical theory predicts the existence of trapped global (or fundamental) sausage fast magnetoacoustic modes in thick and dense coronal loops only, with the periods estimated as the ratio of double the loop length and the Alfvén speed outside the loop. We extend this study to the leaking regime, considering global sausage modes of long loops with small density contrasts.\ud Methods. Anti-symmetric fast magnetoacoustic perturbations (sausage, or m = 0 modes) of a low β plasma slab with the symmetric Epstein profile of plasma density are modelled numerically.\ud Results. It was found that long loops with sufficiently small density contrast can support global sausage leaky modes of detectable quality. The periods of the leaky modes are found to be approximately determined by the loop length and the external Alfvén speed. If the loop length can be estimated from imaging observations, the observed period of this mode provides us with the information about the Alfvén speed outside the loop. For typical flaring coronal loops, the estimated periods of the global sausage modes are\ud about 5−60 s.

The possibility of remote diagnostics of coronal structures with impulsively-generated short-period fast magnetoacoustic wave trains is demonstrated. An initially broad-band, aperiodic fast magnetoacoustic perturbation guided by a 1D plasma inhomogeneity develops into a quasi-periodic wave train with a well-pronounced frequency and amplitude modulation. The quasi-periodicity results from the geometrical dispersion of the modes, determined by the transverse profile of the loop, and hence contains information about the profile. Wavelet images of the wave train demonstrate that their typical spectral signature is of a “crazy tadpole’’ shape: a narrow spectrum tail precedes a broad-band head. The instantaneous period of the oscillations in the wave train decreases gradually with time, with a mean value of several seconds for typical coronal values. The period and the spectral amplitude evolution are determined by the steepness of the transverse density profile and the density contrast ratio in the loop, which offers a tool for estimation of the sub-resolution structuring of the corona.

Aims. We extend recently developed seismological methods to analyse oscillating loops which feature a large initial shift in the equilibrium position and investigate additional observational signatures related to the loop environment and oscillation driver.\ud \ud Methods. We model the motion of coronal loops as a kink oscillation damped by mode coupling, accounting for any change in loop length and the possible presence of parallel harmonics in addition to the fundamental mode. We apply our model to a loop which rapidly contracts due to a post-flare implosion (SOL2012-03-09) and a loop with a large lateral displacement (SOL2012-10-20).\ud \ud Results. The seismological method is used to calculate plasma parameters of the oscillating loops including the transverse density profile, magnetic field strength, and phase mixing timescale. For SOL2012-03-09 the period of oscillation has a linear correlation with the contracting motion and suggests the kink speed remains constant during the oscillation. The implosion excitation mechanism is found to be associated with an absence of additional parallel harmonics.\ud \ud Conclusions. The improved Bayesian analysis of the coronal loop motion allows for accurate seismology of plasma parameters, and the evolution of the period of oscillation compared with the background trend can be used to distinguish between loop motions in the plane of the loop and those perpendicular to it. The seismologically inferred kink speed and density contrast imply sub-Alfvénic (MA = 0.16 ± 0.03) propagation of the magnetic reconfiguration associated with the implosion, as opposed to triggering by a wave propagating at the Alfvén speed.

\ud Quasi-periodic rapidly propagating wave trains are frequently observed in extreme ultraviolet observations of the solar corona, or are inferred by the quasi-periodic modulation of radio emission. The dispersive nature of fast magnetohydrodynamic waves in coronal structures provides a robust mechanism to explain the detected quasi-periodic patterns. We perform 2D numerical simulations of impulsively generated wave trains in coronal plasma slabs and investigate how the behavior of the trapped and leaky components depend on the properties of the initial perturbation. For large amplitude compressive perturbations, the geometrical dispersion associated with the waveguide suppresses the nonlinear steepening for the trapped wave train. The wave train formed by the leaky components does not experience dispersion once it leaves the waveguide and so can steepen and form shocks. The mechanism we consider can lead to the formation of multiple shock fronts by a single, large amplitude, impulsive event and so can account for quasi-periodic features observed in radio spectra.\ud \ud

