# Transverse waves in the solar corona

Transverse waves in the solar corona have been discovered in 1998 using the TRansition And Coronal Explorer (TRACE) as displacement oscillations of loops, excited by a nearby flare and/or coronal mass ejection (Aschwanden et al., 1999; Nakariakov et al., 1999). Since then, many examples of transverse waves have been reported. Transverse waves in loops have been interpreted as the magnetohydrodynamic kink mode. Transverse waves have received much attention because they provide information about coronal loops: the average Alfven speed and magnetic field from equating the measured phase speed to the kink speed (Nakariakov & Ofman, 2001), the transverse structuring from studying the damping mechanism, i.e. resonant mode conversion, wave leakage (e.g. Ruderman & Roberts, 2002; Brady & Arber, 2005) and the longitudinal structuring from studying the period ratios of observed harmonics (Andries et al., 2005). They provide a means of testing wave theories that have been proposed to explain coronal heating. Also, as these waves occur near solar eruptions, they provide an insight into the dynamics of the solar corona.

I have contributed to our understanding of these waves as follows:

Spatial seismology of a large coronal loop arcade from TRACE and EIT observations of its transverse oscillations (2010)
First stereoscopic study of a transverse loop oscillation using STEREO (2009)
Discovery of intensity oscillations associated with variations in the line-of-sight column depth of a transversely oscillating loop (2009)
• Discovery of the role of transverse waves in modulation of long-period pulsations in flares (2005)
Clarification of the interpretation of transverse waves in coronal structures in terms of kink rather than Alfven waves (2008)
Discovery of multiple harmonics in transverse loop oscillations and consequent study of further observational examples (2004,2007)
Unique observational study of sunward propagating transverse waves above a flaring site (2005)
Modelling of the role of lateral leakage in a curved loop (2006)

### Transverse waves in loops have been interpreted as the magnetohydrodynamic kink mode

The basic model of MHD waves in coronal loops is based upon a low plasma-β plasma in a straight cylinder with magnetic field in the z-direction. We can visualise the normal modes for such a model. The transverse wave corresponds to a wave with an azimuthal mode number m=1 and phase speed above the (lowest) internal Alfven speed. In the limit of a thin loop in the zero plasma-β limit, the phase speed of the wave corresponds to the kink speed CK given by

$C_\mathrm{K} \,=\, \sqrt{\frac{2}{1 + \rho_e/\rho_i}}\,V_\mathrm{A}$

where ρei is the ratio of external and internal plasma densities and VA is the internal Alfven speed. The fundamental standing kink mode can be visualised as follows:

### Scalings between transverse loop oscillation parameters

 This applet allows to look at the scaling between various oscillation parameters. For example, the scaling exponent between period and damping time contains information about the damping mechanism (power index 1 indicates resonant mode conversion). Of course, physical processes will depend on more than two variables. Therefore, the scalings here are only a first step in representing the data. The measurements are taken from the following papers: Aschwanden et al. (1999) Nakariakov et al. (1999) Aschwanden et al. (2002) Wang & Solanki (2004) Verwichte et al. (2004) Hori et al. (2007) Van Doorsselaere et al. (2007) De Moortel & Brady (2008) Van Doorsselaere et al. (2009) Verwichte et al. (2009) Verwichte et al. (2010) Mrozek (2011)

#### Recent reviews on transverse waves

Verwichte, E., Foullon, C., Van Doorsselaere, T., Smith, H.M. and Nakariakov, V.M.: 2009, Coronal seismology using transverse loop oscillations, Plasma Phys. Control. Fusion 51, 124019
Nakariakov, V.M. and Verwichte, E.: 2005, Coronal Waves and Oscillations, Living Rev. Solar Phys. 3, 1-63.
Van Doorsselaere, T., Verwichte, E. and Terradas, J.: 2009, The effect of loop curvature on coronal loop kink oscillations, Space Sci. Rev., 149, 299
Andries, J., Van Doorsselaere. T., Roberts, B., Verth, G., Verwichte, E. and Erdélyi, R.: 2009, Coronal Seismology by Means of Kink Oscillation Overtones, Space Sci. Rev. 149, 3

