# Key publications

## I Ballai

B Orza, I Ballai, R Jain and K Murawski: 2012,

**The effect of the environment on the P1/P2 period ratio for kink oscillations of coronal loops**, Astron. Astrophys, 537, 41.

We show that under coronal conditions the effect of the temperature difference between the plasma inside and outside the magnetic structure can change considerably the period ratio of loop oscillations (this change can reach even 40%). We also show that once dispersive effects are taken into account, with oscillation periods shorter than the cut-off period (determined by the density and temperature difference) the domain where the model can be applied is reduced. In the case of prominences embedded in the hot corona, the effect of the environment is negligible given the high density and temperature difference between the chromospheric prominences and corona.

We discuss what implications our model has on seismological (or diagnostics) techniques in the solar corona. Our analysis shows that the period ratio P1/P2 is sensitive to the temperature difference between the loop and its environment and this effect should always be taken into account when estimating the degree of density structuring with period ratio method of coronal loops.

ADSI Ballai, E Forgács-Dajka and M Douglas: 2011,

**Magnetoacoustic surface gravity waves at a spherical interface**, Astron. Astrophys., 527, 12.

We studied the properties of global coronal MHD waves assuming them to be fast magnetoacoustic waves propagating at a spherical interface in a spherical diverging magnetic field. The frequency of waves is shown to increase with decreasing density contrast at the interface. We also show that, for a given azimuthal wavenumber, the magnetic field has a very small effect on the value of the frequency of waves. When plotted against the location of the interface (in the radial direction) the frequency varies inversely proportional to the distance, while for a fixed density ratio and location of the interface the frequency is obtained to be defined in a very narrow region.

ADSI Ballai, D Jess and M Douglas: 2011,

**TRACE observations of driven loop oscillations**, Astron. Astrophys., 534, 13.

We use TRACE EUV observations to study the oscillations of coronal loops. We found two distinct periods belonging to the fundamental kink mode and its first harmonic, but could also reflect the stage of an overdriven loop. Additional scenarios for explaining the two periods are listed, each resulting in a different value of the magnetic field and the intrinsic and sub-resolution properties of the coronal loop.

We show that the observation of two distinct periods in a coronal loop does not necessarily lead to a unique conclusion. Multiple plausible scenarios exist, suggesting that both the derived strength of the magnetic field and the sub-resolution properties of the coronal loop depend entirely on which interpretation is chosen. The interpretation of the observations in terms of a combination of the natural kink mode of the coronal loop, driven by a harmonic EIT wave seems to result in values of the magnetic field consistent with previous findings. Other interpretations, which are realistic, such as kink fundamental mode/first harmonic and the oscillations of two sub-resolution threads result in magnetic field strengths that are below the average values found before.

ADSI Ballai and M.S Ruderman 2011,

**Nonlinear Effects in Resonant Layers in Solar and Space Plasmas**, Space Science Reviews, 158, 412.

We reviewed recent advances in the theory of nonlinear driven magnetohydrodynamic (MHD) waves in slow and Alfvén resonant layers. Simple estimations show that in the vicinity of resonant positions the amplitude of variables can grow over the threshold where linear descriptions are valid. Using the method of matched asymptotic expansions, governing equations of dynamics inside the dissipative layer and jump conditions across the dissipative layers are derived. These relations are essential when studying the efficiency of resonant absorption. Nonlinearity in dissipative layers can generate new effects, such as mean flows, which can have serious implications on the stability and efficiency of the resonance.

ADSJ Terradas, M Goossens and I Ballai: 2010,

**The effect of longitudinal flow on resonantly damped kink oscillations**, Astron. Astrophys., 515, 46.

Using the low ratio of flow and Alfvén speeds we derive simple analytical expressions to the damping rate of kink oscillations. The analytical expressions are in good agreement with the resistive eigenmode calculations. Under typical coronal conditions the effect of the flow on the damped kink oscillations is weak when the characteristic scale of the density layer is similar or lower than the characteristic width of the velocity layer. However, in the opposite situation the damping rates can be significantly altered, specially for the backward propagating wave which is undamped while the forward wave is overdamped.

