∎

11institutetext:  V. M. Nakariakov 22institutetext: Centre for Fusion, Space
and Astrophysics, Department of Physics, University of Warwick, Coventry CV4
7AL, UK,
School of Space Research, Kyung Hee University, Yongin, 446-701, Gyeonggi,
Korea,
St Petersburg Branch, Special Astrophysical Observatory, Russian Academy of
Sciences, St Petersburg 196140, Russia
22email<EMAIL_ADDRESS>33institutetext: S. A. Anfinogentov
44institutetext: Institute of Solar-Terrestrial Physics SB RAS, Lermontov St.
126, Irkutsk 664033, Russia 55institutetext: P. Antolin 66institutetext:
Department of Mathematics, Physics & Electrical Engineering, Northumbria
University, Newcastle Upon Tyne, NE18ST, UK 77institutetext: R. Jain
88institutetext: School of Mathematics and Statistics, University of
Sheffield, Sheffield S3 7RH, UK 99institutetext: D. Y. Kolotkov
1010institutetext: Centre for Fusion, Space and Astrophysics, Department of
Physics, University of Warwick, Coventry CV4 7AL, UK,
Institute of Solar-Terrestrial Physics, Lermontov St., 126a, Irkutsk 664033,
Russia 1111institutetext: E.G. Kupriyanova 1212institutetext: Central
Astronomical Observatory at Pulkovo of RAS, Saint Petersburg, Russia
1313institutetext: D. Li 1414institutetext: Key Laboratory of Dark Matter and
Space Astronomy, Purple Mountain Observatory, CAS, Nanjing, 210033, China
State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing,
100190, China
CAS Key Laboratory of Solar Activity, National Astronomical Observatories,
Beijing, 100101, China 1515institutetext: N. Magyar 1616institutetext: Centre
for Fusion, Space and Astrophysics, Physics Department, University of Warwick,
Coventry CV4 7AL, UK 1717institutetext: G. Nisticò 1818institutetext:
Dipartimento di Fisica, Università della Calabria, via P. Bucci, Cubo 31C,
87036, Arcavacata di Rende (CS), Italy 1919institutetext: D.J. Pascoe
2020institutetext: Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium
2121institutetext: A. K. Srivastava 2222institutetext: Department of Physics,
Indian Institute of Technology (BHU), Varanasi-221005, India
2323institutetext: J. Terradas 2424institutetext: Departamet de Física,
Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain,
Institut d’Aplicacions Computacionals de Codi Comunitari (IAC), Universitat de
les Illes Balears, 07122 Palma de Mallorca, Spain 2525institutetext: S.
Vasheghani Farahani 2626institutetext: Department of Physics, Tafresh
University, Tafresh 39518 79611, Iran 2727institutetext: G. Verth
2828institutetext: Plasma Dynamics Group, School of Mathematics and
Statistics, University of Sheffield, Sheffield, UK 2929institutetext: D. Yuan
3030institutetext: Institute of Space Science and Applied Technology, Harbin
Institute of Technology, Shenzhen, Guangdong 518055, China 3131institutetext:
I. V. Zimovets 3232institutetext: Space Research Institute of the Russian
Academy of Sciences (IKI RAS), 84/32 Profsoyuznaya St., Moscow 117997, Russia

# Kink oscillations of coronal loops

V. M. Nakariakov  S. A. Anfinogentov  P. Antolin  R. Jain  D. Y. Kolotkov  E.
G. Kupriyanova  D. Li  N. Magyar  G. Nisticò  D. J. Pascoe  A. K. Srivastava
J. Terradas  S. Vashegani Farahani  G. Verth  D. Yuan  I. V. Zimovets

(Received: date / Accepted: date)

###### Abstract

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.

###### Keywords:

Sun: corona Sun: waves Magnetohydrodynamics

###### pacs:

52.35.Bj 94.30.cq 96.60.P-

###### Contents

  1. 1 Introduction
  2. 2 Empirical properties of decaying kink oscillations
    1. 2.1 Spectroscopic observations of kink oscillations of coronal loops
  3. 3 The Zaitsev–Stepanov–Edwin–Roberts model
  4. 4 Damping of kink oscillations by resonant absorption
    1. 4.1 Non-exponential damping regime of resonant absorption
    2. 4.2 Observational evidence and seismological application of the two-regime damping
  5. 5 Kink oscillations in twisted cylinders
  6. 6 Transverse oscillations of current-carrying loops due to the electromagnetic interaction with the external electric currents
  7. 7 Kink modes in the presence of parallel shear plasma flows
    1. 7.1 Basic effect of flow on the kink eigen frequency and eigen function
    2. 7.2 Kink waves in jets
    3. 7.3 Negative energy wave effects on kink oscillations
  8. 8 Kink oscillations in loops undergoing cooling
  9. 9 Oscillation period ratios and their seismological inferences
    1. 9.1 Theoretical modelling of the period ratio
      1. 9.1.1 Effect of perpendicular nonuniformity
      2. 9.1.2 Effect of axial non-uniformity
      3. 9.1.3 Effect of stationary axial flow
    2. 9.2 Observational detection of higher parallel harmonics and their seismological applications
  10. 10 Nonlinear effects in kink oscillations
    1. 10.1 Nonlinear generation of fluting perturbations by kink-driven Kelvin–Helmholtz instability
    2. 10.2 Nonlinear coupling of kink and fluting oscillations
    3. 10.3 Field-aligned flows driven by the ponderomotive force in kink waves
  11. 11 Excitation of kink oscillations
  12. 12 Decayless kink oscillations
    1. 12.1 Observations of decayless oscillations
    2. 12.2 Theoretical and numerical models for decayless oscillations
  13. 13 Possible detection of kink oscillations in the microwave and radio bands
  14. 14 Conclusions and outstanding problems

##  1 Introduction

One of the key features which makes the plasma of the solar corona different
from other natural plasma environments, for example, the Earth’s
magnetosphere, is a pronounced field-aligned filamentation of the macroscopic
parameters, such as the density and temperature. In particular, coronal active
regions consist of dense hot plasma loops which are clearly observed as bright
structures in the EUV and soft X-ray, and sometimes in microwaves (see Reale,
2014, for a comprehensive review). The loops are anchored at the chromosphere
at locations known as footpoints. Usually both footpoints of a loop are
located in the same active region, while sometimes loops link different active
regions. The typical minor radii of coronal loops are a few Mm, while major
radii reach several hundred Mm. The longest, transequatorial loops link active
regions situated in different hemispheres. Individual loops are separated by
apparently more rarefied plasma regions. The density contrast inside and
outside a loop is from a few tens percent up to one hundred times or more in
flaring active regions. Coronal loops, being the main building block of active
regions, have been remaining a puzzling, intensively debated plasma structure
for several decades. The key open questions are the mechanisms responsible for
their appearance, typical spatial scales and their lifetimes of several hours
or longer; why the loop’s minor radius is usually constant along the loop;
whether the loop has a fine structure in density and/or temperature, and if
yes, which one (a bundle of threads, a set of co-axial shells, or something
else). As dense and hot plasma objects, coronal loops are directly linked with
the coronal heating problem. In addition, equilibrium solutions describing
coronal loops remain unknown.

Coronal loops are observed to be dynamic objects. The brightness determined by
the temperature and density, and geometry, may vary in time. In addition,
there could be field-aligned plasma flows of various nature, and wave motions.
Repetitive transverse displacements of the loop axis are called kink
oscillations. Kink oscillations of coronal loops, i.e., standing kink waves or
oscillatory bouncing of the loops, predicted theoretically in 1970s (Zajtsev
and Stepanov, 1975; Ryutov and Ryutova, 1976), and discovered observationally
in the high-resolution EUV data obtained with the Transition Region And
Coronal Explorer (TRACE ) (Handy et al., 1999) in late 1990s (Aschwanden et
al., 1999; Nakariakov et al., 1999), have become one of the most intensively
studied magnetohydrodynamic (MHD) wave phenomena in the solar corona. The
major breakthrough in the observational study of kink oscillations is
associated with their detection with the Atmospheric Imaging Assembly (AIA) on
the Solar Dynamics Observatory (SDO) spacecraft. The first detection of kink
oscillations of coronal loops with AIA was reported by (Aschwanden and
Schrijver, 2011). Since that time, several hundred kink oscillation events
have been found in AIA data, usually in the 171 Å passband (Nechaeva et al.,
2019).

Since the discovery, our understanding of kink oscillations has evolved
through several observational and theoretical advances. Almost immediately
after the first detection of decaying oscillations, the decay was linked with
resonant absorption (Ruderman and Roberts, 2002; Goossens et al., 2002), i.e.,
a linear transformation of the observed collective transverse movements of the
loop as a whole to unresolved torsional movements localised at certain
magnetic surfaces. That seminal result allowed the research community to use
the wealth of the theoretical knowledge on resonant absorption for the
interpretation of observations. Ofman and Aschwanden (2002) used the empirical
scaling of the observed damping times and periods to demonstrate its
consistency with the resonant absorption theory. Van Doorsselaere et al.
(2004b) demonstrated that the effect of the curvature of the loop on kink
oscillations and their resonant absorption is weak, and hence justified the
applicability of the straight cylinder model. On the other hand, alternative
interpretations were proposed too, which link kink motions with either
eigenmodes of coronal arcades (Hindman and Jain, 2014), or with a propagating
fast magnetoacoustic wave train (Uralov, 2003; Terradas et al., 2005).

Dymova and Ruderman (2005) developed a mathematical formalism for taking into
account a non-uniformity of the equilibrium plasma parameters along the loop.
Terradas et al. (2008) demonstrated numerically that the sheared layer between
the loop and the external medium formed by the transverse movements of the
loop is subject to Kelvin-Helmholtz instability (KHI). Antolin and Van
Doorsselaere (2019) established that resonant absorption further enhances the
instability. Resonant absorption was shown to work effectively in a cylinder
with an irregular cross-section (Pascoe et al., 2011). It was established that
the oscillation damping could have either an exponential or Gaussian profile
(Pascoe et al., 2012). Analysing a catalogue of decaying kink oscillations,
Zimovets and Nakariakov (2015) established empirically that the oscillations
are preferentially excited by a displacement of the loop by a low coronal
eruption. In addition, the linear correlation of oscillation periods and
lengths of the oscillating loops unequivocally demonstrated that kink
oscillations are eigenmodes of the loops. Moreover, it was found that the
oscillation quality-factor is proportional to the initial amplitude to the
power of minus two thirds (Goddard and Nakariakov, 2016).

In addition to the decaying kink oscillations, a decayless regime of kink
oscillations was observationally found. The first observational evidence is
reported by Wang et al. (2012) as an event of transverse oscillations of EUV
loops, growing in amplitude. Further observations showed that a loop could
oscillate in either decaying or decayless regime in different time intervals,
while the oscillation period remains the same (Nisticò et al., 2013).
Anfinogentov et al. (2015) found that decayless kink oscillations are
ubiquitous in coronal active regions, and occur even in the absence of solar
flares, eruptions or other impulsive energy releases. Oscillation periods of
decayless oscillations correlate linearly with the lengths of the oscillating
loops, and hence the oscillations are natural modes of the loop. Nakariakov et
al. (2016) demonstrated that in the statistical distribution of decayless
oscillation amplitudes there are no peaks associated with a certain period,
which disproved their excitation by 5-min or 3-min oscillations in the lower
layers of the solar atmosphere. It was suggested that the oscillation damping
by resonant absorption could be compensated by either steady or random flows
near footpoints of the loop. In the former case, the oscillations are actually
self-oscillations with the amplitudes determined by the balance between energy
losses and gains. This list of major new results is by all means incomplete
and not exclusive. In parallel, there are intensive studies of propagating
kink waves in loops and streamers, see, for example, (Tomczyk and McIntosh,
2009) and (Chen et al., 2014), respectively, which are out of scope of this
review. An exception was made for kink oscillations of hot plasma jets, as it
is often difficult to disentangle the phase speed and the speed of the
equilibrium plasma flow, and hence the kink oscillatory movements could be
standing.

The confident and frequent detection of kink oscillations makes them a
promising probe of physical conditions in the coronal loops hosting them,
i.e., kink oscillations are a tool for coronal seismology. Kink oscillations
are used for estimating the absolute value of the magnetic field (e.g.,
Nakariakov and Ofman, 2001) and its variation along the loop (Verth and
Erdélyi, 2008), the spatial scale of the density stratification (e.g., Andries
et al., 2005), and give information about transverse profiles of the Alfvén
speed and mass density (Aschwanden et al., 2003), including its steepness
(Pascoe et al., 2016b). More recently, it was demonstrated that the ubiquitous
kink oscillations observed in coronal active regions during the quiet time
periods, allow for the seismological mapping of the Alfvén speed and magnetic
field (e.g., Anfinogentov and Nakariakov, 2019). Kink oscillations in loops of
the sigmoid shape have been shown to carry the information about the free
magnetic energy associated with non-potential field geometry (e.g., Magyar and
Nakariakov, 2020).

In this review we concentrate mainly on the results obtained in the last ten
years, i.e., since 2010. Previous comprehensive reviews which address kink
oscillations of coronal loops include Aschwanden (2009); Andries et al.
(2009b); Ruderman and Erdélyi (2009); Terradas (2009); van Doorsselaere et al.
(2009); De Moortel and Nakariakov (2012); Liu and Ofman (2014). A possible
role of kink oscillations in the coronal heating problem, which has been
intensively studied since their first detection, is reviewed in Van
Doorsselaere et al. (2020).

The review is organised as follows. In Section 2 we describe empirical
properties of decaying kink oscillations, including relationships between
their parameters and also parameters of the host loops. In Section 3 we review
the standard model of kink oscillations of a plasma cylinder. Section 4 is
dedicated to linear mechanisms for the damping of kink oscillations. Section 5
addresses the effect of the magnetic twist. Section 6 presents kink
oscillations of a current-carrying loop, caused by the magnetic mirror force.
In Section 7 we discuss kink oscillations of in the presence of parallel shear
plasma flows, including negative energy wave instabilities. In Section 8, kink
oscillations in loops undergoing plasma cooling are considered. In Section 9
we review period ratios of different parallel harmonics, their detection in
observations, and seismological inferences. In Section 10 nonlinear effects
are discussed. Section 11 describes excitation mechanisms. In Section 12 we
consider observations and theoretical models of decayless kink oscillations.
Section 13 presents possible detections of kink oscillations in the radio band
including microwaves. In Section 14 we summarise outstanding problems and draw
conclusions.

