# Alfvénic waves in the inhomogeneous solar atmosphere

R. J. Morton<EMAIL_ADDRESS>R. Sharma  E. Tajfirouze  H.
Miriyala  Department of Maths, Physics and Electrical Engineering, Northumbria
University, UK

###### Abstract

The solar atmosphere is known to be replete with magneto-hydrodynamic wave
modes, and there has been significant investment in understanding how these
waves propagate through the Sun’s atmopshere and deposit their energy into the
plasma. The waves’ journey is made interesting by the vertical variation in
plasma quantities that define the solar atmosphere. In addition to this large-
scale inhomogeneity, a wealth of fine-scale structure through the chromosphere
and corona has been brought to light by high-resolution observations over the
last couple of decades. This fine-scale sturcture represents inhomogeneity
that is thought to be perpendicular to the local magnetic fields. The
implications of this form of inhomogeneity on wave propagation is still being
uncovered, but is known to fundamentally change the nature of MHD wave modes.
It also enables interesting physics to arise including resonances, turbulence
and instabilities. Here we review some of the key insights into how the
inhomogeneity influences Alfvénic wave propagation through the Sun’s
atmosphere, discussing both inhomogeneities parallel and perpendicular to the
magnetic field.

The Sun (1693), Alfven waves (23), Solar corona (1483), Solar chromosphere
(1479), Magnetohydrodynamics (1964)

##  1 Introduction

Ever since it was established that the coronal plasma had temperatures in
excess of a million degrees (based on the emission lines from forbidden
transitions of highly ionized atoms, Grotian 1939; Edlén 1941; Alfvén 1941;
Edlén 1943), heating of the Sun’s atmosphere or corona has remained an
intriguing problem in solar and stellar astronomy. As one of the promising
answers to this dilemma, Alfvénic waves have long attracted the attention of
the community as a mechanism that could not only answer the coronal heating
but can also explain the acceleration of solar wind (e.g., Arregui 2015; Van
Doorsselaere et al. 2020b). A particular reason for the interest in this wave
mode, as compared to the other magnetohydrodynamic (MHD) waves, is it’s highly
incompressible nature that makes it difficult for the Alfvénic waves to
dissipate their energy. This property enables these waves to transfer the
mechanical energy from the photosphere, where they are generated via the
interaction of the magnetic fields with the convective motions, up into the
corona and out into the heliosphere. However, the journey of the Alfvénic
waves through the Sun’s atmosphere is not straightforward, with the
inhomogeneous nature of the plasma leading to many interesting wave-related
phenomena. Here we do not just refer to inhomogeneities along the magnetic
field, but also the fine-scale perpendicular structuring that is observed
throughout the chromosphere and corona.

An example of such perpendicular structuring is visible in the Extreme Ultra-
Violet (EUV) image of the corona from NASA’s Solar Dynamic Observatory (SDO)
displayed in Figure 1. Present in the image are: a myriad of loop-like
structures that emanate from the Active Regions; comparatively fainter loops
in the quiet Sun (best seen off-limb) that are the base of larger structures
extending into the heliosphere (e.g., streamers, pseudo-streamers); and near-
radially oriented striations extending out of the field of view in the quiet
Sun and polar regions (again best seen off-limb). The visibility of these
structures is thought to be due to an excess of density relative to the
ambient plasma, highlighting collections of magnetic field lines. The density
enhancements are possibly due to local heating events associated with the
magnetic fields (e.g., Cargill 1994; Yokoyama & Shibata 2001; Gudiksen &
Nordlund 2005; Klimchuk et al. 2008).

Figure 1: An EUV image of the Sun’s corona taken with the Solar Dynamic
Observatory. The image reveals the fine-scale striations throughout the upper
layer of the Sun’s atmosphere, indicating the highly inhomogeneous nature of
the coronal plasma. The dynamic range of the image has been altered to enable
the faint plasma emission around the limb to be visible.

Past efforts have placed significant attention upon understanding the
influence of an inhomogeneous plasma on Alfvénic wave propagation. However,
this has, until recently, been largely split into distinct areas of focus. One
of these areas was predominantly interested in modelling the impact of
Alfvénic waves on the solar wind (for reviews see, e.g., Bruno & Carbone 2005;
Cranmer et al. 2017), and hence focused on the inhomogeneity along the
magnetic field, though a number of studies attempt to include the influence of
the lower solar atmosphere as well. However, others were particularly
interested in the role of perpendicular structuring, driven primarily by
interests in wave heating of coronal loops (for reviews see, e.g., Arregui
2015; Nakariakov & Kolotkov 2020). The final camp focused on wave propagation
through partially ionized plasmas, investigating the role of ion-neutral
effects (for a review, see, e.g., Ballester et al. 2018). The division into
these areas is, in part, due to the vast range of scales that would be
required to incorporate all the vital physics from the photosphere to the
corona (and beyond). However modern numerical resources are enabling these
areas to draw closer together. We should mention that a number of works that
have sought to bring together inhomogeneity parallel and perpendicular to the
magnetic field. The studies where largely focused on emulating and predicting
properties of strongly damped resonant coronal loop oscillations (starting
with, e.g., Andries et al. 2005b, a; Verth 2007; Verth et al. 2007), and the
opportunities for magneto-seismology (see, e.g., Andries et al. 2009;
Nakariakov et al. 2021, for related reviews).

In the following we aim to provide an overview on the role inhomogeneities
play in Alfvénic wave propagation. The phenomenon we discuss are likely to be
present across the solar atmosphere, although the specifics (e.g., associated
time-scales, magnitudes) of the wave generation, propagation and dissipation
will depend upon the local plasma conditions. However, to bring together the
details, it is necessary to outline a stage upon which wave propagation and
evolution takes place. Here we will focus on wave propagation in the quiescent
Sun and coronal holes, and largely avoid active regions. This choice is mainly
driven by the fact that the magnetic structure and plasma conditions in the
quiescent Sun and coronal holes are similar, at least in the lower solar
atmosphere. Further, active regions are relatively infrequent features on the
Sun when compared to the quiet Sun. Hence, the description of wave propagation
in the quiet Sun provides a more general picture. This is not to say that the
details of Alfvénic waves in active regions are unimportant or uninteresting,
far from it111There are also plenty of reviews that focus on waves in active
regions (see, e.g., Nakariakov & Kolotkov 2020, for a recent review).. We also
note that in some of the given references, the analytic and numerical results
are tailored to plasma conditions thought to be representative of coronal
loops in active regions. Hence, it should be kept in mind that the numerical
values given in those papers with respect to the phenomenon under study will
need to modified for the quiet Sun.

###  1.1 Basic structure of the solar atmosphere

Figure 2: A descriptive model of the solar atmosphere. The left panel shows
temperature and density profiles. The lower solar atmosphere up to $\sim 2000$
km is based on the FAL quiet Sun model (see text) and an additional coronal
extension. The right hand panel displays an empirical magnetic field model for
a magnetic bright point with a photospheric field strength of 1300 G. The
corresponding Alfvén and sound speeds for the model atmosphere are also shown.

To highlight some of the key inhomogeneities that affect the Alfvénic waves,
it is worth providing a description of the basic structure of the Sun’s
atmosphere. The left panel of Figure 2 shows the temperature and mass density
profiles as a function of height in the lower solar atmosphere, from
photosphere to the coronal base. The atmospheric profile shown is from a
series of models by Fontenla et al. (1993), of which we select the quiet Sun
profile (referred to as the FAL-C model). The FAL model is extended into the
corona following Soler et al. (2017) with an assumption of a fully ionised
corona222The data and source code for generating the atmospheric profiles
shown throughout this review are available at:
https://github.com/Richardjmorton/FAL_models..

As noted by Rutten (2021), it is worth recognising that the FAL models really
only represent the atmosphere of a solar-analog. The atmosphere in these
models is 1D plane parallel with no magnetic field and lacking any of the
dynamics that are observed (e.g. waves, shocks, granulation, spicules).
However, the FAL models (and the earlier VAL versions Vernazza et al. 1981)
are a vital starting point for many investigations into wave propagation
through the atmosphere of a Sun-like star. With this in mind, it can be seen
that the temperature and density are expected to continuously vary with
height. To complement the temperature and density, Figure 2 also shows an
empirical model for the variation in vertical magnetic field strength (Leake
et al. 2005). The shown magnetic field profile should represent the expanding
magnetic field emanating from intense magnetic flux tubes embedded in the
photosphere (known as magnetic bright points due to their appearance in
certain photospheric diagnostics, e.g., Berger & Title 1996; Tsuneta et al.
2008; de Wijn et al. 2009; Ito et al. 2010). In combination with the mass
density, one can estimate the Alfvén speed ($v_{A}=B/\sqrt{\mu_{0}\rho}$)
profile through the atmosphere (and using the prescribed temperature profile
we also provide the sound speed, $c_{s}$, profile).

For the Alfvénic wave propagation, the temperature structure is usually not a
major consideration due to the highly incompressible nature of the waves
(although will play a role in the story, e.g., through its influence on the
ionisation state of the plasma - Section 5.1). Of particular importance is the
mass density, and to a degree the magnetic field, which shapes the details of
the waves’ journey through different layers of the stratified solar
atmosphere. It can be seen in Figure 2 that the mass density has a scale
height of 100-200 km in the lower solar atmosphere, which leads to a rapid
increase in Alfvén speed. To conserve wave energy flux ($F=\frac{1}{2}\rho
v^{2}v_{A}$), this leads to wave amplification, which can then induce non-
linear mode coupling and shocks (discussed in Section 5). Upwardly propagating
waves then encounter the sharp change in density that occurs at the transition
region, which acts as a frequency filter and is a decisive factor in
determining the Alfvénic wave energy that is able to enter the corona. The
change in Alfvén speed in the corona and above is much more gradual, although
it no less interesting; it is a key factor in the development of turbulence
(Section 4.3).

Figure 3: A sketch of the organisation of the magnetic field in the quiet Sun.
The left hand image is the photospheric magnetic field measured from the
Zeeman splitting on an Fe I line (a magnetogram) by the Heliosesmic & Magnetic
Imager onboard SDO. The central feature of the image is relatively large
clusters of magnetic field that highlight the network. The bottom right panel
shows a small portion of the network, revealing the small-scale magnetic flux
features as bright points (image courtesy of V. Henriques, taken with the
Swedish Solar Telescope operated by Institute for Solar Physics, Stockholm,
Sweden).

###  1.2 The large-scale organisation of the magnetic field

The large-scale organisation of the magnetic field plays an important role in
characterising the spatial and temporal evolution of waves in the solar
atmosphere. Hence, we believe it is important to overview critical aspects of
this element here. A complete assessment, however, will require many more
pages though we direct an avid reader to many other excellent texts discuss
the structure, dynamics and evolution, (e.g., Schrijver & Zwaan 2000; de Wijn
et al. 2009; Bellot Rubio & Orozco Suárez 2019).

