# Line-tied boundary conditions can cause resonant absorption models to
generate unphysically large boundary layers

A. P. K. Prokopyszyn School of Mathematics and Statistics, University of St
Andrews, St Andrews, Fife, KY16 9SS, U.K.  A. N. Wright School of Mathematics
and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, U.K.  A.
W. Hood School of Mathematics and Statistics, University of St Andrews, St
Andrews, Fife, KY16 9SS, U.K.

###### Abstract

This paper uses linear magnetohydrodynamics to model resonant absorption in
coronal plasma with a Cartesian coordinate system. We impose line-tied
boundary conditions and tilt the background magnetic field to be oblique to
the transition region. Halberstadt & Goedbloed (1993, 1995); Goedbloed &
Halberstadt (1994); Arregui et al. (2003) show that line-tied boundary
conditions cause their resonant absorption models to produce steep boundary
layers/evanescent fast waves. We aim to study the importance of the boundary
layers and assess their significance in a solar context. We calculate the
solutions in a model where we impose line-tied boundary conditions and compare
this with a model where we include the chromosphere instead. Results are
calculated analytically and then verified numerically. We show that line-tied
boundary conditions can cause the model to overestimate the boundary layers’
amplitude significantly. If the fast waves can propagate in the chromosphere,
then the line-tied model accurately predicts the boundary layers’ amplitude.
However, if the fast waves are evanescent, then the boundary layers’ size is
reduced significantly, and the line-tied model overestimates their amplitude.
This leads to the counter-intuitive result that the length scales tangential
to the transition region can play an essential role in determining line-tied
boundary conditions’ validity. The results suggest that line-tied boundary
conditions can cause the model to generate unphysically large boundary layers.
However, researchers may wish to continue to use them in their models for
their simplicity and ability to significantly reduce computation time if they
understand and are aware of their flaws.

Analytical mathematics, Solar oscillations, Solar coronal waves,
Magnetohydrodynamical simulations, Magnetohydrodynamics, Alfven waves, Solar
atmosphere, Solar corona, Solar physics

††journal: The Astrophysical Journal

tablenum SIXtablenum

##  1 Introduction

Magnetohydrodynamic (MHD) waves are commonplace in the solar atmosphere. They
have been observed over the last two decades as a consequence of new, improved
imaging and spectroscopic instruments (see, for example, Tomczyk et al.
(2007), McIntosh et al. (2011) and De Moortel & Nakariakov (2012)). Goossens
et al. (2011) gives a review of the linear behaviour of MHD waves. The
dissipation of Alfvén waves has been the basis of many coronal heating models
(see review by Arregui (2015) and references therein).

This paper focuses on studying resonant absorption (see, for example, Ionson
1982) which describes the process where magnetoacoustic waves with a given
frequency mode convert to Alfvén / Alfvénic waves at a location where the
background magnetic field lines have the same natural Alfvén frequency. If the
magnetoacoustic waves’ frequency equals the field lines’ natural frequency,
then resonance is produced. Here mode conversion can convert a magnetoacoustic
wave into an Alfvén wave. This is a one-way process that results in a
localised, high concentration of energy, in a process which is analogous to
Barton’s pendulum experiment. Barton’s investigation shows a heavy pendulum
driving a series of lighter pendula/oscillators. One of these lighter
oscillators, called the resonant pendulum, has the same natural frequency as
the driving frequency. Since all the oscillators connect to the same string,
they are all driven at the same frequency. After a while, the system reaches a
steady-state where the whole system oscillates at the driver frequency. At
steady- state, the resonant pendulum oscillates at a significantly larger
amplitude than the other oscillators. The string which connects the
oscillators is analogous to magnetoacoustic waves, and the oscillators are
comparable to standing Alfvén waves. For both cases, energy concentrates in a
narrow region due to resonance.

Resonant absorption plays a crucial role in coronal seismology. Coronal
seismology is becoming an increasingly relevant field due to developments in
observational instruments, computational power and MHD wave theory (De Moortel
& Nakariakov, 2012). It is a technique where mathematical models are used to
infer hard to measure quantities, e.g. the coronal magnetic field strength or
density gradients from easier to measure quantities, e.g. wave frequencies and
damping time. Advances in coronal seismology could provide us with better
knowledge of physical values in the solar atmosphere. This may help us solve
problems such as the coronal heating problem, and it could present us with new
puzzles to solve. Resonant absorption can play a vital role in damping
observed coronal loop oscillations (Nakariakov et al., 1999; Terradas et al.,
2006). Ruderman & Roberts (2002) show that a kink wave in a cylindrical flux
tube can mode convert to a torsional Alfvén wave. This results in the kink
wave amplitude following a decay profile that closely resembles observed decay
profiles. Resonant absorption can also play a key role in triggering the
Kelvin-Helmholtz instability in coronal loops which causes the plasma flow to
become turbulent (see, e.g. Antolin et al. 2016; Howson et al. 2017).

In our models, we drive fast magnetoacoustic waves in from the side for
mathematical convenience. These waves could be generated by, for example, a
nearby flare. However, it is not our aim to investigate how the waves enter
the corona, but instead, we study their dynamics once they are in the corona.
The origin of coronal waves remains an open question. It is difficult for
Alfvén waves generated at the photosphere to enter the corona (Cranmer & van
Ballegooijen, 2005), due to the rapid exponential decay of the density with
height in the chromosphere and the steep decrease at the transition region.
Hollweg (1984) suggests resonances in coronal loops and spicules provide
enough energy flux to the corona to match the observed wave velocity
amplitudes. Cally & Hansen (2011); Hansen & Cally (2012) suggest that mode
conversion from fast waves to Alfvén waves at the transition region enables
sufficient energy flux to enter the corona. It is also possible that the
corona itself generates Alfvén waves via magnetic reconnection (Cranmer, 2018)
and the energy released during reconnection partially feeds into the energy
contained in the Alfvén waves.