Aims. We carry out a statistical study of the inferred coronal loop cross-sectional density profiles using extreme ultraviolet (EUV) imaging data from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO).\ud Methods. We analysed 233 coronal loops observed during 2015/2016. We consider three models for the density profile; the step function (model S ), the linear transition region profile (model L), and a Gaussian profile (model G). Bayesian inference is used to compare the three corresponding forward modelled intensity profiles for each loop. These are constructed by integrating the square of the density from a cylindrical loop cross section along the line of sight, assuming an isothermal cross section, and applying the instrumental point spread function.\ud Results. Calculating the Bayes factors for comparisons between the models, it was found that in 47 % of cases there is very strong evidence for model L over model S and in 45 % of cases very strong evidence for model G over S . Using multiple permutations of the Bayes factor the favoured density profile for each loop was determined for multiple evidence thresholds. There were a similar number of cases where model L or G are favoured, showing evidence for inhomogeneous layers and constantly varying density cross sections, subject to our assumptions and simplifications.\ud Conclusions. For sufficiently well resolved loop threads with no visible substructure it has been shown that using Bayesian inference and the observed intensity profile we can distinguish between the proposed density profiles at a given AIA wavelength and spatial resolution. We have found very strong evidence for inhomogeneous layers, with model L being the most general, and a tendency towards thicker or even continuous layers.

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 consider a coronal loop kink oscillation observed by the atmospheric Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO) which demonstrates two strong spectral components. The period of the lower frequency component being approximately twice that of the shorter frequency component suggests the presence of harmonics.\ud \ud Methods. We examine the presence of two longitudinal harmonics by investigating the spatial dependence of the loop oscillation. The time-dependent displacement of the loop is measured at 15 locations along the loop axis. For each position the detrended displacement is fitted as the sum of two damped sinusoids, having periods P1 and P2, and a damping time τ. The shorter period component exhibits anti-phase oscillations in the loop legs.\ud \ud Results. We interpret the observation in terms of the first (global or fundamental) and second longitudinal harmonics of the standing kink mode. The strong excitation of the second harmonic appears connected to the preceding coronal mass ejection (CME) which displaced one of the loop legs. The oscillation parameters found are P1 “ 5.00 ˘ 0.62 minutes, P2 “ 2.20 ˘ 0.23 minutes, P1{2P2 “ 1.15 ˘ 0.22, and τ{P “ 3.35 ˘ 1.45.

Context. Magnetohydrodynamic waveguides such as dense coronal loops can support standing modes. The ratios of the periods of oscillations for different longitudinal harmonics depend on the dispersive nature of the waveguide and so may be used as a seismological tool to determine coronal parameters.\ud \ud Aims. We extend models of standing sausage modes in low β coronal loops to include the effects of loop curvature. The behaviour of standing sausage modes in this geometry is used to explain the properties of observed oscillations that cannot be accounted for using straight loop models.\ud \ud Methods. We perform 2D numerical simulations of an oscillating coronal loop, modelled as a dense slab embedded in a potential magnetic field. The loop is field-aligned and so experiences expansion with height in addition to being curved. Standing sausage modes are excited by compressive perturbations of the loop and their properties are studied.\ud \ud Results. The spatial profiles of standing sausage modes are found to be modified by the expanding loop geometry typical for flaring loops and modelled by a potential magnetic field in our simulations. Longitudinal harmonics of order n > 1 have anti-nodes that are shifted towards the loop apex and the amplitude of anti-nodes near the loop apex is smaller than those near the loop footpoints.\ud \ud Conclusions. We find that the observation of standing sausage modes by the Nobeyama Radioheliograph in a flaring coronal loop on 12 January 2000 is consistent with interpretation in terms of the global mode (n = 1) and third harmonic (n = 3). This interpretation accounts for the period ratio and spatial structure of the observed oscillations.

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%.

As waves and oscillations carry both energy and information, they have enormous potential as a plasma heating mechanism and, through seismology, to provide estimates of local plasma properties which are hard to obtain from direct measurements. Being sufficiently near to allow high-resolution observations, the atmosphere of the Sun forms a natural plasma laboratory. Recent observations have revealed that an abundance of waves and oscillations is present in the solar atmosphere, leading to a renewed interest in wave heating mechanisms. This short review paper gives an overview of recently observed transverse, propagating velocity perturbations in coronal loops. These ubiquitous perturbations are observed to undergo strong damping as they propagate. Using 3D numerical simulations of footpoint-driven transverse waves propagating in a coronal plasma with a cylindrical density structure, in combination with analytical modelling, it is demonstrated that the observed velocity perturbations can be understood in terms of coupling of different wave modes in the inhomogeneous boundaries of the loops. Mode coupling in the inhomogeneous boundary layers of the loops leads to the coupling of the transversal (kink) mode to the azimuthal (Alfven) mode, observed as the decay of the transverse kink oscillations. Both the numerical and analytical results show the spatial profile of the damped wave has a Gaussian shape to begin with, before switching to exponential decay at large heights. In addition, recent analysis of CoMP (Coronal Multi-channel Polarimeter) Doppler shift observations of large, off-limb, trans-equatorial loops shows that Fourier power at the apex appears to be higher in the high-frequency part of the spectrum than expected from theoretical models. This excess high-frequency FFT power could be tentative evidence for the onset of a cascade of the low-to-mid frequency waves into (Alfvenic) turbulence.