The organisation of the magnetic field depends upon height in the solar
atmosphere, but is ultimately engendered by its distribution in the
photosphere. The gas density and temperature vary by orders of magnitude
between the photosphere and the corona, with the dynamics of the plasma
controlling the evolution of the field in the photosphere (ratio of plasma
pressure to magnetic pressure, or plasma beta, being greater than 1).
Eventually, the rapid decrease in plasma pressure means that the magnetic
field begins to dominate both the appearance and dynamics in the upper
chromosphere and beyond (where plasma beta is less than 1).

At the photospheric level, the magnetic field is found to emerge as (or
coalesce into) small, intense kilogauss elements concentrated in the lanes
between the granules (these magnetic bright points are seen in the Ca II image
in Figure 3). Over time, many of these small elements are advected to the
edges of the so-called network (Zirin 1985; Suárez et al. 2012; Gošić et al.
2014), which is thought to highlight the lanes of a super-granular flow
pattern deep in the interior (Figure 3). This produces a large-scale ($\sim
15-30$ Mm) patchwork of magnetic field (November & Simon 1988; De Rosa &
Toomre 2004), although is still made up from clusters of small-scale magnetic
elements (see the magnetogram in Figure 3). The small-scale elements within
the interior of the network (or internetwork) are generally thought to be the
foot-points of low-lying, closed magnetic fields, extending only as high as
the chromosphere and transition region (González & Rubio 2009; Wiegelmann et
al. 2010). In contrast, those magnetic fields at the network boundaries extend
into the upper atmosphere, spreading out as the gas pressure falls off and
eventually filling the volume of the corona. In the upper right panel of
Figure 3 we sketch this scenario. These magnetic fields are then either
closed, connecting with opposite polarity flux and spanning the internetwork
region; or they are open funnels, connected to the solar wind (more detailed
discussions can be found in, e.g., Peter 2001; Wedemeyer-Böhm et al. 2009).
The separation of network and internetwork fields is likely not so neat, with
the potential for short loops to exist with a foot-point in both (Wiegelmann
et al. 2010) and internetwork fields also contributing to the funnels.

###  1.3 The fine-scale structure

Near the top of the chromosphere, the magnetic field begins to dominate both
the appearance and dynamics of the solar atmosphere. As mentioned, the plasma
beta is much less than one in these regions. This is complemented by high
magnetic Reynold’s number ($\gg 1$), meaning that the magnetic field obeys the
so called ‘frozen-in’ condition which prohibits the cross-field diffusion of
the plasma. Heating events can lead to evaporation or injection of plasma into
the upper parts of the atmosphere and the consequence of the ‘frozen-in’
condition is that the plasma is confined to move along the magnetic field
lines. This leads to enhancements of density along bundles of magnetic field
lines, and provides the fine-scale structure seen in many images of the
chromosphere333It is suggested by Rutten & van der Voort (2017) that some of
the chromospheric fine structure observed in Hydrogen $\alpha$ images is
actually a delayed signature of heating event that took place earlier. Hence,
the observed features may not represent the current topography of the magnetic
field.  and corona (e.g., Figure 1). The fine-structure has many forms, from
elongated chromospheric fibrils and coronal loops, to short and highly dynamic
spicules that originate in the chromosphere. The fibrils, spicules (e.g.,
Rutten 2007; Tsiropoula et al. 2012; Jess et al. 2015; Carlsson et al. 2019)
and active region coronal loops (e.g., Reale 2014) have been well-studied .
However, many of the other density striations seen in Figure 1, e.g.,
quiescent coronal loops and the near radial features in polar coronal holes,
are presently little-studied.

In both the chromosphere and corona, the well-studied structures have widths
down to the resolution of current instrumentation; with typical widths around
$\sim 200-300$ km (e.g. Antolin & van der Voort 2012; Pereira et al. 2012;
Brooks et al. 2013; Morton & McLaughlin 2014). Estimates for the over-density
of some of these features, namely coronal loops in active regions, has been
possible. For example, Aschwanden et al. (2000) suggests the density of warm
coronal loops is an order of magnitude more than that expected from
hydrostatic equilibrium, which the ambient plasma is expected to be in. More
recently, Pascoe et al. (2017) use seismology and estimate over-density values
in the range 1.5-5.

Figure 4: The destruction of simplified loop geometries. The figure shows the
cross-sections of coronal loops, displaying the density. The magnetic field is
orientated perpendicular to the 2D slice shown. The loops are initially over-
dense with a Gaussian density profile in the cross-section (left hand panel),
which gradually becomes distorted over time. The time step between panels is
250 s and the density colour-bar is shown in units of $10^{-12}$ kg m-3.
Figure reproduced from Magyar et al. (2017).

Apart from the density and cross-sectional width of fine-scale structures,
another noteworthy aspect is the cross-sectional geometry of the flux tubes in
the solar atmosphere. Parker (1974a, b) suggested cylindrical geometry of
these features using a simplistic hydrodynamic mechanism involving turbulent
pumping to coalesce magnetic fields into the slender, tube-like structures.
This simplified assumption remained a cornerstone to the wave studies in
localised chromospheric and coronal structures (e.g., Goossens et al. 2011).
However, there has been some work that has ventured from this assumption,
(e.g., Morton & Erdélyi 2009; Pascoe et al. 2011; Morton & Ruderman 2011; Guo
et al. 2020). The observational evidence for the actual cross-sectional
geometry of the chromospheric and coronal structures is scant. Theoretical
work suggests that the shape of the flux tube cross section can vary along the
structure, and is likely oblate (Ruderman 2009; Malanushenko & Schrijver
2013). Observational studies are particularly complex, and appear to show
mixed results (McCarthy et al. 2021). Moreover, recent simulations have
demonstrated that, even when starting with circular cross-sections, the
presence of transverse waves lead to a distortion of the flux tubes cross-
section through instabilities (e.g., Antolin et al. 2014; Magyar & Van
Doorsselaere 2016; Magyar et al. 2017; Antolin & Van Doorsselaere 2019, and
discussed further in Section 4.2). Figure 4 shows simulation results from
Magyar et al. (2017) that demonstrate this process in action.

Furthermore, it has been speculated that the appearance of a near-circular
tube-like geometry of the fine-scale structures could be due to observational
artefacts. Judge et al. (2011) suggested that some observed chromospheric
spicular structures are wrapped current sheets that appear as thin, elongated
features. More recently, Malanushenko et al. (2022) put forward the idea that
some bundles of coronal loops are optical illusions in EUV passbands. The
authors suggest that these features are actually complex and diffuse objects
(‘wrinkles in the coronal veil’) that appear as compact, slender structures
due to artifacts that arise from the inherent integration along the line of
sight over extended regions in optically thin plasma. The authors examined the
coronal structures using highly realistic numerical simulations, and found
that the cross-sections of many coronal structures appear to show a highly
turbulent appearance (similar to the cross-sections from Magyar et al. 2017
shown in Figure 4). However, it is not discussed how or why the corona is
structured in such away.

Importantly, while the actual cross-sectional shape of flux tubes is unknown,
both observations and simulations suggest that there are plasma
inhomogeneities in the direction perpendicular to the magnetic field. As we
shall now discuss, this enables a wealth of interesting wave physics to take
place and is likely a key factor in enabling the dissipation of wave energy in
the chromosphere and corona.

##  2 Linear Alfvén Wave Theory

To appreciate the impact of an inhomogeneous plasma on wave propagation, we
briefly discuss some of the fundamental theoretical aspects of linear Alfvénic
waves in a structured plasma. It turns out that perpendicular inhomogeneity in
the plasma modifies the classical picture of wave behaviour presented in the
textbook examples of homogeneous plasma, and removes the clear division
between Alfvén and magneto-sonic modes. It was noted as early as 1982 by
Hasegawa & Uberoi (1982) that the presence of inhomogeneity leads to coupling
of the total pressure to the dynamics for Alfvén modes. Inhomogeneity also
enables the MHD waves to have a different appearance in different parts of the
plasma, readily modifying their dominant characteristics. The following aims
to be a discussion of the salient points, and we recommend readers refer to
the original texts for deeper discussions (e.g., Hasegawa & Uberoi 1982;
Goossens 2003; Goossens et al. 2012, 2019). We first discuss the case of a
homogeneous plasma before moving onto the more interesting physics that occurs
when this restriction is dropped.

###  2.1 The homogeneous case

To begin, let us assume that we have a plasma with magnetic field that is
oriented in the $z$ direction and is homogeneous in all plasma quantities
along the field. We make no assumptions about the plasma perpendicular to the
field at this moment. The linearised ideal MHD equations can be reduced to
(e.g., Goossens et al. 2012),

$\displaystyle\omega^{2}\xi_{z}-k_{z}c_{s}^{2}Y$ $\displaystyle=$
$\displaystyle 0,$ (1) $\displaystyle
k^{2}v_{A}^{2}k_{z}\xi_{z}+(\omega^{2}-k^{2}(c_{s}^{2}+v_{A}^{2}))Y$
$\displaystyle=$ $\displaystyle 0$ (2)
$\displaystyle(\omega^{2}-k_{z}^{2}v_{A}^{2})Z$ $\displaystyle=$
$\displaystyle 0,$ (3)

for planar, harmonic waves, i.e., wave variables are proportional to
$\exp(ik_{z}z-i\omega t)$, where $\omega$ is the frequency and $k_{z}$ is the
longitudinal wave number. Here the equations have been expressed in terms of
variables that expose the underlying physics. The variables are the magnitude
of the wavevector $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$, the sound speed
$c_{s}$, the Alfvén speed $v_{A}$, and the Lagrangian displacement along the
magnetic field, $\xi_{z}$. Additionally, we have also used the plasma
compression,

$Y=-i\nabla\cdot\vec{\xi},$ (4)

and the vorticity parallel to the magnetic field,

$Z=-i(\nabla\times\vec{\xi})_{z},$ (5)

where $\vec{\xi}$ is the Lagrangian displacement vector. It is also useful to
define the ratio of the horizontal components of the total pressure force and
the magnetic tension force (Goossens et al. 2012), which is given by,

$\Lambda(\omega^{2})=\frac{\omega^{2}}{(k_{z}v_{A})^{2}}-1.$ (6)

In an homogeneous media of infinite extent, the system of equations (1)-(3)
are decoupled into two systems. One of these describes Alfvén waves and
contains only the vorticity along the field, i.e., Eq. 3. There is no
compression or displacement along the magnetic field ($Y=0$, $\xi_{z}=0$). The
solution to this set of equations results in the well known dispersion
relation,

$\omega=k_{z}v_{A}.$

Hence, from Eq. 6 it can be found that $\Lambda(\omega^{2})=0$, and magnetic
tension is the only restoring force. The other system involves the variables
$\xi_{z}$ and $Y$, and defines the magneto-sonic modes, i.e., Eqs. 1 and 2.
Unlike the Alfvén waves, these magneto-sonic modes are compressible and have a
longitudinal displacement but do not propagate parallel vorticity.

###  2.2 Perpendicular inhomogeneities

The addition of an inhomogeneity in the direction perpendicular to the
magnetic field leads to a coupling of the wave variables, which, in the case
of a straight magnetic field, is mediated by Eulerian perturbation of the
total pressure (for a twisted field there is a stronger coupling arising from
the so-called coupling functions, e.g., Sakurai et al. 1991; Goossens et al.
2019). This coupling leads to MHD waves having mixed properties and is present
even in the linear MHD system (e.g., Goossens et al. 2002). There are two
scenarios that are worth discussing as they highlight key aspects of
increasing the geometric complexity.