This paper closely follows the seminal work of Halberstadt & Goedbloed (1993);
Goedbloed & Halberstadt (1994); Halberstadt & Goedbloed (1995). They modelled
linear MHD waves and resonant absorption in a closed domain where the
background magnetic field is tilted to be oblique to the solar surface. They
showed that imposing line-tied boundary conditions causes the Alfvén waves and
fast waves to couple. Suppose the transverse length scales are shorter than
the Alfvén wavelength then evanescent fast waves/boundary layers from at the
line-tied boundaries. Note that resonant absorption typically generates
transverse length scales, which are many times shorter than the Alfvén
wavelength. Goedbloed & Halberstadt (1994) clearly shows the boundary layers
in Figure 6c of their paper for the fast wave components of the Lagrangian
displacements ($\xi_{F}$ and $\eta_{F}$). We recommend reading Goedbloed &
Halberstadt (1994) for an in-depth explanation of how the boundary layers
form, but we also give a brief explanation in the penultimate paragraph of
Section 3. Arregui et al. (2003) confirms the existence of these boundary
layers via a numerical approach. One of the key differences between the work
presented in this paper to that already in the literature is that we test the
validity of imposing line-tied boundary conditions. To test the validity, we
check if a model which includes the chromosphere instead of imposing line-tied
boundary conditions can produce similar boundary layers. We model the corona
and chromosphere using a piecewise constant background density profile with a
discontinuous jump from the corona to the chromosphere.

This paper’s outline is as follows; in Section 2 we present the equations we
will use to model the plasma and then discuss the applicability of some of the
assumptions we make in the corona. In Section 3, we calculate normal mode
solutions in a domain where the Alfvén speed is uniform. We impose an incident
Alfvén wave and calculate the unique solution which ensures the velocity is
zero at $z=0$. Since the Alfvén speed is uniform, resonant absorption cannot
occur. However, we choose the length scales in $x$ to be extremely short to
simulate the conditions near a singularity in a resonant absorption
experiment. Section 4 builds on Section 3 by using a background Alfén speed
which is piecewise constant in $z$. The regions $z>0$ and $z<0$ corresponds to
the corona and chromosphere, respectively. We impose an incident Alfvén wave
and calculate the unique solution which ensures continuity of the velocity and
its derivative at $z=0$. By comparing results from Sections 3 and 4, we can
test how robust the boundary layers produced by line-tied boundary conditions
are when we consider the finite nature of the jump in Alfvén speeds and the
small length scales of a resonance layer. In Section 5 we use a background
Alfvén speed which is a function of $x$ and piecewise constant in $z$. Our
goal here is to test if steep boundary layers form in a resonant absorption
experiment. Finally, in Section 6, a summary of our results and conclusions
are given.

##  2 Model and equations

In this section, we describe the most general model we will use. However,
throughout this paper, we will start with a simple model. In subsequent
sections, we will add complexity to the model as we head towards the most
relevant description of resonance in coronal magnetic fields. For example, in
Sections 3 and 4, the background Alfvén speed, $v_{A}$, will not depend on $x$
but, in Section 5, we include a variation in $x$ as well as a model
chromosphere. We consider linear perturbations on a static background
equilibrium. Thus, we assume that

$\displaystyle u(\boldsymbol{x},t)$ $\displaystyle\ll$ $\displaystyle v_{A},$
(1) $\displaystyle\boldsymbol{B}(\boldsymbol{x},t)$ $\displaystyle=$
$\displaystyle\boldsymbol{B}_{0}(\boldsymbol{x})+\boldsymbol{b}(\boldsymbol{x},t),\
\text{where}\ b\ll B_{0},$ (2) $\displaystyle\rho(\boldsymbol{x},t)$
$\displaystyle=$
$\displaystyle\rho_{0}(\boldsymbol{x})+\rho_{1}(\boldsymbol{x},t),\
\text{where}\ \rho_{1}\ll\rho_{0},$ (3)

and

$u=\absolutevalue{\boldsymbol{u}},\ B_{0}=\absolutevalue{\boldsymbol{B}_{0}},\
b=\absolutevalue{\boldsymbol{b}},$ (4)

where $\boldsymbol{u}$ denotes the plasma velocity, $\boldsymbol{B}$ denotes
the magnetic field strength, $\rho$ denotes the plasma density and $v_{A}$
denotes the background Alfvén speed. $\boldsymbol{B}_{0}$ and $\rho_{0}$
denote the equilibrium magnetic field and density and $\boldsymbol{u}$,
$\boldsymbol{b}$, $\rho_{1}$ denote the perturbation components. We justify
assumption (1) because observational evidence suggests that $u/v_{A}$ lies
approximately in the range $[10^{-2},10^{-1}]$ (McIntosh et al., 2011;
McIntosh & De Pontieu, 2012). We approximate $\partialderivative*{B_{0}}{t}$
by zero, since magnetic structures can last much longer than the typical
Alfvén travel time of waves in coronal loops. If we take the Alfvén speed in
the corona as about $1$\,\mathrm{M}\mathrm{m}\,\mathrm{s}^{-1}$$ and assume
that coronal loops have a characteristic length of about
$100$\,\mathrm{M}\mathrm{m}$$ then this gives an Alfvén travel time of about
$100$\,\mathrm{s}$$. Magnetic structures can last much longer than this, for
example, active region prominences can last a few hours or a day, while
quiescent prominences can last between a few and 300 days (Priest, 2014). We
assume that $\partialderivative*{\rho_{0}}{t}$ approximately holds in coronal
loops which are isothermal. Klimchuk (2015) shows that loops can be
approximated as isothermal if the time interval between heating
events/nanoflares in the loops is sufficiently small.