ADSC T C Clack, I Ballai and M S Ruderman: 2009,

**On the validity of nonlinear Alfvén resonance in space plasmas**, Astron. Astrophys., 497, 317.

Here we demonstrated that the the Alfvén resonance (with isotropic/anisotropic dissipation) wave dynamics can be described by the linear theory with great accuracy.The nonlinear corrections to resonant waves in the Alfvén dissipative layer are derived, and it is proved that they are small.

ADSI Ballai, M Douglas and A Marcu: 2008,

**Forced oscillations of coronal loops driven by EIT waves**, Astron. Astrophys, 488, 1125.

We study the nature of oscillations in a straight magnetic cylinder under the influence of an external driver modelling the force due to the global EIT wave. Results show that for a harmonic driver the dominant period in the generated oscillation belongs to the driver. Depending on the period of driver, compared to the natural periods of the loop, a mixture of standing modes harmonics can be initiated. In the case of a non-harmonic driver (modelling a shock wave), the generated oscillations in the loop are the natural periods only. The amplitude of oscillations is determined by the position of the driver along the tube.

ADSI Ballai: 2007,

**Global coronal seismology**, Sol. Phys., 246, 177-185.

First study on global coronal seismology linking the properties of coronal global and local waves. Using TRACE observational data we construct the first magnetic map of the quiet region of the corona

ADSI. Ballai, R. Erdelyi and J. Hargreaves: 2006,

**Slow magnetohydrodynamic waves in stratified and viscous plasmas**, Phys. Plasmas, 13, 042108.

We derived for the first time the Klein-Gordon-Burgers equation describing the propagation of slow sausage waves in stratified and viscous plasma. The evolutionary equation is solved in limiting cases depending on the characteristic spatial change of variables compared to the gravitational scale-height.

ADSI. Ballai, R. Erdélyi and B. Pintér: 2005,

**On the Nature of Coronal EIT Waves**, Astrophys. J., 633, 145.

We showed here, for the first time, that EIT waves propagating in the low solar corona are indeed fast magneto-acoustic waves with well-defined periods. These findings can open amazing diagnostic possibilities of the quiet Sun, flares/CMEs and coronal loops.

ADS

## R Erdélyi

Allcock, M. and Erdélyi, R. Solar Physics, 292, 35, 20 pages, 2017.

**First paper to study magnetohydrodynamic waves in an asymmetric magnetic slab**. Analytical models of solar atmospheric magnetic structures have been crucial for our understanding of magnetohydrodynamic (MHD) wave behaviour and in the development of the field of solar magneto-seismology. In this neat paper, an**analytical approach is used to derive the dispersion relation for MHD waves in a magnetic slab of homogeneous plasma enclosed on its two sides by non-magnetic, semi-infinite plasma with different densities and temperatures**. This**generalises the classic magnetic slab model**, which is symmetric about the slab. The dispersion relation, unlike that governing a symmetric slab, cannot be decoupled into the well-known sausage and kink modes, i.e. the modes have mixed properties. The eigenmodes of an asymmetric magnetic slab are better labelled as**quasi-sausage**and**quasi-kink modes**. It is shows how**avoided crossings occur between quasi-sausage and quasi-kink surface modes**, allowing modes to adopt different properties for different parameters in the external region.

ADSMorton, R.J., Verth, G., Jess, D.B., Kuridze, D. , Ruderman, M.S., Mathioudakis, M. and Erdelyi, R. Nature (Comm.), 3:1315 (8 pages), doi: 10.1038/ncomms2324, (2012).

This paper detailed the**ubiquitous****dual MHD wave mode**in the lower solar atmosphere, a**first on-disk demonstration of****sufficient MHD wave energy to potentially heat small-scale solar magnetic waveguides.**ADSWedemeyer-Böhm, S., Scullion, E., Steiner, O., Rouppe van der Voort, L., de la Cruz Rodriguez, J., Fedun, V. and Erdelyi, R. Nature, 486, 505–508 (2012.