##  2 Empirical properties of decaying kink oscillations

The abundant observational detection of kink oscillations of coronal loops
with AIA allows for the search for correlations between different parameters
of the oscillations and properties of the oscillating loops. To the date, the
most comprehensive catalogue of impulsively excited decaying kink
oscillations111We use this term to distinguish them from decayless kink
oscillations which are not seen to result from an impulsive event. The
decayless regime is discussed in Section 12., which includes information about
223 oscillating loops in 96 oscillation events has been compiled by Nechaeva
et al. (2019). Statistical properties of kink oscillations were determined by
approximating the evolution of detrended transverse displacements of the
loop’s segments by an exponentially decaying harmonic function (see Figure 1).
The oscillation periods range from 1 to 28 minutes, with 74% of the detections
in the range of 2–10 minutes. About 90% of the oscillations have the apparent
displacement amplitude in the plane of the sky, i.e., the amplitude reduced by
the projection effect, in the range of 1–10 Mm. The lengths of oscillating
loops are 70–600 Mm. Kink oscillations in shorter loops may be missing because
of insufficient time resolution. The typical apparent displacement amplitude
is about 1% of the loop length, although it is higher in terms of the loop
minor radius.

Figure 1: Left: Example of the determination of the oscillation parameters.
(Top panel): the semi-transparent black stars show the instantaneous positions
of the boundary of the oscillating loop, picked by hand for tracking the
oscillation. The power-law background trend is shown by the red line. The
initial displacement $a_{0}$ is also shown. (Bottom panel): the blue dots show
the detrended signal. The red curve shows the best-fitting exponentially
decaying harmonic oscillation. Right: Quality factor of 101 kink oscillations
of coronal loops plotted against the projected oscillation amplitude. The
inset shows the same dependence in the log–log plot. The dotted and dashed
curves show the approximation of the upper-right boundary of the data clouds
with a linear function in the log–log plot and the corresponding power-law
function in the linear plot, respectively. Figures are taken from Nechaeva et
al. (2019).

An important empirical result is the increase in the kink oscillation period
with the length of the oscillating loop, demonstrated by Figure 2. Such a
scaling, together with the synchronous displacement of different segments of
the oscillating loop, seen in movies made by time sequences of 2D EUV images,
indicates that kink oscillations are normal modes of the loop. In a vast
majority of observed cases, the displacement has a single maximum, i.e., an
antinode, at the loop top, and two nodes, at footpoints. Hence, kink
oscillations are fundamental normal modes in those cases. In some rare cases,
second harmonics are detected too, with the third node at the loop top, and
two antinodes in the legs (see, e.g., Andries et al., 2005; Arregui et al.,
2013a; Pascoe et al., 2016a; Li et al., 2017b). The third harmonic has been
detected too (Duckenfield et al., 2019). The harmonics discussed here are
harmonics in the axial direction, along the field. There could also be radial
harmonics. The linear dependence of the oscillation period $P_{\mathrm{k}}$ on
the loop length $L$ allows one to estimate the kink wave speed which for the
fundamental harmonic is $C_{\mathrm{k}}=2L/P_{\mathrm{kink}}$. The histogram
of the empirically determined values of $C_{\mathrm{k}}$ is shown in the right
panel of Fig. 2. The average value of the kink speed is
$C_{\mathrm{k}}=1328\pm 53$ km s-1. The value of $C_{\mathrm{k}}$ is
determined by the densities and magnetic fields in the loop and its
environment (see Section 3). The broadness of the distribution reflects the
broadness of the physical conditions in the oscillating loops.

Figure 2: Left: Empirical scaling of kink oscillation periods in the decaying
regime with the length of the oscillating loop. The “old” and “new” data
denote the data summarised in the old (Zimovets and Nakariakov, 2015; Goddard
et al., 2016) and updated (Nechaeva et al., 2019) catalogues. The solid line
indicates the best-fitting linear function, with the error bars shown by the
dashed curves. Right: The histogram of kink speeds estimated by their periods
and lengths of the oscillating loops. Figures are taken from Nechaeva et al.
(2019).

Analysis of the kink oscillation damping performed under the assumption of an
exponential decay confirmed the linear scaling of the damping time with the
oscillation period, established by Ofman and Aschwanden (2002) on a much
shorter dataset. The linear correlation coefficient was estimated as 0.640. On
average, the exponential damping time is $1.79\pm 0.14$ the oscillation
period. A surprising finding was the dependence of the oscillation quality
factor $Q$ defined as the ratio of the exponential damping time over the
oscillation period on the initial amplitude $A$ (Goddard and Nakariakov, 2016)
(see their Figure 1 right). The amplitude is defined as the projected initial
displacement. As the angle between the oscillation plane and the line-of-sight
is usually unknown, the apparent amplitude is lower than its actual value.
Thus, the actual scaling should be determined by the outer boundary of the
data cloud in the figure, i.e., by several points only. It was found that
$\log Q=-(0.68\pm 0.13)\log(A/\mathrm{Mm})+(2.80\pm 0.37)$. Thus, the quality
factor scales as $A^{-2/3}$.

###  2.1 Spectroscopic observations of kink oscillations of coronal loops

A promising new trend in the observational study of kink oscillations of
coronal loops is their detection in coronal emission spectra. Periodic
movements of a plasma loop along the line-of-sight appear as Doppler shift
oscillations of the corresponding spectral lines. The lack of simultaneous
modulation of the emission intensity could be taken as the evidence of a kink
oscillation which is weakly compressive in the long-wavelength regime.
However, in the optically thin regime, kink oscillations could modulate the
spectral line intensity by the modulation of the column depths along the line-
of-sight (see, e.g., Cooper et al., 2003; Yuan and Van Doorsselaere, 2016a;
Antolin et al., 2017). The amplitude of the intensity modulation can readily
reach several percent (Verwichte et al., 2009).

In contrast to the ubiquitous observations of kink oscillations with EUV
imaging telescopes, detections of kink oscillations in spectral data remain
sporadic. This is probably due to the line-of-sight superposition, which
strongly affects the Doppler velocity measurements as demonstrated by forward
modelling of numerical simulations of kink oscillations (De Moortel and
Pascoe, 2012; Antolin et al., 2017). Li et al. (2017a) presented a detailed
study of the decay oscillation in a hot flare loop on 27 October 2014.
Periodic changes from red to blue shifts were clearly seen in Doppler
velocities of FeXXI (Figure 3, top panel), with a dominant period of about 3.1
minutes. Such an oscillation was not seen in the line-integrated intensity, as
seen in Figure 3, bottom panel. The authors concluded that the hot flare loop
was most likely oscillating in a weakly-compressive MHD mode, such as the
standing kink oscillation. The observational results were consistent with
those from MHD modelling which simulated the manifestation of a standing kink
oscillation of coronal loops in Doppler velocities (e.g., Chen and Peter,
2015; Yuan and Van Doorsselaere, 2016a). A magnetic field strength of around
68 G was estimated in the loop. In another event, on 10 September 2014 Zhou et
al. (2016) found a standing kink oscillation during the precursor phase of a
flare. The precursor oscillation with alternating red and blueshifts was seen
in Doppler velocities of FeXXI. The oscillation period was $\sim$280 s, and
the oscillation lasted for about 13 minutes. The authors stated that the
precursor oscillation was triggered by a periodic energy release via magnetic
reconnection.

Figure 3: Upper: Spacetime image of Doppler velocities at FeXXI line. Lower:
Spacetime image of line-integrated intensities with a logarithmic brightness
scale. Two green lines are the bounds of the selected loop-top region. The
colour bars show the line-of-sight speed in km s-1. Adaptation of Figure 3 in
Li et al. (2017a)

##  3 The Zaitsev–Stepanov–Edwin–Roberts model

The model that has proven to be standard in the kink oscillation modelling is
based on linear MHD perturbations of a plasma cylinder surrounded by a plasma
with different properties, with both the internal and external plasmas being
penetrated by a straight magnetic field. Such a cylinder could also be
considered as a magnetic flux tube. Because of the frozen-in condition, its
displacement is accompanied by perturbation of the magnetic field. The
cylinder is a one-dimensional non-uniformity of MHD parameters of the plasma,
such as the temperature, mass density, the magnetic field, and, possibly,
stationary flows along the boundary of the cylinder. Equilibrium quantities
must satisfy the magnetostatic force balance condition. Kink perturbations of
such a cylinder are transverse, and hence can be of linear, elliptical or
circular polarisation. In the cylindrical frame of reference with the $z$ axis
coinciding with the cylinder’s axis, the kink perturbation has the azimuthal
wave number $|m|=1$, with different signs of $m$ corresponding to the opposite
senses of the circular polarisation. The key difference of the kink mode from
MHD modes with $m\neq 1$ is the displacement of the axis of the cylinder.

Dispersion relations for coronal kink oscillations in the important case of
the radial non-uniformity given by a step function, and a straight and uniform
magnetic field parallel to the axis of the cylinder, and in the absence of
equilibrium mass flows, were independently derived by Zajtsev and Stepanov
(1975) and Edwin and Roberts (1983). We shall refer to this model as the ZSER
model. The ZSER dispersion relation is a transcendental algebraic equation

$\frac{\rho_{0\mathrm{i}}}{\rho_{0\mathrm{e}}}\,\frac{(C_{\mathrm{Ai}}^{2}-V_{\mathrm{ph}}^{2})}{(C_{\mathrm{Ae}}^{2}-V_{\mathrm{ph}}^{2})}\frac{\kappa_{\mathrm{e}}}{n_{\mathrm{i}}}\frac{K^{\prime}_{1}\,(\kappa_{\mathrm{e}}\,a)}{K_{1}\,(\kappa_{\mathrm{e}}\,a)}=\frac{J^{\prime}_{1}\,(n_{\mathrm{i}}\,a)}{J_{1}\,(n_{\mathrm{i}}\,a)},$
(1)

where $\rho_{0\mathrm{i}}$ and $\rho_{0\mathrm{e}}$ respectively represent the
equilibrium internal and external mass densities; $a$ is the radius of the
cylinder in equilibrium, the indices $\mathrm{i}$ and $\mathrm{e}$ indicate
the internal or external media; and the functions $J_{1}$ and $K_{1}$
represent the Bessel function of the first kind and the modified Bessel of the
second kind, both being of the first order, the prime denotes the derivative
with respect to the argument. The Bessel functions describe the radial
structure of the perturbations. The effective radial wave number is

$\displaystyle\kappa_{\alpha}^{2}=\frac{(C_{\mathrm{A}\alpha}^{2}-V_{\mathrm{ph}}^{2})(C_{\mathrm{s}\alpha}^{2}-V_{\mathrm{ph}}^{2})}{(C_{\mathrm{A}\alpha}^{2}+C_{\mathrm{s}\alpha}^{2})(C_{\mathrm{t}\alpha}^{2}-V_{\mathrm{ph}}^{2})}k_{z}^{2},$
(2)

where $k_{z}$ is the axial wave number, $V_{\mathrm{ph}}=\omega/{k_{z}}$ is
the phase speed along the axis of the cylinder, note that for the internal
medium $n_{\mathrm{i}}^{2}=-\kappa_{\mathrm{i}}^{2}>0$; $C_{\mathrm{A}\alpha}$
and $C_{\mathrm{s}\alpha}$ respectively represent the Alfvén and sound speeds;
and the index $\alpha$ stands for either $\mathrm{i}$ or $\mathrm{e}$. In
typical coronal loops, $C_{\mathrm{Ai}}<C_{\mathrm{Ae}}$ and
$\max\\{C_{\mathrm{si}},C_{\mathrm{se}}\\}<\min\\{C_{\mathrm{Ai}},C_{\mathrm{Ae}}\\}$
(e.g. Reale, 2014). The tube speed is
$C_{\mathrm{t}\alpha}=C_{\mathrm{s}\alpha}C_{\mathrm{A}\alpha}/(C_{\mathrm{s}\alpha}^{2}+C_{\mathrm{A}\alpha}^{2})^{1/2}$.
As typically in coronal loops the plasma parameter $\beta$ is much less than
unity, the cold plasma approximation, $\beta=0$, with
$C_{\mathrm{s}\alpha}=C_{\mathrm{t}\alpha}=0$, is often used. In standing kink
oscillations, the axial wave numbers are discrete.

The ZSER dispersion model describes an infinite number of radial harmonics of
kink oscillations. In the long wavelength regime, $k_{z}a\ll 1$, all of them
except the lowest one are leaky, i.e., their radial structure outside the
cylinder is given by a Henkel function which appears because of the negative
sign of $m_{\mathrm{e}}^{2}$. In contrast, the fundamental radial harmonic of
the kink mode is trapped, i.e., its radial structure outside the cylinder is
evanescent for all values of $k_{z}$. The long wavelength limit is often
referred to as the thin tube limit in the consideration of the lowest radial
harmonics. In this limit, its phase speed can be approximated as

$V_{\mathrm{ph}}\approx C_{\mathrm{k}}\left[1-{\cal Q}K_{0}({\cal
T}|k_{z}|a)(k_{z}a)^{2}\right],$ (3)

where

${\cal Q}=\frac{\rho_{\mathrm{0i}}\rho_{\mathrm{0e}}{\cal
T}^{2}(C_{\mathrm{Ae}}^{2}-C_{\mathrm{Ai}}^{2})}{2C_{\mathrm{k}}^{2}(\rho_{\mathrm{0i}}+\rho_{\mathrm{0e}})^{2}},\mbox{\
\ \ \ }{\cal
T}=\left(1-\frac{C_{\mathrm{k}}^{2}}{C_{\mathrm{Ae}}^{2}}\right)^{1/2},$

and

$C_{\mathrm{k}}=\left(\frac{\rho_{\mathrm{0i}}C_{\mathrm{Ai}}^{2}+\rho_{\mathrm{0e}}C_{\mathrm{Ae}}^{2}}{\rho_{\mathrm{0i}}+\rho_{\mathrm{0e}}}\right)^{1/2},$

which is a density-weighted average of the internal and external Alfvén
speeds, is the kink speed first defined by Ryutov and Ryutova (1976). The kink
speed is always lower than $C_{\mathrm{Ae}}$. The phase speed decreases with
the decrease in the axial wavelength. In the limit of a zero-$\beta$ plasma,
for which the internal and external magnetic fields in the cylinder are equal,
the phase speed of the kink mode in the long wavelength limit reduces to

$C_{\mathrm{k}}=C_{\mathrm{Ai}}\sqrt{\frac{2\zeta}{\zeta+1}},$ (4)

where $C_{\mathrm{Ai}}$ is now the Alfvén speed at the axis of the cylinder,
and $\zeta=\rho_{\mathrm{0i}}/\rho_{\mathrm{0e}}$ is the ratio of the internal
and external plasma densities in the equilibrium.

Figure 4 shows the phase speed as a function of the axial wave number for
several different values of the density contrast in the cylinder. The full and
asymptotic solutions are in a good agreement with each other. As in coronal
conditions $m_{\mathrm{i}}^{2}<0$, the perturbation inside the cylinder has an
oscillatory structure, and hence the kink wave can be considered as a fast
magnetoacoustic wave which propagates obliquely to the axis of the cylinder
and experiences reflections at the boundary. At footpoints the kink speed
experiences a very sharp and large jump, and the magnetic field is line-tied,
which leads to the reflection of kink waves back to the coronal part of the
loop. Thus, in loops, there could be standing kink waves with discrete values
of the axial wave number $k_{z}$, referred to as parallel kink harmonics.

