11institutetext: Tom Van Doorsselaere 22institutetext: Centre for mathematical
Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan
200B bus 2400, B-3001 Leuven, Belgium
22email<EMAIL_ADDRESS>33institutetext: Abhishek K.
Srivastava 44institutetext: Department of Physics, Indian Institute of
Technology (BHU), Varanasi-221005, India 55institutetext: Patrick Antolin
66institutetext: Department of Mathematics, Physics and Electrical
Engineering, Northumbria University, Newcastle Upon Tyne, NE1 8ST, United
Kingdom 77institutetext: Norbert Magyar 88institutetext: Centre for Fusion,
Space and Astrophysics, Physics Department, University of Warwick, Coventry
CV4 7AL, UK 99institutetext: Soheil Vasheghani Farahani 1010institutetext:
Department of Physics,Tafresh University, Tafresh 39518 79611, Iran
1111institutetext: Hui Tian 1212institutetext: School of Earth and Space
Sciences, Peking University, Beijing 100871, China
Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, China 1313institutetext: Dmitrii Kolotkov
1414institutetext: Centre for Fusion, Space and Astrophysics, Physics
Department, University of Warwick, Coventry CV4 7AL, UK
Institute of Solar-Terrestrial Physics SB RAS, Irkutsk 664033, Russia
1515institutetext: Leon Ofman 1616institutetext: Department of Physics,
Catholic University of America, Washington, DC, USA
NASA Goddard Space Flight Center, Greenbelt, MD, USA 1717institutetext:
Mingzhe Guo 1818institutetext: Institute of Space Sciences, Shandong
University, Weihai 264209, People’s Republic of China
Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU
Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium
1919institutetext: Iñigo Arregui 2020institutetext: Instituto de Astrofísica
de Canarias, 38205, La Laguna, Tenerife, Spain
Departamento de Astrofísica, Universidad de La Laguna, 38206, La Laguna,
Tenerife, Spain 2121institutetext: Ineke De Moortel 2222institutetext: School
of Mathematics and Statistics, University of St Andrews, North Haugh, St
Andrews, KY16 9SS, UK
Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern,
NO-0315 Oslo, Norway 2323institutetext: David Pascoe 2424institutetext: Centre
for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,
Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium

# Coronal heating by MHD waves

Tom Van Doorsselaere  Abhishek K. Srivastava  Patrick Antolin  Norbert Magyar
Soheil Vasheghani Farahani  Hui Tian  Dmitrii Kolotkov  Leon Ofman  Mingzhe
Guo  Iñigo Arregui  Ineke De Moortel  David Pascoe

(Received: date / Accepted: date)

###### Abstract

The heating of the solar chromosphere and corona to the observed high
temperatures, imply the presence of ongoing heating that balances the strong
radiative and thermal conduction losses expected in the solar atmosphere. It
has been theorized for decades that the required heating mechanisms of the
chromospheric and coronal parts of the active regions, quiet-Sun, and coronal
holes are associated with the solar magnetic fields. However, the exact
physical process that transport and dissipate the magnetic energy which
ultimately leads to the solar plasma heating are not yet fully understood. The
current understanding of coronal heating relies on two main mechanism:
reconnection and MHD waves that may have various degrees of importance in
different coronal regions. In this review we focus on recent advances in our
understanding of MHD wave heating mechanisms. First, we focus on giving an
overview of observational results, where we show that different wave modes
have been discovered in the corona in the last decade, many of which are
associated with a significant energy flux, either generated in situ or pumped
from the lower solar atmosphere. Afterwards, we summarise the recent findings
of numerical modelling of waves, motivated by the observational results.
Despite the advances, only 3D MHD models with Alfvén wave heating in an
unstructured corona can explain the observed coronal temperatures compatible
with the quiet Sun, while 3D MHD wave heating models including cross-field
density structuring are not yet able to account for the heating of coronal
loops in active regions to their observed temperature.

###### Keywords:

Sun: corona Sun: waves

##  1 Introduction

Coronal heating is a long-standing problem. It is quite clear that the energy
for the hot corona comes from the convective motions of the solar photosphere,
and that the magnetic field plays a key role in it. However, how the energy is
transported and dissipated is still not fully understood. Proposed heating
mechanisms are classified based on the comparison of the convective time
scales and the Alfvén transit time in the corona. Slow driving of the magnetic
field that produces reconnection and energy release in current sheets and null
points, is referred to as DC heating mechanisms (Parker, 1986). Fast driving
of the magnetic field, is known as MHD wave heating (e.g. Alfvén, 1947) and AC
heating mechanisms.

DC heating mechanisms consider the slow stressing of the coronal magnetic
field. It leads to current sheets where Ohmic dissipation is at work, or to
Parker’s idea of nanoflares. In the latter idea, the magnetic field is tangled
because of the photospheric motions, so that it evolves into a non-potential
state. This magnetic energy is then released into the plasma by reconnection,
resulting in localised heating and thus nanoflares. DC heating mechanisms have
received a lot of attention over the years in numerical modelling and
observations (e.g. Gudiksen and Nordlund, 2005; Reep et al., 2013; Rempel,
2017; Warnecke et al., 2017; Srivastava et al., 2019).

In this review, we focus on AC heating mechanisms. In these heating
mechanisms, the convective motions launch disturbances that travel into the
corona, usually in the form of magnetohydrodynamic (MHD) waves. In the corona,
the wave energy has to be dissipated to heat the plasma. This is non-trivial,
because many wave modes damp only on resistive time scales in a homogeneous
plasma, and damping times are proportional to the magnetic Reynolds number. In
the solar corona, the Reynolds number is very large, on the order of
$10^{14}$, which would produce unrealistically long heating times, compared to
the coronal cooling timescale. Cross-field inhomogeneity enables physical
processes that produce a cascade of wave energy to small spatial scales where
dissipative processes may act and heat the plasma more rapidly. In recent
years, there has been a drive towards (1) observational characterisation of
wave energy content in the corona, and (2) numerical modelling of wave
heating, focusing on the energy input, the energy propagation and the energy
dissipation. Here, we aim to give an overview of these recent results.

For material beyond the current review, we refer the reader to, for example,
Aschwanden (2019).

###  1.1 Brief historical overview

The idea for heating the solar corona by MHD waves has been around for more
than half a century. An overview of the early ideas before the 80s can be
found in Kuperus et al. (1981). In those years, it was realised that it is
necessary to generate small scales in order to damp the wave energy in a
timely manner. Heyvaerts and Priest (1983) developed the theory of phase
mixing, while resonant absorption (Chen and Hasegawa, 1974) was applied to the
corona for the first time by Ionson (1978). Subsequently, the MHD wave energy
input and transmission into coronal loops was studied in key papers such as
Hollweg (1984). Later on, the efficiency of the heating by resonant absorption
was calculated in 1D numerical models of loops (Poedts et al., 1990, among
others), and in 3D models (Ofman et al., 1994, and follow-up works), even
resulting in forward modelled coronal loops (Belien et al., 1996), motivated
by early high resolution soft X-ray observations of coronal loops that became
available from the Yohkoh satellite.

In hindsight, it is amazing that these early papers worked so well and were so
relevant, given that no direct observational evidence was available back then
on the presence of MHD waves in the solar corona. Indeed, although
substantial, only indirect evidence existed of the potentially important role
of MHD waves in the solar atmosphere. Such evidence was based on the strong
emission and broad non-thermal line widths in the upper chromosphere,
transition region and corona with observations from Skylab (Feldman et al.,
1988) and HRTS (High-Resolution Telescope and Spectrograph, Dere and Mason,
1993). This changed dramatically with the launch of SOHO (Solar and
Heliospheric Observatory) and TRACE (Transition Region And Coronal Explorer).
Data from the former were used to show the presence of slow waves in the
corona (Chae et al., 1998; Ofman et al., 1999; Berghmans and Clette, 1999),
while data from the latter revealed the presence of transverse, post-flare
loop oscillations (Nakariakov et al., 1999; Schrijver et al., 1999; Aschwanden
et al., 1999). An overview of the contextualisation of earlier models with
those observational findings is given in Walsh and Ireland (2003); Ofman
(2005).

LCR models (named after the usual symbols in electric components for
inductance L, capacitance C and resistance R) were also developed extensively
for MHD waves in coronal loops (see Stepanov et al., 2012, and reference
therein). In such models, the magnetic twist in the loop behaves as a current
system with its source in the photosphere and equivalent electric resistivity
and inductance in the corona (see e.g., Spicer, 1977; Carlqvist, 1979).
However, it is still unclear how this LCR model relates to MHD models of
coronal loop oscillations (based on e.g. Edwin and Roberts, 1983). Moreover,
since no development of these models was made in the last years, we will omit
them from this review.

###  1.2 Observational motivation

Starting with the advent of high resolution space-based observations of the
solar corona in soft X-ray and EUV in the 90s, there is now an avalanche of
new MHD wave modes detected in the solar corona, as described in detail by De
Moortel and Nakariakov (2012). Their consequences for heating the corona are
discussed in Arregui (2015).

Another major paradigm change in our perception of MHD waves in the solar
atmosphere came from ground-based observations with CoMP (Tomczyk et al.,
2007), which showed the omnipresence of these waves in the solar corona, and
spectroscopic and imaging from space with Hinode, which, thanks to its high
resolution, allowed to better quantify the amount of energy available for the
corona (De Pontieu et al., 2007). Moreover, SDO/AIA revealed the existence of
many new wave modes and made more detailed observations of previously observed
coronal waves (for a review, see Liu and Ofman, 2014).

The improvement in spatial, spectral and temporal resolution of
instrumentation also meant that MHD waves could be characterised in terms of
their slow, fast or Alfvén nature, as well as their wave numbers (see the
review by De Moortel and Nakariakov, 2012). Indeed, high spatial resolution
and high sensitivity allows to distinguish transverse motions of the
waveguides as well as the compressibility of the gas from density diagnostics,
while high spectral and temporal resolution allows to relate these quantities
to the evolution of the Doppler and non-thermal velocities, thereby
determining the 3D motion of the plasma produced by the waves (see e.g.
Fujimura and Tsuneta, 2009; Kitagawa et al., 2010; Wang, 2016; Hinode Review
Team et al., 2019, chapter 6.1).