The**first evidence of****direct MHD wave energy tunnelling****from photophere to corona;**ADSErdélyi, R. and Fedun, V., Solar Phys., 263, 63, 2010.

**First paper to derive the****dispersion relation****for magneto-acoustic waves in****compressible magnetically twisted flux tubes**, a culmination of a series of our papers about modelling built.

ADSJess, D.B., Mathioudakis, M., Erdélyi, R. et al, Science, 323, 2009.

**First identification of torsional Alfvén waves in solar physics.**AWs were predicted by H. Alfvén in 1942 (for which he received ~30 yrs later a Nobel-prize), and are widely believed to be one of the major candidates to heat the solar corona a problem that has puzzled astrophysicists for over a century.**We have shown that these waves have more than the required energy flux to heat the magnetically active solar atmosphere!**So far this is the only proven physical phenomenon, without any criticisms in the literature, that have enough observed energy flux that covers the necessary heating and the consequent observed radiative losses in the atmosphere. My significant and leading contribution was to propose the main principle of the observations (see Erdélyi & Fedun, Science 318, 2007) and to provide the theoretical interpretations and numerical modelling of the data.

ADSMorton, R.J. & Erdélyi, R., Astrophys. J., 707, 750–760, 2009.

**First study to determine the effect of****cooling****on transverse oscillations of****coronal loops.**ADSBanerjee, D., Erdélyi, R., Oliver, R. & O'Shea, E. Solar Physics, 246, pp.3-29, 2007.

An**Invited Review**paper**to discuss the present and future observing trends in solar atmospheric magnetoseismology.**With modern imaging and spectral instruments observing in the visible, EUV, X-ray, and radio wavelengths, the detection of oscillations in the solar outer atmosphere has become a routine event. These oscillations are considered to be the signatures of a wave phenomenon and are generally interpreted in terms of magnetohydrodynamic (MHD) waves. With multiwavelength observations from ground- and space-based instruments, it has been possible to detect waves in a number of different wavelengths simultaneously and, consequently, to study their propagation properties. Observed MHD waves propagating from the lower solar atmosphere into the higher regions of the magnetized corona have the potential to provide excellent insight into the physical processes at work at the coupling point between these different regions of the Sun. In this review some new**trends in the observational study of waves and oscillations are summarised**, in particular, their**origin and their propagation through the atmosphere is discussed**. We focus on**waves and oscillations in open magnetic structures**(eg solar plumes) and**closed magnetic structures**(eg loops and prominences). We also address observations of**waves in filament fibrils**allied with a better characterization of their propagating and damping properties, the detection of prominence oscillations in UV lines, and the renewed interest in large-amplitude, quickly attenuated, prominence oscillations, caused by flare or explosive phenomena. .

ADSErdélyi, R. & Fedun, V., Science, 318, 2007.

**Shows the first time**how**the****atmospheric coupling of global solar (p-mode) oscillations**may be accountable for chromospheric kink oscillations in magnetic filaments. Identification of the fundamental observable properties of AWs.

ADSErdélyi, R. & Verth, G., Astron. & Astrophys, 462, 743, 2007.

**First demonstration to obtain diagnostic information about the density structure of coronal loops by measuring the amplitude profile of the fundamental standing mode of the fast kink wave.**Laying down the foundations of solar atmospheric spatio-magneto-seismology.

ADSDe Pontieu, B., Erdélyi, R., and James, S.P., Nature, 430, 536, 2004. (also featured on the front cover of Nature)

**Solved**by innovative modelling the long-standing problem (discovered in 1877) of**the origin**and nature**of**powerful solar jets, called**type-I spicules**, shooting plasma into the interplanetary space.**My theoretical explanation, supported by world-class joint satellite and ground-based observations carried out jointly**with my collaborator (Stanford-Lockheed), has been the central theme of major international conferences following our Nature work (eg Roberts WS, 2006; Coimbra WS, 2006; Royal Society DM 2006; NASA/ESA SOHO19 2007; IAUS247 Venezuela, 2007; Evershed Centenary Meeting, India 2008).

ADS

## M S Ruderman

Ruderman, M.S.