Figure 4: The phase speed of the fundamental radial harmonic of the kink mode
of a plasma cylinder as a function of the axial wave number, in the zero
plasma-beta limit. The black, red, green and blue curves correspond to
$C_{\mathrm{Ae}}=10C_{\mathrm{Ai}}$, $C_{\mathrm{Ae}}=5C_{\mathrm{Ai}}$,
$C_{\mathrm{Ae}}=3C_{\mathrm{Ai}}$ and $C_{\mathrm{Ae}}=2C_{\mathrm{Ai}}$,
respectively. The solid curves show solutions to the full dispersion relation,
while the dashed curves show the asymptotic solutions. The phase speed is
normalised to $C_{\mathrm{Ai}}$, while the axial wave number to the cylinder’s
radius.

If dispersive effects given by Eq. (3) are neglected, the oscillation period
for the $n$th parallel harmonic standing kink mode is then

$P_{\mathrm{kink}}^{(n)}\approx\frac{2L}{nC_{\mathrm{k}}}$ (5)

where $L$ is the length of the coronal loop. This definition of the kink
period corresponds to the thin tube thin boundary (TTTB) approximation whereas
parametric studies (Van Doorsselaere et al., 2004a; Soler et al., 2014; Pascoe
et al., 2019) find that the period of oscillation also depends on the width of
the boundary layer when it is sufficiently large. The period of the
fundamental parallel harmonic, with $n=1$, is $P_{\mathrm{kink}}$. One can
introduce a corresponding cyclic frequency, $\omega_{\rm
k}=2\pi/P_{\mathrm{kink}}$. If the kink speed varies along the cylinder,
ratios of the parallel harmonic periods could not be obtained from Eq. (5)
(see Section 9).

A model similar to ZSER has been constructed for a plane plasma slab (Edwin
and Roberts, 1982), see also Yu et al. (2015) and references therein for
recent developments of the model. The key differences between the plane
analogue and the ZSER model are the trigonometric eigenfunctions instead of
the Bessel ones, the phase speed of the kink wave in the long wavelength limit
becoming the external Alfvén speed instead of the kink speed, and the
exponential decrease in the perturbation amplitude in the external medium in
contrast with the super-exponential decay. In a slab with a diffused boundary,
the lower radial harmonic of the kink mode is trapped for all axial
wavelengths, similarly to the cylindrical case. In the corona, the plane model
is applicable to kink oscillations in a number of plasma non-uniformities, for
example, in streamers and pseudo-streamers, plane jets, prominence slabs, etc.

Exact analytical solutions describing kink oscillations in a cylinder with a
smooth transverse profile, e.g., similar to the Epstein profile in the slab
geometry, have not been found. The collective nature of kink oscillations
suggests that the dependence of their periods on the specific shape of the
transverse profile should be weak, at least in the linear regime. However, the
effect of the transverse profile on the oscillation damping is crucial. The
smoothness of the profile determines the effectiveness of the irreversible
transfer of the kink oscillation energy into azimuthal movements of individual
magnetic surfaces in the vicinity of a resonant layer where the local phase
speed of the kink wave coincides with the local Alfvén speed (the “Alfvénic
resonance”). This phenomenon is known as resonant absorption.

##  4 Damping of kink oscillations by resonant absorption

Figure 5: Numerical simulation showing the variation of the amplitude
$\xi_{r}$ (solid line) of a propagating kink wave in a coronal loop with
density profile parameters the inhomogeneous layer width $\epsilon=0.2$ and
density contrast ratio $\zeta=1.3$. The dashed line represents the analytical
solution of Hood et al. (2013) which accurately reproduces the non-exponential
(Gaussian) and exponential damping regimes of damping by resonant absorption.
Figure taken from Hood et al. (2013). Figure 6: Kink oscillation damping
profiles calculated by numerical simulation (solid lines) compared with the
analytical damping profiles corresponding to the exponential (red) and
Gaussian (blue) damping profiles. Green lines represent the general damping
profile of Pascoe et al. (2013) and dashed lines are fitted profiles used by
Pascoe et al. (2019) to generate a lookup table. Panels show the results for a
coronal loop with a density contrast ratio $\zeta=2$ and inhomogeneous layer
width $\epsilon=0.1$ (left), $0.5$ (middle), and $1.0$ (right). Figure taken
from Pascoe et al. (2019).

Figure 7: Estimation of the transverse density profile parameters of a coronal
loop based on the simultaneous analysis of the kink oscillation of the loop
legs (left) and the transverse EUV intensity profile (middle). The right panel
shows the normalised 2D histogram approximating the marginalised posterior
probability density function, with red error bars corresponding to the maximum
a posteriori probability value and 95% credible intervals. Figure taken from
Pascoe et al. (2018).

The observed strong damping of impulsively excited kink oscillations is
attributed to the process of resonant absorption (e.g., Goossens et al., 2011,
2019; De Moortel et al., 2016, and references therein). This has always been
an appealing damping mechanism for kink waves due to its robustness in only
requiring that the transition region between a higher density coronal loop and
the lower density background plasma be diffused, i.e., occurs over a finite
spatial scale. Curiously, the only coronal loops which we would not expect to
exhibit resonant absorption of kink oscillations would be those which have a
discontinuous boundary, i.e., those which are described by the ZSER model (see
Sec. 3). From an observational point of view, this will appear as a damping of
the kink oscillation, accompanied by the growth of the azimuthal motions
manifesting as unresolved Doppler velocity perturbations due to line-of-sight
integration of multiple waves and structures (e.g., De Moortel and Pascoe,
2012; Antolin et al., 2017; Pant et al., 2019). Subsequent phase-mixing of the
Alfvén waves in the inhomogeneous layer can generate small spatial scales
which enhance dissipative processes such as viscosity and resistivity (e.g.,
Pagano and De Moortel, 2017; Pagano et al., 2018).

Initial applications of resonant absorption to account for the strong damping
of large amplitude kink oscillations (Goossens et al., 2002; Ruderman and
Roberts, 2002) were based on analytical derivations for the asymptotic state
of the system, subject to the TTTB approximation, and produced an exponential
damping profile with the form

$A\left(t\right)\propto\exp\left(-\frac{t}{\tau_{\mathrm{D}}}\right),$ (6)
$\frac{\tau_{\mathrm{D}}}{P_{\mathrm{kink}}}=\frac{4}{\pi^{2}\kappa\epsilon},$
(7)

where $\tau_{\mathrm{D}}$ is the exponential damping time and
$\kappa=(\rho_{\mathrm{0i}}-\rho_{\mathrm{0e}})/(\rho_{\mathrm{0i}}+\rho_{\mathrm{0e}})$.
In this geometry, the transition from $\rho_{\mathrm{0i}}$ to
$\rho_{\mathrm{0e}}$ takes place over a cylindrical shell of thickness
$l_{\mathrm{tr}}$ centred on the cylinder’s radius $a$, with
$\epsilon=l_{\mathrm{tr}}/a$ being the normalised inhomogeneous layer width.
The constant of proportionality depends on the shape of the density profile in
the inhomogeneous layer, and here the factor $4/\pi^{2}$ corresponds to a
linear smooth transition between $\rho_{\mathrm{0i}}$ and
$\rho_{\mathrm{0e}}$.

Resonant absorption is a rather universal phenomenon, and depends weakly on
the specific shape of the loop’s cross-section. For example, this effect was
observed in a numerical simulation of a kink oscillation of a bundle of ten
closely packed plasma cylinders, with random positions and density contrasts
(e.g., De Moortel and Pascoe, 2012). Soler and Luna (2015) developed a
mathematical formalism based on the T-matrix theory of scattering, allowing to
compute the periods and damping times of kink oscillations of an arbitrary
configuration of parallel plasma cylinders.

Goossens et al. (2012) have focused on the role of vorticity in MHD waves. The
authors have found that for a piecewise constant density profile the
fundamental radial modes of non-axisymmetric modes, including kink modes, have
the same properties as purely surface Alfvén waves at a true density
discontinuity. But when the discontinuity is replaced with a continuous
variation of density, vorticity is spread out over the whole in-homogeneous
region. Along the same line, Goossens et al. (2020) have used both compression
and vorticity to characterise the spatial evolution of the kink MHD wave. The
most surprising result is the huge spatial variation in the vorticity
component parallel to the magnetic field. In the non-uniform part of the tube,
parallel vorticity increases to values that are several orders of magnitude
higher than those attained by the transverse components of the vorticity,
i.e., in planes normal to the straight magnetic field.

Note that the theoretical study of kink MHD waves in solar plasma waveguides
is usually based on the simplification that the transverse variation of
density is confined to a nonuniform layer much thinner than the radius of the
tube, i.e., the TTTB approximation. Soler et al. (2013) have developed a
general analytic method to compute the dispersion relation and the
eigenfunctions of ideal MHD waves in flux tubes with transversely nonuniform
layers of arbitrary thickness. Interestingly, the results for thick nonuniform
layers deviate from the behaviour predicted in the thin boundary approximation
and strongly depend on the density profile used in the nonuniform layer.

Goossens et al. (2014) have shown that kink waves do not only involve purely
transverse motions of the waveguiding plasma cylinders, but the velocity field
is a spatially and temporally varying sum of both transverse and rotational
motions. These rotational motions are not necessarily signatures of the
classic axisymmetric torsional Alfvén wave alone, because of the contribution
of the kink motion itself. This essentially means that in observations and
depending on the line of sight, the interpretation of the Doppler velocity can
be either very similar or very different to that from a purely torsional
Alfvén wave. Specifically, near the resonant surface, where the kink speed
equals the local Alfvén speed, we expect the Doppler signal to be like that of
an $m=1$ torsional Alfvén wave, while further from the resonant position the
signal would look like a kink wave (see Srivastava and Goossens, 2013, for the
discussion of an observational manifestation of this effect).

Finally, Soler and Terradas (2015) have theoretically investigated the
generation of small scales in nonuniform solar magnetic flux tubes due to
phase mixing of MHD kink waves. Using a modal expansion these waves are
written as a superposition of Alfvén continuum modes that are phase mixed as
time evolves. This analysis describes both the damping of global kink motions
and the building up of small scales due to phase mixing.

###  4.1 Non-exponential damping regime of resonant absorption

Pascoe et al. (2012) performed numerical simulations of kink oscillations
propagating in a coronal loop and found poor agreement with the classical
damping behaviour, described by Eqs. (6) and (7). These simulations were
carried out with a relatively wide inhomogeneous layer in order to reproduce
the strong damping rates observed for both standing and propagating kink waves
in the corona. Since the analytical derivations for the resonant absorption
damping rate are based on the thin boundary approximation, it would be
reasonable to expect numerical simulations with a wide boundary to have a
damping rate which differs from the analytical prediction, with differences of
up to 25% already reported by Van Doorsselaere et al. (2004a). However, Pascoe
et al. (2012) also found that the shape of the damping profile was
inconsistent with the analytical prediction, and proposed, empirically, that a
Gaussian damping profile was more appropriate than an exponential one.

Hood et al. (2013) accounted for the existence of this Gaussian damping regime
by producing an analytical description which considered the initial behaviour
of the kink mode, not just its asymptotic state. Their Eq. (47) describes the
amplitude of the damping profile for all times, though again subject to the
TTTB approximation. Figure 5 shows the variation of the amplitude $\xi_{r}$
(solid line) of a propagating kink wave in a coronal loop with density profile
parameters $\epsilon=0.2$ and $\zeta=1.3$. The dashed line represents the
analytical solution which accurately describes the variation of the damping
profile. The use of a logarithmic scale (right panel) demonstrates that the
damping profile is initially non-exponential, before switching to exponential
after several cycles. The initial stage was characterised for propagating
waves in terms of the normalised parameter $Z=\kappa kz/2$ being small, for
which the non-exponential damping regime could be approximated with a Gaussian
function, i.e., $\propto\exp(-t^{2})$, consistent with the empirical profile
suggested by Pascoe et al. (2012). This condition also implies an inverse
dependence of the location of the switch on the density contrast ratio $\zeta$
and the wavelength of the oscillation. The parametric study by Pascoe et al.
(2013) supported this dependence and proposed the general damping profile
(GDP) as a combination of the two approximations for the Gaussian and
exponential regimes, switching from one to another at a particular switch
height $h$ for propagating waves, or equivalently a switch time
$t_{\mathrm{switch}}$ for standing waves, given by

$\frac{h}{\lambda}=\frac{t_{\mathrm{switch}}}{P_{\mathrm{kink}}}=\frac{1}{\kappa}.$
(8)

The observational detection of this switch in damping profiles was proposed as
a means to estimate the density contrast ratio $\zeta$, whereas the particular
damping rate for each regimes depends on both $\zeta$ and the inhomogeneous
layer width $\epsilon$, and so neither rate alone could be used to constrain
both density profile parameters (e.g., Arregui and Goossens, 2019, and
references therein).

While this Gaussian damping behaviour was initially simulated and derived in
the context of propagating kink waves, its applicability to standing kink
waves (with the appropriate change in variable) has been demonstrated in
numerical simulations (e.g., Ruderman and Terradas, 2013; Magyar and Van
Doorsselaere, 2016; Pagano et al., 2018; Pascoe et al., 2019). For the
exponential damping regime, the relationship between damping length scales and
timescales for propagating and standing kink waves has been demonstrated
explicitly, see, e.g., derivations by Goossens et al. (2002) and Terradas et
al. (2010b).

The GDP proposed by Pascoe et al. (2013) combined both of the TTTB
approximations for the Gaussian and exponential damping regimes into a single
profile of the form

$A\left(t\right)=\left\\{\begin{array}[]{rl}\displaystyle\exp\left(-\frac{t^{2}}{2\tau_{\mathrm{g}}^{2}}\right)&\mbox{$t\leq
t_{\mathrm{switch}}$}\\\ \displaystyle
A_{\mathrm{s}}\exp\left(-\frac{t-t_{\mathrm{switch}}}{\tau_{\mathrm{D}}}\right)&\mbox{$t>t_{\mathrm{switch}}$}\end{array}\right.,$
(9)

$\frac{\tau_{\mathrm{g}}}{P_{\mathrm{kink}}}=\frac{2}{\pi\kappa\epsilon^{1/2}}.$
(10)

where the Gaussian damping time $\tau_{\mathrm{g}}$ here again corresponds to
a linear transition, as for the exponential damping time in Eq. (6).

Figure 6 shows the results of numerical simulations of standing kink
oscillations, performed as part of a parametric study by Pascoe et al. (2019)
to investigate the damping behaviour for loops with wide inhomogeneous layers.
Each panel corresponds to simulation with $\zeta=2$ while $\epsilon$ increases
from $0.1$ (left) to $1.0$ (right). For kink oscillations in low density
contrast loops such as these, the Gaussian damping profile (blue curves)
provides a much better description than the exponential damping profile (red
curves), and the GDP (green curves) further improves the description for later
times. Since the exponential, Gaussian, and general damping profiles are all
based on the thin boundary approximation, they each become poorer as
$\epsilon$ increases. The dashed curves correspond to damping profiles found
by fitting the results of the numerical simulations using spline interpolation
of the amplitude measured every half cycle of the oscillation (plus symbols).
The results of $300$ simulations were compiled by Pascoe et al. (2019) into a
lookup table to provide a convenient means of estimating the damping profile
beyond the applicability of the thin boundary approximation.