### Spatial seismology of a large coronal loop arcade from TRACE and EIT observations of its transverse oscillations

 Top: South-West quadrant of an EIT image, showing an active region near the West limb. The used TRACE field of view is shown as a black rectangular box. The paths of four loops are indicated. Bottom: TRACE field of view of the northern leg of the active region loop arcade. The parallel dashed and solid curves show the central path and width of the loop data cut used, respectively. Top: transverse cut at s = 187 Mm showing the loop intensity as a function of time and transverse coordinate, x. The automatically determined time series of the transverse location loop axis is shown. Middle: Transverse location loop axis as a function of time. The dashed line is a quadratic trend fitted to the time series in the time range indicated by the two vertical dashed lines. Bottom: The loop displacement time series, ξ(t), as a function of time. The thick curve is the fitted damped oscillation. We present a study of transverse loop oscillations in a large coronal loop arcade, using ob- servations from the Transition Region And Coronal Explorer (TRACE) and Extreme-ultraviolet Imaging Telescope (EIT). For the first time we reveal the presence of long-period transverse oscillations with periods between 24 minutes and 3 hours. One loop bundle, 690 Mm long and with an oscillation period of 40 min., is analysed in detail and its oscillation characteris- tics are determined in an automated manner. The oscillation quality factor is similar to what has been found earlier for oscillations in much shorter loops. This indicates that the damping mechanism of transverse loop oscillations is independent of loop length or period. The dis- placement profile along the whole length of the oscillating loop is determined for the first time and consistently between TRACE and EIT. By comparing the observed profile with models of the three-dimensional geometry of the equilibrium and perturbed loop, we test the effect of lon- gitudinal structuring (spatial seismology) and find that the observations cannot unambigiously distinguish between structuring and non-planarity of the equilibrium loop. Associated intensity variations with a similar periodicity are explained in terms of variations in the line of sight col- umn depth. Also, we report intensity oscillations at the loop foot point, which are in anti-phase with respect to the intensity oscillations in the loop body. Lastly, this observation offers the first opportunity to use the transverse oscillations of the arcade to model the Alfven speed profile in the global corona. This work has been published as Verwichte, E., Foullon, C. and Van Doorsselaere, T.: 2010, Spatial Seismology of a Large Coronal Loop Arcade from TRACE & EIT Observations of its Transverse Oscillations, Astrophys. J. 717, 458 Transverse displacement amplitude, ξ0, as a function of distance along the loop, s/L. The solid line is the measurement using EIT. The grey area indicates an error of one EIT pixel either side of the curve. The thick solid line are the measurements using TRACE. a) The long-dashed line is the (projected) displacement amplitude for a horizontally polarised fundamental mode which is constructed by inclining the unperturbed loop by a constant angle of Δθ = -10°. The dotted lines either side of this curve represent the solutions for Δθ = -9° (lower curve) and Δθ = -11° (upper curve). b) Value of f3 in a model that optimally fits the observed loop displacement, as a function of loop inclination θ0. The value of θ0 = -43° is indicated by a vertical dashed line. c) Curves using a non-planar loop for three values of ρ are shown. Alfven speed profile as a function of loop length. The measured Alfven speed from four loops are shown. The error bars take into account uncertainties in loop length, oscillation period and loop density contrast.

### Seismology of a large solar coronal loop from EUVI/STEREO observations of its transverse oscillation

 Difference images for three times and for STEREO A (left) and STEREO B (right). The difference image is contructed as the difference between the image at the given time and the image five minutes earlier. We have performed the first analysis of a transverse loop oscillation observed by both STEREO spacecraft, for an event on the 27th of June, 2007 as seen by the Extreme UltraViolet Imager (EUVI). The three-dimensional loop geometry is determined using a three-dimensional reconstruction with a semi-circular loop model, which allows for an accurate measurement of the loop length. The plane of wave polarisation is found from comparison with a simulated loop model and shows that the oscillation is a fundamental horizontally polarised fast magnetoacoustic kink mode. The oscillation is characterised using an automated method and the results from both spacecraft are found to match closely. The oscillation period is 630±30 s and the damping time is 1000±300 s. Also, clear intensity variations associated with the transverse loop oscillations are reported for the first time. They are shown to be caused by the effect of line-of-sight integration. The Alfv´en speed and coronal magnetic field derived using coronal seismology are discussed. This study shows that EUVI/STEREO observations achieve an adequate accuracy for studying long-period, large amplitude transverse loop oscillations. This work has been published as Verwichte, E., Aschwanden, M.J., Van Doorsselaere, T., Foullon, C. and Nakariakov, V.M.: 2009, Seismology of a large solar coronal loop from EUVI/STEREO observations of its transverse oscillation, Astrophys. J. 698, 397-404 The coronal loop as seen by both STEREO spacecraft with a 3d reconstruction of the loop shown on the right The top figure shows the intensity profiles of the selected path with STEREO B as a function of time and projected distance along the path. The solid line is the centre position of the fitted Gaussian profile. The dashed lines indicate the error. The vertical long-dashed line shows the time when the loops are first perturbed. The bottom figure shows the detrended loop top displacement as a function of time with a fitted damped oscillation superimposed. The time for STEREO B has been corrected for light travel time w.r.t. STEREO A.