Lastly, multi-wavelength observations with instruments such as those of Hinode
and IRIS, or through coordinated observations with space and ground-based
observatories allowed to simultaneously scan several layers of the solar
atmosphere. Although a complicated task, this provides the propagation history
of the wave, which is essential to determine wave processes such as reflection
and refraction, and in particular mode conversion and dissipation of MHD waves
for coronal heating (Arregui, 2015).

##  2 Observations

Below, we describe recent detections of wave power in the solar atmosphere. We
summarise the energy fluxes in Table 1 to give a clear overview.

Table 1: Overview of detected wave energy fluxes. The first column shows the structure in the solar atmosphere with the observed wave mode (second column) and the observing instrument (third column). The fourth column shows the estimated energy flux, as found by the reference mentioned in the fifth column. If a cell is left empty, the value from the previous line are meant. Structure | Wave mode | Instrument | Energy flux (W m-2) | Reference
---|---|---|---|---
coronal arcade | propagating kink | CoMP | 100 | Tomczyk et al. (2007)
coronal funnels | quasi-periodic fast wave trains | AIA | $(0.1-2.6)\times 10^{4}$ | Liu et al. (2011)
|  | simulations | $3.7\times 10^{5}$ | Ofman et al. (2011)
|  | AIA | $1.8\times 10^{2}$ | Ofman and Liu (2018)
magnetic pores | Alfvén waves | ROSA | 1.5$\times$104 (locally) | Jess et al. (2009)
|  |  | 240 (globally averaged) |
| propagating slow sausage | DST | 3.5$\times$104 | Grant et al. (2015)
fibrils | propagating kink | ROSA | $4.3\times 10^{3}$ | Morton et al. (2012)
| propagating fast sausage |  | $1.17\times 10^{4}$ |
spicules | Alfvén waves | SST | $10^{5}$ | Srivastava et al. (2017)
| kink |  | $10^{2}-10^{4}$ | this review, based on De Pontieu et al. (2012)

###  2.1 Impulsively excited standing kink waves

Often flares and low coronal eruptions (LCEs) are seen to excite standing kink
waves in coronal loops (Zimovets and Nakariakov, 2015). Goddard et al. (2016a)
and Nechaeva et al. (2019) performed a statistical study of these types of
events and found that the loops oscillate with an amplitude between 1-10 Mm
and periods between 1-28 min. While the coronal impulsive event that causes
the oscillations is apparent, the exact mechanism for their excitation is not
well understood. Several possibilities were investigated in the literature:
internal and external (gas or magnetic) pressure drivers (Terradas et al.,
2007; McLaughlin and Ofman, 2008; Pascoe et al., 2009; Selwa and Ofman, 2009;
Pascoe and De Moortel, 2014), loop contraction (Russell et al., 2015; Pascoe
et al., 2017), or collision of flows (Antolin et al., 2018a; Pagano et al.,
2019, even though these would result in propagating waves). More information
about observations of impulsively excited standing waves can be found in the
review by Nakariakov et al. (2021).

The energy available in impulsively excited standing waves and the fraction
that is dissipated remains to be characterised from observations. An energy
analysis based on bulk plane Alfvén waves is too simplistic because kink mode
energy is localised in space (Goossens et al., 2013). Magnetic and plasma
structures act as frequency filters, trapping part of the available energy and
distributing it along and across the coronal field. Wave energy propagation,
once filtered by the structure is localised in space. Energy flows into the
resonance because of the jump in the radial component of the Poynting vector
(Arregui et al., 2011), as shown in Fig. 1. Then, it propagates and dissipates
along the field in a way determined by the density profile and dissipative
coefficients.

Figure 1: Spatial distribution of the Poynting vector in the ($r,z$)-plane
around the resonant position at $r/a=1$ for a standing kink mode in a loop
with a density contrast of 10 and a non-uniform layer of length $l/a=0.2$. The
magnetic Reynolds number is $R_{\rm m}=10^{6}$. Figure modified from Arregui
et al. (2011).

Terradas and Arregui (2018) considered each part of the sequence of physical
processes that would enable to heat a typical coronal waveguide from the
energy contained in a typical transverse oscillation. Each part of the
sequence has its characteristic time and spatial scales. The damping time is
determined by the cross-field plasma and field structuring. The energy cascade
to small scales is determined by a phase mixing length that also depends on
the cross-field variation of the Alfvén speed. The onset of resistive
dissipation and its duration depend on the Reynolds number and the cross-field
plasma variation. Considering typical values of these quantities we cannot
expect resistive diffusion to operate during the oscillation process. Any
observational evidence about wave heating by resistive damping of impulsively
excited standing waves will come from the observation of indirect
consequences. According to (Terradas and Arregui, 2018), for a loop
displacement of the order of the radius and typical loop parameters, all the
kinetic and magnetic energy of a typically observed single kink mode is of the
order of $10^{19}$ J. Simple energy conservation calculations indicate that,
even if we were able to concentrate in a typical resonant layer and transform
into internal energy all the kinetic and magnetic energy of a typically
observed single kink mode, we could just obtain a temperature increase of
about 105 K (Terradas and Arregui, 2018).

When an ensemble of mini-tubes are considered, filling factors can be employed
(Van Doorsselaere et al., 2014), which leads to an energy flux reduction
directly related to the filling factor. Although resonant damping and mode
coupling are robust in coronal loop models with rather arbitrary continuous
plasma distributions, the multi-strand structure of loops could be quickly
destroyed due to instabilities in individual strands and their interaction
(Magyar and Van Doorsselaere, 2016).

###  2.2 CoMP waves

Prevalent propagating waves were reported (Tomczyk et al., 2007; McIntosh et
al., 2011) from observations of both the Coronal Multi-channel Polarimeter
(CoMP, Tomczyk et al., 2008) and the Atmospheric Imaging Assembly (AIA, Lemen
et al., 2012) on board the Solar Dynamics Observatory (SDO). Using CoMP
observations, Tomczyk et al. (2007) found ubiquitous upward propagating
disturbances along off-limb coronal loops in the Doppler shift of the Fe xiii
10747 Å line (Figure 2). Since the Doppler shift refers to the motion along
the line of sight (LOS) and the magnetic field lines in the off-limb corona
are largely perpendicular to the LOS, these disturbances are essentially
signatures of transverse MHD waves. The propagation speed was found to be from
a few hundred km s-1 to 2000 km s-1, which is on the order of the coronal
Alfvén speed. Based on these characteristics, Tomczyk et al. (2007) and
Tomczyk and McIntosh (2009) interpreted these transverse waves as Alfvén
waves. They found a power-law spectrum of the Doppler shift with a spectral
index of about 1.5, consistent with isotropic MHD turbulence. The power
spectrum of the observations shows a peak around 3.2 mHz (period of 5
minutes). The velocity amplitude is only 0.5 km s-1, and the energy flux of
these waves appears to be at least three orders of magnitude lower than that
required to balance the radiative losses of the quiet corona. The low-
amplitude is likely due to the low spatial resolution of CoMP ($\sim$9′′) and
the line-of-sight integration resulting in spatial averaging of the waves.
Similar results were also found in open-field regions (Morton et al., 2015).
By carefully analyzing the displacement of coronal structures observed by the
much higher-resolution AIA instrument ($\sim$1.5′′), McIntosh et al. (2011)
found much larger velocity amplitudes, i.e., $\sim$20 km s-1, for these
transverse waves. They estimated the energy flux of the waves as $\sim 100\
{\rm W\,m}^{-2}$ and found that the waves have sufficient energy to power the
quiet corona and fast solar wind. However, the observed wave energy flux is
still much lower than that required to heat the active region corona
($\sim$2000 W m-2).

Figure 2:  Images of the Fe xiii 10747 Å line intensity and the wave
propagation angles derived from the Doppler shift of Fe xiii 10747 Å in part
of the corona observed by CoMP. Adapted from Tomczyk and McIntosh (2009).

The interpretation of these waves as Alfvén waves was controversial. As argued
by Van Doorsselaere et al. (2008), in cylindrical plasma structures these
waves appear to be more appropriately interpreted as fast-mode kink waves
rather than Alfvén waves. However, others claimed that these waves could be
called Alfvénic waves as the major restoring force for these waves is
definitely magnetic tension (Goossens et al., 2009; McIntosh et al., 2011),
they carry parallel vorticity (Goossens et al., 2012) and they are inherently
coupled to the local azimuthal Alfvén waves (Pascoe et al., 2010). This mode
is also called surface Alfvén wave or resonantly damped kink mode. At the end,
the heating will come from azimuthal Alfvén wave dissipation. The global
eigen-mode just plays the role of trapping/chanelling the energy from
around/below. A well defined structure is not needed, as shown by Terradas et
al. (2008b); Rial et al. (2010); Pascoe et al. (2011).
Despite the debate on the nature and terminology for these waves, the
discovery of these propagating transverse waves has caught the attention of
many researchers in the field. There are at least two reasons for this. First,
they allow us to map the global coronal magnetic field based on the technique
of coronal seismology (Magyar and Van Doorsselaere, 2018; Anfinogentov and
Nakariakov, 2019; Yang et al., 2020a, b). Second, their ubiquity means they
could potentially play an important role in coronal heating. The observed
waves show a discrepancy between the outward and inward wave power (Tomczyk
and McIntosh, 2009), a property that resonantly damped wave models seem to
capture well (Verth et al., 2010; Pascoe et al., 2010), and that is also
observed in-situ in the solar wind (see the review by Bruno and Carbone,
2013). A recent study by Montes-Solís and Arregui (2020) considers a distinct
power generated at loop foot-points as an additional source of discrepancy. It
shows that, if present, it would affect obtaining quantitative evidence for
resonant damping.

The origin of the waves detected by CoMP has recently been linked to the Sun’s
internal acoustic oscillations, and in particular to p-modes. Indeed, Morton
et al. (2019) have shown that there is a distinct excess of wave power at the
usual p-mode frequency of $3-5$ mHz, across the solar cycle and different
coronal structures (or regions) in the Sun (Morton et al., 2016). The authors
further show that the usual Alfvénic fluctuations observed with SDO/AIA
(McIntosh et al., 2011) have the same characteristics in their power spectra
as the CoMP waves. Evidently, p-modes contain orders of magnitude more power
than required for coronal heating. The physical mechanism proposed to explain
the ubiquity of these waves is the double mode conversion process of p-modes
during their propagation across the low atmospheric layers, during which their
energy is transferred to Alfvénic modes (Cally and Hansen, 2011; Felipe, 2012;
Cally, 2017). A strong consequence of this is that solar and stellar coronae
have a non-negligible energy source in their internal acoustic oscillations.
Although direct observations of this process are still lacking, these results
also suggest that p-modes may play a role in the onset of standing or
propagating Alfvénic modes.