We simplify further by modelling the background magnetic field strength as
uniform and given by

$\boldsymbol{B}_{0}=B_{0}\boldsymbol{\hat{B}}_{0}.$ (5)

We define the unit vectors $\boldsymbol{\hat{B}}_{0}$,
$\boldsymbol{\hat{\perp}}$ as

$\displaystyle\boldsymbol{\hat{B}}_{0}$ $\displaystyle=$
$\displaystyle\sin\alpha\,\boldsymbol{\hat{y}}+\cos\alpha\,\boldsymbol{\hat{z}},$
(6) $\displaystyle\boldsymbol{\hat{\perp}}$ $\displaystyle=$
$\displaystyle\cos\alpha\,\boldsymbol{\hat{y}}-\sin\alpha\,\boldsymbol{\hat{z}}.$
(7)

as the unit vectors parallel and perpendicular to the equilibrium magnetic
field.

The background Alfvén speed is given by $v_{A}(x,z)=\hat{v}_{A}(x)v_{A}(0,z)$,
where

$v_{A}(0,z)=\begin{cases}v_{A-},&z<0,\\\ v_{A+},&z\geq 0,\end{cases}$ (8)

and $\hat{v}_{A}(x)$ is an arbitrary function of $x$ satisfying
$\hat{v}_{A}(0)=1$. Note that we have neglected the exponential stratification
due to gravity for mathematical convenience.

We model the plasma as cold and ideal. Therefore, the linearised momentum
equation is given by

$\partialderivative{\boldsymbol{u}}{t}=\frac{1}{\mu\rho_{0}(x,z)}\quantity[(\boldsymbol{B}_{0}\dotproduct\gradient)\boldsymbol{b}-\gradient(\boldsymbol{b}\dotproduct\boldsymbol{B}_{0})],$
(9)

and the induction equation by

$\partialderivative{\boldsymbol{b}}{t}=(\boldsymbol{B}_{0}\dotproduct\gradient)\boldsymbol{u}-\boldsymbol{B}_{0}\divergence{\boldsymbol{u}},$
(10)

where

$v_{A}(x,z)=\frac{B_{0}}{\sqrt{\mu\rho(x,z)}}.$ (11)

Since we set the plasma beta equal to zero, the velocity perturbations
parallel to the background magnetic field decouple from the velocity
perturbations perpendicular to the field. Hence, the velocity is given by

$\boldsymbol{u}=u_{x}\boldsymbol{\hat{x}}+u_{\perp}\boldsymbol{\hat{\perp}}$
(12)

and the magnetic field is given by

$\displaystyle\boldsymbol{\hat{b}}$ $\displaystyle=\boldsymbol{b}/B_{0}$ (13)
$\displaystyle=\hat{b}_{x}\boldsymbol{\hat{x}}+\hat{b}_{\perp}\boldsymbol{\hat{\perp}}+\hat{b}_{||}\boldsymbol{\hat{B}}_{0}\;.$

Hence, Equations (9) and (10) simplify to give

$\displaystyle\partialderivative{u_{x}}{t}$ $\displaystyle=$ $\displaystyle
v_{A}^{2}(x,z)\quantity[\nabla_{||}\hat{b}_{x}-\partialderivative{\hat{b}_{||}}{x}],$
(14) $\displaystyle\partialderivative{u_{\perp}}{t}$ $\displaystyle=$
$\displaystyle
v_{A}^{2}(x,z)\Big{[}\nabla_{||}\hat{b}_{\perp}-\nabla_{\perp}\hat{b}_{||}\Big{]},$
(15) $\displaystyle\partialderivative{\hat{b}_{x}}{t}$ $\displaystyle=$
$\displaystyle\nabla_{||}u_{x},$ (16)
$\displaystyle\partialderivative{\hat{b}_{\perp}}{t}$ $\displaystyle=$
$\displaystyle\nabla_{||}u_{\perp},$ (17)
$\displaystyle\partialderivative{\hat{b}_{||}}{t}$ $\displaystyle=$
$\displaystyle-\quantity[\partialderivative{u_{x}}{x}+\nabla_{\perp}u_{\perp}],$
(18)

where

$\displaystyle\nabla_{\perp}$ $\displaystyle=$
$\displaystyle\cos\alpha\partialderivative{y}-\sin\alpha\partialderivative{z},$
(19) $\displaystyle\nabla_{||}$ $\displaystyle=$
$\displaystyle\sin\alpha\partialderivative{y}+\cos\alpha\partialderivative{z},$
(20)

are the derivatives perpendicular and parallel to the equilibrium magnetic
field respectively. We can eliminate $\hat{b}_{x}$, $\hat{b}_{\perp}$,
$\hat{b}_{||}$ to give

$\displaystyle\quantity[\partialderivative[2]{t}-v_{A}^{2}\quantity(\partialderivative[2]{x}+\nabla_{||}^{2})]u_{x}$
$\displaystyle=$ $\displaystyle
v_{A}^{2}\nabla_{\perp}\partialderivative{u_{\perp}}{x},$ (21)
$\displaystyle\quantity[\partialderivative[2]{t}-v_{A}^{2}\quantity(\nabla_{\perp}^{2}+\nabla_{||}^{2})]u_{\perp}$
$\displaystyle=$ $\displaystyle
v_{A}^{2}\nabla_{\perp}\partialderivative{u_{x}}{x}.$ (22)

In this paper, we will derive exact analytic solutions to the equations.
However, the formulas we derive are often cumbersome, and so our discussion
will focus on approximate (but less cumbersome) solutions in our narrower
parameter space. The parameter space we will focus on is

$\displaystyle k_{x}\gg k_{||+},$ (23) $\displaystyle k_{||-}\gg k_{||+},$
(24) $\displaystyle 0<k_{y}\leq O(k_{||+}),$ (25) $\displaystyle
0<\tan\alpha\leq O(1),$ (26)

where

$k_{||\pm}=\frac{\omega}{v_{A\pm}},$ (27)

gives the Alfvén wavenumber and $\omega$ gives the wave frequency. We model
$k_{x}\gg k_{||+}$ as this approximates the conditions near a singularity in a
resonant absorption experiment. The results should also apply to other cases
where $k_{x}\gg k_{||+}$, for example, during the nonlinear development of an
instability. Our condition for $\alpha$ prevents the background magnetic field
from being tangential to the $z=0$ plane.