### The interpretation of transverse waves in the corona in terms of kink instead of Alfven waves

 A series of high-profile papers announced the discovery of Alfven waves in the corona based upon observational evidence of transverse periodic motions. These observations are really exciting and represent a step-up in coronal wave physics, However, we show that in the corona where the plasma is structured into coronal loops, the wave that satisfies the observations is in fact the kink mode and not the Alfven wave. We can see the kink mode as the equivalent of the shear Alfven wave for a structured medium. In a coronal loop, the Alfven wave appears as a torsional mode where the motions are purely in the azimuthal direction amd hence does not perturb the loop axis transversely. Also, Alfven waves do not show coherent behavior, being subject to phase mixing; i.e., Alfvén waves associated with neighboring flux surfaces propagate with different speeds and rapidly get out of phase. This affects their observability. In contrast, kink modes are genuinely collective modes. Furthermore, the kink and Alfvén waves fundamentally differ in many respects; e.g., their phase speeds, propagation, compressibility, nonlinear evolution, damping mechanisms, and excitation are essentially different. This work has been published as Van Doorsselaere, T., Nakariakov, V.M. and Verwichte, E.: 2008, Detection of Waves in the Solar Corona: Kink or Alfvén? , Astrophys. J. Lett. 676, L73-L75. Van Doorsselaere, T., Brady C.S., Nakariakov, V.M. and Verwichte, E.: 2008, Seismological demonstration of perpendicular density structuring in the solar corona, Astron. Astrophys. 491, L9.

### Coronal loop seismology using multiple transverse loop oscillation harmonics

 In this study lead by Dr Tom Van Doorsselaere, TRACE observations (23/11/1998 06:35:57-06:48:43UT) in the 171 Å bandpass of an active region are studied. Coronal loop oscillations are observed after a violent disruption of the equilibrium. The oscillation properties are studied to give seismological estimates of physical quantities, such as the density scale height. A loop segment is traced during the oscillation, and the resulting time series is analysed for periodicities. In the loop segment displacement, two periods are found: 435.6 ± 4.5 s and 242.7 ± 6.4 s, consistent with the periods of the fundamental and 2nd harmonic fast kink oscillation. The small uncertainties allow us to estimate the density scale height in the loop to be 109 Mm, which is about double the estimated hydrostatical value of 50 Mm. Because a loop segment is traced, the amplitude dependence along the loop is found for each of these oscillations. The obtained spatial information is used as a seismological tool to give details about the geometry of the observed loop. This work has been published as Van Doorsselaere, T., Nakariakov, V.M. and Verwichte, E.: 2007, Coronal loop seismology using multiple transverse loop oscillation harmonics, Astron. Astrophys., 473, 959-966.

### Fast magnetoacoustic waves in curved coronal loops

The three A&A papers Fast magnetoacoustic waves in curved coronal loops I. Trapped and leaky modes, II. Tunneling modes and Seismology of curved coronal loops with vertically polarised transverse oscillations by Verwichte, Foullon & Nakariakov present work on vertically polarised fast magnetoacoustic waves in a curved coronal loop.

The loop is modeled as a semi-circular magnetic slab in the zero plasma-beta limit. The governing equations for linear waves have been derived. We show that the wave mode behaviour depends on the slope of the equilibrium density profile, which is modeled as a piece-wise continuous power law curve of index alpha. For all profiles, except for alpha=-4, wave modes are not trapped in the loop and leak out into the external medium through wave tunneling.

The particular case of alpha=-4, which corresponds to a constant Alfven frequency profile (linearly increasing Alfven speed profile), is examined in more detail in Paper I as this is the only model that can support trapped wave modes. The results are compared with a straight slab model and similar behaviour is found. Coupling between sausage and kink wave modes has not been found in the model.