The first case of interest is two uniform plasma separated by a discontinuity
in the Alfvén velocity, which we can assume arises due to a discontinuity in
the density, i.e.,

$\rho(x)=\begin{cases}\rho_{i}&\hskip 14.22636ptx\leq x_{0},\\\
\rho_{e}&\hskip 14.22636ptx>x_{0}.\end{cases}$

At such an interface, Alfvén surface modes are able to exist (Wentzel 1979b,
a; Roberts 1981). In the limit of ‘nearly perpendicular propagation’
($k_{y}^{2}>>k^{2}_{z}$), the dispersion relation for the surface mode is
given by:

$\omega^{2}=k_{z}^{2}\frac{\rho_{i}v^{2}_{Ai}+\rho_{e}v^{2}_{Ae}}{\rho_{i}+\rho_{e}}\equiv\omega_{k}^{2},$
(7)

which lies between the Alfvén frequencies for both plasmas. Hence, $\Lambda$
(given in Eq. 6) is small on either side of the discontinuity and magnetic
tension dominates the restoring forces. The surface mode is also largely
insensitive to the value of the sound speed. Furthermore, it has the property
that parallel vorticity is zero everywhere except at the discontinuity,
contrary to the classical Alfvén wave in uniform plasma (Goossens et al.
2012). For such a mode, an appropriate term is Alfvénic444The term Alfvénic
was already used by Ionson (1978) to describe the surface Alfvén wave., which
highlights such modes are different from the classical Aflvén wave, and has
mixed properties.

The second case is that of a 1D magnetic cylinder, which we describe in
cylindrical coordinates, ($r$, $\theta$, $z$). The magnetic field is still
oriented in the $z$ direction. Since the background plasma depends only on
perpendicular inhomogeneities, which are in the radial ($r$) direction, then
the perturbations, independent of azimuth ($\theta$) and $z$, are proportional
to

$\exp(i(m\theta+k_{z}z-\omega t)),$

where $m$ is the azimuthal wavenumber. In this case, Eqs. 4 and 5 become

$Y=\frac{1}{r}\frac{\partial(r\xi_{r})}{\partial
r}+im\xi_{\theta}+ik_{z}\xi_{z},\hskip
28.45274ptZ=\frac{1}{r}\frac{\partial(r\xi_{\theta})}{\partial
r}-im\frac{\xi_{r}}{r}.$

For the case of $m=0$, it can be seen from these expressions for $Y$ and $Z$
that Eqs. 1-3 are decoupled. The solutions to the two sets of equations are
axisymmetric modes, namely the torsional Alfvén mode and the sausage magneto-
sonic mode. The Alfvén mode depends only upon $\xi_{\theta}$, as each magnetic
surface oscillates at its own frequency. As noted by Goossens et al. (2019),
this is the only pure Alfvén mode in a non-uniform 1D cylindrical plasma. The
nature of the coupling is further elucidated by the coupling function derived
by Sakurai et al. (1991) for a straight magnetic field, namely,

$C_{A}=\frac{m}{r}B_{z}P^{\prime},$ (8)

where $P^{\prime}$ is the total pressure perturbation. It can be seen that a
total pressure perturbation is also required to be non-zero in order for
coupling, as well as $m\neq 0$.

Under the simplifying assumption of a magnetic flux tube with a piece-wise
constant density (e.g., Spruit 1982; Edwin & Roberts 1983), solutions exist
for any $m$ that can be identified with Alfvén waves (i.e., solutions with
$\nabla\cdot\vec{v}=0$). More generally, for any geometry, a pure Alfvén mode
can exist if there is an ignorable transverse direction. However, in the non-
uniform 1D cylinder, modes with $m\neq 0$ have mixed properties and all wave
variables are coupled. All eigenmodes of the system now have all perturbed
quantities that are non-zero (Goossens et al. 2002, 2019). Here, the modes
cannot be separated by the distinctions found in the homogeneous case. Also of
interest, it turns out all wave modes in inhomogeneous plasma are efficient
generators of vorticity (Goossens et al. 2019).

Before moving on, it is worth mentioning the non-axisymmetric kink mode (when
$m=1$) in detail. This particular wave mode has been the subject of great
interest over the last two decades. Transverse oscillations of coronal loops,
first noticed in observations back in the 1990’s, were interpreted as the kink
mode (Aschwanden et al. 1999; Nakariakov et al. 1999). Then, after
observations of transverse waves in the chromosphere (De Pontieu et al. 2007;
Okamoto et al. 2007) and corona (Tomczyk et al. 2007) in 2007, the kink mode
was subject to an intense discussion about the wave properties and
nomenclature. In one of the seminal papers on MHD waves in flux tubes, Edwin &
Roberts (1983) used the prefix ‘fast’ to describe the mode. As mentioned, the
solar atmosphere is highly inhomogeneous and MHD waves in this environment
portray mixed properties and it is misleading for the modes to be classified
using the labels derived from the homogeneous case (Goossens et al. 2019).

The characteristics of the kink mode were examined and discussed in detail by
Goossens et al. (2009), where they demonstrated that the kink mode is in fact
very robust and cares little for the plasma environment when $k_{z}R<1$ (here,
$R$ is the flux tube’s radius). Moreover, the kink mode was shown to be highly
incompressible, with negligible pressure perturbations. The dispersion
relation for the kink mode in a cylinder with a piece-wise constant density is
given by Eq. 7, while the ratio given by Eq. 6 is

$\Lambda_{i}(\omega)=-\Lambda_{e}(\omega)=\frac{\rho_{i}-\rho_{e}}{\rho_{i}+\rho_{e}},$

in the interior and exterior of the cylinder. Hence, for any value of density
contrast, this ratio is always less than one and magnetic tension is the
dominant restoring force. This led Goossens at al. to suggest the descriptor
for these waves should be Alfvénic, alluding to the dominant properties being
close to classical Alfvén wave. In a follow up work, Goossens et al. (2012) go
onto to demonstrate that the fundamental radial modes for any non-axisymmetric
wave mode ($m>0$) can be described as a surface Alfvén wave. In this case, the
waves display the same properties as a surface Alfvén wave at a true
discontinuity555Other radial overtones have properties that Goossens et al.
(2012) call fast-like.. However, if the discontinuity is replaced by a region
of non-uniform, continuously varying density, then the vorticity is present
throughout this region and the modes are Alfvénic global/quasi modes(Goossens
et al. 2002). The presence of vorticity and the continuously varying density
profile leads to a number of interesting effects that impact wave propagation,
damping, and dissipation; which are discussed in section 4.

##  3 Wave excitation

Recent theoretical models have shown that the inhomogeneous nature of the
Sun’s atmosphere could also play a key role in exciting Alfvénic waves. We
first overview the existing paradigm surrounding the excitation of Alfvénic
waves and then discuss mode conversion from inhomogeneities.

###  3.1 Convection

The typical picture of Alfvénic wave generation begins at the photosphere and
was suggested many decades ago (e.g., Osterbrock 1961). The turbulent patterns
of granulation are the most common feature of the photospheric topology
(Figure 3), that manifest from the convective motions deep within the solar
interior. The average lifetime of granular cells is 5-10 minutes and have
horizontal scales of about 1 Mm (e.g., Schrijver et al. 1997). As mentioned in
Section 1, the boundaries of these granular cells appear as dark lanes where
strong plasma downflows are observed. These intergranular lanes form
favourable sites for harbouring intense concentrations of magnetic flux known
as magnetic bright points (MBPs). The MBPs are seen to be passively advected
with the large-scale super-granule flow but are also subject to buffeting from
the granulation (Berger & Title 1996; Berger et al. 1998; van Ballegooijen et
al. 1998; Nisenson et al. 2003; Chitta et al. 2012). The associated horizontal
motions are turbulent and rapid, with an average root-mean-squared (RMS)
velocity of $\sim 1-1.5$ km/s and correlation times of 20-30 s (Chitta et al.
2012), while the line-of-sight (LOS) Doppler velocity magnitudes are of the
order $\sim$1.2 km/s (Nordlund et al. 2009; Oba et al. 2017).

These motions correspond to the both horizontal and vertical shaking of the
magnetic bright points and can induce various types of MHD wave-like
fluctuations at the foot-points of thin magnetic flux tube structures (Spruit
1981; Steiner et al. 1998). While vertical motions can drive sausage or
longitudinal modes (Defouw 1976; Roberts & Webb 1978), the horizontal motions
can induce transverse waves, namely the kink mode, if using a flux tube
description for the bright points (Spruit 1981; Choudhuri et al. 1993;
Musielak & Ulmschneider 2002). Cranmer & van Ballegooijen (2005) suggested
such kink modes would convert to ‘classical’ Alfvén waves in the low
chromosphere when the individual magnetic elements expand and merge with one
another. This implies that the upper atmosphere is homogeneous perpendicular
to the magnetic field (as is the case of the Cranmer & van Ballegooijen
model). However, as discussed in the introduction, the chromosphere and corona
apparently possess a significant amount of inhomogeneity. Hence, the wave
energy should propagate to upper layers as Alfvénic modes instead - although
we are unaware of any model/theory that addresses the details of how this may
happen.

Closely related to the horizontal motions are photospheric vortices, which are
convective down-drafts in the intergranular lanes. They were initially
detected through the observed motions of MBPs (Bonet et al. 2008) and have
been found to be ubiquitous across the photosphere (e.g., Wedemeyer-Böhm &
Rouppe van der Voort 2009; Morton et al. 2013; Giagkiozis et al. 2018; Liu et
al. 2019) and in numerical simulations (e.g., Shelyag et al. 2011; Moll et al.
2011). Such vortices appear to be capable of exciting rotational Alfvénic
modes in numerical simulations (Battaglia et al. 2021; Breu et al. 2022), and
have also been suggested to enable distortions of flux elements shape and
intermixing of field lines, leading to MHD wave excitation (van Ballegooijen
et al. 2011).

###  3.2 Mode conversion

Figure 5: A sketch of the double mode conversion in the lower solar atmosphere
that ends in the production of Alfvénic waves. The details are discussed in
the text. The blue diagonal lines represent the magnetic field. Reproduced
from Khomenko & Cally (2012) permission of the AAS.

More recently, there has been interest in alternative pathways for convective
energy to reach the corona as Alfvénic waves. In part this interest has been
driven by some fundamental theoretical results (Cally & Goossens 20008), but
also by the apparent high reflectivity of the transition region to
photospheric driven Alfvénic waves (suggested to be $\sim 95$ % reflection
rate by Cranmer & van Ballegooijen 2005). The essence of this pathway is shown
in Figure 5.