**Rayleigh-Taylor instability of a magnetic tangential discontinuity in the presence of flow**Astron. Astrophys.,**580**, A37, 2015.

The effect of slow on the Rayleigh-Taylor instability of a magnetic tangential discontinuity was studied. The criterion for the transition from absolute to convective instability was obtained.

ADSRuderman, M.S. and Terradas, J.

**Damping of coronal loop kink oscillations due to mode conversion**Astron. Astrophys.,**555**, A13, 2013.

This paper aimed to verify the correctness of the standard theory of resonant damping of kink oscillations. The standard theory is based on the assumption that the observed kink oscillations are described by eigenmodes of dissipative MHD. This assumption is definitely not applicable to observed kink oscillations of coronal magnetic loops. The integro-differential equation describing the evolution of the initial perturbation in the form of standing kink wave is derived. Its numerical solution showed that the standard theory of resonant damping of kink oscillations underestimate the damping time. However, the error related to using the standard theory is relative small and usually does not exceed 20%.

ADSRuderman, M.S. Astron. Astrophys.

**534**, A78, 2011.**Resonant damping of kink oscillations of cooling coronal magnetic loops.**Resonant damping of kink oscillations of cooling coronal magnetic loops was studied. It was shown that cooling amplifes kink oscillations. The balance between the amplifcation due to cooling and damping due to resonant absorption can result in the existence of decayless kink oscillations.

ADSRuderman, M.S., Goossens, M. and Andries, J. Phys. Plasmas

**17**, art. 082108, 2010.**Nonlinear propagating kink waves in thin magnetic tubes.**In this paper the equation describing the propagation of nonlinear waves along a thin magnetic tube homogeneous in the axis direction and inhomogeneous in the radial direction was derived. The numerical solution of this equation showed that nonlinearity can strongly enhance the damping of propagating kink waves due to resonant absorption.

ADSRuderman, M.S. and Erdelyi, R.

**Transverse Oscillations of Coronal Loops**, Space Sci. Rev.,**149**, 199–228, 2009.

This a review paper presenting all the theoretical results related to kink oscillations of coronal loops that were known at that time.

ADSRuderman, M.S., Verth, G. and Erdélyi, R. Astrophys. J.

**686**, 694, 2008.**Transverse oscillations of longitudinally stratified coronal loops with variable cross-section.**This work is a generalization of the result obtained by Dymova and Ruderman (2005). It was shown that kink oscillations of expanding magnetic tubes can be described by the wave equation with the variable kink speed. The theoretical results are applied to coronal seismology.

ADSVan Doorsselaere, T., Ruderman, M.S. and Robertson, D.

**Transverse oscillations of two parallel coronal loops**, Astron. Astrophys.,**485**, 849–857, 2008.

Kink oscillations of a system of two parallel thin magnetic tubes were studied using bicylindircal coordinates.

ADSDymova, M. and Ruderman, M.S.

**Non-axisymmetric oscillations of thin prominence Fibrils**, Solar Phys.,**229**, 79–94, 2005.

It was shown that kink oscillations of a thin magnetic tube with the density varying along the tube can be described by the wave equation with the variable kink speed. This equation was used to study oscillations of a prominence thread with piecewise constant density.

ADSRuderman, M.S. and Roberts, B. Astrophys. J.

**577**, 475, 2002.**Damping of coronal loop oscillations.**In this paper the initial value problem for kink oscillations of a magnetic tube is solved. The decrement of oscillation damping due to resonant absorption is calculated. The theoretical results are used to obtain information about the transverse structure of a coronal loop.

ADSRuderman, M.S. J. Plasma Phys.

**63**, 43, 2000.**Interaction of sound waves with inhomogeneous magnetised plasma in strongly nonlinear resonant slow wave layer.**The solution describing the wave motion in a slow resonant layer was obtained under the condition that nonlinearity strongly dominates dissipation. It was shown that, in this case, wave energy is dissipated not in the whole resonant layer like in the linear theory, but in slow shocks. The connection formulae giving the jumps of pressure and perpendicular velocity across the resonant layer were derived.