##  3 Uniform background Alfvén speed

In this section, we will calculate the solution in a domain where the
background Alfvén speed is uniform, i.e. $v_{A}(x,z)=v_{A+}$ and we impose an
incident Alfvén wave from $z>0$. We will impose line-tied boundary conditions
at $z=0$, i.e. we will ensure that $u_{x}=u_{\perp}=0$ at $z=0$. The model we
use here is very similar to that used in Section 2 of Halberstadt & Goedbloed
(1993) and Section 5 of Goedbloed & Halberstadt (1994). The only significant
difference is that they use a closed system with line-tied boundary conditions
imposed at both ends in $z$ whereas we model an open system with line-tied
boundary conditions at $z=0$. We find that modelling an open-loop makes the
maths a little bit easier. Goedbloed & Halberstadt (1994) shows that imposing
line-tied boundary conditions forces the fast and Alfvén waves to couple due
to the impossibility to reconcile the requirements of the vanishing of both
$u_{x}$ and $u_{\perp}$. Therefore, pure Alfvén wave solutions and pure fast
wave solutions cannot exist. The fast waves can be evanescent and form
boundary layers at $z=0$. Our goal in this section is to reproduce these
boundary layers to explain how they form and motivate why testing the validity
of line-tied boundary conditions is necessary.

By assuming the variables are of the form

$f(\boldsymbol{x},t)=f_{0}\exp[i(k_{x}x+k_{y}y+k_{zn}+\omega t)],$ (28)

where $f$ can be $u_{x}$, $u_{\perp}$, $b_{x}$, $b_{\perp}$ or $b_{||}$, we
can derive the following dispersion relation

$\Big{[}\omega^{2}+v_{A+}^{2}\nabla_{||n}^{2}\Big{]}\Big{[}\omega^{2}-v_{A+}^{2}(k_{x}^{2}+k_{y}^{2}+k_{zn}^{2})\Big{]}=0,$
(29)

where

$\nabla_{||n}=i(k_{y}\sin\alpha+k_{zn}\cos\alpha),$ (30)

and $n$ is an integer used to distinguish the different solutions. Solutions
to Equation (29), where the left bracket equals zero, correspond to Alfvén
waves that propagate parallel to the background magnetic field. Solutions,
where the right bracket equals zero, correspond to fast waves, and these waves
propagate isotropically. Hence, the possible values of $k_{zn}$ are

$\begin{pmatrix}k_{z1}\\\ k_{z2}\\\ k_{z3}\\\
k_{z4}\end{pmatrix}=\begin{pmatrix}k_{||+}/\cos\alpha-k_{y}\tan\alpha\\\
-k_{||+}/\cos\alpha-k_{y}\tan\alpha\\\
i\sqrt{k_{x}^{2}+k_{y}^{2}-k_{||+}^{2}}\\\
-i\sqrt{k_{x}^{2}+k_{y}^{2}-k_{||+}^{2}}\end{pmatrix}\;,$ (31)

for the upward and downward propagating Alfvén waves and the upward and
downward propagating fast waves. Note that, as
$k_{x}/k_{||+}\rightarrow\infty$, the fast waves are evanescent ($k_{z3}$) and
exponentially growing ($k_{z4}$), instead of propagating.

We assume the solutions are of the form

$f(\boldsymbol{x},t)=\sum_{n=1}^{4}f_{n}\exp[i(k_{x}x+k_{y}y+k_{zn}z+\omega
t)].$ (32)

We impose an incident Alfvén wave from above ($z>0$) by setting $u_{\perp
1}=u_{0}$, where $u_{0}$ gives the incident wave amplitude and set the
amplitude of the incident fast waves to zero, $f_{4}=0$. From Equation (21) we
know that

$u_{xn}=\hat{u}_{xn}u_{\perp n},$ (33)

where

$\displaystyle\hat{u}_{xn}$ $\displaystyle=$
$\displaystyle\frac{-ik_{x}\nabla_{\perp n}}{\mathcal{L}_{n}-k_{x}^{2}},$ (34)
$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$
$\displaystyle\nabla_{||n}^{2}+k_{||+}^{2},$ (35) $\displaystyle\nabla_{\perp
n}$ $\displaystyle=$ $\displaystyle i(k_{y}\cos\alpha-k_{zn}\sin\alpha).$ (36)

The boundary conditions require $u_{x}=u_{\perp}=0$ at $z=0$ and this gives
the following matrix equation

$\begin{pmatrix}1&1\\\
\hat{u}_{x2}&\hat{u}_{x3}\end{pmatrix}\begin{pmatrix}u_{\perp 2}\\\ u_{\perp
3}\end{pmatrix}=-u_{0}\begin{pmatrix}1\\\ \hat{u}_{x1}\end{pmatrix}.$ (37)

Hence,

$\begin{pmatrix}u_{\perp 1}\\\ u_{\perp 2}\\\ u_{\perp
3}\end{pmatrix}=u_{0}\begin{pmatrix}1\\\
-(\hat{u}_{x1}-\hat{u}_{x3})/(\hat{u}_{x2}-\hat{u}_{x3})\\\
(\hat{u}_{x1}-\hat{u}_{x2})/(\hat{u}_{x2}-\hat{u}_{x3})\\\ \end{pmatrix}.$
(38)

From Equations (16), (17) and (18), we know that

$\displaystyle v_{A+}\hat{b}_{xn}$ $\displaystyle=$
$\displaystyle\frac{\nabla_{||n}}{ik_{||+}}u_{xn},$ (39) $\displaystyle
v_{A+}\hat{b}_{\perp n}$ $\displaystyle=$
$\displaystyle\frac{\nabla_{||n}}{ik_{||+}}u_{\perp n},$ (40) $\displaystyle
v_{A+}\hat{b}_{||n}$ $\displaystyle=$
$\displaystyle-\frac{1}{ik_{||+}}\quantity[ik_{x}u_{xn}+\nabla_{\perp
n}u_{\perp n}].$ (41)