It is well established that the Sun’s interior (and the interior of many other
stars with convective envelopes) posses a wide range of acoustic oscillations,
known as p(pressure)-modes. The p-modes are largely confined to the solar
interior but can leak out into the atmosphere. The acoustic modes can be
absorbed by the magnetic fields of sunspots (Bogdan et al. 1993) and convert
to magneto-acoustic modes (e.g., Spruit & Bogdan 1992; Cally & Bogdan 1997).
This conversion occurs in the region of the equipartition layer (where
$c_{s}=v_{A}$) and requires the waves to have frequency greater than the
acoustic cut-off frequency. Here, field inclination also proves an important
ingredient in enabling p-mode absorption (Cally 2000; Crouch & Cally 2003).
The inclination modifies the acoustic cut-off frequency (Bel & Leroy 1977)
which enables lower frequency acoustic modes to propagate into the atmosphere
(termed the ‘ramp’ effect by Cally) and reach the equipartition layer. The
variation of acoustic cut-off frequency with magnetic field inclination has
been observed in sunspots numerous times (e.g., McIntosh & Jefferies 2006;
Löhner-Böttcher et al. 2016; Morton et al. 2021).

The absorption of p-modes has also been demonstrated for slender magnetic flux
tubes (Bogdan et al. 1996), with similar mode transmission/conversion
processes expected to take place (Bogdan et al. 2003). Observations by
Jefferies et al. (2006) revealed the presence of low-frequency magneto-
acoustic modes at supergranule boundaries indicates the ramp effect is also in
action, with the authors terming the network magnetic fields as ‘magneto-
acoustic portals’.

For those waves reaching the equipartition layer, the initial fast acoustic
mode is converted to a slow mode; sometimes referred to as transmission as the
energy of the wave maintains its basic character. This conversion is largely
favoured in regions of vertical magnetic field, where the attack angle between
the wave vector and magnetic field is small. However, when the attack angle is
large, the p-modes predominantly get converted to fast magneto-acoustic waves,
which have a magnetic character. Due to the significant increase in the Alfvén
speed in the upper chromosphere (Figure 2), the fast wave is then refracted.
Subsequently, around the height of refraction, another linear mode conversion
takes places. This conversion is from fast to Alfvén, which is inherently 3
dimensional and requires the wavevector to be in a different plane to the
magnetic field lines (Cally & Goossens 20008; Cally & Hansen 2011; Hansen &
Cally 2012; Khomenko & Cally 2012). This pathway appears to enable a
significant fraction of the p-mode energy to reach the corona as Alfvénic
waves (Khomenko & Cally 2012), with Hansen & Cally (2012) estimating that
around 30 % of the p-mode flux gets carried through the transition region as
Alfvénic waves. Hansen & Cally (2012) also provide a crude estimate for the
energy flux in the 3-5 mHz band of 800 Wm-2, which would mean that this
process could provide a substantial contribution to meeting the energy
requirements for maintaining a hot corona.

While the just referenced works focus on plasma configurations with no
perpendicular structuring, the influence of a inhomogeneous corona is
investigated in Cally (2017) and Khomenko & Cally (2019). It is found that an
incident fast wave on a collection of flux tubes largely converts to Alfvén
waves. The incident wave is also subject to scattering in Fourier space,
exciting the kink mode and fast waves in the corona.

At present, the observational evidence for the generation of coronal Alfvénic
waves by mode conversion is still tentative. It has been found that the power
spectra of coronal Doppler velocity fluctuations, interpreted as the kink
mode, presents an excess of power at around $\sim 3-4$ mHz (Tomczyk et al.
2007; Morton et al. 2016, 2019). This excess power appears to be inconsistent
with expectations from driving by horizontal granular motions, and the
location in frequency space is coincident with the peak in p-mode power.
Morton et al. (2019) demonstrated this enhanced power is present throughout
the corona and also seemingly present through various stages of the Sun’s
magnetic activity cycle, implying the process is global and also somewhat
insensitive to the global magnetic field structure. Recent one dimensional
simulations have attempted to incorporate the role of longitudinal waves in
wave-driven solar wind models (Shimizu et al. 2022). The results show an
enhanced Alfvén wave flux in the corona leading to greater mass loss rates.
While Shimizu et al. 2022 make a case for a linear mode conversion being
responsible for the coronal Alfvénic waves in the model. However, it is not
evident (to us at least) that the physical process in action in the model is a
linear one, given the requirements for three dimensions expressed in previous
works (e.g., Cally & Goossens 20008). Further, the result that the mode
conversion rate is dependent upon amplitude (Section 3.3 in Shimizu et al.
2022) suggests that the process may be non-linear. Regardless, of the
mechanism the work support a relationship between p-modes and the observed
coronal Alfvénic waves.

##  4 Wave damping

There are a number of mechanisms for energy exchange between the MHD waves,
other wave modes, and the ambient medium. In a uniform plasma, the damping of
Alfvén waves occurs on time scale $\tau_{d}={\lambda^{2}}/\eta$, where $\eta$
is the diffusivity and on length scale $L_{d}=v_{A}\tau_{d}$. However this
dissipation is weak due to magnetic Reynolds number, $R_{m}$, being large,
i.e., $R_{m}={L_{d}}/{\lambda}\gg 1$. Theory shows that the presence of
inhomogenity introduces additional mechanisms that enable the waves to damp
more effectively. These damping/dissipation processes are strongly influenced
by the inhomogeneity of the plasma and background magnetic field conditions,
and act with varying levels of efficacy in transferring Alfvénic wave energy
to the plasma (or other modes or scales). Perhaps unsurprisingly, it turns out
that the interplay of multiple processes influences the efficacy of each
mechanism in the solar atmosphere.

###  4.1 Phase mixing and Resonant absorption

The most well-known process for damping Alfvénic waves is probably phase
mixing (Heyvaerts & Priest 1983), which requires a gradient in the Alfvén
speed perpendicular to the direction of wave propagation. The Alfvén waves
propagating along individual magnetic field lines (or magnetic surfaces)
become out of phase as they propagate, generating small scales and thus
increasing local gradients perpendicular to the field. The previously
discussed equations for Alfvénic waves, i.e., Eqs. 1-3, are based on the ideal
MHD equations. Heyvaerts & Priest (1983) demonstrated that including viscosity
($\nu$) and magnetic diffusivity ($\eta$) leads to an equation for shear
Alfvén waves (under some restrictions) of the form,

$\frac{\partial^{2}v}{\partial
t^{2}}=v_{A}^{2}(x)\frac{\partial^{2}v}{\partial
z^{2}}+(\nu+\eta)\frac{\partial^{2}}{\partial x^{2}}\frac{\partial v}{\partial
t},$ (9)

where $v$ is the velocity perturbation and the magnetic field is oriented in
the $\hat{z}$ direction. It is clear from this equation that the increase in
gradients due to phase mixing enhances the viscous and Ohmic dissipation of
the wave energy. The length-scale over which phase mixing occurs is given by
(e.g., Mann et al. 1995),

$L_{ph}=\frac{2\pi}{t|d\omega_{A}(x)/dx|},$ (10)

and can be seen to depend on the gradients of the variation in Alfvén speed.
Here $\omega_{A}\approx v_{A}k_{\parallel}$, with $k_{\parallel}$ the
wavenumber parallel to the magnetic field.

Closely related to phase mixing is resonant absorption (Ionson 1978). If we
assume that some region of plasma has a continuous variation in density or
magnetic field, then the Alfvén speed is a continuous quantity and there
exists a continuum of Alfvén modes with $\omega_{A}(x)$. If Alfvén waves are
driven with a frequency, $\omega_{D}$, that lies within the continuum, then a
resonance occurs at locations where $\omega_{A}(x)=\omega_{D}$. Hence, wave
energy can be transferred from large-scale motions to localised, small-scale
motions. This is also true for surface Alfvén waves, where the discontinuous
region discussed in Section 2.2 is replaced by a region with continuously
varying Alfvén speed (Lee & Roberts 1986; Hollweg & Yang 1988). As with phase
mixing, the inclusion of viscous and Ohmic dissipation enables the wave energy
at the resonance to be dissipated effectively. Interestingly, as well as
enabling the coupling of the wave variables (Sections 2, 2.2), the total
pressure plays an important role in resonant absorption. Hollweg & Yang (1988)
noted that ‘[resonant] absorption can occur in any situation where the total
pressure perturbations are imparted to field lines satisfying the Alfvén …
resonance conditions’. Moreover, resonant absorption is able to modify the
nature of classical Alfvén waves, introducing non-zero total pressure
variations and compressibility (Poedts et al. 1989, 1990; Goossens & Poedts
1992).

Due to the appearance of localised flux tubes in images of the corona (e.g.,
Figure 1), many studies in the last two decades have focused on examining wave
propagation in a cylindrical wave guide with a continuous density profile
perpendicular to the magnetic field (see, e.g., Goossens et al. 2005, 2011,
for reviews). The region of continuous density has often been limited to a
narrow annulus at the boundary of the wave guide666Mainly for the sake of
mathematical simplicity., and referred to as the inhomogeneous layer. In the
case of the kink mode, the kink frequency lies in the Alfvén continuum, i.e.
$\omega_{Ai}<\omega_{k}<\omega_{Ae}$ and the global energy of the kink mode is
transferred to other Alfvénic modes in the inhomogeneous layer (rotational
modes in the case of the cylinder, e.g., Poedts et al. 1989; Sakurai et al.
1991; Pascoe et al. 2012; Goossens et al. 2013; Giagkiozis et al. 2016).
Similar to phase mixing, the rate of resonant absorption is found to depend on
the degree of the inhomogeneity. For propagating kink waves777A similar
expression for damping rate is also obtained for standing modes Ruderman &
Roberts (2002). it was shown by Terradas et al. (2010) that, under the
approximation of thin tube and thin boundary, the damping length of the kink
mode due to resonant absorption is given by:

$L_{D}=\lambda
R\frac{4}{\pi^{2}}\frac{\rho_{i}+\rho_{e}}{(\rho_{i}-\rho_{e})^{2}}\left|\frac{d\rho}{dr}\right|_{r_{A}},$
(11)

where $R$ is the flux tube radius and the term
$\left|{d\rho}/{dr}\right|_{r_{A}}$ is the gradient of density across the
resonant layer. This expression clearly reveals that the density difference
between the wave guide and the ambient plasma is inversely proportional to the
damping length, which results in a higher rate of resonant absorption when the
inhomogeneity contrast is larger. In terms of observed wave damping in the
solar atmosphere, direct evidence for resonant absorption is ambiguous.
However, the theory appears to give a good account of the observed damping of
standing (e.g., Aschwanden et al. 2003; Verwichte et al. 2013) and propagating
kinks modes (Verth et al. 2010; Tiwari et al. 2019, 2021; Morton et al. 2021).

Figure 6: An example of resonant absorption in an inhomogenous corona. The
left panel shows the contours of density in a cross-section perpendicular to
the magnetic field. The largest density is a factor of 2 greater than the
external plasma. The right panel shows the concentration of wave energy at
regions where the local Alfvén speed equals the kink speed of the local
density maxima. Figure adapted from Pascoe et al. (2011) with permission of
the AAS.