###  2.3 Decayless oscillations

Decayless oscillations normally refer to standing waves without obvious
observed damping in coronal loops. In coronal lines111Small amplitude
decayless oscillations were also observed in a prominence with Hinode/SOT
(Ning et al., 2009), and of significant and increasing amplitude in a coronal
loop with rain (Antolin and Verwichte, 2011). Due to the presence of
chromospheric material and the associated processes specific to the formation
of prominences and coronal rain, the processes responsible for such
oscillations may likely be different than those observed in coronal lines.,
these oscillations were first reported independently by Wang et al. (2012) and
Tian et al. (2012) through imaging and spectroscopic observations,
respectively. The transverse oscillations reported by Wang et al. (2012) from
SDO/AIA observations appear to be triggered by a coronal mass ejection. They
lasted for more than ten cycles and even revealed growing amplitudes (Figure
3). Tian et al. (2012) performed a survey of decayless (referred to as
“persistent” in their paper) Doppler shift oscillations using three-month
spectroscopic observations from the EUV Imaging Spectrometer (EIS, Culhane et
al., 2007) on board Hinode. They found that such decayless oscillations, with
a period of 3–6 minutes and a velocity amplitude of 1–2 km s-1, are very
common in quiet coronal loops and can be observed in several coronal emission
lines formed in the temperature range of 1.3–2 MK (Figure 4). These
oscillations generally reveal no obvious damping during the whole observed
time interval, which often last for a few hours. They were interpreted as
standing kink oscillations by these authors. Since 2012, these decayless
oscillations have been frequently reported for a variety of different
observations (e.g., Nisticò et al., 2013, 2014), and it was found that their
periods scale with the loop length (Anfinogentov et al., 2015), strongly
supporting the standing wave interpretation.

Figure 3:  Decayless oscillations of coronal loops seen in the loop
displacement, peak flux and loop width estimated from the AIA 171 Å images.
Adapted from Wang et al. (2012). Figure 4:  Decayless oscillations of quiet
coronal loops seen in the Doppler shift of four coronal emission lines. The
Y-axis is roughly aligned with the corona loop. Adapted from Tian et al.
(2012).

The decayless behavior of these waves suggests a continuous supply of energy
to the system. Several different ideas have been proposed to maintain these
decayless or persistent oscillations. For instance, Nakariakov et al. (2016)
interpreted these decayless kink oscillations as a self-oscillatory process (a
process in which the driver is a consequence of the oscillation itself, see
Jenkins, 2013), driven by interaction between the loops and quasi-steady flows
at the loop footpoints. Antolin et al. (2016) have shown that the decayless
pattern could also be an apparent effect produced by the combination of low
instrumental resolution and the periodic brightening generated by TWIKH rolls
(Transverse Wave Induced Kelvin-Helmholtz rolls, cf. Sec. 3.3). In this
interpretation, the decayless character reflects the low damping character of
the azimuthal Alfvén waves resonantly coupled to the kink mode at the boundary
layer. Another theory is through continued footpoint driving (Karampelas et
al., 2017) providing an upward Poynting flux balancing the resonant and non-
linear damping of the kink waves. Recent numerical experiments (Karampelas et
al., 2019b; Guo et al., 2019b) reproduce the observations of decayless
oscillations very well. More recently, Afanasyev et al. (2020) developed a
one-dimensional and time-dependent analytical model by considering kink
oscillations of coronal loops driven by random motions of loop footpoints.
They have managed to reproduce a number of observational facts about these
decayless kink oscillations, such as the quasi-monochromaticity, period-length
relationship and excitation of multiple harmonics.

###  2.4 Available energy in quasi-periodic fast mode wave trains

The high cadence and spatial resolution of SDO/AIA in EUV enabled the
discovery of quasiperiodic fast mode propagating wave trains (QFPs) (see, Liu
et al. (2010, 2011), the review Liu and Ofman (2014). The waves were observed
to be associated with flares and propagate at a high speed of 1000-2000 km s-1
with periods in the range of minutes and intensity variations $\delta I/I\sim
5$%. The waves were modeled first by Ofman et al. (2011) using a 3D MHD model
of a realistic bipolar active region structure and identified as fast-mode
magnetoacoustic waves. Since the initial discovery, these waves were observed,
studied, and modeled in many events associated with flares (see, e.g. Liu et
al., 2012; Shen and Liu, 2012; Shen et al., 2013, 2017, 2018; Yuan et al.,
2013; Kumar and Manoharan, 2013; Nisticò et al., 2014; Zhang et al., 2015;
Goddard et al., 2016b; Qu et al., 2017; Ofman and Liu, 2018). Evidently, the
QFPs carry some of the energy flux produced by the flares, and the observed
dissipation of the QFP waves should result in coronal heating. The energy flux
of the waves was estimated first using the parameters observed by SDO/AIA on
2010/08/01 by Liu et al. (2011) and was found to be in the range
$(0.1-2.6)\times 10^{4}$ W m-2, which is of the same order as the steady-state
heating requirement of active region loops. The QFP wave energy flux estimate
was based on the WKB approximation given by $E=\rho\delta
v^{2}v_{ph}/2\geq\rho(\delta I/I)^{2}v_{ph}^{3}/8$ (Aschwanden, 2004), where
the intensity of the line emission, $I\propto\rho^{2}$, and the phase speed
$v_{ph}$ were determined directly from SDO/AIA observations ($v_{ph}=1600$ km
s-1), and the number density is taken to be the typical coronal electron
density $\sim 10^{8}$ cm-3. The QFP wave trains lasted on the order of $\sim$
0.5 hr and were repeatedly produced in the 2010/08/01 event. However, the
divergence of the magnetic funnel in the active region and apparent
dissipation of the waves reduced the apparent energy flux away from the
flaring source.

Ofman et al. (2011) used the 3D MHD model results in an idealized bipolar
active region of QFP wave trains combined with the WKB approximation to
estimate the energy flux as $E=\rho\delta v^{2}V_{f}/2=3.7\times 10^{5}$ W
m-2, where $\rho$ is the coronal density, $\delta v$ is the wave velocity
amplitude, and $V_{f}$ is the fast magnetoacoustic speed. The model energy
flux was an order of magnitude greater than the direct estimate by Liu et al.
(2011). However, the model estimate was based on idealized AR parameters in
qualitative agreement with observations. In a recent study by Ofman and Liu
(2018) of the double QFP event observed by SDO/AIA on 2013-5-22 the lower
limit of the wave energy flux was estimated as $1.8\times 10^{2}$ W m-2 high
in the corona. The authors concluded that, taking into account coronal loop
expansion, the wave energy flux could be at least one to two orders of
magnitude higher near the source of the QFPs.

Statistical study of QFP waves indicates that these waves are quite common and
often observed to be associated with C-class flares, although they can also
result from stronger and weaker flares (Liu et al., 2016). Thus, combined with
the challenging detection of these waves for lower energy flares, the
contribution of QFP wave trains to coronal heating and to the energy flux
requirement may be more significant than initially estimated. Thus, the
problem of QFP wave train heating parallels the coronal heating problem by
flares, where small energy flares (i.e., undetected nanoflares) are required
for coronal heating and possibly their associated QFP waves are not directly
detected (Aschwanden, 2004).

###  2.5 Observations and energy estimates of MHD waves in the lower solar
atmosphere

The lower solar atmosphere (photosphere and chromosphere) plays an important
role in the transmission of wave energy from the solar interior to the corona.
Therefore, the assessment of wave energy content in this layer is crucial in
understanding the coronal heating problem. Moreover, it is well established
that the solar chromosphere requires substantially more energy flux than the
corona (103–104 W m-2) to compensate for its huge radiative losses (Withbroe
and Noyes, 1977).

Recent observational findings from high-resolution ground (e.g., 1m-Swedish
Solar Telescope, ROSA – Rapid Oscillations in the Solar Atmosphere) and space-
based (e.g., IRIS – Interface Region Imaging Spectrometer, SDO – Solar
Dynamics Observatory) observations reveal the presence of a high amount of
energy flux associated with different MHD modes in a variety of magnetic
structures coupling the various layers of the solar atmosphere.

In the photosphere, wave behaviour is found in pores and sunspots. Jess et al.
(2009) reported the detection of Alfvén waves with periods of the order of
126-400 s associated with a large bright-point group. The energy flux
associated with these wave modes is found to be 1.5$\times$104 W m-2 in the
chromosphere, which partially fulfills the chromospheric energy requirements.
Using that 1.6% of the solar surface is observed to be covered by bright-
points and assuming that they support such torsional waves with a 42%
transmission coefficient (which may be on the high side), Jess et al. (2009)
have estimated the global average energy in the corona as 240 W m-2, which is
sufficient to heat it locally.
Grant et al. (2015) have detected upwardly propagating slow sausage waves in
magnetic pores, which initially carry an energy flux of 3.5$\times$104 W m-2.
The sausage wave energetics show a substantial decrease up to the
chromosphere. These observations make it evident that magnetic pores transport
waves to the higher layers, while also releasing energy in the local
chromospheric plasma. However, it is not well quantified how much wave energy
undergoes mode conversion, reflection or refraction. Aside from the frequently
observed sausage modes, higher order magnetoacoustic oscillations ($m\geq 1$)
have also been observed in photospheric waveguides (e.g. Jess et al., 2017;
Stangalini et al., 2018, and references cited there), including body and
surface modes (Keys et al., 2018). Furthermore, Grant et al. (2018) presented
evidence of Alfvén wave heating of the chromospheric plasma in an active
region sunspot umbra. They showed the presence of mode conversion and the
formation of magnetoacoustic shocks.
In conclusion, the random buffeting motions in the photosphere generate many
MHD modes in the photospheric magnetic flux tubes along with a sufficient
amount of energy flux. However, one has to bear in mind that these magnetic
flux concentrations constitute only a small part of the photosphere, and it is
unclear if these wave motions also propagate to the corona in quiet regions,
where such photospheric magnetic flux tubes are less prevalent.