###  4.2 Observational evidence and seismological application of the two-
regime damping

Investigations of the shape of the damping profile of kink oscillations by De
Moortel et al. (2002) and Ireland and De Moortel (2002) suggested non-
exponential behaviour, though the time resolution of TRACE was insufficient
for conclusive evidence. In the statistical study of $223$ standing kink
oscillations by Goddard et al. (2016); Nechaeva et al. (2019), the authors
attempted to classify visually whether the damping profile appeared to be non-
exponential, exponential, or containing both profiles. Several clear examples
of non-exponential damping behaviour were observed, motivating a follow up
study by Pascoe et al. (2016c) which aimed to quantitatively test whether a
Gaussian or exponential damping profile best described the damping behaviour.
For six of the highest quality oscillations observed, the instantaneous
position of the oscillating loop as a function of time was fitted with a
sinusoidal oscillation with an exponential and then a Gaussian damping
profile, and the two models compared by their corresponding $\chi^{2}$ values.
Three of the six cases favoured the Gaussian damping profile, two favoured the
exponential, and the remaining one was inconclusive. Further analysis of one
of the loop oscillations by Morton and Mooroogen (2016) also supported the
damping profile being Gaussian rather than exponential, and demonstrated that
the period of oscillation varies in time with the variation of the loop’s
parameters (a feature of kink oscillations also observed by Wang et al., 2012;
White et al., 2013; Nisticò et al., 2013; Russell et al., 2015).

The purely Gaussian or exponential profiles considered in the above studies
represent the limiting cases of the two regimes of damping by resonant
absorption, but demonstrate that kink oscillations can be measured with
sufficient accuracy to discriminate between different damping behaviour. Thus,
the regime switching time $t_{\mathrm{switch}}$ can be used as an additional
observable for seismological diagnostics. The seismological method using both
regimes (Eq. 9) was first applied by Pascoe et al. (2016b) to estimate the
density profile parameters $\zeta$ and $\epsilon$. The seismologically-
inferred density profile was also compared with the transverse EUV intensity
profile of the loop to estimate the radius of the loop and hence the physical
width of the transition region $l_{\mathrm{tr}}=\epsilon a$. The method was
subsequently improved by Pascoe et al. (2017a) to include the additional
effects of a time-dependent period of oscillation, the presence of additional
longitudinal harmonics, and any decayless component of the oscillation. The
improved method also included the use of Bayesian inference (e.g., Arregui et
al., 2013b, 2015; Arregui, 2018) to improve the calculation of parameter
values and uncertainties. The shape of the damping profile is sensitive to the
level of the noise in the oscillation data (see Figure 3 of Pascoe et al.,
2018). Furthermore, the dependence of the damping rate on both $\zeta$ and
$\epsilon$ means the extent to which each of these parameters is constrained
depends on the particular value, and in general a strong constraint on one
parameter corresponds to a weak constraint on the other. The use of Bayesian
inference to calculate the joint posterior distribution is a convenient way of
representing these uncertainties.

Since the damping behaviour of kink oscillations is not always sufficient on
its own to strongly constrain the loop density profile parameters, it is
desirable to complement the seismological method with additional information.
Pascoe et al. (2017b) and Goddard et al. (2017) used the EUV intensity profile
of coronal loops to provide an independent estimation of $\epsilon$, and this
method was combined in Pascoe et al. (2018) to produce a diagnostic method
combining both seismological (kink oscillation) and spatial (EUV profile)
information, as shown in Figure 7. One of the results of these studies is that
coronal loops are observed to have a range of inhomogeneous layer widths such
that the thin boundary approximation cannot generally be considered
applicable. In particular, the inferred value of $\epsilon\sim 0.9$ for the
loop in Figure 7 is not consistent with the thin boundary approximation
assumed by the GDP (c.f. Figure 6). This motivated the parametric study of
Pascoe et al. (2019) to produce a seismological technique based on the results
of numerical simulations which is not restricted by the thin boundary
approximation (although it does not include nonlinear effects such as the
modification of the coronal loop density profile by KHI, see Section 10.1).

The above seismological studies are based on assumption of the linear density
profile in the inhomogeneous layer. As previously mentioned, a different
density profile would modify the constants of proportionality for the damping
rates due to resonant absorption. However, this effect represents a relatively
small uncertainty in modern seismology (see further discussion in Section 6.2
of Pascoe et al., 2018).

##  5 Kink oscillations in twisted cylinders

Eruptive phenomena occurring the corona, such as flares and mass ejections,
release free, or non-potential magnetic energy stored in active regions.
Coronal loops with a non-potential field have either a sigmoid shape or the
field twisted around the loop’s axis (e.g. Magyar and Nakariakov, 2020).
Coronal loops with a free magnetic energy can be considered as plasma
cylinders with an axially twisted magnetic field. In such loops the
equilibrium magnetic field has axial and azimuthal components, with the latter
referred to as $B_{\phi}$. The rate of the twist should not be very high, as
otherwise the equilibrium is unstable (see, e.g., Mei et al., 2018, and
references therein). At the axis of the cylinder the component $B_{\phi}$ is
zero and the magnetic field is purely axial. The main effect of magnetic twist
on transverse kink oscillations is to break the symmetry with respect to the
propagation direction. This property has relevant consequences for standing
modes that require in general the superposition of propagating waves. For
example, when there is no twist, the frequency of the mode with an axial
wavenumber $k_{z}$ is the same as for the mode $-k_{z}$ ($\omega_{\rm k}=\pm
k_{z}\,C_{\mathrm{k}}$ in the thin tube limit). It is straightforward to
construct the standing solution in this case. However, under the presence of
twist the situation is more complicated since the modes with wave numbers
$k_{z}$ and $-k_{z}$ have different frequencies. To solve properly this
problem we have to combine more than one wave to satisfy the boundary
conditions at footpoints, as it was shown in detail by Ruderman and Terradas
(2015) (see also references therein) for a particular choice of the dependence
of the azimuthal component of the field on the transverse coordinate (see also
Terradas and Goossens (2012)).

Interestingly, the oscillation period of standing kink waves is unaffected by
the presence of twist (for a weak twist and in the thin tube approximation).
Therefore the cylinder oscillates transversally at the characteristic kink
frequency, $\omega_{\rm k}$. Second order modifications to this frequency have
been calculated analytically in Ruderman and Terradas (2015). Nevertheless,
the most important effect of magnetic twist on transverse oscillations is
related to the polarisation of the movements. It can be shown that the change
in the direction of the polarisation is linearly proportional to the amount of
twist. This was studied in detail by Terradas and Goossens (2012); Ruderman
and Terradas (2015); Terradas et al. (2018). For linear twist profile,
$B_{\phi}=A_{\mathrm{twist}}r$ in $r<a$, the polarisation along the axis
(pointing in $z-$direction) changes according to the following simple
expression

$\displaystyle\xi_{x}$ $\displaystyle\approx$
$\displaystyle\cos\left(\frac{A_{\mathrm{twist}}}{B_{0}a}\frac{1}{4}\left(2z-L\right)\right)\,\sin\left(\frac{\pi}{L}z\right),$
$\displaystyle\xi_{y}$ $\displaystyle\approx$
$\displaystyle\sin\left(\frac{A_{\mathrm{twist}}}{B_{0}a}\frac{1}{4}\left(2z-L\right)\right)\,\sin\left(\frac{\pi}{L}z\right),$

where the variables $\xi_{x}$ and $\xi_{y}$ represent the displacements in the
$x$ and $y-$directions, the axial coordinate $z$ varies from 0 to $L$, and
$B_{0}$ is the axial magnetic field at the axis (Ruderman and Terradas, 2015)
.

Note that in those notations, in the zero-twist limit, $A_{\mathrm{twist}}=0$,
the axis displacement is in the $x-$direction only and purely sinusoidal along
the $z-$coordinate (satisfying line-tying conditions at the footpoints).
Nevertheless, when $A_{\mathrm{twist}}\neq 0$ the polarisation is in general
mixed and the previous expressions show that it depends on the position $z$
along the tube axis. The important result here is that a weak twist can
produce displacements in any direction perpendicular to the unperturbed tube
axis. In some cases the apparent displacement of a loop, produced by a
fundamental mode, may resemble that of the second parallel harmonic, i.e.,
with a node near the apex. From the seismological point of view, the
signatures of twist on observed standing kink oscillations could be used as a
way to infer the value of the azimuthal component of the magnetic field. But
since real coronal loops are in many cases non-planar and non-circular the
comparison between theory and observations is not straightforward. These
results are based on the assumption of very weak twist, allowing to avoid
difficulties that appear when the $B_{\phi}$ component of the magnetic field
is increased. Alfvénic resonances cannot be avoided for moderate twist even if
the transverse profile of the density is infinitely sharp. The effect of
magnetic twist on the nonlinear evolution of kink oscillations is discussed in
Sec. 10.1.

For propagating waves the effect of twist (and also flow) on the transverse
kink modes has been investigated analytically and numerically in, e.g., Karami
and Bahari (2010); Bahari (2017a, 2018). Bahari and Jahan (2020) have
concluded that the asymmetry of the wave about the apex point is not affected
much by the magnetic twist, but the magnetic twist causes an overestimation of
both the flow speed and kink speed in the oscillating loop.

##  6 Transverse oscillations of current-carrying loops due to the
electromagnetic interaction with the external electric currents

Figure 8: Left: A loop segment with a horizontal line-current
$i_{\mathrm{c}}$, situated at the height $h$ above the surface of the Sun, in
the magnetic environment formed by two external currents $I_{0}$ separated by
the distance $2d$. A virtual mirror current describing the interaction of the
loop current $i_{\mathrm{c}}$ with the electrically conductive surface of the
Sun is situated at the height $2h$ strictly below $i_{c}$ (not shown in the
sketch). The angle $\alpha$ shows the inclination of the apparent loop plane
(the red dashed line) to the vertical axis (adapted from Kolotkov et al.
(2016, 2018)). Right: Dependence of the vertically-polarised oscillation
period $P_{y}$ (12) upon the loop current $i_{c}$ for the loop length $L=250$
Mm, inclination angle $\alpha=\pi/6$, minor radius $a=1$ Mm, particle
concentration $n_{e}=1.5\times 10^{9}$ cm-3, and $I_{0}=10^{9}$ A (red),
$10^{10}$ A (green), and $10^{11}$ A (blue). The black solid line shows the
characteristic period $P_{\mathrm{mir}}$ (13), typical of vertically-polarised
oscillations in a shallowed magnetic dip with $h\ll d$.

An alternative mechanism for transverse oscillations of current-carrying
magnetic flux tubes was proposed in a series of works Kolotkov et al. (2016,
2018), based on the interaction of the electric current inside the flux tube
with the magnetic coronal environment and the electrically conductive
photosphere. In this section we discuss on how this model could be adapted to
transverse oscillations of coronal loops with electric current and assess, in
particular, the applicability and importance of this effect in modelling the
kink oscillations.

Consider a coronal loop segment with the horizontal electric current
$i_{\mathrm{c}}$ embedded in the background magnetic field with a dip formed
by two horizontal current sources $I_{0}$, situated at the lower layers of the
solar atmosphere and directed oppositely to $i_{\mathrm{c}}$ (see the
schematic sketch in Fig. 8). Also in the model, the loop segment is considered
to interact with the electrically conductive photosphere through the inclusion
of a so-called _mirror current_ , which is a virtual current situated strictly
symmetrically with respect to $i_{\mathrm{c}}$, has the same magnitude and
opposite direction (not shown in Fig. 8). Thus, the dynamics of the loop
segment in such a low-dimensional model is governed by the mutual effect of
the magnetic forces ${F}_{1}={F}_{2}=k_{1}/\sqrt{d^{2}+h^{2}}$ (between the
loop current $i_{\mathrm{c}}$ and two external currents $I_{0}$, with
$k_{1}=\mu_{0}I_{0}i_{\mathrm{c}}/2\pi$ and the characteristic spatial scales
$h$ and $d$ shown in Fig. 8 ) and ${F}_{\mathrm{m}}=k_{2}/2h$ (between
$i_{\mathrm{c}}$ and the mirror current, with
$k_{2}=\mu_{0}i_{\mathrm{c}}^{2}/2\pi$), and the gravity force
${F}_{\mathrm{g}}={\cal R}g$ (with ${\cal R}$ being a linear mass density of
the loop, measured in kg m-1). All forces in this model are taken per unit
length. We note that due to integrating over the loop cross-sectional area the
internal structure of the loop does not affect the discussed transverse
oscillations. Also, neglecting the line-tying boundary conditions for the
guiding magnetic field of the loop does not allow for taking the effects of
the magnetic hoop force and magnetic tension force into account, see e.g.,
Cargill et al. (1994); Vršnak (2008). On the other hand, a low-dimensional
nature of the model allows for its straightforward analytical treatment and
revealing the explicit relationships between the basic parameters of the
oscillations and the loop.

The equilibrium state of such a loop segment is maintained by the vertical
force balance

$\frac{2k_{1}h}{d^{2}+h^{2}}+\frac{k_{2}}{2h}={\cal R}g,$ (11)

connecting the loop parameters (${\cal R}$, $h$, and $i_{\mathrm{c}}$) with
those of the external field ($I_{0}$ and $d$). For example, Eq. (11) implies
that increase in the external current $I_{0}$, keeping the loop parameters
${\cal R}$, $h$, and $i_{\mathrm{c}}$ constant, would lead to the
corresponding increase of the distance $d$ thus decreasing the dimensionless
parameter $h/d$ and mitigating the effect of the external field dip on the
dynamics of the loop.

In the linear regime, i.e. for a small displacement of the loop from its
initial equilibrium position, the oscillations in the vertical and horizontal
directions are independent of each other and hence can be considered
separately. In particular, the period of small-amplitude vertically polarised
oscillations of the loop segment takes the following elegant form (cf. the
expression for the oscillation period derived for curved current-carrying
loops in Eq. (B14) of Cargill et al. (1994))

$P_{y}=\frac{P_{\mathrm{mir}}\sqrt{k_{1}k_{2}}}{[(2{\cal R}gh-
k_{2})^{2}-2k_{2}(2{\cal R}gh-k_{2})+k_{1}k_{2}]^{1/2}},$ (12)

where

$P_{\mathrm{mir}}=2\pi\sqrt{\frac{4{\cal R}h^{2}}{k_{2}}}\approx 2.6\times
10^{-2}\times\frac{a[\mathrm{Mm}]L[\mathrm{Mm}]\cos\alpha\sqrt{n_{e}[10^{9}\,\mathrm{cm}^{-3}]}}{i_{\mathrm{c}}[10^{10}\,\mathrm{A}]}[\mathrm{min}]$
(13)

is the characteristic value of the vertical oscillation period for the regime
of weak interaction with the external field dip, $h/d\ll 1$, independent of
the external currents $I_{0}$ (Kuperus and Raadu, 1974). In this regime, the
magnetic field dip is shallowed and the vertically polarised oscillation is
mainly sustained by the magnetic mirror force $F_{\mathrm{m}}$. For practical
purposes, the period $P_{\mathrm{mir}}$ (13) was rewritten in terms of the
loop length $L$, angle $\alpha$ between the loop plane and the vertical axis,
loop volume number density $n_{e}$, minor radius $a$, and the loop current
$i_{\mathrm{c}}$. As shown by Fig. 8, the values of the period $P_{y}$ are
about a few to several minutes for typical loop parameters (see, e.g., Zaitsev
et al. (1998); Khodachenko et al. (2009) for seismological estimations of the
loop current $i_{\mathrm{c}}$) and tend to $P_{\mathrm{mir}}$ for increasing
$I_{0}$ (lessening $h/d$), as prescribed by equilibrium condition (11). For
example, for the parameters used in Fig. 8 and $i_{\mathrm{c}}=10^{11}$ A, one
obtains $P_{\mathrm{mir}}\approx 1$ min. The possibility for the effective
damping of oscillations in terms of a similar model with $h/d\ll 1$ was
analytically demonstrated by Zaitsev and Stepanov (2018) through accounting
for the drag force between the oscillating flux tube and the ambient plasma.
In the same regime with $h/d\ll 1$, the period of horizontal oscillations
tends to infinity and hence is not discussed here.