The presence of a continuous Alfvén speed profile showcases the first example
of the joint influence of different wave damping mechanisms; in this case,
resonant absorption and phase mixing. The resonant Alfvénic waves within the
inhomogeneous layer (arising from the transfer of energy from the kink mode)
can then propagate along magnetic surfaces at different Alfvén speeds, hence
are subject to phase mixing (Terradas et al. 2008; Soler & Terradas 2015). The
expression for phase mixing lengths, Eq. 10, has been shown to provide a good
description of the development of small-scales in analytical and numerical
models of kink mode damping (e.g., Pascoe et al. 2013; Soler & Terradas 2015).
This dual process has been demonstrated to be robust in multi-stranded
structures (Terradas et al. 2008) and with arbitrary inhomogeneities (Pascoe
et al. 2011). Figure 6 shows numerical results from Pascoe et al. (2011),
where a collection of over-dense magnetised plasma structures were driven as
to excite the kink mode. After some time in the simulation, the wave energy of
the global mode can be seen to be concentrated towards the boundaries of the
structures (Figure 6 right panel), with concentrations close to the locations
where $\omega_{A}({x},y)=\omega_{k}$. The details of the dual process are
similar at a plasma interface (Lee & Roberts 1986; Hollweg & Yang 1988), which
is not so surprising given the fact that the kink mode can be described as a
surface Alfvén wave. This would suggest that resonant absorption and phase
mixing are key features of Alfvénic wave damping throughout the solar
atmosphere, even if coronal structures are collections of magnetic fields
associated with diffuse and complex density structures (as suggested by, e.g.,
Magyar & Van Doorsselaere 2016; Malanushenko et al. 2022).

In terms of contributing to the coronal heating it appears that the actual
heating rates, due to the combination of the resonant absorption of kink modes
and subsequent phase mixing of rotational motions, are inadequate to
compensate for the radiative losses (Pagano & Moortel 2017; Howson et al.
2020), even when a realistic broadband driver is employed (Pagano & De Moortel
2019; Pagano et al. 2020). Further, observational estimates of the frequency-
dependent damping rate of propagating kink waves in the quiet Sun suggest that
resonant wave damping is likely small (Tiwari et al. 2021; Morton et al.
2021). In coronal holes the Alfvénic waves are found to largely undamped
(e.g., Dolla & Solomon 2008; Banerjee et al. 2009; Morton et al. 2015). This
pattern of weak damping and no damping could reflect that the density
differences across the inhomogeneities are small in the quiet Sun and smaller
in coronal holes. Further, it would appear that the transfer of energy to the
rotational modes is small and any subsequent phase mixing is also weak.

We note that there has been some evidence for wave damping in coronal holes
above 1.2 $R_{\odot}$ (Bemporad & Abbo 2012; Hahn et al. 2012; Hara 2019),
inferred from a decrease in the non-thermal widths of spectral lines (which is
often taken as an indicator of Alfvénic waves). The suggested rapid damping is
somewhat surprising and there is difficulty in explaining it through known
wave damping mechanisms. The results have only been found in Hinode EIS data
so far, with the suggestion that uncertainties in scattered light may account
for the observed decrease in line widths (Zhu et al. 2021). Further, Gilly &
Cranmer (2020) suggest that failure to take into account non-equilibrium
ionisation and line broadening from the solar wind can lead to misleading
inferences from spectral line widths. There have also been other suggestions
for the observed decrease in line width (see, e.g., Cranmer 2018).

While the situation does not appear to be favourable for wave heating via
phase mixing, the effects pertinent to the stability of the resonance layer
can influence energy transfer to small scales.

###  4.2 Instabilities

The role of wave-driven instabilities has received significant focus in the
last few years and appears to be an important ingredient for effective wave
energy dissipation in the solar atmosphere. The instabilities are localised
processes that are entangled with the dynamics and inhomogeneity of the
plasma. While there are many instabilities that can occur in a plasma, there
has been a particular focus on the Kelvin-Helmholtz (KH) instability in
conjunction with propagating Alfvénic waves in a cylindrical waveguide.

The KH instability is a fluid instability formed by two fluids undergoing
differential shearing motion across an interface, with the instability
destroying the shear layer through the development of vortices. The KH
instability can also occur in magnetized plasma if the velocity is
perpendicular to the magnetic field, although its properties are modified by
the magnetic field due to the frozen-in condition. However, for a velocity
shear parallel to the magnetic field, the instability cannot develop as the
magnetic tension has a stabilizing effect (Chandrasekhar et al. 1961). If the
magnetic field is sheared across the instability, this can have a stabilizing
or destabilizing effect on the KH mode, depending on the properties of
magnetic shear (Ofman et al. 1991). KH has been studied theoretically in the
context of coronal plasmas, where it is believed to play an important role in
the transition to turbulence and heating (e.g., Heyvaerts & Priest 1983; Ofman
et al. 1994; Karpen et al. 1994).

Figure 7: Development of the Kelvin Helmholtz instability at the boundary of a
magnetised plasma cylinder. In descending order, the rows show the number
density, temperature, flow velocity vectors, parallel vorticity and parallel
current. Towards the end of the simulation, the small-scales present are much
better observed in the parallel vorticity and current. Reproduced from Antolin
& Van Doorsselaere (2019).

With respect to wave motions, it was suggested that because Alfvén waves can
oscillate on neighbouring magnetic surfaces without interacting, hence can
have an arbitrary phase with respect to each other. This would result in shear
flows between the surfaces, which could be susceptible to the KH instability.
The instability can then enhance the efficiency of the wave dissipation by
more rapidly generating small-scales than phase mixing alone (Heyvaerts &
Priest 1983; Hollweg 1984; Hollweg & Yang 1988). The development of the KH
instability was shown to be present in numerical non-linear studies of the
resonant absorption of Alfvén waves in inhomogeneous plasma (e.g., Ofman et
al. 1994; Karpen et al. 1994).

More recently, the development of KH was also shown to be present at the
boundary of cylindrical flux tubes undergoing kink motions (an example of this
is shown in Figure 7, but see also, e.g., Terradas et al. 2008; Soler et al.
2010; Antolin et al. 2014; Magyar & Van Doorsselaere 2016; Antolin et al.
2017; Karampelas et al. 2019). The transverse motions of the flux tube through
an ambient plasma naturally leads to a shear layer at the tube boundary, which
is at an angle to the magnetic field (see the flow velocity in Figure 7).
Soler et al. (2010) demonstrated that for a non-twisted cylindrical flux tube
with a discrete change in density at the boundary, the dispersion relation (in
the thin tube approximation) is given by

$\omega\approx\frac{\rho_{i}}{\rho_{i}+\rho_{e}}m\delta
v\pm\left[\frac{(\rho_{i}v^{2}_{Ai}+\rho_{e}v^{2}_{Ae})}{(\rho_{i}+\rho_{e})}k^{2}_{z}-\frac{\rho_{i}\rho_{e}}{(\rho_{i}+\rho_{e})^{2}}m^{2}\delta
v^{2}\right]^{1/2},$ (12)

where $\delta v$ is the velocity jump at the shear layer. In the absence of
magnetic twist, the shear flow is perpendicular to the magnetic field. The two
terms within the square brackets represent the suppression term (which is
essentially the square of the kink speed) and the driving term, respectively.
The stability of the shear layer depends upon relative sizes of these two
terms. A similar term can be obtained for surface Alfvén waves (Hillier et al.
2018).

As discussed, the resonant conversion of the kink mode to local Alfvén modes
in the inhomogeneous layer also leads to phase mixing and further increases
the shear velocity (Antolin et al. 2014; Antolin & Van Doorsselaere 2019). The
KH instability is found to develop more slowly in the presence of an
inhomogeneous layer (Terradas et al. 2008). It appears the reason for this is
not yet clear (P. Antolin, private communication), but simulations show that
the onset times are longer for increasing inhomogeneity length-scales. Antolin
& Van Doorsselaere (2019) suggest this is due to the fact that the larger
inhomogeneous length scales preferentially excite unstable modes with larger
wavelengths, which have slower growth rates. Further, the KH instability
distorts the inhomogeneous layer, thereby widening it and enabling the
Alfvénic motions to become present in a greater portion of the loop (Antolin
et al. 2014). This process could be severe, and result in complete distortion
of an initial circular cross-section (as discussed in Section 1.3 Magyar & Van
Doorsselaere 2016). The development of the KH instability can be delayed by
the presence of magnetic twist (Soler et al. 2010; Terradas et al. 2018) or
large values of dissipative coefficients (Howson et al. 2017), but the
boundary of the oscillating loop is always unstable (Barbulescu et al. 2019).

The presence of the KH instability is also prominent when a cylindrical flux
tube is perturbed as to only excite torsional Alfvén modes (Guo et al. 2019;
Díaz-Suárez & Soler 2021). The development of the instability occurs in much
the same manner as the kink mode, suggesting the initial shear from the
swaying motion of the kink mode is not a necessary condition to generate the
instability in the wave guide. For the KH instability from torsional modes, it
is the phase mixing of the Alfvén waves that leads to the velocity shears, as
suggested by Heyvaerts & Priest (1983). Díaz-Suárez & Soler (2021) undertook a
parameter study showing how the onset of the KH instability is influenced by a
number of factors, highlighting the size of the inhomogeneous layer across the
flux tube and the density contrast can play key roles. Here, the authors
suggest the onset of KH instability is delayed for wider inhomogeneous layers
as the phase mixing takes longer to develop and, hence, so does the build-up
of shear velocities. For density contrast, the development of the instability
occurs faster when the density contrast is larger (although after a certain
value, the rate of phase mixing is insensitive to the density contrast).

The development of the KH instability appears to be able to provide a
substantial heating of the plasma (Karampelas et al. 2017), unlike the case of
resonant absorption and phase mixing. The KH instability can increase the rate
at which small-scales are developed from phase mixing alone, hence expediting
the transfer of energy to dissipative scales. The development of the KH
instability and subsequent heating can be increased when multiple modes are
initially present (Guo et al. 2019). It was shown by Shi et al. (2021) that
the development of KH instability in simulations of kink oscillations can
provide heating rates, that under specific conditions are sufficient enough to
balance radiative losses in the quiet Sun.

It should be noted that the majority of these studies focus on standing kink
modes, which have, to date, only been identified in active regions. In
contrast, observations of Alfvénic waves in the quiet Sun and coronal holes
appear to show only propagating modes (Morton et al. 2015; Tiwari et al. 2021;
Morton et al. 2021). It would appear that the KH instability is also able to
develop in numerical simulations of a system containing only propagating
Alfvénic waves, although the low resolution of the simulations prevented
detailed study of the instability (Pagano & De Moortel 2019). However,
Zaqarashvili et al. (2015) suggest propagating waves are stable to the KH
instability as the magnetic and velocity perturbations of Alfvénic modes are
in anti-phase, meaning that that magnetic field stabilises the system. Hence,
there is still some ambiguity around what conditions are required for the KH
instability to develop in a system primarily containing propagating waves.

###  4.3 Turbulence

Turbulent cascades have long been considered a promising candidate for
dissipation of Alfvén waves to heat coronal loops (e.g., Hollweg 1983;
Rappazzo et al. 2007; van Ballegooijen et al. 2011; Verdini et al. 2012) and
accelerate the solar wind (e.g., Hollweg 1986; Velli et al. 1989; Verdini et
al. 2010). Inhomogeneity is essential in many models of Alfvénic wave
propagation from the corona to the solar wind. However, it is predominately
assumed to be along the field only. More recently the influence of
perpendicular inhomogeneity has become of interest. So, to set the scene for
this contemporary take, we briefly discuss salient aspects of the long-studied
Alfvén wave turbulence (the work in this area is extensive and spans many
decades, and an excellent review can be found in Bruno & Carbone 2005).