Figure 5: Simultaneous wave modes in a chromospheric magnetic flux tube using
ROSA Hα observations. Time-distance map (‘b’) exhibits the displacement of the
tube’s axis (red-dotted line) as a whole (kink waves), while its cross-section
variation (sausage wave) is depicted by the yellow bars measuring the
variation of tube’s width. All measurement is done by Morton et al. (2012)
from where this figure is adopted and displayed here. Kink wave respectively
have period, upward propagation velocity, amplitude, and energy flux as
$\sim$232 s, $\sim$71 km s-1, $\sim$5 km s-1, 4300 W m-2. While, the same for
the observed sausage wave are given respectively as $\sim$197 s, $\sim$67 km
s-1, $\sim$1-2 km s-1, 11700 W m-2

In the chromosphere, magnetic structures play a major role in guiding MHD
waves. They are thus the prime structures in which waves are detected. In
particular, Fig. 5 displays the results of Morton et al. (2012) who have
observed the simultaneous presence of fast kink and sausage waves in mottles
and fibrils using ROSA Hα observations, and found that they carry an average
energy of respectively 4300 W m-2, and 11700 W m-2, once again stressing the
large amount of energy present in the lower atmosphere.

Going from the chromosphere to the transition region, spicules and their TR
counterparts show ample evidence of wave dynamics. Observations of swaying
motions in spicules with high-resolution imaging instruments such as
Hinode/SOT and IRIS have long been attributed to propagating Alfvénic waves
(De Pontieu et al., 2007; Tian et al., 2014). The amplitudes of the swaying
motions are usually of the order of $10-25$ km s-1, although a factor of 5
larger motions have also been observed (Antolin et al., 2018b). In general, 1
or 2 oscillations of the spicules can be captured before the structure
disappears, and a wide range of reported periods of $100-500$ s and also high-
frequency ($20-50$ s Okamoto and De Pontieu, 2011). Srivastava et al. (2017)
showed the ubiquitous presence of high frequency ($\approx$12–42 mHz)
torsional motions in spicular-type structures in the chromosphere (Fig. 6,
left box). Their numerical model showed that these observations resemble
torsional Alfvén waves associated with high frequency drivers containing a
huge amount of energy ($\approx$105 W m-2) in the chromosphere (Fig. 6, right-
panel). It is important to note, however, that the observational signatures of
TWIKH rolls can also explain these observations (Antolin et al., 2018b). Even
after partial reflection from the transition region, Srivastava et al. (2017)
found that a significant amount of energy ($\approx$103 W m-2) is being
transferred into the overlying corona, which is sufficient to compensate the
coronal radiative losses (Fig. 6, right panel). The propagation speeds of
waves in spicules are often difficult to measure, due to their short
lifetimes, the combination of (upward/downward) propagating and standing waves
(Okamoto and De Pontieu, 2011) and also due to the rapidly increasing Alfvén
speed at the observing heights. On average, speeds of $200-300$ km s-1 are
reported in these works.

Figure 6: The spicular-type structure (top panel of the left-box) shows
periodic reversal of the velocity (snapshot shown in the bottom-panel of the
left-box) indicating the presence of torsional Alfvén wave. The numerical
simulation (top-panel of the right-box) exhibits similar torsional motions in
the model fluxtube, that carry substantial energy (bottom-panel of the right-
box) through the solar chromosphere, TR, and corona (Credit: Srivastava et
al., 2017).

The swaying motions of spicules suggest a transverse displacement of a
waveguide and therefore such motions are often interpreted as kink waves.
However, caution is required when only imaging information is available, since
torsional Alfvén waves may also produce strand-like, swaying structures such
as spicules (Antolin et al., 2018b). Moreover, additional to the swaying,
torsional and longitudinal motions are observed in spicules (De Pontieu et
al., 2012; Sekse et al., 2013), and are likely strongly connected to their
generating mechanisms and to the nature of the wave. The nature of the
Alfvénic wave (kink or torsional Alfvén wave) is important when trying to
estimate the energy flux carried by the wave, since the collective or local
nature distributes the wave energy differently across the waveguide (Van
Doorsselaere et al., 2014). Assuming a filling factor of 1 (a best case
scenario), a typical spicule mass density of $8-16\times 10^{-14}~{}$g cm-3
and based on the reported values, one can estimate the energy flux in swaying
spicules as $0.1-3\times 10^{3}~{}$W m-2, with the strongest cases at
$1-4\times 10^{4}~{}$W m-2. Usually, $3\%$ of the energy is assumed to enter
the corona, leading to average energy flux values of $100$ W m-2 available for
coronal heating, which is sufficient for the quiet Sun and for the
acceleration of the solar wind (Tian et al., 2014).

###  2.6 Heating function assessment by slow waves

The presence of waves in the solar atmosphere may not only provide energy for
heating the corona, but could also provide us with a tool to seismologically
estimate the coronal heating function and the related dissipation
coefficients, such as thermal conduction and viscosity (see the review by Wang
et al., 2021, this issue). Here we forego that, more importantly, even the
most basic property of the magnetized coronal plasma such as the magnetic
field strength needed for any magnetically based heating function estimate is
difficult to determine directly, and in some cases can only be estimated from
coronal seismology. However, we keep this subject for the review of Nakariakov
et al. (2021), this issue.

The impact on the dynamics of MHD waves of thermodynamic activity of the
corona, i.e. processes of its continuous cooling via the optically thin
radiation and thermal conduction, and resupply of energy by some yet unknown
heating mechanism, was recently investigated by Nakariakov et al. (2017);
Claes and Keppens (2019); Kolotkov et al. (2019, 2020). This allows for
developing a new approach for a remote diagnostics of thermodynamic properties
of the corona, including processes of its cooling and heating, by coronal
seismology. For example, additional restrictions on the coronal heating
mechanism can be obtained via accounting for perturbations of the thermal
equilibrium of the corona by compressive, e.g. slow magnetoacoustic, waves.
Indeed, assuming the plasma heating and cooling processes are some different
functions of the plasma thermodynamic parameters, i.e. density and
temperature, and potentially of the magnetic field too, both of them can be
perturbed by the waves. Such a wave-caused destabilisation of the initial
thermal equilibrium leads to the onset of a _heating/cooling misbalance_
acting as an additional natural mechanism for the energy exchange between the
plasma and the wave.

Kolotkov et al. (2019) showed that it is convenient to use specific thermal
misbalance time scales $\tau_{1}$ and $\tau_{2}$ connected to the rates of
change of the net energy gain $H(\rho,T)$, and loss $L(\rho,T)$ including
radiative cooling and field-aligned thermal conduction, through the function
$Q(\rho,T)=L-H$ with the plasma density $\rho$ and temperature $T$,
$Q_{T}\equiv\left(\partial Q/\partial T\right)_{\rho}$ and
$Q_{\mathrm{\rho}}\equiv(\partial Q/\partial\rho)_{T}$, as $\tau_{1}={\gamma
C_{\mathrm{V}}}/\left[{Q_{T}-(\rho_{0}/T_{0})Q_{\rho}}\right]$ and
$\tau_{2}={C_{\mathrm{V}}}/{Q_{T}}$ with $C_{\mathrm{V}}$ being the specific
heat capacity. With this, the dispersion relation for linear slow waves
describes the evolution of two acoustic and one thermal (entropy-related)
modes. The 3rd order dispersion relation can be solved numerically for the
real, $\omega_{\mathrm{R}}$, and imaginary, $\omega_{\mathrm{I}}$, parts of a
complex frequency $\omega$, while allowing the value of $\omega_{\mathrm{I}}$
to be comparable to $\omega_{\mathrm{R}}$. The _q-factor_ (also known as
quality factor) is computed as $q=\tau_{\mathrm{D}}/P$, using the oscillation
period $P=2\pi\omega_{\mathrm{R}}^{-1}$ and damping time
$\tau_{\mathrm{D}}=\omega_{\mathrm{I}}^{-1}$.

Using the radiative cooling $L(\rho,T)$ from the CHIANTI atomic database (Dere
et al., 1997, 2019), and a parametrised heating function
$H(\rho,T)\propto\rho^{a}T^{b}$, Kolotkov et al. (2019) compute numerically
the oscillation quality factor $q$ as a function of the heating indices $a$
and $b$.

Figure 7: Dependence of the oscillation quality factor $q$ on the density and
temperature power indices used for parametrising the heating function as
$H(\rho,T)\propto\rho^{a}T^{b}$, for the following values of the equilibrium
parameters of the plasma: $T_{0}=6.3\times 10^{6}\,\mathrm{K}$,
$\rho_{0}=10^{-11}\,\mathrm{kg\,m}^{-3}$, loop length $L=180\times
10^{6}\,\mathrm{m}$, Spitzer conductivity
$\kappa=10^{-11}T_{0}^{5/2}\,\mathrm{W\,m}^{-1}\,\mathrm{K}^{-1}$, mean
particle mass $m=0.6\times 1.67\times 10^{-27}\,\mathrm{kg}$,
$k_{\mathrm{B}}=1.38\times
10^{-23}\,\mathrm{m}^{2}\,\mathrm{kg}\,\mathrm{s}^{-1}\,\mathrm{K}^{-1}$, and
$\gamma=5/3$. The black and red contours show the regimes of damping (positive
$q$) and amplification (negative $q$), respectively. The blue-shaded region
shows values of $a$ and $b$ where the time scales of the heating/cooling
misbalance $\tau_{1}$ and $\tau_{2}$ become negative, for which other thermal
instabilities may occur (see Field, 1965). The hatched region shows values of
$q<2$ for slow-mode oscillations detected in observations (Nakariakov et al.,
2019). The yellow-shaded region outlines heating models which are stable to
thermal instability and for which $q<2$. Figure modified from Kolotkov et al.
(2019).

The results are displayed in Fig. 7 for different values of the heating power
indices $a$ and $b$ and for the equilibrium parameters corresponding to hot
and dense post-flare coronal loops. Direct comparison of the values of $q$
shown in Fig. 7 with those usually seen in observations of slow
magnetoacosutic oscillations of coronal loops (see e.g. Nakariakov et al.,
2019, for the most recent statistical survey) by e.g. SOHO/SUMER and
Yohkoh/BCS, shows that the heating models with $a$ and $b$ approximately
delineated by a triangle with vertices $(-3.5,-4)$, $(0,-0.5)$, and $(5,-4)$
in Fig. 7, excluding the regions of the thermal mode instability, could be
responsible for the observed rapid damping of slow waves in the corona (Wang,
2011). Thus, the proof-of-concept of Kolotkov et al. (2019) show that slow
waves can indeed serve to put limits on (power indices of) the coronal heating
function.