This gives the full set of solutions to the equations. However, we do not
write out the solutions in full here as they are cumbersome. We find the
equations are easier to interpret if we construct series solutions by taking
appropriate limits. Resonant absorption cannot occur because the background
Alfvén speed is uniform. Therefore, singularities cannot form in our model. To
simulate the conditions near a singularity in a resonant absorption experiment
where $\partialderivative*{v_{A}}{x}\neq 0$, we calculate the asymptotic
expansions for $k_{x}/k_{||+}\rightarrow\infty$. The leading order asymptotic
expansions for the Alfvén wave coefficients is given by

$\displaystyle\hat{b}_{||1}=\hat{b}_{||2}$ $\displaystyle=$ $\displaystyle 0,$
(42) $\displaystyle\frac{u_{x1}}{u_{0}},\ \frac{u_{x2}}{u_{0}},\
\frac{v_{A+}\hat{b}_{x1}}{u_{0}},\ \frac{v_{A+}\hat{b}_{x2}}{u_{0}}$
$\displaystyle=$ $\displaystyle O\quantity(\frac{k_{||+}}{k_{x}}),$ (43)
$\displaystyle\frac{u_{\perp 1}}{u_{0}},\ \frac{u_{\perp 2}}{u_{0}},\
\frac{v_{A+}\hat{b}_{\perp 1}}{u_{0}},\ \frac{v_{A+}\hat{b}_{\perp 2}}{u_{0}}$
$\displaystyle=$ $\displaystyle O(1),$ (44)

where the full leading order expressions are given in Appendix A.1. For large
$k_{x}/k_{||+}$, the fast wave coefficients are given by

$\displaystyle\frac{u_{x3}}{u_{0}},\ \frac{u_{\perp 3}}{u_{0}}$
$\displaystyle=$ $\displaystyle O\quantity(\frac{k_{||+}}{k_{x}}),$ (45)
$\displaystyle\frac{v_{A+}\hat{b}_{x3}}{u_{0}},\ \frac{v_{A+}\hat{b}_{\perp
3}}{u_{0}},\ \frac{v_{A+}\hat{b}_{||3}}{u_{0}}$ $\displaystyle=$
$\displaystyle O(1).$ (46)

This shows that the leading order solution for $u_{x}$, $b_{x}$, $b_{\perp}$
and $b_{||}$ contain evanescent fast wave terms and this gives rise to the
boundary layers seen in, for example, Halberstadt & Goedbloed (1993, 1995);
Arregui et al. (2003).

Now, we give a brief explanation as to why the boundary layers form. Imposing
$\boldsymbol{u}=0$ at $z=0$ forces the normal component of the magnetic field,
$b_{z}$, to be zero. For proof, consider the $z$-component of the induction
equation,

$\partialderivative{b_{z}}{t}=\boldsymbol{\hat{z}}\cdot\curl(\boldsymbol{u}\crossproduct\boldsymbol{B}_{0}).$

The curl in the $z$-direction only depends on derivatives in $x$ and $y$.
Since $\boldsymbol{u}=0=$ constant at $z=0$, the $x$ and $y$ derivatives are
zero. Therefore, if $b_{z}$ is initially zero, it will remain zero for all
time. Therefore, at $z=0$,

$b_{z}=\cos\alpha b_{||}-\sin\alpha b_{\perp}=0,$

rearranging gives

$b_{||}=\tan\alpha\,b_{\perp}.$

Here, $b_{\perp}$ gives the magnetic field component of the Alfvén wave and
$b_{||}$ gives the magnetic pressure component of the fast wave. Hence, for
$\alpha\neq 0$, if there is a large-amplitude Alfvén wave, then there must
also be a large-amplitude fast wave (for $\alpha\neq 0$). If
$v_{A}^{2}(k_{x}^{2}+k_{y}^{2})>\omega^{2}$, this forces the associated
$k_{z}$ to be imaginary, which means the fast wave must be evanescent and so
boundary layers will form.

Note that for $k_{x}\rightarrow\infty$, $k_{z3}\rightarrow ik_{x}$. Therefore,
the time-averaged energy associated with the evanescent fast waves is
proportional to $\exp(-2k_{x}z)$ and the total fast wave energy is
proportional to

$k_{||+}\int_{0}^{\infty}\exp(-2k_{x}z)dz=\frac{k_{||+}}{2k_{x}}.$ (47)

Thus, the evanescent fast waves’ energy tends to 0 as
$k_{x}\rightarrow\infty$, for $u_{0}$ fixed. Hence, the evanescent fast waves
will have a limited effect on the rate of resonant absorption. Since the
energy associated with the boundary layers does not grow as the length scales
in $x$ get shorter, they cannot be absorbing energy. Moreover, the boundary
layers cannot directly transport energy away as they are evanescent.

##  4 Piecewise constant background Alfvén speed

In this section we expand upon the previous section by keeping the Alfvén
speed uniform in $x$ but using a piecewise constant profile in $z$ instead,
i.e. the Alfvén speed is given by

$v_{A}(z)=\begin{cases}v_{A-},&z<0\\\ v_{A+},&z\geq 0.\end{cases}$ (48)

The region $z>0$ models the corona and $z<0$ models the chromosphere,
therefore, $v_{A+}>v_{A-}$. We impose an incident Alfvén wave from the region
$z>0$ and then calculate the unique solution which ensures continuity of
$u_{x}$, $u_{\perp}$, $\partialderivative*{u_{x}}{z}$,
$\partialderivative*{u_{\perp}}{z}$ at $z=0$. By checking if our solutions
agree with the previous section, we can test the validity of imposing line-
tied boundary conditions.