The incompressible MHD equations that describe the evolution of the magnetic
field and the velocity fluctuations can be written in terms of the so-called
Elsässer variables, $\vec{z}^{~{}\pm}=\vec{v}\mp\vec{v_{A}}$ (Elsasser 1950),
where $\vec{v_{a}}=\vec{B}/\sqrt{\mu_{0}\rho}$. It can be shown that they can
be reduced to the following equation:

$\frac{\partial\vec{z}^{~{}\pm}}{\partial
t}+(\vec{z}^{~{}\mp}\cdot\nabla)\vec{z}^{~{}\pm}=-\frac{1}{\rho}\nabla
P-\vec{v_{A}}(\nabla\cdot\vec{v_{A}}),$ (13)

where $P=p+\frac{B^{2}}{2\mu_{0}}$ is the total pressure. The second term on
the left hand side describes the non-linear interaction of counter propagating
waves along the mean magnetic field, and is the key factor in enabling a
turbulent cascade. The non-linear interaction deforms the counter-propagating
wave packets and generates the small-scale structures. A typical assumption
for Alfvénic turbulence in the corona and solar wind is that there is strong
mean magnetic field, such that the amplitudes of perturbations are small in
comparison to the background magnetic field (Strauss 1976, referred to as
Reduced MHD). With such a strong anisotropy, the cascade is primarily in the
perpendicular wavenumbers with parallel cascades suppressed, compared to,
e.g., a case with a weaker background field (Oughton et al. 2003).

Typically, the inhomogeneity is only assumed to be along the magnetic field
(due to gravitational stratification), and the Alfvénic waves are subject to
non-WKB reflection due to the gradient in the Alfvén speed (Heinemann & Olbert
1980; Velli 1993). In the presence of density inhomogeneities along the
magnetic field, the final term on the right-hand side of Eq. 13 leads to the
reflection of Alfvén waves888Perhaps unsurprisingly, there is only reflection
in an incompressible plasma if there is a gradient density. One may have
thought this should be the case for variations in Alfvén speed due to the
magnetic field gradients, however this is not the case (Magyar et al.
2019b).(Heinemann & Olbert 1980; Velli 1993). This term is vital for the
development of turbulence in the solar wind, given it is expected that
Alfvénic waves are launched predominantly at the Sun. Without this reflection,
there would be no counter-propagating waves to interact. Much of the solar-
focused literature on Alfvénic wave turbulence generally only includes this
form of reflection to drive the turbulence, with some compressible simulation
also able to include Alfvén wave reflection to due to parametric decay
instability (e.g., Sagdeev & Galeev 1969; Goldstein 1978; Shoda et al. 2018b,
a; Réville et al. 2018).

A compelling development in the discussion of Alfvénic wave turbulence is
recent work that has included the presence of perpendicular density
inhomogeneities across the magnetic field. Largely this has been neglected in
turbulence studies. However, the presence of the inhomogeneity enables a self-
cascade of the Alfvénic waves, given the name ‘uni-turbulence’ (Magyar et al.
2017, 2019b). As discussed in Section 2.2, the presence of inhomogeneities
perpendicular to the magnetic field leads to the existence of surface Alfvén
modes in the presence of a discontinuity, or a quasi/global Alfvénic waves in
the presence of a continuous variation (e.g., Goossens et al. 2002, 2012).
Magyar et al. (2017) showed via numerical simulations of unidirectional
Alfvénic modes that both Elsässer components are present, propagating in the
same direction. This leads to the cascade of wave energy in the perpendicular
direction, with the timescale for the cascade dependent on the size of the
density contrast (Van Doorsselaere et al. 2020a), namely:

$\tau=\sqrt{5\pi}\frac{1}{\omega
a}\frac{\rho_{i}+\rho_{e}}{\rho_{i}-\rho_{e}},$ (14)

where $a$ is the normalised oscillation amplitude. The presence of both
components is due to the fact that, as emphasised already, in inhomogeneous
media MHD waves have mixed properties. Magyar et al. (2019a) demonstrate that
when compressibility and inhomogeneity of the plasma are taken into account,
the Elsässer formalism cannot be used to separate parallel and anti-parallel
propagating waves. In a homogeneous, compressible plasma the slow and fast
modes are necessarily described by both components. Hence, waves with mixed
properties should also have to be described using both Elsässer components. We
refer readers to Magyar et al. (2019b) for a discussion of the subtleties of
this phenomena applied to Alfvénic modes.

The study of this turbulence phenomenon and its role in plasma heating is
still in its infancy but it would appear to be an exciting avenue to aid in
the transfer of wave energy to dissipation scales. Application to the damping
of standing kink modes in active regions suggest that it could explain an
observed non-linear relationship between wave amplitude and damping times
(Goddard & Nakariakov 2016; Van Doorsselaere et al. 2021). This phenomenon
could also be a dominant source of energy transfer in open field regions,
i.e., coronal holes, where the Alfvénic wave propagation appears pre-
dominantly uni-directional (i.e., anti-sunward Morton et al. 2015). For
closed, loop-like magnetic fields, observations suggest that counter-
propagating waves are the norm (Tiwari et al. 2021). These waves are likely
driven at both foot-points of the loops (by granulation or mode conversion)
and suffer strong wave reflection at the transition region; ensuring a
constant supply of counter-propagating waves. Hence, uni-turbulence would play
a complementary role. However, Morton et al. (2021) examined a number of loops
in the quiescent corona and inferred that the density difference across the
inhomogeneities must be small to explain the apparently weak frequency-
dependent wave damping of the observed kink modes (i.e., due to resonant
absorption). This would also imply that time-scales for energy transfer by
uni-turbulence may also be long given Eq. 14.

##  5 Waves in the chromosphere

In order to reach the corona, the waves originating in the photosphere, must
pass through the gauntlet of the chromosphere. This layer remains one of the
least understood regions of the Sun’s atmosphere due to many complex physical
phenomena that arise there. However, there is a general agreement that the
chromosphere is highly dynamic and continuously exchanging mass and energy
with the corona (for reviews see, e.g., Judge 2006; Carlsson et al. 2019).
Moreover, this complex region is populated by fine-scale structures (see
review by Tsiropoula et al. 2012), that bridge the photosphere with the corona
and act as conduits for plasma and wave energy transfer across the transition
region. A peculiar aspect of this layer is associated with moderate rise in
its thermal profile (Fig. 2), as compared to the photosphere. This maintains
temperatures of the order of $\sim$104 K in the chromosphere, which enables
much of the hydrogen and helium to remain neutral up until the transition
region (Figure 8). In these conditions, partial ionisation of plasma becomes a
critical factor that can have a strong effect on the dynamics (see reviews by,
e.g., Leake et al. 2014; Martínez-Sykora et al. 2015). In combination with
large gradients along the magnetic field, it indicates that most of the non-
thermal energy remains in the chromosphere.

The observed dynamics of chromospheric features reflect the presence of
confined MHD waves that emanate from the photosphere. Alfvénic waves are
routinely observed and are often classified as a particular wave mode
depending upon their observational characteristics (see review by Verth & Jess
2016). They carry mass, energy and momentum across this layer. However, in the
lower solar atmosphere there are two major factors that influence Alfvénic
wave propagation, Namely the presence of partial ionisation (which leads to
wave damping) and the gradient of the Alfvén speed (Fig. 3, leading to
reflection and non-linear effects).

###  5.1 Partial ionisation

Figure 8: Properties of the partially ionised solar atmosphere based on the
FAL-C atmosphere. The top left panel displays the relative fraction of
different components of the plasma. The top right plasma composition in terms
of the relative fractions of ions and neutrals fractions. The bottom row shows
the resistivity through the lower solar atmosphere. The left panel displays
the classical electrical resistivity (i.e., electron collisions). The right
panel shows the values of resistivity for electrical (including neutrals) in
blue dash and Cowling resistivity in solid orange.

The process of ionisation primarily depends on the (inelastic) collisions
between the different constituent particles of the plasma. In Figure 8, we
illustrate the ionisation degree for the dominant components required for a
multi-fluid solar plasma (i.e., hydrogen and helium), along with the fractions
of ions and neutrals. The main influence that partial ionisation has on wave
propagation arises through the generalised Ohm’s Law, namely;

$\frac{\partial\vec{B}}{\partial t}=-\nabla\times(-\vec{v}\times\vec{B}+\eta
j_{\parallel}+\eta_{p}j_{\perp}+\frac{1}{en_{e}}\vec{j}\times\vec{B})$ (15)

where, $j_{\parallel}$ and $j_{\perp}$ are the parallel and perpendicular
current respectively. All other symbols have the usual meaning. The second
term on the right is the Joule heating (with $\eta$ the electrical
resistivity) that describes the electron collisions with ions and neutrals.
The third term is due to the additional collisions of ions and neutrals (and
$\eta_{p}$ is the Pedersen resistivity), while the fourth term is the Hall
term. The Pedersen resistivity can be decomposed into the perpendicular
components of the electrical resistivity and the Cowling resistivity999Also
referred to as the coefficient of ambipolar diffusion, $\eta_{A}$. , i.e.
$\eta_{p}=\eta+\eta_{C}$. The Cowling resistivity for a Hydrogen-Helium plasma
is given by (e.g. Zaqarashvili et al. 2013),

$\eta_{C}=\frac{B^{2}}{\mu_{0}}\frac{\alpha_{He}\xi_{H}^{2}+\alpha_{H}\xi_{He}^{2}+\alpha_{HeH}(\xi_{H}+\xi_{He})^{2}}{(\alpha_{H}\alpha_{He}+\alpha_{H}\alpha_{HeH}+\alpha_{He}\alpha_{HeH})},$

where, $\xi_{n}$ is the fraction of species $n$, and $\alpha_{n_{1}n_{2}}$ is
the coefficient of friction between different species $n_{1}$ and $n_{2}$
(note that a subscript of a single species means the sum of coefficients for
ion-neutral collisions associated with the given species).

Studies to evaluate the impact of ion-neutral effects in the solar atmosphere
were pioneered by Piddington (1956) and Osterbrock (1961), where these authors
first highlighted their importance as a wave damping mechanisms in the
chromosphere. Decades later, their initial research was further extended for
the analysis of Alfvén wave propagation in a partially ionised single-fluid
(i.e., hydrogen) MHD model (De Pontieu & Haerendel 1998; Leake et al. 2005),
incorporating more realistic estimates for plasma parameters. It was suggested
that wave damping due to Ohmic diffusion is dominant over ion-neutral damping
in low photosphere and chromosphere, but can be neglected due to the long
damping lengths. However, in the upper chromosphere, the damping is sensitive
to ion-neutral effects (which can be inferred the magnitude of the Cowling
resistivity in Figure 8) and is effective for high-frequency waves ($f>0.01$
Hz). However, low frequency waves remain unaffected by the presence of
neutrals and experience no damping due to this mechanism. Further, Leake et
al. (2005) also demonstrated that increasing the magnetic field strength can
weaken the damping of waves due to ion-neutral effects. Further, weak damping
of Alfvénic waves in partially ionised atmosphere can also create drag forces
that can lift the cold/dense plasma against the gravity and generate spicule-
like structures in the vertical direction (Haerendel 1992; De Pontieu &
Haerendel 1998; James et al. 2003).