The discussed misbalance between heating and cooling processes in plasma can
cause additional phase shifts between density and temperature perturbations in
slow waves, thus affecting, for example, estimates of the effective polytropic
index. In particular, Zavershinskii et al. (2019) demonstrated analytically
that the polytropic index, a coefficient linking the slow wave phase speed
with the plasma temperature, can vary non-monotonically with temperature due
to the effect of the thermal misbalance, so that it deviates from the
adiabatic value 5/3 to 1.4-3.2. This can be a natural cause for higher values
of the polytropic index at hotter plasma temperatures observed by Krishna
Prasad et al. (2018), that cannot be explained by the classical Spitzer
thermal conduction. Implication of these theoretical results for probing the
corona is a promising future research avenue.

##  3 Models

###  3.1 Phase mixing models

Figure 8: Snapshot of Alfvén wave velocity component out of the plane, for a
periodic driver on the bottom boundary. The figure displays the turning of the
wave fronts due to phase mixing in the middle region with the largest
variation of the Alfvén speed, and the associated phase mixing damping in the
region where the turning is the strongest, is indicated with the vertical
white dashed line. Figure taken from McLaughlin et al. (2011).

As already pointed out by Cowling (1953) and Piddington (1956), classical
resistive or viscous dissipation of the Alfvén wave energy is not efficient in
a homogeneous magnetized plasma and this slow dissipation rate remains a key
obstacle in more advanced present day wave-based heating models (see e.g.
Arregui, 2015; Cargill et al., 2016). The main problem is to rapidly transfer
the wave energy from large to small length scales, where (classical)
dissipation is effective. The concept of phase mixing as described by
Heyvaerts and Priest (1983) aims to enhance this dissipation rate through the
creation of small length scales in an inhomogeneous medium. Phase mixing
occurs when a local gradient in the Alfvén speed is present; as waves
propagate along field lines with different Alfvén speeds, the wave front turns
and the waves become increasingly out of phase. This is shown graphically in
Fig. 8, displaying results of McLaughlin et al. (2011). This process generates
increasingly smaller length scales (across the background magnetic field) and
hence, more efficient dissipation. Although resonant absorption and phase
mixing are often discussed as individual mechanisms, there is a natural link
between these mechanisms, as both rely on the presence of a variation in the
local Alfvén speed profile (Soler and Terradas, 2015). Hence, the small-scale
oscillations in the resonant layer will naturally undergo phase mixing due to
local variation in the Alfvén speed (see e.g. Ruderman et al., 1997a, b).
However, the damping caused by phase mixing depends on the actual value of the
resistivity, while the damping of resonant absorption does not (as resonant
absorption itself is an ideal process). An extensive body of literature on
both resonant absorption and phase mixing exist, for which we refer the
interested reader to reviews by e.g. Aschwanden (2004); Goedbloed and Poedts
(2004); Goossens et al. (2011).

Most theoretical (including computational) studies of phase mixing start from
an equilibrium setup where a pre-existing profile in the Alfvén speed is
present. Most often, this setup consists of a uniform background magnetic
field, where a gradient in the density is balanced by a temperature variation
to maintain pressure balance. Waves are then considered as perturbations of
this equilibrium or injected through boundary driving. The process of phase
mixing will lead to the most efficient dissipation of the Alfvén waves (and
hence strongest heating) where the gradient in the density (Alfvén speed)
profile is steepest.

The self-consistency of heating by phase mixing of Alfvén waves was
investigated in more detail by Cargill et al. (2016) by analysing the
evolution of the density profile. These authors showed that although phase
mixing increases the efficiency of the wave energy dissipation where gradients
in the local Alfvén speed occur, the resulting heating in this basic model
cannot self-consistently sustain the required density profile in closed loops.
As the local density in closed loops is related to the magnitude of the
heating (see e.g. Klimchuk, 2006; Reale, 2014), the phase mixing heating
profile is not consistent with the heating profile to sustain the density
profile. Cargill et al. (2016) also investigated whether feedback of the
heating on the density profile through evaporation would be able to modify the
local density (see also Ofman et al., 1998). Although some local structuring
of the density profile occurred, it only happened on timescales longer than
the cooling and draining timescales and hence, does not help address the
efficiency problem of wave-heating. In addition, the authors point out that
transport coefficients need to be substantially enhanced to obtain effective
heating in the first place. Using MHD simulations including thermal conduction
and optically thin radiation, Van Damme et al. (2020) for the first time
modelled the feedback process through evaporation entirely self-consistently
(i.e. without the use of scaling laws) and also found that Alfvén wave phase
mixing only leads to modest heating in the shell regions of the loop, where
the mass increase through evaporation is not sufficient to modify the phase
mixing process.

If the imposed density structure is not compatible with the heating profile
resulting from phase mixing, how then is this assumed equilibrium structure
supported? Is an alternative heating mechanism present that is compatible with
the density structuring? And if so, does that immediately imply that the wave-
based heating (through phase mixing) is comparatively small?

In addition to requiring the presence of a variation in the local Alfvén speed
profile, the phase mixing model as it was originally introduced by Heyvaerts
and Priest (1983) implies the presence of an ignorable coordinate, a setup
which might not be representative of the highly inhomogeneous solar atmosphere
(Parker, 1991; Ofman and Davila, 1995), although approximately similar
conditions may occur in coronal holes.

Pagano and De Moortel (2017) investigated the heating by phase mixing in a 3D
numerical model of a boundary-driven flux tube. The flux tube is modelled as a
cylindrical density enhancement in a uniform magnetic field and transverse
displacements of the footpoint of the cylinder generate kink modes which, as
they propagate along the flux tube, couple to azimuthally polarised Alfvén
modes in the boundary shell of the cylindrical flux tube. Due to the density
gradient in the inhomogeneous boundary layer, the Alfvén waves phase mix but,
even using (excessively) large values of magnetic resistivity and large-
amplitude footpoint driving, the heating due to phase mixing was found to be
insufficient to be relevant for coronal heating (i.e. to balance the expected
losses through radiation and conduction), a conclusion similar to the remark
made by Cargill et al. (2016) about the need to substantially enhance
transport coefficients. By varying parameters such as the length of the non-
uniform layer, the density structure, and the persistence of the driver,
Pagano and De Moortel (2017) find that phase mixing of these propagating waves
leads to temperature increases of the order of 105K or less, a figure that is
in agreement with the analytical estimate by Terradas and Arregui (2018), even
though these were for standing waves.

When simulations are performed using 3 observed coronal loop oscillation
harmonics as input, Pagano et al. (2018) found that the presence of these
multiple harmonics causes drifting of the location of the heating. Still, the
mechanism did not seem to provide enough energy to maintain the full thermal
structure constrained by the observed coronal properties, and the multiple
harmonics inhibited the formation of small scales. Pagano and De Moortel
(2019) included an observed spectrum of transverse waves, using a boundary
driver which consists of a series of 1000 superimposed random pulses was used
drawn from a reconstructed spectrum. The results again indicate that it is
unlikely that phase mixing of Alfvén waves generated by the observed power
spectrum heats coronal loops, although, the waves could be important in the
generation of small scales.

###  3.2 Alfvén wave heating models

####  3.2.1 Alfvén wave induced shock heating

Alfvén waves have long been particularly attractive coronal heating candidates
due to their ability to carry large amounts of energy throughout the solar
atmosphere (Alfvén, 1947; Uchida and Kaburaki, 1974) and can potentially also
lead to solar wind acceleration (Hollweg, 1990; Ofman, 2010). The Poynting
flux upward from convective motions in magnetic concentrations at photospheric
level is expected to be on the order of $10^{6}$ W m-2 (Parnell and De
Moortel, 2012), although the strong magnetic expansion in the upper layers
leads to an effective Poynting flux of $10^{4}$ W m-2 into the chromosphere.
The ion-neutral friction due to the partial ionisation of the plasma in the
chromosphere strongly damps the Alfvén waves, in particular the high-frequency
spectrum. Combined with reflection, these effects lead to only a $1-3$ %
effective transmission rate into the corona (Soler et al., 2017, 2019),
similar to estimated footpoint leakage of coronal waves (De Pontieu et al.,
2001). Overall, the energy budget from torsional Alfvén waves generated in the
photosphere is estimated to be on the order of $10^{2}$ W m-2 in the corona,
which is just enough for the quiet Sun or coronal hole. The double mode
conversion process from p-mode waves into Alfvénic waves in the chromosphere
and transition region is expected to contribute similar energy rates (Morton
et al., 2019).
Various models for Alfvén wave propagation in the solar atmosphere have a
different nature depending on the region and wave guide under consideration.
The geometry of the waves has been selected to be either planar (a.k.a linear,
e.g., Suzuki and Inutsuka, 2005; Murawski and Musielak, 2010) or azimuthal
(a.k.a. circular, e.g., Zhugzhda, 1996; Vasheghani Farahani et al., 2010;
Wójcik et al., 2017). For instance Alfvén wave propagation of waves with
linear polarisation may be more appropriate in the relatively diffuse coronal
holes, while the circular polarisation may be more applicable for the
propagation in structures showing strong density contrast, such as solar jets,
spicules, tornadoes, and loops (Vasheghani Farahani et al., 2011).

Alfvén waves need an efficient dissipation mechanism in order to play a
dominant role in coronal heating. A first successful dissipative model was
based on nonlinear mode conversion of these waves into compressive modes
(evolving into shocks) due to the flux tube expansion (and associated
centrifugal force) and the ponderomotive force (Hollweg et al., 1982; Lau and
Siregar, 1996; Ofman and Davila, 1998; Antolin and Shibata, 2010). This idea
has persisted through the decades thanks to the accompanying ability to
generate spicule-like excursions of material into the corona (Kudoh and
Shibata, 1999; Matsumoto and Shibata, 2010; Arber et al., 2016; Brady and
Arber, 2016) and accelerate the solar wind (Matsumoto and Suzuki, 2014; Suzuki
and Inutsuka, 2006). Furthermore, this model has demonstrated that waves can
lead to small nanoflare-like intensity bursts during dissipation (Moriyasu et
al., 2004; Antolin et al., 2008), thus placing caution when interpreting
observations of these events or associating the nanoflare term solely to
reconnection-based models. Although highly self-consistent in its ability to
explain various features of the solar atmosphere, this model relies on the
presence of density fluctuations, and sufficiently large transverse wave
fluctuations (compared to the local Alfvén speed) for the relevant nonlinear
effects to become important. A natural question is whether this matches
observations and the answer yet unknown. Parker Solar Probe does indicate,
however, that the density fluctuations in the open corona is far greater than
previously thought (Bale et al., 2019).