ADS

## G Verth

Morton, R. J., Verth, G., Jess, D.B., Kuridze, D, Ruderman, M. S., Mathioudakis, M. and Erdélyi, R., Nature (Comm.), 3, 1315, 2012.

**First identification of ubiquitous and concurrent MHD kink and sausage wave modes in the Sun’s chromosphere.**The estimated energy flux associated with these waves suggested that the chromosphere is actually a vast reservoir of wave energy that has the potential to meet both chromospheric and coronal heating requirements.

ADSFedun, V., Verth, G., Jess, D.B, and Erdélyi, R., Astrophys. J., 740, L46, 2011.

**First demonstration of exploiting frequency filtering of torsional Alfvén waves by chromospheric magnetic field in order to reconstruct the real 3D structure of solar flux tubes.**This innovative technique was successfully illustrated using data from the Solar Optical Universal Polarimeter (SOUP) based at the Swedish Solar Telescope (SST).

ADSSoler, R., Terradas, J., Verth, G. and Goossens, M., Astrophys. J., 736, 10, 2011.

**First theoretical study of the properties of propagating kink waves in solar waveguides where there is both continuous longitudinal and transverse variation in the equilibrium Alfvén speed.**It was found that the resultant amplitude is a product between the two effects (i) the variation of amplitude due to longitudinal stratification (no frequency dependence) (ii) the damping due to resonant absorption caused by continuous variation of Alfvén speed in transverse direction (frequency dependent).

ADSVerth, G., Goossens, M. and He, J.-S., Astrophys. J., 733, L15, 2011.

**First magnetoseismological inversion of Hinode/Solar Optical Telescope (SOT) data to determine both the height variation of magnetic field strength and plasma density along a solar spicule exploiting the high resolution observation of a propagating kink wave.**ADSTerradas, J., Goossens, M. and Verth, G., Astron. and Astrophys, 524, 23, 2010.

**First theoretical work to show that the resonant damping of propagating kink waves has a frequency dependent damping length, i.e., higher frequency kink waves have a shorter damping length than their lower frequency counterparts.**This important result was used in a subsequent paper which found observational evidence of this frequency dependent damping in propagating coronal kink waves (Verth, G., Terradas, J. and Goossens, M., Astrophys. J., 718, L102, 2010).

ADSVerth, G., Terradas, J. and Goossens, M. 2010

**First estimate of the frequency dependent damping length due to resonant absorption of propagating coronal kink waves observed using data from the Coronal Multi-Channel Polarimeter (CoMP).**ADSRuderman, M.S., Verth, G. and Erdélyi, R. Astrophys. J. 686, 694, 2008.

**Transverse oscillations of longitudinally stratified coronal loops with variable cross-section.**The wave equation for kink oscillations of a thin expanding magnetic tube inhomogeneous in the longitudinal direction is derived. This equation is used to calculate the ratio of periods of the fundamental mode and first overtone of kink oscillations. It is shown that, while the density stratification decreases this ratio, the tube expansion increases it. Hence, the two effects are acting in the opposite directions. This important theoretical result was used for the interpretation of Transition Region And Coronal Explorer (TRACE) data in Verth, G., Erdélyi, R. & Jess, D. B., Astrophys. J., 687, L45, 2008.

ADSVerth, G. and Erdélyi, R, Astron. and Astrophys, 486, 1015, 2008.

**First demonstration to determine the combined effect of magnetic and density stratification on transversal coronal loop oscillations.**ADSVerth, G., Erdélyi, R. and Jess, D. B., Astrophys. J., 687, L45, 2008.

**First magnetoseismological estimate of the density scale height in a coronal loop using a Transition Region And Coronal Explorer (TRACE) observation of a standing kink oscillation of fundamental mode and first overtone, correcting for the important effects of loop inclination, geometry and magnetic field stratification.**ADSErdélyi, R. and Verth, G., Astron. and Astrophys, 462, 743, 2007.

**First demonstration to obtain diagnostic information about the density structure of coronal loops by measuring the amplitude profile of the fundamental standing mode of the fast kink wave.**Laying down the foundations of solar atmospheric spatio-magnetoseismology.

ADS