Naturally, one should also expect to have additional viscosity effects due to
ion-neutral interactions, arising through the momentum equation. Khodachenko
et al. (2004) provided a quantitative comparison of the relative magnitudes of
damping due to viscous and collisional terms, based on the analytic
expressions for the wave damping times. It was demonstrated that ion-neutral
effects are more efficient compared to viscous forces for Alfvén waves. Soler
et al. (2015b) extended this line of work by incorporating multiple potential
damping mechanisms into an analytic investigation, and confirmed that
viscosity plays no important role in wave damping throughout the chromosphere.
Using a single fluid description, Soler et al. (2015b) included the effects of
partial ionisation involving both neutral hydrogen and helium. For Alfvén
waves, they showed that Ohmic diffusion dominated the damping at low heights,
while ambipolar diffusion became more important at greater heights in the
chromosphere. This result was in qualitative agreement with De Pontieu &
Haerendel (1998) and Leake et al. (2005), although Soler et al. (2015b)
suggested that Ohmic diffusion is actually an important damping mechanism in
the low chromosphere (see also Goodman 2011; Tu & Song 2013; Arber et al.
2016). The cause of this difference is mainly due to taking into account the
role of electron-neutral collisions when calculating the Ohmic resistivity,
which leads to a significant increase in the magnitude of the coefficient
(bottom left panel, Figure 8). Soler et al. (2015b) also highlighted the role
of the magnetic field strength, which determines at which heights ambipolar
damping overtakes Ohmic damping.

Although insightful, the use of single fluid approximations to understand the
wave behaviour in the photosphere and chromosphere has been shown to be
incorrect. This aspect was examined in detail by Zaqarashvili et al. (2011b),
who investigated MHD wave propagation in a two fluid model consisting of
neutral hydrogen and ions. They showed that for time-scales less than the ion-
neutral collision time, both fluid species tend to behave independently and a
single-fluid approximation does not remain valid any more. Importantly, the
damping of high frequency waves is a lot weaker than suggested by the single
fluid approach. Their results again confirmed that low-frequency waves were
not strongly influenced by partial ionisation of the plasma. More generally,
Soler et al. (2013) showed that the damping of waves is most efficient when
the wave frequency is comparable to the ion-neutral frequency.

Given the sizable fraction of neutral helium in the lower solar atmosphere,
Zaqarashvili et al. (2011a, 2013), extended the analysis with the inclusion of
three fluids. The addition of helium species was found to have an additional
impact on the damping of high frequency Alfvén waves. However, this mechanism
is strongly dependent on the plasma temperature, with larger wave damping
rates at higher temperatures ($1-4\times 10^{4}$ K), when ionisation of
hydrogen becomes prevalent (see Figure 8). Further, Soler et al. (2015a) found
that there exists a particular wavelength range for which Alfvén waves are
overdamped (10$\lesssim\lambda\lesssim$1000 m), resulting in localised energy
dissipation and thus restricting the propagation as Alfvén wave.

In an effort to incorporate perpendicular inhomogenities into studies of wave
propagation in a partially ionised plasma, Soler et al. (2012) investigate the
influence of ion-neutral collisions on kink wave propagation in a cylindrical
flux tube in a two-fluid study. The influence of the ion-neutral interaction
was incorporated in the momentum equation through frictional terms. With this
formalism, Soler et al. (2012) found that ion-neutral collisions appear to
have little impact on kink wave propagation and resonant absorption remained
the dominant damping mechanism. Interestingly, they found that the damping
lengths due to resonant absorption, for kink waves with periods less than 10
s, are smaller than or of the same order as the height of the chromosphere.
Hence, they suggested that only waves with periods greater than 10 s are able
to reach the coronal level in the form of kink motions, while the shorter
period waves would reach the corona as rotational motions.

Finally, we note that use of the multi-fluid approach has been suggested to be
invalid for short wavelengths. This was first highlighted by Soler et al.
(2015a) who demonstrated that the multi-fluid approach, which essentially
treats each plasma species separately, is not applicable to high-frequency
waves below of heights $\sim 900$ km. They suggest a kinetic description of
the plasma is required as both ions and electrons collide too frequently with
the neutrals, meaning that the thermal properties of each species are not
independent.

Figure 9: The ratio of wave velocity amplitude to Alfvén speed is shown as a
function of height. The curves are calculated using a WKB approximation for
wave amplification in an ideal plasma using the density from the FAL model and
the magnetic field from the empirical model, see Figure 2.

###  5.2 Alfven speed gradient

Aside from the longitudinal homogeneity in ionisation state, there is also a
longitudinal inhomogeneity in the Alfvén speed through the chromosphere due to
the decrease in density (Figure 2). This variation leads to amplification of
the waves propagating from the photosphere to the corona. The relationship
between wave amplitude and density can be obtained under a WKB approximation,
namely $\mid\delta v\mid\propto\rho^{-1/4}$. Hence, the upwardly propagating
Aflvén waves can become non-linear, demonstrated by the increase of the ratio
of the wave amplitude to the Alfvén speed ($\mid\delta v\mid/v_{A}$) with
height101010We note the form of this profile is dependent upon the functional
form of the magnetic field used and can decrease in the corona (c.f. Figure 4
in Wang & Yokoyama 2020). (shown in Figure 9). Given that the profiles in
Figure 9 are found using the WKB approximation, they are not valid for waves
with wavelengths comparable to or greater than the density scale height in the
lower solar atmosphere. Moreover, the calculation of the profiles neglects any
physics that would act to damp these waves, e.g. ion-neutral effects, Ohmic
diffusivity, and mode conversion (of which we now discuss). Hence, in reality
the increase in wave amplitude is less rapid than shown in the figure.

The large amplitude of the Alfvén waves leads to an increase in the
ponderomotive force, which is able drive motions parallel to the magnetic
field. This enables the mode conversion of Alfvénic modes to both fast and
slow magnetoacoustic modes (e.g., Hollweg et al. 1982; Hollweg 1992). Both the
fast and slow waves also develop into shocks in the upper chromosphere and low
corona. When the magneto-acoustic shock occurs in the chromosphere, there is
an upward displacement of the chromosphere and transition region creating
spicule-like structures (e.g., Hollweg et al. 1982; Hollweg 1992). The slow
shocks driven by Alvén waves are found to be very effective at heating the
lower solar atmosphere (Matsumoto & Suzuki 2012; Arber et al. 2016; Wang &
Yokoyama 2020). Arber et al. (2016) suggest that the relative heating rates
due to shock heating are four orders of magnitude greater than the Pederson
resistivity. However, the heating rate in their 1.5D simulations may be
enhanced by shock coalescence. The fast and slow mode waves produced by Alfvén
waves are also capable of heating the corona (Hollweg 1992; Kudoh & Shibata
1999; Moriyasu et al. 2004; Antolin et al. 2008; Antolin & Shibata 2010).

The variation in Alfvén speed also leads to significant reflection of the
waves in the chromosphere, and especially at the transition region. The
reflection of waves occurs if the wavelength of the wave is comparable or
greater than the length-scale over which there is appreciable variation in
Alfvén speed (e.g., Ferraro & Plumpton 1958; Velli 1993). Hence there is
frequency filtering effect, with low frequency Alfvén waves affected to a
greater extent than higher frequency waves (see Figure 10 panel c for a curve
of reflection versus frequency; details of calculation given in Soler et al.
2017, 2019). The wave reflection also sets up a system of counter propagating
waves in the chromosphere, which, as discussed in Section 4.3, enables
Alfvénic wave turbulence to develop (e.g., van Ballegooijen et al. 2011).

In general, the transition region provides a substantial barrier to all waves,
significantly influencing the transmission to the corona (Hollweg 1981;
Verdini et al. 2012), with Cranmer & van Ballegooijen (2005) suggesting the
reflection rate is on the order of 95%. Fully dynamical simulations suggest
the wave reflection might be lower, but still substantial, with 85% of the
initial wave energy flux reaching the corona (Suzuki & Inutsuka 2005; Suzuki &
ichiro Inutsuka 2006). In the Suzuki & Inutsuka (2005); Suzuki & ichiro
Inutsuka (2006) model, the transition region is dynamic due to intermittent
shock heating in the chromosphere and the sharp density gradient is smoothed
out. Arber et al. (2016) estimate of the efficiency of wave transmission to
the corona by comparing the Poynting flux at the photosphere against the
Poynting flux entering the corona. In their incompressible simulation, the
transmission to the corona is a relatively smooth function of frequency. The
transmission is most effective for intermediate frequencies, peaking at around
40% of the initial wave flux for frequencies of 0.1-1 Hz, with the reduction
in transmission for higher frequencies related to wave damping. The degree of
magnetic expansion can also influence the transmission curve, with larger
rates of expansion shifting the transmission peak to lower frequencies
(Similon & Zargham 1992; Soler et al. 2017). In Figure 10 we show the
reflection, transmission (to corona) and absorption (i.e., dissipation due to
wave damping) curves for a model of Alfvén wave propagation from Soler et al.
(2019) model (which show a qualitatively similar behaviour to that in Arber et
al. 2016).

Figure 10: The structure of the magnetic field from the model of torsional
Alfvén wave propagation through the lower solar atmosphere from Soler et al.
(2019). Panel a shows the expanding magnetic field in 3D. Panel b displays a
cross-sectional profile of the magnetic field in the radial direction. Panel c
is the horizontally averaged profiles for reflection, transmission and
absorption as a function of wave frequency. Panel d shows the contribution to
the wave heating rates as a function of height in the model atmosphere.
Figures reproduced from Soler et al. (2019) with permission of the AAS.

###  5.3 Perpendicular inhomogeneity

From the previous discussions, it is clear that Alfvénic wave propagation is
strongly influenced by the perpendicular inhomogeniety of plasma, hence it is
worth touching upon its influence in a partially ionised plasma. While there
are a number of 2D and 3D models of wave propagation through the lower
atmosphere that include or develop some perpendicular inhomogeniety (e.g.,
Shelyag et al. 2016; Kuźma et al. 2020; Battaglia et al. 2021; Breu et al.
2022), a fascinating insight can be found in Soler et al. (2019). The model
focuses on the propagation of torsional Alfvén modes from photosphere to
corona but is quite restrictive; it is an incompressible, 2.5D model of linear
waves. However, it is an insightful culmination of a series of works that
carefully increases the models complexity and isolates the impact of aspects
of wave physics (see Soler et al. 2013, 2015b, 2017, 2019, 2021). In Soler et
al. (2019), the perpendicular inhomogeniety arises through the implementation
of a 3D magnetic field, which is shown in Figure 10. Through the chromosphere
the magnetic field expands but that expansion rate varies as a function of
radius. As such, there is a variation of the Alfvén speed in the perpendicular
direction, as well as the parallel direction. This leads to phase mixing of
the Alfvén waves and the development of small-scale shears in the velocity and
magnetic perturbations across the flux tube. In turn, this leads to an
enhancement of local currents and increases the rate of Ohmic diffusion.