We assume a solution of the form

$f=\sum_{n=1}^{4}f_{n+}\exp[i(k_{x}x+k_{y}y+k_{zn+}z+\omega t)],$ (49)

for $z\geq 0$ and

$f=\sum_{n=1}^{4}f_{n-}\exp[i(k_{x}x+k_{y}y+k_{zn-}z+\omega t)],$ (50)

for $z<0$ where

$\begin{pmatrix}k_{z1\pm}\\\ k_{z2\pm}\\\ k_{z3\pm}\\\ k_{z4\pm}\\\
\end{pmatrix}=\begin{pmatrix}k_{||\pm}/\cos\alpha-k_{y}\tan\alpha\\\
-k_{||\pm}/\cos\alpha-k_{y}\tan\alpha\\\
i\sqrt{k_{x}^{2}+k_{y}^{2}-k_{||\pm}^{2}}\\\
-i\sqrt{k_{x}^{2}+k_{y}^{2}-k_{||\pm}^{2}}\end{pmatrix}.$ (51)

We impose an incident Alfvén wave from $z>0$ by setting $u_{\perp 1+}=u_{0}$,
where $u_{0}$ gives the incident wave amplitude. We impose the amplitude of
the incident fast waves to be zero. Therefore, $f_{4+}=0$. There are no
incident waves from $z<0$, hence, $f_{2-}=f_{3-}=0$. From Equation (21) we
know that

$u_{xn\pm}=\hat{u}_{xn\pm}u_{\perp n\pm},$ (52)

where

$\displaystyle\hat{u}_{xn\pm}$ $\displaystyle=$
$\displaystyle\frac{-ik_{x}\nabla_{\perp
n\pm}}{\mathcal{L}_{n\pm}-k_{x}^{2}},$ (53) $\displaystyle\mathcal{L}_{n\pm}$
$\displaystyle=$ $\displaystyle\nabla_{||n\pm}^{2}+k_{||\pm}^{2},$ (54)
$\displaystyle\nabla_{\perp n\pm}$ $\displaystyle=$ $\displaystyle
i(k_{y}\cos\alpha-k_{zn\pm}\sin\alpha),$ (55) $\displaystyle\nabla_{||n\pm}$
$\displaystyle=$ $\displaystyle i(k_{y}\sin\alpha+k_{zn\pm}\cos\alpha).$ (56)

We require continuity of $u_{x}$, $\partialderivative*{u_{x}}{z}$, $u_{\perp}$
and $\partialderivative*{u_{\perp}}{z}$ at $z=0$. This gives the following
matrix equation

$\begin{pmatrix}\hat{u}_{x1-}&\hat{u}_{x4-}&-\hat{u}_{x2+}&-\hat{u}_{x3+}\\\
k_{z1-}\hat{u}_{x1-}&k_{z4-}\hat{u}_{x4-}&-k_{z2+}\hat{u}_{x2+}&-k_{z3+}\hat{u}_{x3+}\\\
1&1&-1&-1\\\
k_{z1-}&k_{z4-}&-k_{z2+}&-k_{z3+}\end{pmatrix}\begin{pmatrix}u_{\perp 1-}\\\
u_{\perp 4-}\\\ u_{\perp 2+}\\\ u_{\perp
3+}\end{pmatrix}=u_{0}\begin{pmatrix}\hat{u}_{x1+}\\\ k_{z1+}\hat{u}_{x1+}\\\
1\\\ k_{z1+}\end{pmatrix}.$ (57)

From Equations (16), (17) and (18), we know that

$\displaystyle v_{A+}\hat{b}_{xn\pm}$ $\displaystyle=$
$\displaystyle\frac{\nabla_{||n\pm}}{ik_{||+}}u_{xn\pm},$ (58) $\displaystyle
v_{A+}\hat{b}_{\perp n\pm}$ $\displaystyle=$
$\displaystyle\frac{\nabla_{||n\pm}}{ik_{||+}}u_{\perp n\pm},$ (59)
$\displaystyle v_{A+}\hat{b}_{||n\pm}$ $\displaystyle=$
$\displaystyle-\frac{1}{ik_{||+}}\quantity[ik_{x}u_{xn\pm}+\nabla_{\perp
n\pm}u_{\perp n\pm}].$ (60)

Figure 1: This figure shows plots of the real part of $u_{x}$ (top row),
$u_{\perp}$ (middle row) and $\hat{b}_{||}$ (bottom row) at $x=y=t=0$ as a
function of $z$ for $k_{x}=10k_{||+}$ (left column) and $k_{x}=1000k_{||+}$
(right column). The case where the background Alfvén speed is uniform and
line-tied boundary conditions is imposed is shown in blue and the case where
the Alfvén speed is piecewise constant and continuity of $\boldsymbol{u}$ is
imposed is shown in dashed orange. A description of this figure is continued
in the caption of Figure 2. Figure 2: This figure is similar to Figure 1
except it plots the imaginary part of each variable instead. The blue and
orange curves show good agreement in the left-hand column for both figures.
The small differences occur due to the blue curves’ waves undergoing perfect
reflection and the orange waves undergoing imperfect reflection at $z=0$. The
right-hand column shows less agreement between the plots, and this is because
the fast waves are evanescent, whereas in the left-column the fast waves can
propagate for $z<0$. Equations (42)-(46) and (65)-(69) show that the fast wave
terms in the blue solutions are of a higher order than the orange solutions.

This gives the exact solution to the equations. However, the solution is too
cumbersome to present directly. To build our intuition for how the solutions
change with $k_{x}$ we plot the real/imaginary part of $u_{x}$, $u_{\perp}$
and $\hat{b}_{||}$ as a function of $z$ at $x=y=t=0$ in Figure 1/2
respectively. We show the parameters we use in the top-right of each figure.
The blue curve shows the solutions from Section 3 where line-tied boundary
conditions ($\boldsymbol{u}=0$) are imposed at $z=0$. The dashed orange curve
shows the solution with continuity of $\boldsymbol{u}$ and
$\partialderivative*{\boldsymbol{u}}{z}$ imposed at $z=0$ with a piecewise
constant background Alfvén speed. In the left-hand column, $k_{x}=10k_{||+}$,
this means that the fast waves are evanescent for $z>0$ and can propagate for
$z<0$. It shows that the solutions are in good agreement and suggests that
line-tied boundary conditions give a good approximation for the solution where
$z>0$. The small disagreement occurs due to line-tied boundary conditions
causing perfect reflection, whereas, a finite jump in the background Alfvén
speed results in partial reflection. In the right column, $k_{x}=1000k_{||+}$,
this means that the fast waves are evanescent everywhere. The plots show poor
agreement near $z=0$ and good agreement for $z\gg 0$. They show that the model
with line-tied boundary conditions can significantly overestimate the
evanescent fast waves’/boundary layers’ amplitude.