In Paper II the profiles with alpha not equal to -4 have been explored. The waves are shown to be all damped due to lateral leakage. It is demonstrated that waves either leak straight out into the external medium or have to overcome an evanescent barrier, which is linked to wave tunneling. The wave solutions consist of alternating vertically polarised kink and sausage branches. Fast kink oscillations may have a non-zero density perturbation when averaged across the loop. The calculated damping rate of fast magnetoacoustic kink oscillations is shown to be consistent with related numerical simulations and show that lateral leakage may explain the observed damping of (vertically polarised) fast magnetoacoustic kink oscillations.

The third paper takes the model of Paper II and explores its potential for the application of coronal seismology. Two types of observational examples are investigated. The Alfven speed and equilibrium density profile are determined from these observations. It is shown that the mechanism of lateral leakage of fast magnetoacoustic kink oscillations described in this model is efficient. In fact, the damping is so efficient that in order to match predicted values with observational ones, either the loop needs to be highly contrasted or the transverse Alfven speed profile needs to be close to linear. Possible improvements to make the modeling of lateral wave leakage in loops more realistic, allowing a lower damping efficiency, are discussed.

The paper Leakage of waves from loops by wave tunneling by Brady, Verwichte and Arber builds further on the work by Brady and Arber (2005), to better understand the decay of vertically polarised fast kink modes of coronal loops by the mechanism of wave tunneling. Simulations are performed of fast kink modes in straight flux slabs which have Alfven speed profiles which include a tunneling region. The decay rates are found to be determined by the mode number of the trapped mode and the thickness of the tunneling region. Two analytical models are suggested to explain the observed decay. The decay rates for these straight slabs are found to be slower than in observations and those found by Brady and Arber (2005) using curved magnetic slabs. It is found that the difference between straight and curved slabs can be represented as a geometric correction to the decay rate.

This work is based on four papers:

Verwichte, E. Foullon, C. and Nakariakov: 2006, Fast magnetoacoustic waves in curved coronal loops I. Trapped and leaky modes, Astron. Astrophys. 446, 1139-1149.
Verwichte, E. Foullon, C. and Nakariakov: 2006, Fast magnetoacoustic waves in curved coronal loops II. Tunneling modes, Astron. Astrophys., 449, 769-779.
Verwichte, E., Foullon, C. and Nakariakov, V.M.: 2006, Seismology of curved coronal loops with vertically polarised transverse oscillations, Astron. Astrophys., 452, 615-622.
Brady, C.S., Verwichte, E. and Arber, T.D.: 2006, Leakage of waves from loops by wave tunneling, Astron. Astrophys., 449, 389-399.

### Transverse waves in a post-flare supra-arcade

TRACE 195Å image of AR 9906 on April 21st, 2002 at 02:00:41 UT

#### Press Release

For the first time a detailed analysis of observations of propagating transverse waves in an open coronal structure is presented. We focus on TRACE observations of transverse waves in the post-flare supra-arcade of NOAA active region 9906 on the 21st of April 2002, associated with dark tadpole-like sunward moving structures. The waves are interpreted as propagating fast magnetoacoustic kink surface waves.

This work has been published as

Verwichte, E., Nakariakov, V.M. and Cooper, F.C.: 2005, Transverse waves in a post-flare supra-arcade, Astron. Astrophys. Lett. 430, L65-L68.

Data slice of spatial coordinate across the tadpole-ray structures as a function of time at a fixed height above the solar surface. The transverse wave-trains of the tadpole tails are clearly visible

### Transverse loop oscillations in a post-flare loop arcade

TRACE image of post-flare loop arcade with analysed loop path superimposed
Characteristics of transverse oscillations in a coronal loop arcade, SoHO 15 Workshop Coronal Heating, 6-9 September 2004, St Andrews, Scotland

TRACE observations from April 15th, 2001 of transverse oscillations in coronal loops of a post-flare loop arcade are investigated. They are considered to be standing fast kink oscillations. Oscillation signatures such as displacement amplitude, period, phase and damping time are deduced from 9 loops as a function of distance along the loop length.

For the first time multiple oscillation modes are found with different amplitude profile along the loop length, suggesting the presence of a second harmonic. This discovery has triggered a string of theoretical studies that exploit the period ratio to deduce the longitudinal structuring of coronal loops.

The damping times are consistent with the hypothesis of phase mixing and resonant absorption, although there is a clear bias towards longer damping times compared with previous studies. The coronal magnetic field strength and coronal shear viscosity in the loop arcade are derived.

This work has been published as

Verwichte, E., Nakariakov, V.M., Ofman, L. and DeLuca, E.E.: 2004, Characteristics of transverse oscillations in a coronal loop arcade, Solar Physics 223, 77-94.

Transverse loop displacement as a function of time.