Some of key results from the Soler et al. (2019) are shown in Figure 10, which
also nicely summarises aspects of the previous discussions. In panel c, the
reflection, transmission and absorption (i.e. wave damping) of the waves are
shown as a function of frequency. There is clear frequency dependency for all
quantities. It can be immediately seen that the transmission of wave energy to
the corona is drastically reduced compared to the estimates from the lower
dimensional models discussed previously. The peak of the transmission curve is
close to 5 mHz, and only $\sim$1% of the photospheric wave energy flux is able
to reach the corona. The reason for the decrease in transmission is due to the
increase in Ohmic heating arising from the phase mixing. Ohmic heating now
dominates the frictional heating (due to ion-neutral collisions) through the
majority of the lower atmosphere (panel d), and it is only surpassed in the
upper chromosphere ($\sim 1700$ km). In this model, the Ohmic heating rate has
increased while the frictional heating rate is similar to previous lower
dimensional models (Soler et al. 2017). For an injected wave flux at the
photosphere with a realistic total energy of $~{}10^{9}$ erg cm-2 s-1, the
wave flux entering the corona turns out to be only a factor of two smaller
than the expected energy flux required to sustain coronal temperatures in the
quiet Sun.

We note that the sensitivity of this result to the magnetic geometry, namely
the height of the most significant expansion, is not entirely clear but could
play a role in determining the relative contributions of the two heating
mechanisms. Additionally, given the strong influence of shock heating in the
1.5D models of Arber et al. (2016), it remains to be seen if the findings from
Soler et al. (2019) are still valid for non-linear wave propagation.

Another interesting work comes from González-Morales et al. (2020), who take a
different approach to examining the impact of inhomogeneities on wave
propagation. The framework is single fluid 3D MHD but incorporates the
generalised Ohm’s Law (Eq. 15) and additional terms in the energy equations to
describe a partially ionised plasma. Instead of focusing on a single flux-
tube, a small region of the quiet Sun is modelled and includes solar magneto-
convection with a near self-generated magnetic field. This naturally generates
magnetic flux tubes through the concentration of flux in the intergranular
lanes. The model does not reach the corona but is confined to the lower solar
atmosphere. The realistic nature of the magneto-convection means that there
are gradients perpendicular to the magnetic field direction throughout the
simulation domain. The paper does not discuss phase mixing of Alfvén waves or
associated instabilities (although they should be present) but does highlight
the important role of two other effects, ambipolar diffusion and the Hall
term. In common with Soler et al. (2019), González-Morales et al. (2020) find
that ion-neutral effects can lead to efficient absorption of Poynting flux.
Due to the highly inhomogeneous nature of the plasma, MHD waves and associated
instabilities (e.g., Kelvin-Helmholtz) enhance vorticity (e.g., Goossens et
al. 2019; Howson et al. 2019), which in turn increases the magnitude of the
currents (e.g., Howson et al. 2021) and increasing the Ohmic and ambipolar
heating rates. Futhermore, the inclusion of the Hall term is found to result
in a doubling of the Poynting flux at the upper chromosphere (compared to
without its presence), which is partially due to a Hall-induced mode coupling
of fast and Alfvén waves (see also Cally & Khomenko 2015; González-Morales et
al. 2019).

##  6 Summary

The solar atmosphere is a complex plasma environment that is characterised by
the networks of magnetic field that thread it, and the large variations in
plasma parameters with height. The temperature, density and magnetic field
vary by orders of magnitude from the photosphere to the corona and solar wind,
and makes the Sun’s atmosphere an interesting laboratory for studying MHD wave
propagation. In addition to the large-scale variations, the plasma is also
observed to be highly structured on small-scales ($\sim$100’s km)
perpendicular to the magnetic field (although there could be much finer
structuring below the current limits of instrumentation), likely the result of
localised heating events that are believed to take place throughout the
atmosphere. As we have hopefully highlighted through the review, the presence
of plasma inhomogenities in the solar atmosphere produces a fantastic array of
physics that impacts upon wave propagation through the magnetised plasma.

The main motivation for many investigations of Alfvénic waves is to understand
their role in the heating of the atmospheric plasma and the acceleration of
the solar wind (they are also proving useful for magneto-seismology and
estimates of the coronal magnetic field, e.g., Anfinogentov & Nakariakov 2019;
Yang et al. 2020). For the waves to contribute to these processes, they need
to propagate upwards from the photosphere into, and through, the partially
ionised, magnetically-dominated chromosphere. A fraction of the Alfvénic waves
are able to cross the barrier provided by the dramatic drop in density at the
transition region, reaching the corona and beyond. We have discussed various
details of the journey, but typically in isolation. This somewhat reflects how
the research has been undertaken to date. As mentioned at the beginning of
this review, it is challenging to include all the necessary physics into a
single atmospheric model. Hence, to date, many studies have focused on
individual problems in isolation in order to make progress (this, of course,
is always the way). So to wrap up this review, we highlight a few
representative state-of-the-art numerical studies and note the current
tensions in the modelling.

There are many large-scale models that examine wave propagation, focusing on,
e.g., wave heating/acceleration of the solar wind over 10’s of solar radii, or
global simulations of coronal heating. While demonstrating the feasibility of
wave-based plasma heating/wind acceleration via turbulence, one of the main
sacrifices these models make is that they largely ignore the role of
perpendicular inhomogeneity; missing resonances, phase mixing and
instabilities. There are also other restrictions required to make these
numerical simulations feasible. For example, the latest Alfvén wave solar wind
simulations have been able to incorporate compressibility into a 3D system to
examine the non-linear coupling to slow modes, but only starting at the
coronal base (e.g., Shoda et al. 2019) or includes a simple lower atmosphere
with a reduced numerical resolution (Matsumoto 2020). Another examines the
propagation of Alfvén waves from the lower atmosphere and the onset of
parametric instabilities, but using a 1D single fluid approach ignoring
partial ionisation (e.g., Réville et al. 2018). Moreover, a global model of
coronal heating based on Alfv́en wave turbulence (the Alfvén wave solar model
- AW-SoM, van der Holst et al. 2014) is able to reproduce global EUV emission
reasonably well, but only includes wave propagation from the upper
chromosphere and uses Reduced MHD and WKB wave propagation (with correction
terms) to predict wave heating rates. Also, smaller-scale Alfvénic-wave-driven
turbulence models, i.e. focusing on single flux tubes, appear to be able to
reproduce necessary heating rates for chromospheric/coronal plasma, but only
include a basic photosphere/chromosphere without dynamics or multi-fluid
effects (e.g., Matsumoto 2018), and are often based on Reduced MHD (e.g., van
Ballegooijen et al. 2011; Verdini et al. 2012).

There are also a wealth of studies, some of which we discussed here, that
concentrate on the impact of non-ideal phenomena and instabilities on wave
propagation; requiring high-resolution simulations to capture the relevant
scales. Hence the models are typically confined to the lower atmosphere or are
single, highly idealised flux tubes, with many of the latter models confined
to the coronal section or include only a basic lower atmosphere (i.e., no
dynamics or partial ionisation). For example, as discussed in Section 5.3,
González-Morales et al. (2020) undertook hydrodynamic convection driven 3D
simulations of lower atmosphere including a generalised Ohm’s law and
examining the impact of ambipolar diffusion and the Hall term on Alfvénic
waves. However, the simulation domain was confined to heights below 1.5 Mm. To
include a corona, Soler et al. (2019) model a single flux tube with a
temporally stationary atmosphere based on the FAL atmosphere, solving
linearised multi-fluid wave equations. The linearisation obviously means that
shocks, mode conversion and momentum transfer are not present, all of which
are representative of the observed chromosphere. Moreover, most models with a
multi-fluid chromosphere do not include time-dependent hydrogen ionisation,
which acts over relatively long time-scales (e.g., Leenaarts et al. 2007).
Hence the chromospheric ionisation fraction and temperature are different from
equilibrium which would influence the efficacy of various multi-fluid
phenomena, e.g., Cowling resistivity.

The final note to end on is a comment on the observations of Alfvénic waves in
the lower solar atmosphere and corona. To date, the observational studies have
been relatively few (excluding work on the standing kink modes in active
regions), and little is known from an observational perspective. Over the last
three decades there have been a number of indirect measurements of their
presence through spectral line broadening (e.g., Banerjee et al. 1998, 2009;
Hahn et al. 2012) in coronal holes and the quiet Sun corona. It took the
advent of high-resolution observations to measure the Alfvénic waves directly
(Tomczyk et al. 2007; De Pontieu et al. 2007; Jess et al. 2009; McIntosh et
al. 2011; Hillier et al. 2013; Morton et al. 2012), but the number of studies
focused on analysing the details of propagating waves is few. Those that have
been performed have largely focused on measuring the typical properties (e.g.,
amplitude, period) in a single region and providing energy estimates of the
waves in the structures examined (e.g., coronal loops, spicules). However, the
details of excitation, evolution and dissipation are scant. Part of this issue
is due to the challenge of connecting the observed dynamics through the
different regions of the atmosphere. Some of the issue lies in observing the
chromosphere, which is complicated by extended formation heights of the
resonance lines (requiring narrow spectral filters to isolate different
heights, e.g., Judge & Carlsson 2010) and non-local and non-equilibrium
contributions to radiative transfer that make interpreting the measured
diagnostics formidable (Leenaarts 2020). For the corona, the task is somewhat
simpler and there is a wealth of suitable data from the likes of the Coronal
Multi-Channel Polarimeter (Tomczyk et al. 2008) and the Solar Dynamic
Observatory (Pesnell et al. 2012). It is unclear why more attention hasn’t
been given to analysing the coronal Alfvénic waves present in the data. The
measurements are challenging (e.g., Thurgood et al. 2014; Weberg et al. 2018;
Tiwari et al. 2021) but opportunities exist to compare the theoretical
predictions discussed in preceding sections with observed wave behaviour. It
is surely an area worthy of focus in the coming years. More attention may be
focused on the observational behaviour of Alfvénic waves in the near future
when DKIST (Rimmele et al. 2020; Rast et al. 2021) and Solar Orbiter (Müller
et al. 2020) provide images of the chromosphere and corona at unprecedented
spatial and temporal resolution. We wait with anticipation to see what wave
physics these new facilities might uncover.

The authors are supported by a UKRI Future Leader Fellowship
(RiPSAW—MR/T019891/1). We also would like to thank P. Antolin for patiently
discussing a number of issues, and thank I. Arregui, M. Goossens, A. Hillier,
R. Soler for valuable discussions. Further thanks are required to the two
anonymous referees, whose insightful comments greatly improved the review. RJM
would like to dedicate this review to the memory of Deenah Morton, without
whose love and support this work and many others would not have been possible.
For the purpose of open access, the author(s) has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
Data and code used within this work is available at
https://github.com/Richardjmorton/FAL_models. Figures have been created with
the help of NumPy (Harris et al. 2020), matplotlib (Hunter 2007) and IPython
(Perez & Granger 2007). On behalf of all authors, the corresponding author
states that there is no conflict of interest.

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