In the nonlinear regime with the loop displacements comparable to the
characteristic spatial scales $h$ and $d$, the horizontal and vertical
oscillations were found to be strongly coupled between each other, with the
horizontal perturbation effectively exciting the vertical mode. The effect of
coupling was shown to be more pronounced for smaller angles between the
direction of the initial perturbation and the horizontal axis. As such, it
demonstrates the lack of a simple elliptical trajectory of the loop segment in
the nonlinear regime. Likewise, a metastable equilibrium of the loop was
revealed, which is stable to small-amplitude perturbations and may become
unstable if the oscillation amplitude exceeds a certain threshold. The
nonlinear oscillation periods were shown to acquire a dependence on the
oscillation amplitude, varying by up to 10–30% with respect to the linear
regime.

The presented model should be considered as a simple, essentially low-
dimensional model, not taking the magnetic tension force typically associated
with kink oscillations into account. Despite this, the obtained periods of
vertically polarised oscillations of the loop segment, driven by the
electromagnetic interaction of the loop current with the external field and
electrically conductive photosphere, are seen to be well consistent with
typical periods of kink oscillations. This indicates a clear need for
accounting for this mechanism in modelling and interpreting their
manifestations in observations. In particular, in the magnetic configurations
without a dip (or with a sufficiently suppressed dip satisfying the condition
$h\ll d$), the discussed vertically polarised oscillations may still occur due
to the mirror current effect.

##  7 Kink modes in the presence of parallel shear plasma flows

In the majority of observed plasma structures, stationary flows222Here we
consider the flows which last longer than several oscillation periods as
stationary. are field-aligned, as a consequence of the low-$\beta$ plasma. The
flow can be driven by pressure imbalances, as in the case of siphon flows
between the two footpoints of a loop, heating or cooling, leading to coronal
loop filling or draining, or be induced by magnetic reconnection (Reale,
2014).

###  7.1 Basic effect of flow on the kink eigen frequency and eigen function

Ruderman (2010) derived general expressions for the modification of the kink
frequency in standing waves due to the presence of an axial stationary flow,
$U_{0}$. Interestingly this author found that contrary to the static case,
different positions along the tube oscillate with a different phase. This
effect was further investigated by Terradas et al. (2011). Under the
assumption that the flow speed is much smaller than the Alfvén speed in the
cylinder, the kink frequency is modified as

$\displaystyle\omega=k_{z}\,C_{\mathrm{k}}{\left(1-\frac{\rho_{\mathrm{i}}}{\rho_{\mathrm{i}}+\rho_{\mathrm{e}}}\frac{U_{0}^{2}}{C_{\mathrm{k}}^{2}}\right)}\left[{1-\frac{\rho_{\mathrm{i}}\rho_{\mathrm{e}}}{\left(\rho_{\mathrm{i}}+\rho_{\mathrm{e}}\right)^{2}}\frac{U_{0}^{2}}{C_{\mathrm{k}}^{2}}}\right]^{-1/2},$
(14)

where

$\displaystyle k_{z}=n\frac{\pi}{L},\,\,\,n=1,2,\ldots.$ (15)

From the comparison with the eigen frequencies for the static case,
$\omega=k_{z}\,C_{\mathrm{k}}$, it turns out that flow always leads to a
frequency reduction, i.e., an increase in the oscillation period.

In the presence of an axial stationary flow, the displacement along the axial
axis has the following form

$\displaystyle\xi(t,z)=A\sin k_{z}z\cos\left(\omega t+k_{\mathrm{U}}z\right),$
(16)

where $A$ is an arbitrary constant, and we have introduced the wavenumber
modified by the flow,

$\displaystyle
k_{\mathrm{U}}=k_{z}\frac{\rho_{\mathrm{i}}}{\rho_{\mathrm{i}}+\rho_{\mathrm{e}}}\frac{U_{0}}{C_{\mathrm{k}}}\left[{{1-\frac{\rho_{\mathrm{i}}\rho_{\mathrm{e}}}{\left(\rho_{\mathrm{i}}+\rho_{\mathrm{e}}\right)^{2}}\frac{U_{0}^{2}}{C_{\mathrm{k}}^{2}}}}\right]^{-1/2}.$
(17)

Therefore, there is a linear phase dependence of the standing kink mode along
the loop, and in one full period, the eigenfunctions exhibit an asymmetry
about the loop top. This information about the eigenfunction was used by
Terradas et al. (2011) to infer, a stationary flow in an oscillating loop.

Bahari (2017a) extended the theory of standing waves in the presence of a
stationary flow to twisted magnetic tubes without resonant damping. Ruderman
and Petrukhin (2019) have addressed the standing wave problem under the
presence of a stationary flow and resonant damping. These authors obtained
expressions for the ratio of decrements and oscillation frequencies for
particular flow profiles, and concluded that for typical values of a
stationary flow in coronal loops the effect of the flow on parameters of kink
oscillations is weak.

For propagating waves the effects of stationary flows on resonantly damped
kink oscillations have been investigated analytically and numerically by
Terradas et al. (2010a); Soler et al. (2011); Bahari (2018).

###  7.2 Kink waves in jets

Kink oscillations have been detected in plasma structures with collimated
flows, for example in coronal jets and loops with siphon flows. Coronal jets
are intensively studied in the context of impulsive energy releases and the
mass supply to the solar wind (Raouafi et al., 2016). Coronal X-ray jets are
the largest among jets of other types (Shibata et al., 2007). Their widths are
reported between 2 and 20 Mm. Hot jets constitute high-speed outflows which
are observed to live from over a minute up to almost an hour. Perpendicular
non-uniformity of the equilibrium flow leads to waveguiding effects on fast
magnetoacoustic waves, in particular, on kink waves. As in static plasma
structures, the oscillatory displacement of the jet axis is an indication of a
kink oscillation. Kink oscillations of jets are an important tool for their
diagnostics (Vasheghani Farahani et al., 2009; Karamimehr et al., 2019).
Cirtain et al. (2007) reported kink displacements of a soft-X-ray jet, with
the oscillation period of 200 s and apparent propagation speeds about $800$ km
s-1. Similar oscillations with amplitudes around $800$ km and periods about
$14$ min have been observed as Doppler oscillations of the Hi Ly$\alpha$
coronal emission line at 1.43 $R_{\odot}$ above the limb, and interpreted as
kink oscillations of a narrow, jet-like ejection observed higher up in the
white-light corona (Mancuso and Raymond, 2015). A related topic is kink
oscillations of coronal streamers (Decraemer et al., 2020).

Under the assumption that the jet’s life time is much longer than the kink
oscillation period, Vasheghani Farahani et al. (2009) described linear kink
oscillations of hot coronal jets by adapting the ZSER model (see Section 3,
and also Goossens et al., 1992). Dispersion relation (1) modified by a uniform
axial stationary flow $U_{\mathrm{i}}$ becomes

$\frac{\rho_{\mathrm{0i}}}{\rho_{\mathrm{0e}}}\,\frac{(C_{\mathrm{Ai}}^{2}-\Omega_{i}^{2}/k_{z}^{2})}{(C_{\mathrm{Ae}}^{2}-\omega^{2}/k_{z}^{2})}\frac{\kappa_{\mathrm{e}}}{n_{\mathrm{i}}}\frac{K^{\prime}_{1}\,(\kappa_{\mathrm{e}}\,a)}{K_{1}\,(\kappa_{\mathrm{e}}\,a)}=\frac{J^{\prime}_{1}\,(n_{\mathrm{i}}\,a)}{J_{1}\,(n_{\mathrm{i}}\,a)},$
(18)

where $\Omega_{\mathrm{i}}=\omega-U_{\mathrm{i}}k_{z}$, and the expression for
$n_{\mathrm{i}}$ (see the text under Eq. (1) has the Doppler-shifter phase
speed $V_{\mathrm{ph}}-U_{\mathrm{i}}$ instead of $V_{\mathrm{ph}}$. Using
this model, Vasheghani Farahani et al. (2009) concluded that if the only
damping mechanism is due to resonant absorption, kink oscillations with the
observed period of about 200 s in soft X-ray jets last several oscillation
cycles before completely damping out. Regarding the origin of the
oscillations, Vasheghani Farahani et al. (2009) proved that neither KHI caused
by the equilibrium flow shear at the jet’s boundary, nor instabilities
connected with negative energy waves (see Section 7.3) could be responsible
for the generation of the oscillations.

###  7.3 Negative energy wave effects on kink oscillations

Shear flows, for example, in plasma jets, could lead to an interesting
phenomenon of negative energy waves (e.g., Ryutova, 1988; Ruderman and
Goossens, 1995). When a shear flow speed, i.e., the difference between the
flows internal and external to a plasma structure that hosts MHD waves exceeds
the phase speed of any of guided waves, that wave becomes a backward wave
(e.g., Nakariakov and Roberts, 1995). In other words, a backward wave is the
wave which propagates in the direction opposite to the direction in the
absence of a flow. If the source of the stationary flow is excluded from the
considered system, the energy of backward waves becomes negative (e.g.,
Joarder et al., 1997). Any decrease in the energy, for example, due to
dissipation or leakage, leads to the increase in the amplitude of a wave with
the negative energy. It results in so-called negative energy wave
instabilities.

In the linear regime of a negative energy wave instability, the negative
energy wave amplitude experiences the exponential growth, while in the
nonlinear regime a much faster explosive instability is possible, when the
amplitude reaches an infinite value in a finite time333Obviously, in reality,
the infinite amplitude is not reached, as the amplitude growth above a certain
level is counteracted by additional nonlinear effects. An important feature of
negative energy instabilities is their occurrence for flow shears much lower
than the threshold of KHI (Joarder et al., 1997).

To provide further insight, consider a plasma cylinder with a straight
equilibrium magnetic field along the tube axis embedded in a viscous plasma
medium with a straight magnetic field. The equilibrium is similar to the ZSER
model (Section 3), but, in addition, there is a field-aligned equilibrium flow
inside the cylinder with the stationary speed $U_{0}$. In addition, the plasma
viscosity $\nu_{\mathrm{e}}$ is taken into account in the external medium. The
equilibrium flow creates a shear at the boundary of the cylinder. In the
incompressible limit, the propagation of kink waves guided by the cylinder is
described by the dispersion relation

$\displaystyle\left[\left(\frac{\rho_{\mathrm{0i}}I_{1}}{k_{z}I^{\prime}_{1}}-\frac{\rho_{0\mathrm{e}}K_{1}}{k_{z}K^{\prime}_{1}}\right)\omega^{2}-2k_{z}U_{0}\frac{\rho_{\mathrm{0i}}I_{1}}{k_{z}I^{\prime}_{1}}\omega+\frac{\rho_{\mathrm{0i}}I_{1}}{k_{z}I^{\prime}_{1}}(k_{z}U_{0})^{2}\right.$
$\displaystyle\left.-\frac{\rho_{\mathrm{0i}}I_{1}}{k_{z}I^{\prime}_{1}}\omega_{\mathrm{Ai}}^{2}+\frac{\rho_{0\mathrm{e}}K_{1}}{k_{z}K^{\prime}_{1}}\omega_{\mathrm{Ae}}^{2}\right]$
$\displaystyle+i\omega\rho_{0\mathrm{e}}\nu_{\mathrm{e}}\left[\frac{K_{1}}{k_{z}K_{1}^{{}^{\prime}2}}\left({\cal
D}K^{\prime}_{1}-\frac{K_{1}^{\prime}}{a^{2}}\right)+\frac{2K_{1}^{2}}{k_{z}^{2}a^{3}K_{1}^{{}^{\prime}2}}-2k_{z}\frac{K_{1}^{\prime\prime}}{K_{1}^{\prime}}\right]=0,$
(19)

where, $\omega_{\mathrm{Ae}}=k_{z}C_{\mathrm{Ae}}$ and
$\omega_{\mathrm{Ai}}=k_{z}C_{\mathrm{Ai}}$. As in the ZSER model,
$I_{1}(k_{z}a)$ and $K_{1}(k_{z}a)$ represent the modified Bessel functions of
the first order and of the first and second kinds, respectively. Note that the
argument $k_{z}a$ has been omitted, and ${\cal D}$ is a differential operator
defined in (Yu and Nakariakov, 2020). We need to point out that in the
considered incompressible limit, the internal structure of the perturbations
is given by the modified Bessel function $I_{1}$ in contrast with the ZSER
model which is essentially compressible. The dispersion relation is a
quadratic equation with respect to $\omega$. Because of the viscosity,
coefficients of this equation include the imaginary unity. Thus, its solution
could be written as a complex frequency constituted by real
$(\omega_{\mathrm{r}})$ and imaginary $(\omega_{\mathrm{i}})$ parts. The
explicit expressions regarding the phase speed, and the damping/growth
determined by $\omega_{\mathrm{i}}$ are

$\displaystyle
V_{\mathrm{ph}\pm}=\left({U_{0}\pm\sqrt{\frac{\rho_{0\mathrm{e}}I_{1}^{\prime}K_{1}}{\rho_{\mathrm{0i}}I_{1}K_{1}^{\prime}}U_{0}^{2}+{\cal
A}{\cal B}}}\right)/{\cal B},$ (20)
$\displaystyle\frac{\omega_{\mathrm{i}}}{\omega_{\mathrm{r}}}=\pm\frac{\nu_{\mathrm{e}}\frac{\rho_{0\mathrm{e}}I^{\prime}_{1}K_{1}}{\rho_{\mathrm{0i}}I_{1}K_{1}^{\prime}}\left[\frac{(a^{2}\nabla^{2}K_{1}^{\prime}-K_{1}^{\prime})}{K_{1}^{\prime}}-\frac{2k_{z}^{2}a^{2}K_{1}^{\prime\prime}}{K_{1}}+\frac{2K_{1}}{k_{z}aK_{1}^{\prime}}\right]}{C_{\mathrm{Ai}}k_{z}({\cal
B}V_{\mathrm{ph\pm}}-U_{0})},$ (21) $\displaystyle{\cal
A}=\left(C_{\mathrm{Ai}}^{2}-\frac{\rho_{0\mathrm{e}}I_{1}^{\prime}K_{1}}{\rho_{\mathrm{0i}}I_{1}K_{1}^{\prime}}C_{\mathrm{Ae}}^{2}\right),\,\,\,\,\,{\cal
B}=\left(1-\frac{\rho_{0\mathrm{e}}I_{1}^{\prime}K_{1}}{\rho_{\mathrm{0i}}I_{1}K_{1}^{\prime}}\right).$
(22)