A well established model is that of Suzuki and Inutsuka (2005) who describe
the propagation of a low frequency Alfvén wave in coronal holes from the
photosphere to an altitude of 0.3AU. The granular motion of the photosphere
results in steady transverse motions perturbing the magnetic field lines and
exciting Alfvén waves (Cranmer and van Ballegooijen, 2005). The 1D model
provided by Suzuki and Inutsuka (2005) resembles a superradial open magnetic
flux tube conserving magnetic flux, where the expansion is a two step
superradial function with the use of two expansion factors. Their choice for a
two step superradial function was due to the height dependence of the magnetic
field strength reconstructed by Tu et al. (2005). They included the effect of
field line curvature, radiative cooling and thermal conduction in the MHD
equations. The solutions to these MHD equations enabled Suzuki and Inutsuka
(2005) to compute the variations of the radial and tangential speeds together
with the temperature and density as a function of altitude (see Fig. 9 for a
sketch of the configuration). The results proved adequate for atmospheric
heating due to two effects; the first stage of heating is due to the
dissipation of low frequency Alfvén waves mainly heating the inner solar
atmosphere, while the second stage of heating is due to the induction of
compressive perturbations (especially slow waves) due to the nonlinear efects
connected with the Alfvén wave propagation that causes wave steepening ending
up in shock formation that mainly contributes towards coronal heating. The
nonlinear effects connected with outward propagating Alfvén waves that
experience shocks contributes in two ways; the first aspect is the heating due
to the increase of wave amplitudes that is confirmed by the non-thermal
broadening of the emission lines, the second aspect is the creation of shocks
which rapidly dissipates the waves. The contribution of the second aspect
towards heating is greater than the first aspect (Nakariakov et al., 2000). As
a matter of fact, the efficiency of damping and hence heating due to low
frequency Alfvén waves also depends on the activity of the region. In
particular, the damping of surface Alfvén waves occurs on a shorter scale in
active regions compared to quiet solar regions. This means that in quiet
regions, the surface Alfvén wave is able to contribute towards heating the
corona at higher altitudes (Evans et al., 2009). In addition, the period of
the Alfvén wave itself is key in determining its contribution to coronal
heating. This statement is backed by the simulations of Matsumoto and Shibata
(2010) where nonlinear Alfvén waves were driven by photospheric convection
towards the transition region in the presence of gravity and empirical
chromospheric cooling.

Figure 9: Left: Sketch on the configuration of Alfvén wave heating models.
Right: Alfvén wave energy flux as a function of period. The figures are taken
from Matsumoto and Shibata (2010).

Their model showed that the region bounded between the photosphere and the
transition region acts as a resonant cavity, a fact that has been recently
observed above sunspots for the first time (Jess et al., 2019). It can be
readily noticed from the right panel of Fig. 9 that waves with periods between
$100$ s to $500$ s are able to transport flux to the corona, and that the
maximum energy flux is carried when the period of the Alfvén wave is around
$400$s coinciding with the resonant frequencies in the cavity (Matsumoto and
Shibata, 2010).

Coronal resonances of Alfvén wave propagating in solar loops have also been
reported in the model proposed by Antolin and Shibata (2010). The loop is a
gravitationally stratified semitorus, with magnetic field expansion, similar
to the left panel of Fig. 9. The advantage of this model is that it sheds
light on the efficiency of heating taking into account the loop geometry in
which Alfvén waves propagate. This enabled Antolin and Shibata (2010) to state
that Alfvén waves provide efficient and uniform heating when propagating in
thick loops where the area expansion between the photosphere and corona is
greater than 2.5 orders of magnitude with lengths at least as long as 80 Mm.
Since in the solar quiet regions the loops are both longer and wider compared
to loops in active regions, the efficiency of Alfvén wave heating in quiet
solar regions is more pronounced compared to active regions. Regarding the
period of the waves, Antolin and Shibata (2010) concluded that the shocks
connected with long-period waves increase the average temperature of the
corona, while shocks connected with short-period waves are unable to further
heat the corona despite being more numerous (Suzuki and Inutsuka, 2006). The
short-period waves are only just strong enough to maintain the coronal
temperature. Heating by the resonance cavity with a monochromatic driver
matching the eigenfrequency of the loop was found to lead to temperatures
close to 5 MK, but the loops were not in a state of thermal equilibrium.

Regarding the contribution of the compressive shocks towards coronal heating,
Matsumoto (2016) stated that shock compressive heating is very efficient below
the altitude of 4 Mm, while above this height the incompressive heating due to
direct dissipation of magnetic and velocity shear in Alfvén waves is dominant.
Moreover, Matsumoto and Suzuki (2014) showed that these chromospheric
compressive shocks generate curved wedge shaped Alfvén waves, which could play
a role in heating the higher layers. Thus, the back reaction of the induced
compressive perturbations on the Alfvén wave results in Alfvén wave shocks
that also contribute towards coronal heating. The induced compressive
perturbations are due to the nonlinear forces connected with Alfvén waves,
namely the magnetic tension, centrifugal, and ponderomotive forces. It is
worth noting that the ponderomotive coupling of Alfvén waves to slow modes
creates shocks that dominates the heating due to resistive dissipation (Arber
et al., 2016; Brady and Arber, 2016). The non-linear behaviour of Alfvén waves
was described by e.g. Suzuki (2008); Vasheghani Farahani et al. (2011); Suzuki
(2012); Vasheghani Farahani and Hejazi (2017). In particular, Vasheghani
Farahani et al. (2012) implemented the second order thin flux tube
approximation (Zhugzhda, 1996) and studied the parallel nonlinear cascade of
torsional and shear Alfvén waves in open magnetic fields. They showed that the
shock formation for shear Alfvén waves comes into play earlier than torsional
waves in the lower solar atmopshere.
Another mechanism playing a role in chromospheric heating is Ohmic diffusion
and ion-neutral collisions which strongly affects torsional Alfvén waves
during their propagation in the chromosphere (Soler et al., 2019; Wójcik et
al., 2020) and leads to heating. In any case, the chromosphere plays a crucial
role for coronal heating, because of the resulting evaporation of material due
to coronal heating and thus providing the mass source. The localised heating
could result in thread structuring in the corona (Copil et al., 2008), which
is in contradiction with the recent results of Cargill et al. (2016); Van
Damme et al. (2020).

####  3.2.2 Alfvén wave turbulence heating

Turbulent heating models have gained popularity recently in the context of
coronal heating (see the review by Bruno and Carbone, 2013, of this mechanism
in the solar wind context). However, there is still no direct evidence that
the corona is turbulent. This might change in the near future thanks to the
Parker Solar Probe, which could confirm turbulence at heights around or below
the Alfvén critical point. Still, the non-thermal broadening of the coronal
spectral lines could already be indicative of turbulent fluctuations (Banerjee
et al., 1998; Singh et al., 2006; Hahn and Savin, 2013, 2014). More indirect
evidence is present in the CoMP observations through the measurement of a
$1/f$ spectrum in closed loops (Morton et al., 2016, 2019). Turbulent heating
models are best categorized as wave or alternative-current (AC) mechanisms in
the limit of weak turbulence, as the turbulent energy cascade is thought to be
generated by nonlinear (self) interactions of waves. The underlying idea of
energy conversion is similar to other wave-based heating mechanisms:
increasingly smaller length scales are created, until dissipation becomes
important and converts kinetic and magnetic energy into heat. The main
difference compared to other mechanisms is that in turbulence small scales are
created nonlinearly (unlike in phase mixing, resonant absorption, etc.),
producing an energy cascade to smaller scales (although, in some scenarios
inverse cascade can also take place). We differentiate three main scales or
ranges in turbulence, as depicted schematically in Fig. 10. The largest scale
is the energy containing scale, at which the forcing of the plasma takes
place, e.g. the size of convective cells in the photosphere (depicted in blue
in Fig. 10). The inertial range is an intermediate scale where the actual
energy cascade is initiated. The dynamics are self-similar and independent of
both the nature of forcing and dissipation (shown in green in Fig. 10),
expected to correspond to Kolmogorov power law indices of -5/3. Lastly, at
scales where dissipative terms are on the order of advective terms (Reynolds
and/or magnetic Reynolds numbers $\approx 1$), turbulence enters the
dissipation range, where heat is generated (assuming collisional dissipation),
displayed with the red zone in Fig. 10. In this last range, the power law
index depends on the particular dissipation mechanism (e.g. electron or proton
dissipation) that is considered, growing ever steeper.

Figure 10: A schematic graph of the power law behaviour of turbulent
fluctuations in the solar atmosphere and wind. The left, blue range shows the
energy input range with a power law index of -1. The middle, green range shows
the inertial range with a power law index of -5/3, where the energy cascade
rate is independent of the scale. The right, red range shows the dissipation
range, where the ever steeper power law slope depends on the specific
dissipation process.

The most researched turbulence-generating wave interaction is that of counter-
propagating pure Alfvén waves in the incompressible limit. In this scenario,
the Alfvén wave wavefronts are deformed in successive collisions, leading to a
cascade towards higher wavenumbers (see Figure 11).

Figure 11: Illustrating the mutual deformation of two counter-propagating
Alfvénic wave-packets along some background magnetic field $\mathbf{B}$. The
undisturbed wave-packets must collectively vary along both perpendicular
directions for nonlinear interaction to occur (Howes and Nielson, 2013),
depicted by the different orientation of the ellipsoid making up the packets.
These ellipsoids can be considered isosurfaces of velocity perturbations.
After the collision (on the upper left), the wave-packets are mutually
deformed, leading to ellipsoids scattered to higher wavenumbers, i.e.
appearing thinner.