D.J.P. acknowledges financial support from STFC. I.D.M. acknowledges support of a Royal Society University Research Fellowship. The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/ 2007−2013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/solspanet). Context. Observations such as those by the Coronal Multi-Channel Polarimeter (CoMP) have revealed that broadband kink oscillations are ubiquitous in the solar corona. Aims. We consider footpoint-driven kink waves propagating in a low β coronal plasma with a cylindrical density structure. We investigate the excitation and damping of propagating kink waves by a broadband driver, including the effects of different spatial profiles for the driver. Methods. We employ a general spatial damping profile in which the initial stage of the damping envelope is approximated by a Gaussian profile and the asymptotic state by an exponential one. We develop a method of accounting for the presence of these different damping regimes and test it using data from numerical simulations. Results. Strongly damped oscillations in low density coronal loops are more accurately described by a Gaussian spatial damping profile than an exponential profile. The consequences for coronal seismology are investigated and applied to observational data for the ubiquitous broadband waves observed by CoMP. Current data cannot distinguish between the exponential and Gaussian profiles because of the levels of noise. We demonstrate the importance of the spatial profile of the driver on the resulting damping profile. Furthermore, we show that a small-scale turbulent driver is inefficient at exciting propagating kink waves Publisher PDF

Magnetohydrodynamic waves interact with structured plasmas and reveal the internal magnetic and thermal structures therein, thereby having seismological applications in the solar atmosphere. We investigate the evolution of fast magnetoacoustic pulses in randomly structured plasmas, in the context of large-scale propagating waves in the solar atmosphere. We perform one dimensional numerical simulations of fast wave pulses propagating perpendicular to a constant magnetic field in a low-$\beta$ plasma with a random density profile across the field. Both linear and nonlinear regimes are considered. We study how the evolution of the pulse amplitude and width depends on their initial values and the parameters of the random structuring. A randomly structured plasma acts as a dispersive medium for a fast magnetoacoustic pulse, causing amplitude attenuation and broadening of the pulse width. After the passage of the main pulse, secondary propagating and standing fast waves appear in the plasma. Width evolution of both linear and nonlinear pulses can be well approximated by linear functions; however, narrow pulses may have zero or negative broadening. This arises because a narrow pulse is prone to splitting, while a broad pulse usually deviates less from their initial Gaussian shape and form ripple structures on top of the main pulse. A linear pulse decays at almost a constant rate, while a nonlinear pulse decays exponentially. A pulse interacts most efficiently with a random medium which has a correlation length of about half of its initial pulse width. The development of a detailed model of a fast MHD pulse propagating in highly structured medium substantiates the interpretation of EIT waves as fast magnetoacoustic waves. Evolution of a fast pulse provides us with a novel method to diagnose the sub-resolution filamentation of the solar atmosphere., Comment: 25 pages, 8 figures

Context. We present a new event of quasi-periodic wave trains observed in EUV wavebands that rapidly propagate away from an active region after a flare.\ud Aims. We measured the parameters of a wave train observed on 7 December 2013 after an M1.2 flare, such as the phase speeds, periods and wavelengths, in relationship to the local coronal environment and the energy sources.\ud Methods. We compared our observations with a numerical simulation of fast magnetoacoustic waves that undergo dispersive evolution and leakage in a coronal loop embedded in a potential magnetic field.\ud Results. The wave train is observed to propagate as several arc-shaped intensity disturbances for almost half an hour, with a speed greater than 1000 km s-1 and a period of about 1 min. The wave train followed two different patterns of propagation, in accordance with the magnetic structure of the active region. The oscillatory signal is found to be of high-quality, i.e. there is a large number (10 or more) of subsequent wave fronts observed. The observations are found to be consistent with the numerical simulation of a fast wave train generated by a localised impulsive energy release.\ud Conclusions. Transverse structuring in the corona can efficiently create and guide high-quality quasi-periodic propagating fast wave trains.