Figure 3: This figure plots the relative difference (see Equation 61)
introduced by using line-tied boundary conditions to estimate the size of the
boundary layers as a function of $k_{x}$. Unless stated otherwise, we use the
parameters shown in the top-left. In the top plot, the different colours
correspond to different values of $\alpha$, in the middle plot they correspond
to different values of $k_{y}$ and in the bottom plot they correspond to
different values of $k_{||-}$. If $k_{x}$ is to the left/right of the vertical
dashed lines, then the fast waves can propagate/are evanescent for $z>0$
respectively. Similarly, if $k_{x}$ is to the left/right of the vertical
dotted lines, the fast waves can propagate/are evanescent for $z<0$
respectively.

We have seen that depending on the value of $k_{x}$, our models can sometimes
agree on the evanescent fast waves’ amplitude and sometimes disagree strongly.
Our goal now is to quantify how well the model with line-tied boundary
conditions approximates the amplitude of the evanescent fast wave/boundary
layer as a function of $k_{x}$, for the parameter space given by Equations
(23)-(26). Figure 3 plots the relative difference, given by,

$\text{relative difference}=\absolutevalue{\frac{u_{\perp 3}-u_{\perp
3+}}{u_{\perp 3+}}},$ (61)

where $u_{\perp 3}$ gives the amplitude of the fast wave component of
$u_{\perp}$ at $z=0$ in the model where line-tied boundary conditions are
imposed and $u_{\perp 3+}$ gives the amplitude in the model where a piecewise
constant background Alfvén speed is used. Equations (33), (39), (40), (41)
show that $u_{xn}$, $b_{xn}$, $b_{\perp n}$, $b_{||n}$ are equal to scalar
multiples of $u_{\perp n}$, similarly, Equations (52), (58), (59), (60) show
that $u_{xn+}$, $b_{xn+}$, $b_{\perp n+}$, $b_{||n+}$ are equal to the same
scalar multiples of $u_{\perp n+}$ respectively. Therefore,

$\absolutevalue{\frac{u_{\perp 3}-u_{\perp 3+}}{u_{\perp
3+}}}=\absolutevalue{\frac{u_{x3}-u_{x3+}}{u_{x3+}}}=...=\absolutevalue{\frac{b_{||3}-b_{||3+}}{b_{||3+}}}.$

We show all the plots as functions of $k_{x}$, and the parameters take values
shown in the top-left (unless stated otherwise). The vertical dashed lines
show where

$k_{x}=\real\quantity(\sqrt{k_{||+}^{2}-k_{y}^{2}})$

and the dotted lines show where

$k_{x}=\real\quantity(\sqrt{k_{||-}^{2}-k_{y}^{2}})=k_{||-}+O\quantity(\frac{k_{||+}}{k_{||-}}),$

where we assume $k_{y}$ satisfies Equation (25). If $k_{x}$ is to the right of
these vertical lines, then the fast waves are evanescent, and if $k_{x}$ is to
the left, then the fast waves can propagate. The plots show that the relative
difference between the models is dependent on whether the fast waves are
evanescent or propagating. If $k_{x}\gg k_{||-}$, then the models show poor
agreement and the model with line-tied boundary conditions overestimates the
size of the boundary layers. On the other hand, if $k_{x}\ll k_{||-}$ then the
models show good agreement.

Since $k_{||-}\gg k_{||+}$ it is not clear if $k_{x}\gg k_{||-}$, even if the
waves undergo resonant absorption. Our goal now is to check if it is possible
for $k_{x}\gg k_{||-}$ in the corona by calculating the steady-state
wavenumber, $k_{x}^{*}$, for driven standing phase-mixed Alfvén waves in a
viscous and resistive domain (note that waves during resonant absorption
undergo a similar phase-mixing process). Heyvaerts & Priest (1983); Priest
(2014) calculate this value as

$\displaystyle k_{x}^{*}$
$\displaystyle=\quantity[\frac{6\omega}{(\eta+\nu)a}]^{1/3}$ (62)
$\displaystyle\approx 1.82\times
10^{-3}\quantity[\frac{\omega}{10^{-2}$\,\mathrm{s}^{-1}$}\frac{10$\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$}{\eta+\nu}\frac{10^{6}$\,\mathrm{m}$}{a}]^{1/3}$\,\mathrm{m}^{-1}$$

for typical coronal values, where $\eta+\nu$ denotes the sum of the magnetic
diffusivity and kinematic viscosity coefficients and $a$ is given by

$a=\frac{v_{A}}{\partialderivative*{v_{A}}{x}}.$ (63)

The Alfvén wavenumber parallel to the background magnetic field in the
chromosphere, $k_{||-}$, is given by

$k_{||-}=10^{-5}\quantity(\frac{\omega}{10^{-2}$\,\mathrm{s}^{-1}$})\quantity(\frac{10^{3}$\,\mathrm{m}\,\mathrm{s}^{-1}$}{v_{A-}})$\,\mathrm{m}^{-1}$.$
(64)

for typical values. Note that our value for $k_{x}^{*}$ represents an upper
bound for $k_{x}$ in the corona. Waves may not be able to phase-mix to such
short-length scales due to them leaking out of the corona (Prokopyszyn & Hood,
2019) or because of the plasma’s thermodynamic response (Cargill et al.,
2016). Comparing Equations (62) and (64), they suggest that $k_{x}^{*}$ can
greatly exceed $k_{||-}$ for waves undergoing resonant absorption in the
corona. However, it takes time for waves to phase-mix to such short length
scales. Therefore, for early times $k_{x}\ll k_{||-}$ is valid and after the
waves have sufficiently phase-mixed $k_{x}\gg k_{||+}$ is valid. Our line-tied
boundary condition solutions from Section 3 provide a good approximation for
the case where $k_{x}\ll k_{||-}$ as imposing line-tied boundary conditions is
equivalent to modelling the chromospehre as infinitely denser than the corona
which causes $k_{||-}\rightarrow\infty$. In Appendix B we check that the
asymptotic expansions from Section 3 are recovered if $k_{||+}\ll k_{x}\ll
k_{||-}$.