In the corona, counter-propagating transverse or Alfvénic waves can exist in
both closed structures, such as coronal loops, and open structures, due to
wave reflection. Wave reflection occurs when the Alfvén speed varies along the
propagation direction, i.e. along the background magnetic field, linearly
coupling the outward and inward-propagating Alfvén waves (Leroy, 1980; Hollweg
and Isenberg, 2007). Alfvén speed variations along the field exist in the
corona because of gravitational stratification of the plasma, and the
spherical/dipolar expansion of the magnetic field with height. In the
turbulent heating models, the source of input energy is usually the convective
flows in the photosphere, which generate waves and eventually turbulence by
shuffling the magnetic field lines. There are numerous models of coronal
heating based on turbulence for both coronal loops (van Ballegooijen et al.,
2011; Downs et al., 2016; van Ballegooijen et al., 2017), and open structures
(Perez and Chandran, 2013; Woolsey and Cranmer, 2015; Chandran and Perez,
2019), and also global models (van der Holst et al., 2014; Oran et al., 2017).
The models for Alfvén wave heating of active regions (van Ballegooijen and
Asgari-Targhi, 2018) show that temperatures reach maximally 2.5 MK, and that
other heating mechanisms are needed to go beyond those temperatures.
The ‘local’ models mostly use a reduced MHD treatment, which neglects density
perturbations (i.e., are incompressible) and transverse inhomogeneities in the
plasma, among other simplifications. For the global models, Alfvén wave
equations are employed in van der Holst et al. (2014), containing heavily
approximated terms for reflection and turbulent dissipation, and Reynolds-
averaged mean flow equations with turbulent transport in Usmanov et al.
(2018). All these models conclude that a multi-million K corona can be
maintained by turbulent heating, while balancing radiative cooling and
conductive losses. A common discrepancy between turbulent heating models and
observed properties of the corona is the excessive non-thermal line broadening
that results from the models (Brooks and Warren, 2016).

Recently, it was realized that the existence of counterpropagating waves is
not the only way to generate turbulence (Magyar et al., 2017, 2019). In the
presence of transverse inhomogeneities, propagating Alfvénic waves can
nonlinearly self-deform, generating a cascade to smaller scales, called
uniturbulence (see Figure 12).

Figure 12: Same as in Figure 11, but having only the wave-packet propagating
parallel to the magnetic field. The background plasma is inhomogeneous across
the field. This is causing the initial Alfvénic wave-package to self-deform
nonlinearly as it propagates, leading to elongated ellipsoids, i.e. cascading
to higher wavenumbers.

This can constitute an additional channel towards a turbulent cascade and can
potentially enhance the dissipation rate, leading to more heating. However,
there are no models of coronal heating incorporating the effects of
uniturbulence, as of yet. Still, forward modelling of simulations with
uniturbulence (Pant et al., 2019) can elegantly and self-consistently explain
the observed correlation between wave Doppler shift and spectral line
broadening (McIntosh and De Pontieu, 2012, as also explained in Nakariakov et
al.,2021).

Full 2.5D and 3D MHD models are now available (e.g. Dahlburg et al., 2012;
Shoda et al., 2019) that take into account both the nonlinear Alfvén wave
heating via compressive effects, and the Alfvén wave turbulence, and initial
results seem to point to the dominance of Alfvén wave turbulence over the
compressive heating in the corona, while the opposite is true at the lower
heights of the magnetic canopy (Matsumoto, 2016, 2018). Since these models
target quiet Sun conditions for the loops it is not yet clear whether the same
holds for the denser and more dynamic active region loop conditions.

To date, all Alfvén wave heating models have been mostly successful at
generating and sustaining a lower energy budget corona as in the quiet Sun or
coronal holes, while they all seem to fall short of the $10^{4}$ W m-2 average
heating rate needed for active region coronal loops (see chapter 6.1 Hinode
Review Team et al., 2019, for a more thorough discussion). In particular, the
hot loops at the core of active regions with temperatures of 5 MK or more seem
to be unexplained by wave models in general (van Ballegooijen and Asgari-
Targhi, 2018). Furthermore, these models seem to generate preferentially
uniformly heated coronae in line with the RTV scaling law (Rosner et al.,
1978), rather than footpoint heating. In turn, Alfvén wave heating models do
not seem able to explain thermal non-equilibrium and the associated coronal
rain (Antolin et al., 2010). Comparison of results from the Alfvén wave
turbulence model with Hinode/EIS observations of active region loops indicates
that longitudinal flows are an important ingredient, suggesting that either
compressive effects may become important or that a different mechanism is
present (Asgari-Targhi et al., 2014). It is important to note, however, that
the very demanding computational requirements needed in fully consistent wave
models strongly limit the current predictions (see section 4)

###  3.3 Kink-mode driven turbulent heating models

Standing transverse waves can be considered as superposition of
counterpropagating waves, thus these can generate a turbulent cascade as well.
In closed loops, initial perturbations such as flares, low coronal eruptions
or shock waves can easily excite standing kink oscillations. Due to the shear
motions between the loops and external corona, the Kelvin-Helmholtz
instability can thus be induced near the loop boundary (Terradas et al.,
2008a; Antolin et al., 2014, 2017; Magyar et al., 2015). Moreover, the
generation of azimuthally polarised Alfvén waves due to resonant absorption
increases the velocity shear with the external plasma, which can also enhance
the instability (Antolin and Van Doorsselaere, 2019). The effect of the
instability is to generate transverse wave induced Kelvin-Helmholtz rolls
(TWIKH rolls), extended along the magnetic field, as shown in Fig. 13. The
wave energy can dissipate efficiently if the turbulent structures are
sufficiently developed. Direct comparison of such a model with observations of
a prominence (Okamoto et al., 2015; Antolin et al., 2015), gives an indication
that the emission is indeed shifting from a temperature of $10^{4}$ K to
$10^{5}$ K after the passing of the transverse wave, in accordance with the
numerical model (see Fig. 14). Recent studies by Antolin and Van Doorsselaere
(2019) revealed that resonant absorption plays a key role in energizing and
spreading the TWIKH rolls throughout a coronal loop.

Figure 13: A tomography of TWIKH rolls. Cross-sections of density (in yellow)
are shown at different locations along half the length of a prominence
oscillating with a standing kink mode after roughly 2 periods. The coloured
lines correspond to selected magnetic field lines traced from the bottom. The
TWIKH rolls drag the field lines, generating twisting and strong changes in
magnetic field pressure in a wide layer around the boundary. The simulation
corresponds to the high resolution run discussed in Antolin et al. (2015).
Figure 14: Evidence for resonant absorption and heating in a prominence. The
left column provides a set of observables obtained with coordinated
observations (top panel) between Hinode/SOT and IRIS of an active region
prominence off-limb. (Left-middle) Prominence threads were seen oscillating
transversely and fading in the Ca II H line of SOT ($10^{4}$ K) and
subsequently appearing in the hotter SJI 1400 channel of IRIS (dominated by Si
IV emission at $10^{4.8}$ K approximately). (Left-bottom) The prominence
threads oscillate in the POS (yellow with green curve) out-of-phase with the
Doppler velocity in Si IV captured by the IRIS slit (purple). The right column
shows the forward modelling of a numerical simulation of a prominence thread
with the observed characteristics. The thread oscillates transversely with a
kink mode and triggers the KHI at the boundary (top-right). Resonant
absorption is initially highly localised in the boundary, which fuels the KHI.
TWIKH rolls are excited, whose dynamics reflect the resonant absorption and
phase mixing dynamics but at a larger, observable scale. The TWIKH rolls mix
the cold prominence plasma with the hot coronal surroundings and widen the
boundary of the loop. A turbulent cascade is established which leads to wave
dissipation and heating. As the prominence plasma is heated, its emission in
the synthesised Ca II H line fades out and becomes stronger in the hotter Si
IV line (right-middle). The resonant absorption and phase mixing dynamics
become observable due to the KHI, leading to an out-of-phase relation between
the POS motion of the thread and the Doppler velocity of the heated prominence
plasma. A very good match is thus obtained with the observations. A sketch of
the observed physical processes is given on the right. Adapted from Okamoto et
al. (2015) and Antolin et al. (2015).

The most basic setup that was used was a cylindrical loop with an initial
velocity pulse, setting up a damped standing wave. This setup was used to show
that the non-linearly developing KHI introduces an amplitude dependent damping
(Magyar and Van Doorsselaere, 2016), which matches reasonably well with
observations (Goddard et al., 2016a). As intuitively understood, the magnetic
twist provides a stabilising effect (Howson et al., 2017a; Terradas et al.,
2018), which delays the development of the instability without completely
suppressing it. This corresponds well with the predictions from analytical
models (Hillier et al., 2019; Barbulescu et al., 2019). Moreover, the
stabilising effect of viscosity and resistivity was investigated numerically
(Howson et al., 2017b; Karampelas et al., 2019a). However, as explained in
Sec. 2.1, simple calculations using basic estimates show that the directly
observed standing waves in the solar corona do not seem to contain sufficient
energy for heating it.

It was soon realised that the TWIKH rolls could play an important role if
waves are driven at a loop footpoint. In that configuration, the TWIKH rolls
provide an elegant mechanism to move the energy from the input scales, to the
smaller length scales of the inertial range, and on to the dissipation scales
at which the plasma is heated, just as Alfvén wave turbulence would do. This
damping mechanism in the inertial range is independent of transport
coefficients. In doing so, the energy cascade balances the input energy,
creating a (statistically) steady state, indirectly feeding the driving energy
into plasma heating. The steady state that is obtained in the numerical models
could be a potential model for the observed decayless observations.

In the setup with the footpoint driver in a closed loop, Karampelas and Van
Doorsselaere (2018) obtained a fully deformed loop cross-section. Although the
turbulence significantly changes the initial equilibrium, the loop structure
can still be observed in forward models, meaning that turbulent loops
probability exist in the real solar corona. Further studies (Karampelas et
al., 2019a) considering gravity in a loop, stressed the importance of
longitudinal stratification, which improves the efficiency of wave heating.
The TWIKH roll formation dependence on the driver frequency was investigated
by Afanasyev et al. (2019), who subsequently also considered the driving with
a turbulent spectrum (Afanasyev et al., 2020).