Eqs. (20) and (21) provide quantitative and qualitative information regarding
the influence of the shear flow and viscosity on various aspects of the kink
wave regarding its phase speed, damping, instability thresholds, and negative
energy wave excitation. In particular, it could be noticed from Eq. (21) that
the sum of the three terms of the numerator, which dictates whether the growth
rate is positive or negative, is positive for longer wavelengths as the
denominator of the first and third terms (terms with positive sign) possess
the square of the wave number that makes them large compared to the second
term that has the wave number in its numerator. Negative energy wave
instability appears for all axial wave numbers. In general, the onset of the
negative energy wave instability of the kink perturbation occurs when the
shear flow speed exceed the critical speed

$U_{\mathrm{cr}}=C_{\mathrm{Ai}}\sqrt{1-\frac{\rho_{\mathrm{0e}}}{\rho_{\mathrm{0i}}}\frac{C_{\mathrm{Ae}}^{2}}{C_{\mathrm{Ai}}^{2}}\frac{I_{1}^{\prime}K_{1}}{I_{1}K_{1}^{\prime}}}.$
(23)

More rigorously, negative energy wave instabilities occur when the shear flow
speed is between $U_{\mathrm{cr}}$ and the KHI threshold. In particular, in
the long-wavelength regime, the flow speed shear allowing for the onset of
negative energy wave instabilities is several times lower than the KHI
threshold. Hence the threshold of the negative energy wave instability tends
to longer wavelengths. The lowest value of the shear flow, allowing for the
negative energy wave instability, is reached for the axial wavelength
comparable to the diameter of the cylinder. This result is readily applicable
to standing waves, as Ruderman (2018) showed that the growth rate of a
standing wave is equal to the growth rate of the propagating wave with
negative energy minus the damping rate of the propagating wave with positive
energy.

Zhelyazkov et al. (2016) carried out case studies on observed coronal jets and
stated that the onset of KHI regarding kink waves depends on whether the
plasma jet is incompressible or compressible, as the incompressibility
elevates the KHI threshold speed. The magnetic twist of the jet decreases the
KHI threshold speed. Nonetheless, a weak twist in case of an incompressible
plasma jet still has a threshold higher than the compressible case. However,
for the incompressible limit, the shear speed may not need to be be more than
40% faster than the internal Alfvén speed to create negative energy waves. It
is worth noting that the viscosity of the external medium itself provides
information regarding the temperature of the external medium proving adequate
for the background temperature to play a direct role on the negative energy
kink wave excitation (Yu and Nakariakov, 2020).

##  8 Kink oscillations in loops undergoing cooling

Even though most coronal loops live for much longer than their characteristic
cooling time (Reale, 2014), they appear to undergo continuous thermal
evolution, for most of the time being in a cooling phase (Viall and Klimchuk,
2012). There have been numerous observations of kink oscillations in loops
undergoing evolution, manifesting in EUV variability, mostly consistent with
cooling (e.g., Aschwanden and Terradas, 2008). In some particular cases,
undamped large-amplitude kink oscillations were observed in these apparently
cooling loops (Aschwanden et al., 2002; Aschwanden and Schrijver, 2011).
Therefore, it seemed likely that cooling of the plasma during kink
oscillations might have an impact on the oscillation properties. In what
followed, numerous studies exploited the influence of time-dependent coronal
loop models on the properties of kink oscillations. The first such studies
incorporating loop cooling indicated that cooling contributes to the damping
of kink oscillations (Morton and Erdélyi, 2009). However, Ruderman (2011b)
showed that neglecting the flow generated by the density decrease due to
cooling led to wrong conclusions in Morton and Erdélyi (2009) about the effect
of cooling, leading to an amplification of the oscillations instead.
Additionally, the decrease in density leads to a higher kink speed of the
loop, thus cooling reduces the kink oscillation period. Taking into account
resonant absorption, Ruderman (2011a) showed that the amplification due to
cooling is not very efficient in counterbalancing resonant damping of kink
oscillations, except for very short cooling times. This was verified using 3D
numerical simulations by Magyar et al. (2015), who found an even weaker
amplification of the oscillations due to cooling, when accounting properly for
the density evolution at the loop footpoints. This decrease in the efficiency
of damping when considering time dependent density at the loop footpoints was
confirmed theoretically by Bahari (2017b). It was concluded that cooling alone
cannot explain the observed undamped amplitude kink oscillations (see Section
12), especially when taking into account enhanced damping due to nonlinearity
(see Section 10) (Magyar and Van Doorsselaere, 2016). An additional effect of
loop expansion was considered by Ruderman et al. (2017) and Shukhobodskiy et
al. (2018). They showed that loop expansion reduces the damping rate due to
resonant absorption, and accounting for the amplification due to cooling could
result in this case in undamped oscillations, for specific parameters. In
Figure 9 the amplitude evolution of the fundamental kink oscillation of a
coronal loop from Magyar et al. (2015) (without expansion) is presented,
compared with the analytical results described above.

Figure 9: Normalized displacement amplitudes in a fundamental kink oscillation
at the apex over time, from three different numerical simulations: linear
perturbation with cooling (red dots), without cooling (black dots) and
nonlinear perturbation with cooling (green dots). The red curve represents the
best-fit exponential decay, the blue curve is the analytically predicted
displacement from Ruderman (2011a), and the brown curve is accounting for
expansion (Ruderman et al., 2017). Adapted from Magyar et al. (2015).

These theoretical models were tested and applied for coronal seismology on
observed oscillations of cooling coronal loops by Nelson et al. (2019).

##  9 Oscillation period ratios and their seismological inferences

###  9.1 Theoretical modelling of the period ratio

Theoretically, the ratios of the fundamental kink mode period
$P^{(1)}_{\mathrm{kink}}$ to the $n$-th parallel harmonic period,
$P^{(1)}_{\mathrm{kink}}/(nP^{(n)}_{\mathrm{kink}})$, have been shown to
depend on the wave dispersion, i.e., the effects of finite $k_{z}$,
gravitational stratification, axial and perpendicular density structuring,
loop cross-sectional ellipticity and geometry, and the axial magnetic field
non-uniformity due to field expansion (e.g., Andries et al., 2009b, and
references therein). With regards to the density non-uniformity, this also
depends upon the loop’s temperature, curvature, inclination, etc., that affect
the density scale-height and thus the harmonic period ratios (Orza et al.,
2012; Orza and Ballai, 2013). For example, Ruderman et al. (2016) showed that
the period ratio $P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}}$ is lower
or greater than unity when the kink speed increases or decreases with height,
respectively. The former case represents loops where the plasma density is
gravitationally stratified and the magnetic field is approximately constant
(weakly expanding with height), while the latter case is applicable to loops
with a significant enough expansion with height such that the rate of decrease
in $B^{2}$ from the loop footpoints to apex dominates over the rate at which
$\rho$ decreases. A similar approach has also been used to predict kink mode
period ratios in prominence threads (e.g., Soler et al., 2015, and references
therein), where the plasma density is assumed to be greater at the apex than
at the footpoints, i.e. the kink speed decreases towards the apex.

Karami and Bahari (2012) considered the effect of magnetic twist, considering
the twisted field to be localised in an annulus region of the cylinder, which
also resulted in the deviation of the period ratio of first two harmonics
($P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})$) from unity. As the
twist parameter increases, it was found that the period ratio decreases from
unity, and achieves a minimum value about 0.9 at approximately
$B_{\phi}/B_{z}=0.01$, and for stronger twist increases to unity again.

Thus, the departure of the normalised kink period ratios from unity can be
employed as a diagnostics tool of coronal loops, and provide information
crucial for understanding this enigmatic plasma structure (e.g., Ballai et
al., 2011; Srivastava and Goossens, 2013; Pascoe et al., 2016a; Li et al.,
2017b; Duckenfield et al., 2018, 2019).

####  9.1.1 Effect of perpendicular nonuniformity

The most basic reason for the departure of the
$P^{(1)}_{\mathrm{kink}}/(nP^{(n)}_{\mathrm{kink}})$ ratios from unity is the
effect of waveguide dispersion. Indeed, as the phase speed of the kink wave
decreases with the increase in the axial wave number $k_{z}$, different
parallel harmonics which have the wavelengths $2L/n$ have different phase
speeds $\omega/k_{z}=2L/(nP^{(n)}_{\mathrm{kink}})$ (see Figure 4), and thus
the increase in the mode number does not decrease the oscillation period by an
integer $n$. Exact expressions for dispersion relations could be obtained for
a handful of transverse density profiles only. Macnamara and Roberts (2011)
studied the effect of waveguide dispersion on the period ratios in a
zero-$\beta$ plasma slab with a symmetric Epstein density profile, and
obtained

$\displaystyle{\left({\frac{P^{(1)}_{\mathrm{kink}}}{2P^{(2)}_{\mathrm{kink}}}}\right)}^{2}=\displaystyle{\frac{1}{4}}{\left(\frac{\displaystyle
2\frac{\pi^{2}d^{2}}{4L^{2}}-\frac{C^{2}_{\mathrm{Ai}}}{C^{2}_{\mathrm{Ae}}}+\sqrt{\frac{C^{4}_{\mathrm{Ai}}}{C^{4}_{\mathrm{Ae}}}+4\frac{\pi^{2}d^{2}}{4L^{2}}-4\frac{C^{2}_{\mathrm{Ai}}}{C^{2}_{\mathrm{Ae}}}\frac{\pi^{2}d^{2}}{4L^{2}}}}{\displaystyle\frac{\pi^{2}d^{2}}{8L^{2}}-\frac{C^{2}_{\mathrm{Ai}}}{C^{2}_{\mathrm{Ae}}}+\sqrt{\frac{C^{4}_{\mathrm{Ai}}}{C^{4}_{\mathrm{Ae}}}+\frac{\pi^{2}d^{2}}{4L^{2}}-\frac{C^{2}_{\mathrm{Ai}}}{C^{2}_{\mathrm{Ae}}}\frac{\pi^{2}d^{2}}{4L^{2}}}}\right)},$
(24)

where $d$, $C_{\mathrm{Ai}}$, $C_{\mathrm{Ae}}$ are respectively the half-
width, and the Alfvén speeds at the slab’s axis and infinity. For thin and
long or short and fat loops the period ratio
($P^{(1)}_{\mathrm{kink}}/2P^{(2)}_{\mathrm{kink}}$) is close to one. In
particular, in the long wavelength limit, $k_{z}d\to 0$, the ratio tends to
unity. However, this ratio rapidly decreases with the increase in $k_{z}d$,
and reaches 0.75 at $k_{z}d\approx 0.05$ in a loop with
$C_{\mathrm{Ae}}/C_{\mathrm{Ai}}=10$. This effect is even more pronounced for
$C_{\mathrm{Ae}}/C_{\mathrm{Ai}}>10$, the condition which can be seen in
flaring loops.

Yu et al. (2015) performed a comprehensive study of dispersive properties of
kink waves in a plasma slab with the linear, parabolic, inverse-parabolic and
sine profiles of the density, sandwiched in between two regions with constant
densities. Interestingly, for those profiles, kink oscillation periods were
found to differ by $<13$% from each other. Chen et al. (2018) developed that
study accounting for finite-$\beta$. It was established that for parameters
typical for the coronal plasma, the finite-$\beta$ effect on kink oscillation
is at most marginal.

####  9.1.2 Effect of axial non-uniformity

For a coronal loop of length $L$, assuming a constant axial magnetic field and
a gravitationally stratified plasma density with scale height $H$, if $L\ll
H$, then the first three kink mode harmonic periods can be approximated as

$\displaystyle P^{(1)}_{\mathrm{kink}}$ $\displaystyle=$ $\displaystyle
P_{\mathrm{kink}}\left(1+\frac{L}{3\pi^{2}H}\right)^{-1},$ (25) $\displaystyle
2P^{(2)}_{\mathrm{kink}}$ $\displaystyle=$ $\displaystyle
P_{\mathrm{kink}}\left(1+\frac{L}{15\pi^{2}H}\right)^{-1},$ (26)
$\displaystyle 3P^{(3)}_{\mathrm{kink}}$ $\displaystyle=$ $\displaystyle
P_{\mathrm{kink}}\left(1+\frac{L}{35\pi^{2}H}\right)^{-1},$ (27)

where $P_{\mathrm{kink}}$ is the fundamental kink mode period of a coronal
loop of constant kink speed with a value equivalent to that of the kink speed
at the footpoint of a gravitationally stratified loop (e.g. Andries et al.,
2009b). For a loop inclined from vertical, the scale height $H$ should be
multiplied by a cosine of the inclination angle.

Assuming the density to be constant along the loop, and taking magnetic field
expansion with height into account, results in the period ratio,

$\frac{P^{(1)}_{\mathrm{kink}}}{2P^{(2)}_{\mathrm{kink}}}\approx
1+\frac{(3\Gamma^{2}-1)}{2\pi^{2}},$ (28)

where $\Gamma=a_{\mathrm{apex}}/a_{\mathrm{f}}$ is the magnetic expansion
factor such that $a_{\mathrm{apex}}$ and $a_{\mathrm{f}}$ are the minor radii
at the loop apex and footpoints, respectively (Verth and Erdélyi, 2008). Note
that Eq. (28) was derived assuming a weakly expanding potential magnetic field
with a straight axis, i.e., a “magnetic bottle” type configuration. Since
$\Gamma>1$ for a loop that expands in with height Eq. (28) gives a period
ratio $P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})>1$. Therefore, if
the magnetic field strength is decreasing with height, i.e., there is an
expanding magnetic flux tube, then it has the opposite effect on the period
ratio to that of density stratification. This can be understood mathematically
by considering the governing Sturm–Liouville problem in the long wavelength
limit where both the density and magnetic field strength vary along the loop,

$\frac{d^{2}}{dz^{2}}\left(\frac{\xi_{\perp}}{a(z)}\right)+\frac{\omega^{2}}{C_{\mathrm{k}}^{2}(z)}\left(\frac{\xi_{\perp}}{a(z)}\right)=0,\;\;\xi_{\perp}=0\;\;\mathrm{at}\;\;z=0\;\;\mathrm{and}\;\;z=L,$
(29)

where $\xi_{\perp}(z)$ is the displacement perturbation perpendicular to the
magnetic field, $a(z)$ is the minor radius varying along the field, and
$C_{\mathrm{k}}^{2}(z)=2B^{2}(z)/[\mu(\rho_{0\mathrm{i}}(z)+\rho_{0\mathrm{e}}(z))]$.
Since the kink speed squared in Eq. (29) has the terms $B^{2}(z)$ and
$\rho_{0\mathrm{i}}(z)+\rho_{0\mathrm{e}}(z)$ reciprocal to each other, if the
magnetic field strength and plasma density both decrease with height then they
will have an opposing effect on the solution. This is readily apparent when
considering the effect of magnetic and density stratification on the
eigenfunction of the second parallel harmonic. Loop expansion shifts the anti-
nodes towards the apex of the loop but gravitational density stratification
shifts them in the opposite direction towards the footpoints (see Figs. 6 and
7 in Andries et al. (2009b)). Note also from Eq. (29), that in the particular
case when $B^{2}(z)\propto\rho_{0\mathrm{i}}(z)+\rho_{0\mathrm{e}}(z)$ we will
have $P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})=1$ since the kink
speed is constant along the loop with both effects exactly cancelling each
other. Results of full-scale numerical simulations of kink oscillations in a
stratified 3D coronal loop performed by (Afanasyev et al., 2019) agree well
with the estimations made by (Andries et al., 2005; Verth and Erdélyi, 2008).