I.D.M. acknowledges support from a Royal Society University Research Fellowship. The computational work for this paper was carried out at the joint STFC and SFC (SRIF)-fundedclusterattheUniversityofStAndrews(UK). The research leading to these results has also received funding from the European Commissions Seventh Framework Programme (FP7/2007-2013) under the grant agreement SOLSPANET (project No. 269299;www.solspanet.eu/solspanet). So far, the straight flux tube model proposed by Edwin & Roberts is the most commonly used tool in practical coronal seismology, in particular, to infer values of the (coronal) magnetic field from observed, standing kink mode oscillations. In this paper, we compare the period predicted by this basic model with three-dimensional (3D) numerical simulations of standing kink mode oscillations, as the period is a crucial parameter in the seismological inversion to determine the magnetic field. We perform numerical simulations of standing kink modes in both straight and curved 3D coronal loops and consider excitation by internal and external drivers. The period of oscillation for the displacement of dense coronal loops is determined by the loop length and the kink speed, in agreement with the estimate based on analytical theory for straight flux tubes. For curved coronal loops embedded in a magnetic arcade and excited by an external driver, a secondary mode with a period determined by the loop length and external Alfvén speed is also present. When a low number of oscillations is considered, these two periods can result in a single, non-resolved (broad) peak in the power spectrum, particularly for low values of the density contrast for which the two periods will be relatively similar. In that case (and for this particular geometry), the presence of this additional mode would lead to ambiguous seismological estimates of the magnetic field strength. Publisher PDF

Context: Fast magneto-acoustic waves are highly dispersive in waveguides, so they can generate quasi-periodic wave trains if a localised, impulsive driver is applied. Such wave trains have been observed in the solar corona and may be of use as a seismological tool since they depend upon the plasma structuring perpendicular to the direction of propagation.\ud \ud Aims. We extend existing models of magnetoacoustic waveguides to consider the effects of an expanding magnetic field. The funnel geometry employed includes a field-aligned density structure.\ud \ud Methods: We performed 2D numerical simulations of impulsively generated fast magneto-acoustic perturbations. The effects of the density contrast ratio, density stratification, and spectral profile of the driver upon the excited wave trains were investigated.\ud \ud Results: The density structure acts as a dispersive waveguide for fast magneto-acoustic waves and generates a quasi-periodic wave train similar to previous models. The funnel geometry leads to generating additional wave trains that propagate outside the density structure. These newly discovered wave trains are formed by the leakage of transverse perturbations, but they propagate upwards owing to the refraction caused by the magnetic funnel.\ud \ud Conclusions: The results of our funnel model may be applicable to wave trains observed propagating in the solar corona. They demonstrate similar properties to those found in our simulations.

We present a Bayesian seismology inversion technique for propagating magnetohydrodynamic (MHD) transverse waves observed in coronal waveguides. The technique uses theoretical predictions for the spatial damping of propagating kink waves in transversely inhomogeneous coronal waveguides. It combines wave amplitude damping length scales along the waveguide with theoretical results for resonantly damped propagating kink waves to infer the plasma density variation across the oscillating structures. Provided the spatial dependence of the velocity amplitude along the propagation direction is measured and the existence of two di erent damping regimes is identified, the technique would enable us to fully constrain the transverse density structuring, providing estimates for the density contrast and its transverse inhomogeneity length scale.

Aims. To investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented. Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple di.erential-integral equation was used. Approximate analytical solutions to this equation are presented. Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(-z2 /L2 g))/2, where L2g = 16/ǫκ2 k2 . Although the actual value of 16 in Lg depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough. Publisher PDF

Aims. We investigate the damping process for propagating transverse velocity oscillations, observed to be ubiquitous in the solar corona, due to mode coupling. Methods. We perform 3D numerical simulations of footpoint-driven transverse waves propagating in a low β coronal plasma with a cylindrical density structure. Mode coupling in an inhomogeneous layer leads to the coupling of the kink mode to the Alfven mode, observed as the decay of the transverse kink oscillations. Results. We consider the spatial damping profile and find a Gaussian damping profile of the form exp(−z/Lg) to be the most congruent with our numerical data, rather than the exponential damping profile of the form exp(−z/Ld) used in normal mode analysis. Our results highlight that the nature of the driver itself will have a substantial influence on observed propagating kink waves. Conclusions. Our study suggests that this modified damping profile should be taken into account when using coronal seismology to infer local plasma properties from observed damped oscillations.