Considering the turbulent motions at the footpoint of a coronal loop anchored
in the photosphere, Guo et al. (2019b) employed a mixed transverse and
torsional driver at the loop footpoint, going beyond the implicitly excited
Alfvén waves in the simulations of Pascoe et al. (2011). Comparing with a pure
kink driven loop and a pure torsional driven loop, they found that the KHI
eddies are fully developed in the whole loop cross-section and that internal
energy and temperature in the mixed driven loop are higher. This means that
the mixed modes lead to a more efficient dissipation in the turbulent state of
plasma and that the KHI acts as an agent to dissipate energy in both wave
modes. In addition, forward models showed very similar images to the
observations of decayless oscillations. This means that the ubiquitous
decayless oscillations could play an important role in coronal heating.
However, as is shown in Figure 15, due to the limited resolution of
instruments, neither Alfvén modes nor the fine structures are observable. This
means that although difficult to capture, Alfvén modes probably can coexist
with kink modes, leading to enhanced heating.

Figure 15: Forward-modelling results for the numerical models in the Fe IX 171
$\AA$ line at the apex with an LOS angle of $45^{\circ}$. The upper row is
obtained with the full numerical resolution and the lower row with a degraded
resolution comparable to SDO/AIA. Figure taken from Guo et al. (2019b).

A further study by Guo et al. (2019a) examined how such a mixed mode driver
affects a tightly packed multi-stranded loop. By depositing almost identical
input energy flux at the footpoint of a multi-stranded loop and a density
equivalent monolithic loop, Guo et al. (2019a) found that the multi-stranded
loop is more efficient in starting the heating process due to the quickly
established turbulent state in this model.
Again using a model with footpoint driven loops, Karampelas et al. (2019b)
computed the dependence of the wave amplitude as they would be observed from
forward modelling, on the driver amplitude. They found that there is only a
weak dependence between both. Thus, they showed that the amplitude of the
decayless waves is actually an unrealiable measure of the wave energy that is
being dissipated in the solar corona, as studied in more detail by Hillier et
al. (2020). Karampelas et al. (2019b) concluded that the spectral line
broadening is a much more accurate quantity to give a measure of the wave
energy that is being dissipated. This may influence assessments of energy
content and flux of standing or propagating kink waves (as e.g. in Sec. 2.1).

Even though these models seem promising, at the time of submission, there is
no published paper in which the non-linearly evolving, footpoint-driven
transverse waves can compensate the radiative or thermal conduction cooling of
the coronal plasma in a self-consistent model, contrary to DC heating models
of the solar corona (e.g. Hansteen et al., 2015). But, it should be noted that
the 3D MHD DC models usually employ unrealistically high resistivity and
viscosity, which are the sources for the heating in them.

##  4 Critical Assessment & Conclusions

This review provides an overview of the progress made in (roughly) the last
decade in observations and models of wave heating in the solar corona. A lot
of progress has been made in wave-based heating mechanisms using a combination
of advanced high-resolution satellite and ground-based solar observations, and
advances in computational MHD modeling.
From an observational point of view, we now have detailed observations of
several wave modes that could play a role in heating the corona. Recent multi-
wavelength studies of coronal loops have not found evidence of any significant
heating following large-amplitude kink oscillations (Goddard and Nisticò,
2020; Pascoe et al., 2020). However, decayless kink oscillations and quasi-
periodic propagating fast waves could potentially account for coronal losses
in certain regions. We now have several observational characterisations of the
wave energy content in the solar corona, based on imaging, spectroscopy or
spectropolarimetry. In the chromosphere, a very large energy flux is found in
waves, and if even only a fraction of that is dissipated in the corona, it is
heated efficiently. In the corona itself, wave observations find energy fluxes
comparable to the required heating rate, but also one or two orders of
magnitude below it, particularly in the case of active regions. This wide
disparity in observational estimates of coronal wave energy points to a
variety of wave activity manifestations, and probably indicates that the issue
of coronal wave energy requires further study. The latter point is also
underlined by the discrepancy between the energy flux in numerical models and
their forward modelling (De Moortel and Pascoe, 2012; Van Doorsselaere et al.,
2014; Karampelas et al., 2019b).

From the modelling point of view, the computational models have been lifted to
a new level in recent years thanks to advancement in computational power. Now,
3D MHD simulations are the norm. In addition to previously considered AC
heating mechanisms by non-linear Alfvén waves, these 3D MHD simulations show
that turbulence and development of small scales is important. Simulations of
Alfvén wave turbulence in (perpendicularly) uniform plasmas are able to
maintain coronal temperatures, even in 3D configurations. In order to take
into account the perpendicular density structuring, studies have focused on
phase mixing and the transverse wave induced Kelvin-Helmholtz instability.
Despite the strong effort in these last two subjects, still no self-consistent
simulation exists in which a density enhanced coronal loop is maintained
against cooling by radiation or conduction, with resolved wave energy flux
from the photosphere.

Self-consistent wave modelling studies are extremely demanding in
computational power. The photospheric source (or sub-photopheric), the
chromosphere (rich in wave processes due to its high- to low-$\beta$ variable
conditions), the transition region (in which the Alfvén speed steepens leading
to reflection, refraction and mode conversion) and the corona (in which small
density inhomogeneities and wave-to-wave interaction are important) need to be
included in the model. In addition, the resolution needs to be high enough to
allow wave-associated instabilities and the turbulent spectrum to set in with
the inertial range established for at least a decade in wave number. Through
the decades, these limitations have favoured the investigation of wave
dissipation in waveguides that are already well-defined (such as coronal
loops) without the connection to other atmospheric layers (the latter is only
recently changing, e.g. Van Damme et al., 2020). Although limiting, this
approach has lead to an essential categorisation of wave processes and wave
dissipation locally, and to phase relations between observables for a direct
comparison with high resolution observations (Hinode Review Team et al., 2019,
chapter 6.1). It remains to be seen how these models connect to more advanced
models of wave propagation in the lower atmosphere (e.g. Felipe et al., 2011).

This approach has also allowed to investigate the relative importance of
specific wave mechanisms and the available energy at the coronal base (e.g.
Soler et al., 2019). Improvements in numerical techniques and increased
computation power are coming close to allowing self-consistent modelling which
incorporates all the relevant wave processes in a large scale model (such as a
whole active region). This would allow to fully account for nonlinear
processes and, in particular, global observables of wave heating models for an
easier comparison with observations.

These recent developments have made it clear that the omnipresent coronal
waves also strongly influence DC heating models.
Firstly, due to the observed continuous influx of Alfvénic waves into the
corona, it is expected that the resulting heating is more continuous and less
impulsive than that obtained from magnetic reconnection alone. This long-
standing assumption is only true if waves and reconnection act in a
sufficiently independent way from each other. However, we can think of
scenarios in which one facilitates the other. For instance, reconnection-based
nanoflares in the corona are based on the braiding scenario, through which the
magnetic field is stressed to a certain degree before releasing these stresses
through component (small-angle) reconnection, thus leading to impulsive,
middle to low-frequency events (see chapter 6.2 in Hinode Review Team et al.,
2019). Moreover, transverse wave induced Kelvin-Helmholtz rolls (see Sec. 3.3)
create local twists and small-angle misalignments between the strands and, as
in the magnetopause (Nykyri and Otto, 2004), magnetic reconnection is
expected. In this scenario, the heating from reconnection is expected to be
impulsive and act only when the KHI is triggered. Although twists delay the
formation of the KHI, they do not inhibit the instability (see Sec. 3.3).
Further investigation of this scenario is therefore needed.

Secondly, DC heating models are often based on stranded loop models, in which
each magnetic field line is modelled independently from each other (Reale,
2014). The turbulence appearing in Alfvén wave and kink wave models of loops
are very efficient at mixing plasma perpendicular to the magnetic field
(Magyar and Van Doorsselaere, 2016). This mixing induces adiabatic expansion
and contraction of plasma elements, leading to (apparent) cooling and heating
on particular field lines, which is not taken into account if they are
completely decoupled. Moreover, additional magnetic pressure enhancements are
introduced, leading to a restructuring of the plasma in the single fieldline
models.

Thirdly, the nonlinearly induced perturbations due to Alfvén waves are not
restricted to funnels and flux tubes with transverse or parallel structuring,
but also feature in magnetic reconnection sites when Alfvén waves interact
with a magnetic null point. As the Alfvén wave nears the magnetic null-point
it induces magnetoacoustic waves which in turn contribute towards heating at
the null point (Galsgaard et al., 2003; Thurgood and McLaughlin, 2013; Sabri
et al., 2018). The fast and slow magnetoacoustic waves experience shocks in
the vicinity of the null-point leading to enhanced dissipation there
(McLaughlin et al., 2009; Sabri et al., 2018). Thus, despite the Alfvén wave
being unable to reach the null point while travelling along the separatrices,
their coupling to the magnetoacoustic waves provides an agent to their gradual
dissipation and contribution towards heating near X-points (McLaughlin and
Hood, 2004, 2006).

Overall, we conclude this review with the following one-liner: Motivated by
plentiful recent observations of coronal waves, AC heating models have taken a
leap forward in recent years, but have not yet managed to self-consistently
maintain loops at the observed million degree temperatures. These models make
it worthwile to have a renewed look at DC heating models while including the
cross-field effects induced by wave dynamics.

## Acknowledgements

This paper originated in discussions at ISSI-BJ. T.V.D. was supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 724326) and the C1 grant
TRACEspace of Internal Funds KU Leuven (number C14/19/089). H.T. is supported
by NSFC Grants No. 11825301 and No. 11790304(11790300). P.A. acknowledges
funding from his STFC Ernest Rutherford Fellowship (No. ST/R004285/2).
Numerical computations were carried out on Cray XC50 at the Center for
Computational Astrophysics, NAOJ. D.J.P. was supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 724326). L.O. acknowledges support by
NASA grants NNX16AF78G, 80NSSC18K1131 and NASA Cooperative Agreement
NNG11PL10A to CUA. I.A. was supported by project PGC2018-102108-B-I00 from
Ministerio de Ciencia, Innovacion y Universidades and FEDER funds. I.D.M.
acknowledges support from the UK Science and Technology Facilities Council
(Consolidated Grant ST/K000950/1), the European Union Horizon 2020 research
and innovation programme (grant agreement No. 647214) and the Research Council
of Norway through its Centres of Excellence scheme, project number 262622.
D.Y.K. acknowledges support from the STFC consolidated grant ST/T000252/1 and
the budgetary funding of Basic Research program No. II.16.

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