####  9.1.3 Effect of stationary axial flow

As it is discussed in Section 7.2, a shear of an equilibrium steady flow leads
to the modification of dispersive properties of kink waves, and hence the
period ratios. Li et al. (2013) studied the effect of flows on the period
ratio of standing kink modes in a plasma slab. It was found that the flow
reduces $P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})$ up to 23%
compared with the static case modelled by Macnamara and Roberts (2011). Chen
et al. (2014) generalised this theory and found that the flow significantly
reduces $P^{(1)}_{\mathrm{kink}}/(nP^{(n)}_{\mathrm{kink}})$ from unity for
higher $n$ and weaker density contrast between the waveguide and its
surrounding. In the high density contrast case, the period ratio has a
substantial deviation from unity, up to 13.7 % when the Alfvénic Mach number
reaches 0.8. Yu et al. (2016) confirmed that conclusion by showing that a
steady flow can reduce $P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})$ by
up to 17% relative to the static case even when the density contrast
approaches infinity.

In summary, the combined effects of field-aligned flow, longitudinal density
and magnetic stratification and transverse structuring all affect kink mode
period ratios (Yu et al., 2015). Hence those effects could be estimated
seismologically by this observable, provided there is a procedure allowing one
to estimate them separetely.

###  9.2 Observational detection of higher parallel harmonics and their
seismological applications

Observations of higher parallel harmonics of the kink mode remain a
challenging task. The detection of multiple parallel harmonics in the
displacement signal requires the fitting of a background trend simultaneously
with the oscillatory components, since detrending the time series would bias
subsequent results. In particular, the assumption of an oscillation being
equally distributed about the equilibrium position is satisfied by a weakly
damped harmonic oscillation, but not necessarily for a strongly damped
oscillation with or without additional harmonics present. Pascoe et al.
(2017c) have considered background trends describing with the change in
equilibrium associated with the kink mode excitation mechanism for contracting
and displaced coronal loops.

White et al. (2012) used the hot channels of SDO/AIA instrument and observed
an oscillating coronal loop off the east solar limb in the 131 Å and 94 Å
bandpasses suggesting a temperature in the range of 9–11 MK. They detected a
kink mode period of $302\pm 14$ s and a damping time of $306\pm 43$ s, and
confirmed that this transverse oscillation was more likely to be a higher
harmonic mode since they detected a spatial phase shift of about $180^{\circ}$
in the opposite legs of the loop. It was suggested that the oscillation
excitation mechanism was directly related with reconnection processes that
resulted in the formation of a post-flare loop, and that the excitation
mechanism was not due to a low coronal eruption, as in the majority of
decaying kink oscillations (Section 11).

Transequatorial loop systems are large-scale loop structures that are prone to
be perturbed non-uniformly. Long et al. (2017) used joint imaging and spectral
observations and studied such a system as it was impacted by an EIT wave. They
found that the transverse oscillation amplitude was very small at the loop
apex, so they judged that this loop system oscillated in the second parallel
harmonic mode. Using a seismological approach they estimated a magnetic field
strength of $5.5\pm 1.5$ G which was consistent with a Potential Field Source
Surface (PFSS) magnetic field extrapolation.

Since kink modes can be vertically or horizontally polarised, and observations
have projection effects due to the line of sight and loop inclination, these
cause ambiguities that bring additional difficulties in the accurate spatial
identification of higher harmonics. Yuan and Van Doorsselaere (2016b)
developed a forward model technique to address these issues. They modelled the
manifestation of standing kink modes of coronal loops in specific EUV
bandpasses, allowing the plasma parameters, loop widths and viewing angles to
be adjusted to match observed events of kink loop oscillations (Figure 10). It
was demonstrated that these forward models could be used to effectively
identify kink oscillation harmonics, their polarisations, reproduce the
general profile of oscillation amplitudes and phases, and also show signatures
of multiple harmonic periodicities in the associated EUV emission intensity.

Li et al. (2017b, 2019) analysed kink modes in a pair of coronal loops that
were only about 10” apart from each other. It was found that these two loops
oscillated differently after the excitation by a C-class solar flare. One loop
oscillated with a fundamental mode of 4-min periodicity, whereas the
oscillation of the second loop could be considered as a superposition of two
oscillations, with periods of 4 min and 2-min. According to the amplitude
distribution along the loop (see Figure 4 in Li et al. (2017b)), the 2-min
oscillation node was located at the loop top, so the authors interpreted this
to be the second parallel harmonic. The 4-min oscillation was interpreted as a
fundamental kink mode. This observation shed light on the dynamics of the
excitation and energy propagation of transverse loop oscillations, since the
two oscillating loops were so very closely spaced.

Guo et al. (2015) detected a fundamental kink mode and its second harmonic,
with the periods $530.2\pm 13.3$ s and $300.4\pm 27.7$ s (or $334.7\pm 22.1$
s), respectively. In this study,
$P^{(1)}_{\mathrm{kink}}/(2P^{(2)}_{\mathrm{kink}})<1$, and it was concluded
that the density stratification had a stronger effect on the period ratio than
the axial non-uniformity of the magnetic field variation.

The inversion procedure which allows to deduce parameters of an oscillating
loop from the period ratio measurement is rather subjective, and is based on
certain hypotheses of the main mechanism affecting the ratio. Arregui et al.
(2013a) suggested to use methods of Bayesian statistics to compare the
plausibility of two competing hypotheses, the density stratification and
magnetic field expansion. If additional information is available, e.g., the
damping profile, such a method could be extended to obtain more accurate
seismology. Pascoe et al. (2017a) analysed a kink oscillation with a
reasonable damping function, and extracted a set of three parallel harmonics.
This information was used for the estimation of the density ratio in the
oscillating loop (Figure 11).

Figure 10: (a) Active region 11121 observed at the south-east solar limb by
the SDO/AIA in the 131 Å channel. (b) Difference images made by subtracting
two images taken at about half an oscillation cycle apart. (c)-(f) Difference
images of synthesised oscillating loop model for various modes. Adaptation of
Figure 7 in Yuan and Van Doorsselaere (2016b)

Figure 11: Bayesian inference with higher harmonic modes. Top left: loop
displacement as a function of time. The blue line represents a background
trend, and the green is the model based on the assumption that the signal
includes several decaying harmonic oscillations. The red shaded region
represents the 99% credible intervals for the loop position predicted by the
model. Top right: wavelet analysis of the loop oscillation, with colours
representing the normalised spectral amplitude. The dashed lines show the
time-dependent periods of oscillation described by our model. Middle left:
detrended loop position (symbols) with the first (green), second (blue), and
third (red) longitudinal harmonics. Middle right: density profile parameters
determined by the oscillation damping envelope. The red bars are based on the
median values and the 95% credible intervals, indicated by the dotted and
dashed lines, respectively, in the histograms (bottom panels). The solid
curves are fits to the histogram data using the exponentially modified
Gaussian function. Adaptation of Figure 6 in (Pascoe et al., 2017a).

##  10 Nonlinear effects in kink oscillations

###  10.1 Nonlinear generation of fluting perturbations by kink-driven
Kelvin–Helmholtz instability

At high amplitudes, the kink mode evolution can be entirely different than in
the small-amplitude linear regime. The nonlinear regime is well characterised
by the nonlinearity parameter $\nu_{\mathrm{NL}}\approx{AL}/{a}$, where
$A={v_{0}}/C_{\mathrm{Ai}}$, is the normalised amplitude given by the ratio of
the initial velocity amplitude $v_{0}$ to the internal Alfvén speed
$C_{\mathrm{Ai}}$, and $L$ and $a$ correspond to the length and the radius of
the loop, respectively (Magyar and Van Doorsselaere, 2016; Ruderman and
Goossens, 2014). A nonlinear evolution is obtained for $\nu_{\mathrm{NL}}$
larger than $1$. We shall be focusing on this regime here444Another common
measure of nonlinearity is the ratio $\xi/a$ of the displacement of the flux
tube to its radius. This is related to our definition as
$\xi/a=\nu_{\mathrm{NL}}/\pi\sqrt{(1+\rho_{\mathrm{0e}}/\rho_{\mathrm{0i}})/2}\approx
0.25\nu_{\mathrm{NL}}$, taking usual values for the external and internal
densities $\rho_{\mathrm{0e}}$ and $\rho_{\mathrm{0i}}$.. Nonlinear regime of
the kink mode may lead to the development of KHI and the Rayleigh–Taylor
instability (RTI). The most common and important of the dynamic instabilities
in terms of influence on the overall structure of the loop is the KHI. Recent
analytical and numerical studies have expanded our understanding of the wave
modes that accompany the kink mode in its nonlinear evolution.

Figure 12: Variation of different harmonics of the azimuthal wave number
spectra $G(m)$ of the radial velocity in a kink oscillation, measured near the
boundary of a simulated coronal loop, in time. Different colours and line
styles (see legend) denote different azimuthal modes along a circle fitting
the edge of the loop prior to the kink mode. The $m=1$ mode denotes the
fundamental mode, and we show up to $m=5$. The numerical simulations
corresponding to these results are described in (Antolin and Van Doorsselaere,
2019).

The flows which are strongly sheared in the radial direction, resulting from
resonant absorption (see Sec. 4) would lead to the rapid development of KHI.
The instability is not stabilised by the magnetic field, as the shear flows
appear across it. The KHI causes a mixing of internal and external plasmas and
hence generates or broadens the transition region within which resonant
absorption would take place. The KHI associated with standing kink waves
manifests in several different ways (see, e.g., Antolin et al., 2017) for its
observable signatures. The characteristic KH vortices occur first at the
velocity shear region at the loop interface with the external corona and are
therefore azimuthal and exist in the plane perpendicular to the magnetic field
(Terradas et al., 2008). Since the amplitude of a standing kink oscillation
changes gradually with height, the Kelvin–Helmholtz (KH) vortices become the
so-called transverse wave induced Kelvin–Helmholtz (TWIKH) rolls in 3D, with a
self-similar shape of slowly varying amplitude (Antolin et al., 2014). The
distortion of the transition region in the azimuthal direction, caused by KHI
could be considered as the nonlinear generation of fluting modes with
increasingly growing azimuthal wave numbers $m$, i.e., the azimuthal nonlinear
cascade.

The number of TWIKH rolls excited at any one time is therefore determined by
their azimuthal wave numbers and also by the thickness of the loop boundary
layer at the time of the kink mode excitation. Small azimuthal wave numbers
have high growth rate and therefore will be excited first if the boundary
layer thickness (which determines the thickness of the velocity shear layer)
is smaller than the wavelength of the unstable mode. One can understand why
this is so with the following argument. A vortex is characterised by the top
part of the crest moving within one of the layers generating the shear, while
the bottom is anchored to the layer moving in the opposite direction. Hence,
the size of the vortex cannot be smaller than the thickness of the shear
layer. After a few oscillation cycles the result is a mix-up of unstable
modes, whose combined effect is a very efficient mixing of the plasma. Since
TWIKH rolls are compressive, the mixing leads to a continuous perturbation of
the boundary layer. We can therefore quantify the generation of the fluting
modes by measuring the power of each azimuthal wave number in the radial
velocity along the boundary of the loop (Terradas et al., 2018; Antolin and
Van Doorsselaere, 2019). The distribution of the energy by the azimuthal wave
numbers $m$ for the case of a coronal loop of length $L=200~{}$Mm, radius
$a=1~{}$Mm, with initial transition layer of width $l_{\mathrm{tr}}/a=0.4$ and
subject to an initial velocity perturbation of $v_{0}=16.6~{}$km s-1,
corresponding to a nonlinearity parameter $\nu=3.3$, is shown in Fig. 12. The
two simulations only differ on the viscosity in the numerical model, leading
to a Lundquist number of $10^{4}$–$10^{5}$ for the low viscosity (and more
realistic) case, and 10–100 for the highly viscous case (Antolin and Van
Doorsselaere, 2019). The high viscosity values in the second case effectively
inhibit the KHI. In the case without KHI, i.e., when the viscosity is high, we
see the doubling of the frequency, and the fact that the amplitude of the
$m=2$ mode is quadratically smaller than that of the $m=1$ mode, thereby
matching the theoretical results (Ruderman and Goossens, 2014). All other,
higher azimuthal wave modes have negligible amplitude. The case with KHI
presents a very different picture. In this case, the KHI occurs at $t\approx
6~{}$min, which matches with the time of significant excitation of all
azimuthal wave numbers. The amplitudes of the $m=2$ and $m=3$ become
comparable to the amplitude of the fundamental mode.

The increase of amplitude of the $m=2$ mode in the presence of the KHI can be
understood as follows. At times of maximum displacement the squashing of the
waveguiding cylinder leads to the acceleration of material in the boundary
layer in the opposite direction (i.e. backwards), a flow that is further
enhanced by resonant absorption (see the top row in Fig. 13). When the KHI is
triggered, since this happens at both sides of the cross-section with respect
to the axis of the (linearly polarised) oscillation, we end up with 2
azimuthal flows that collide with each other and generate two compressive
vortices. Because of the compression, a magnetic pinch is produced on the
plasma at the centre of the loop, which is then accelerated in the opposite
direction when the loop starts moving backwards (bottom row in Fig. 13). This
produces a faster leading edge that further enhances the squashing effect,
leading to a stronger amplitude for this mode.

Figure 13: Sketch of the cross-section of a cylinder with the m=2 mode with
(bottom row) and without (top row) KHI. During the transverse oscillation
($m=1$ mode, in the horizontal direction in the sketch), field lines in the
leading edge of the tube have slightly stronger magnetic tension than those at
the trailing edge, leading to a stronger (nonlinear) deceleration at the
leading edge compared to to the trailing edge, thereby squashing the flux tube
(dashed circles in the sketch denote the original shape of the cylinder’s
cross-section). Since the motions are incompressible, this leads to a slightly
elongated cross-section in the vertical direction. Due to the squashing, the
magnetic pressure increases, leading to a fast acceleration in the opposite
direction. This motion (which is partly azimuthal and symmetric with respect
to the oscillation axis) is further enhanced by resonant absorption. When the
KHI is triggered, the combined effect of the azimuthal resonant flow and the
KHI flow (compressive vortices) leads to a compression at the trailing edge of
the flux tube (magnetic pinch) which accelerates the plasma in the opposite
direction, thereby increasing the amplitude of the $m=2$ mode.

It is important to note that, because of resonant absorption at the boundary,
the perturbations and TWIKH rolls are not confined to the boundary layer.
Indeed, as shown in (Antolin and Van Doorsselaere, 2019), the resonant flow
produces a velocity shear in a neighbouring shell that is closer to the axis
of the cylinder, which will, in turn, become KH unstable. This process repeats
inwardly until most of the loop is covered by TWIKH rolls, even in the case of
a single initial perturbation.

Besides the KHI, the RTI can also manifest in nonlinear kink mode
oscillations. When during a linearly polarised kink oscillation, the loop
moves into the background plasma, the front edge of the cylinder, which is
locally perpendicular to the direction of the motion, is subject to RTI. The
characteristic finger-like RTI structures have been detected observationally
by Antolin et al. (2018b); Antolin and Van Doorsselaere (2019), and attributed
to the higher magnetic tension force and lower density region of the external
medium pressing unto the lower magnetic field strength but higher density
region of the loop. The RTI perturbations would also produce compression at
the wake, thereby contributing to fluting mode generation.

Figure 14: Ratio of KHI onset time to the damping time as a function of twist
for three different widths of the inhomogeneous layer ($l_{\mathrm{tr}}/a$).

Magnetic twist has significant effects inhibiting KHI induced by a nonlinear
kink oscillation. This has been investigated numerically in Howson et al.
(2017a); Terradas et al. (2018). The results indicate that magnetic twist
delays the onset of the KHI instability but the thickness of the inhomogeneous
layer also plays an important role (as in the untwisted situation). It has
been established that in the regime of weak damping, when the inhomogeneous
layer is thin, a weak twist does not delay significantly the onset of the
instability. On the contrary, when the inhomogeneous layer is wide, and
therefore the damping time small, then twist has a strong stabilising effect,
see Terradas et al. (2018). It is therefore interesting to compare the
timescales of the attenuation, $\tau_{\mathrm{D}}$ with the onset times of the
KHI in the presence of the twist. The results are shown in Fig. 14. The
dependence $\tau_{\mathrm{KHI}}/\tau_{\mathrm{D}}$ with twist is weak for thin
layers but significantly strong for thick layers ($l_{\mathrm{tr}}/a$ large).

Figure 15: The left panel shows the inhomogeneous layer width calculated from
the transverse intensity profile of a simulated coronal loop undergoing KHI.
The inferred value increases with time and line-of-sight angle. The right
panel is a histogram of observed values of inhomogeneous layer width based on
the statistical study of Goddard et al. (2017). The dashed line represents a
model based on the assumption that loops initially have a thin inhomogeneous
layer which is broadened due to KHI from ubiquitous decayless kink
oscillations. Figure adapted from Pascoe et al. (2020a).

Figure 16:  The left panel shows the position of a loop (black line) perturbed
by two flares approximately 1 hour apart. The modelled position (green line)
is composed of a background trend (blue line) and two damped oscillations
(grey lines). The time dependence of the signal quality (right panel) is
consistent with evolution of the loop due to KHI. Figure adapted from Pascoe
et al. (2020b).

Howson et al. (2017b) demonstrated that the increase in the viscosity and
resistivity acts to suppress the KHI in kink oscillation, delaying its onset
or completely preventing it. Viscosity was found to have a greater effect on
the development of the KHI than resistivity.

Goddard et al. (2018) modelled the evolution of the loop transverse intensity
profile during the decay of a kink oscillation, as manifested in EUV imaging
observations. The results of numerical simulations of kink oscillations with
KHI were used as input parameters of the forward modelling. The strongest
observational evidence for the KHI is found to be a widening of the loop
boundary. With SDO/AIA this effect may be inferred for sufficiently broad
loops, e.g., with more than fifteen data points across the loop. However, such
a theoretically predicted widening of the loop boundary has not been detected
in direct observations of decaying kink oscillations (see e.g. Fig. 9 in
Pascoe et al., 2018). Additional signatures of KHI are a decreasing intensity,
an unchanged minor radius, and visible fine transverse structuring. The
appearance of these signatures is delayed for loops with broader inhomogeneous
layers, and quicker for kink oscillations of higher amplitudes. Pascoe et al.
(2020a) demonstrated that the widening of the loop boundary also depends on
the LOS angle of the observation (Figure 15), and how the distribution of
widths measured in the statistical study of non-oscillating loops by Goddard
et al. (2017) is consistent with KHI due to ubiquitous decayless oscillations
acting to increase the detected values. Pascoe et al. (2020a) also calculated
the increase in damping due to resonant absorption owing to the broadening of
the boundary layer by KHI and found the actual damping to be stronger,
indicating that the fine structures generated by KHI also have a role in
attenuating kink oscillations. Pascoe et al. (2020b) analysed oscillations in
a coronal loop perturbed by two solar flares approximately 1 hour apart,
noting that such multiflare observations are more suited to revealing the
evolution of the loop profile since the time required for KHI to develop is
comparable to, or greater than, the damping time of kink oscillations. The
second oscillation was found to have a lower signal quality (Figure 16),
consistent with KHI generated in the loop by the earlier large amplitude
oscillation.

Finally, it is worth mentioning that Howson et al. (2017a) have concluded that
the presence of twist may increase the ohmic dissipation due to the larger
currents that are produced in this case. However, the presence of magnetic
twist will likely make the instability more difficult to detect in the corona,
but at the same time it will increase its contribution to heating of the solar
atmosphere.

###  10.2 Nonlinear coupling of kink and fluting oscillations

As the generation of TWIKH rolls is a nonlinear cascade in the azimuthal
direction, it is of interest to consider nonlinear interaction of a mother
kink mode with the daughter fluting modes. The first analytical studies of the
kink mode assumed no boundary perturbation, which is equivalent in assuming
that the cross-section remains circular. In turn, this means that there is no
interaction between the kink modes and the fluting modes. Instead, the first
studies on nonlinearity focused on the interaction between the transverse and
field-aligned perturbations (Ryutova and Sakai, 1993).

For the case of a uniform cylinder with uniform magnetic field, the
deformation of the flux tube’s cross-section due to the fluting modes can be
helical in the case of propagating waves (Ruderman et al., 2010), but is
symmetric in the case of standing waves. Another main characteristic is that
their frequency is double that of the kink mode, regardless of whether the
wave is propagating (Ruderman et al., 2010; Ruderman and Goossens, 2014) or
standing (Ruderman, 2017). As explained by Terradas et al. (2018), the
deformation of the cross-section due to the first harmonic and the doubling of
the frequency can be easily understood from a physical point of view by taking
the standing wave case. As the cylinder moves sideways it is decelerated due
to the magnetic tension from the line-tying condition. Due to nonlinearity,
this deceleration is slightly stronger at the head of the cylinder than at the
rear. Also, the higher inertia for the denser, middle parts of the tube
implies a less strong deceleration. These effects lead to a squashing of the
cylinder along the direction of the oscillation at times of the maximum
displacement. Because of the negligible compressibility of the kink wave in
the long-wavelength limit, this squashing necessarily means a proportional
enlargement of the flux tube’s width in the direction perpendicular to the
oscillation in a way to keep the cross-sectional area unchanged. Similarly, at
times of zero displacements coinciding with the maximum velocity, the front of
the cylinder travels at a slightly faster speed than the rear, leading to an
overall oval shape of the cross-section of the cylinder at those times.

The generation of fluting modes enhances the resonant damping of the wave.
This is because in the thin-tube approximation the kink mode and the fluting
modes propagate at the same speed (the kink speed), and thereby resonate
strongly (even in the absence of a resonant layer in the case of an
inhomogeneous boundary layer). This resonance transfers energy from the
fundamental mode into the higher azimuthal harmonics (fluting modes), which
damp faster due to their shorter wavelengths (Ruderman et al., 2010). The
transfer of energy is increasingly more efficient for larger nonlinearity.
Hence, the stronger the nonlinearity is, the stronger is the damping. Still,
the amplitude of the overtones is quadratically dependent on the amplitude of
the fundamental mode, which means that the amplitude of the fundamental mode
is always expected to be much larger than that of the overtones regardless of
the nonlinearity (Ruderman and Goossens, 2014; Terradas et al., 2018). Further
characteristics include a sausage mode perturbation in addition to the fluting
perturbation and a loop apex location for the maximum of the fluting
perturbation (Ruderman, 2017).

On the other hand, in the case of a longitudinally stratified loop (and as
long as the amplitudes are not too high555Formally,
$\nu_{\mathrm{NL}}\lesssim|2-\omega_{2}/\omega_{1}|$, where $\omega_{1}$ and
$\omega_{2}$ correspond to the frequencies of the fundamental mode and first
harmonic.), the nonlinearity does not cause damping (Ruderman and Goossens,
2014). This is because a change in the oscillation frequency is obtained,
which destroys the resonance between the kink mode and the fluting modes, thus
preventing the transfer of energy.

###  10.3 Field-aligned flows driven by the ponderomotive force in kink waves

Apart from siphon or reconnection-driven flows, there also could be field-
aligned flows induced by MHD waves. In the case of kink waves, axial flows of
the plasma could be driven because of nonlinear effects, due to the
ponderomotive force. A distinction must be made between “genuine” flows that
transfer the plasma in a certain direction and do not return it back, and
velocity components of waves along the field. Slow magnetoacoustic waves are
well-known to manifest in field-aligned velocity perturbations. However, when
propagating through a plasma which is inhomogeneous across the field, any
transverse wave has field-aligned velocity perturbations, as the components of
velocity perturbations are linearly coupled (see, e.g., Magyar et al., 2019).
The distinction between flows and waves is clear in theory, but it is less so
in observations (see Verwichte et al., 2010; De Moortel et al., 2015).

The ponderomotive force arises nonlinearly whenever there is a gradient of
magnetic pressure, which generates a plasma flow. The force density vector is
defined by

$\mathbf{F}=-\nabla\left(B^{2}_{\perp}\right)/\mu_{0},$ (30)

where $B_{\perp}$ is the magnetic field perturbation perpendicular to the
background magnetic field. The nonlinear nature of this force is evident,
given the quadratic dependence of magnetic pressure on field perturbations.
Kink waves as other transverse waves readily induce magnetic pressure
variations, thus inducing field-aligned flows. An exception is the case of
circularly-polarised and propagating kink waves, for which the magnetic
pressure perturbation is constant along the field. Nevertheless, circularly-
polarised standing kink waves still induce a ponderomotive force, as magnetic
field perturbation nodes are still present (at the apex in the case of the
fundamental mode). In coronal loops, the ponderomotive force of standing kink
waves is known to lead to flows towards the apex, causing density enhancement
there (Terradas, 2009). However, for typically observed standing kink wave
amplitudes, this effect is small (less than 3%), explaining the lack of its
detection (Terradas and Ofman, 2004). Gas pressure counteracts the continuous
mass accumulation, even in the limit $\beta\ll 1$, leading to the saturation
of this effect (see Rankin et al., 1994, for some quantitative estimates). The
flows induced by the ponderomotive force are therefore periodic (with double
the period of the inducing wave) and point in the same direction in both
phases of the wave. This direction is the kink wave propagation direction
(Shestov et al., 2017; Mozafari Ghoraba and Vasheghani Farahani, 2018), and
from magnetic field antinodes to nodes in the case of standing kink waves.
Figure 17 shows the flow induced by the ponderomotive force in a numerical
simulation of a fundamental standing kink mode of a plasma cylinder.

Figure 17: The variation of transverse velocity perturbation (blue),
transverse magnetic field perturbation (green), and axial flow (red) along the
middle of a plasma cylinder, from the footpoint ($s=0$) to the apex ($s=6$),
at some intermediary displacement. The initial velocity perturbation amplitude
is 10% of the internal Alfvén speed. The axial flow speed is multiplied by 10.
Data from a numerical simulation presented in Magyar and Van Doorsselaere
(2016).

For propagating waves, the flow induced is also referred to as “Alfvénic
wind“.

In large loop oscillations, the ponderomotive force could affect the
phenomenon of the coronal rain (Verwichte et al., 2017), but it is probably
not the primary reason for the observed sub-ballistic motions of the plasma
blobs (Antolin and Rouppe van der Voort, 2012). The ponderomotive force could
be a possible cause of the first ionisation potential (FIP) effect, related to
the enhancement or depletion, respectively, in coronal abundance with respect
to photospheric values of elements with FIP below about 10 eV (Laming, 2015).
Besides its effects in the case of kink oscillations, the ponderomotive force
has been shown to be omnipresent in numerical simulations of turbulent coronal
loops, and it acts as a source of flows, alongside reconnection (Dahlburg et
al., 2016). Additionally, it was shown that spicules might be driven by
propagating Alfvén and kink waves via the steepening and ultimately shock
heating of the flows driven by the induced ponderomotive force in the upper
chromosphere (e.g., Brady and Arber, 2016).

##  11 Excitation of kink oscillations

Decaying kink oscillations are usually detected as an impulsively excited
oscillation with the decay phase being longer than the growing phase. The
initial amplitude of kink oscillations is typically several Mm or several
minor radii of the loop. The excitation has been associated with impulsive
releases of energy and shown to be caused by several mechanisms. Analysis of
58 kink-oscillation events, performed by Zimovets and Nakariakov (2015),
showed that 57 events (98%) were accompanied by lower coronal
eruptions/ejections. Thus, it is natural to link the onset of the oscillation
with an impulsive666Here, by “impulsive”, we mean a process with a
characteristic time much shorter than the period of the excited oscillation.
deposition of the energy associated with the eruption. On the other hand, it
remains unclear whether the initial perturbation provides the loop with a
push, i.e., kinetic energy, or a displacement from the equilibrium, i.e., the
potential energy, or their combination. The estimation of the speed of the
agent which carries the energy from the site of its release to the loop is
usually lower than 500 km s-1. This value is at least two times lower than the
expected fast magnetoacoustic speed in a coronal active region. Thus, most
likely this agent is not a fast magnetoacoustic wave or a fast blast wave. The
latter option is also inconsistent with the observed appearance of Type-II
radio bursts in only 40% of kink oscillation events, which shows that coronal
shock waves appear in less than a half of kink oscillation events. A remaining
option is a displacement of the loop from an equilibrium by a slowly moving
plasma blob or a magnetic rope. In this case, as the main displacement is
experienced by the loop top, the preferentially excited kink mode is the
fundamental one. For example, Shen et al. (2018) studied an observation of a
simultaneous excitation of a kink oscillation with the period of 428 s by an
impingement of a coronal jet on the loop. If the eruption is sufficiently
localised in space and touches a segment of the loop in one of its legs, the