In Appendix A.2 we calculate the solution for the case where $k_{x}\gg
k_{||-},k_{||+}$, as this simulates the conditions for waves undergoing
resonant absorption after they have phase-mixed to sufficiently short length
scales. They show that the leading order asymptotic expansions for the Alfvén
wave coefficients are given by

$\displaystyle\hat{b}_{||1-}=\hat{b}_{||1+}=\hat{b}_{||2+}$ $\displaystyle=$
$\displaystyle 0,$ (65) $\displaystyle\frac{u_{x1-}}{u_{0}},\
\frac{u_{x1+}}{u_{0}},\ \frac{u_{x2+}}{u_{0}},\
\frac{v_{A+}\hat{b}_{x1-}}{u_{0}},\ \frac{v_{A+}\hat{b}_{x1+}}{u_{0}},\
\frac{v_{A+}\hat{b}_{x2+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
O\quantity(\frac{k_{||-}}{k_{x}}),$ (66) $\displaystyle\frac{u_{\perp
1-}}{u_{0}},\ \frac{u_{\perp 1+}}{u_{0}},\ \frac{u_{\perp 2+}}{u_{0}},\
\frac{v_{A+}\hat{b}_{\perp 1-}}{u_{0}},\ \frac{v_{A+}\hat{b}_{\perp
1+}}{u_{0}},\ \frac{v_{A+}\hat{b}_{\perp 2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle O(1),$ (67)

where the full leading order expansions are given in Appendix A.2. The leading
order asymptotic expansions for the coefficients of the fast wave are given by

$\displaystyle\frac{u_{x4-}}{u_{0}},\ \frac{u_{x3+}}{u_{0}},\ \frac{u_{\perp
4-}}{u_{0}},\ \frac{u_{\perp 3+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$ (68)
$\displaystyle\frac{v_{A+}\hat{b}_{x4-}}{u_{0}},\
\frac{v_{A+}\hat{b}_{x3+}}{u_{0}},\ \frac{v_{A+}\hat{b}_{\perp 4-}}{u_{0}},\
\frac{v_{A+}\hat{b}_{\perp 3+}}{u_{0}},\ \frac{v_{A+}\hat{b}_{||4-}}{u_{0}},\
\frac{v_{A+}\hat{b}_{||3+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
O\quantity(\frac{k_{||-}}{k_{x}}),$ (69)

Comparing Equations (65)-(69) with Equations (42)-(46), we see that the Alfvén
wave terms retain the same order and the fast wave terms are reduced here by a
factor $k_{x}$. These results suggest that imposing line-tied boundary
conditions can cause the model to significantly overestimate the amplitude of
the evanescent fast waves / boundary layers for $\alpha\neq 0$. Equations
(65)-(69) show that the leading order velocity does not contain any fast-wave
terms. Compare this with Equations (42)-(46), which show that $u_{x}$ does
contain a fast wave component to leading order. Therefore, for a steady-state
solution in a resonant absorption experiment, the fast wave components for
$u_{x}$ and $u_{\perp}$ should be negligible compared with Alfvén wave
components near the singularities, provided line-tied boundary conditions are
not imposed. In the next section we verify this claim.

It is quite counter-intuitive that the length scales parallel to the
chromosphere/corona interface, e.g. $k_{x}$, can have such a large effect on
the validity of line-tied boundary conditions. Our goal now is to give an
intuitive explanation for why this is. Imposing line-tied boundary conditions
is in some sense equivalent to modelling the chromosome as infinitely denser
than the corona, i.e. setting $k_{||-}\rightarrow\infty$. Therefore, imposing
line-tied boundary conditions forces the fast waves in the chromosphere to be
propagating waves. Propagating waves have a sinusoidal profile while
evanescent waves have an exponential profile. This has important implications
for ensuring continuity of velocity and its derivative at the
chromosome/corona interface and explains why the line-tied solution can be so
different to the solution with a piecewise constant density profile at the
interface.

##  5 Resonant absorption model

In this section, we expand on the previous section by allowing the Alfvén
speed to depend on $x$. Therefore, the Alfvén speed is now given by
$v_{A}(x,z)=\hat{v}_{A}(x)v_{A}(0,z)$ where $v_{A}(0,z)$ is given by Equation
(8). This means that resonant absorption can now occur. At the end of the
previous section we suggested that the fast wave component of $u_{x}$ and
$u_{\perp}$ should be negligible near the singularities in resonant absorption
experiments and our goal now is to check this claim is valid.

We also change the boundary conditions in $z$ here. We impose periodic
boundary conditions in $z$ with period $2L_{z}$, where the region $0<z<L_{z}$
corresponds to the corona and the region $-L_{z}<z<0$ corresponds to the
chromospehre, with jumps in the background Alfvén speed at $z=-L_{z}$, $z=0$,
and $z=L_{z}$.

###  5.1 Normal mode equations

We seek normal mode solutions and assume the variables are of the form

$f(\boldsymbol{x},t)=f^{\prime}(x,z)\exp\\{i[k_{\perp}(\cos\alpha\,y-\sin\alpha\,z)+\omega
t]\\},$ (71)

where

$\omega=\omega_{r}+i\omega_{i}.$

Note that

$\nabla_{\perp}f=e^{i[k_{\perp}(\cos\alpha\,y-\sin\alpha\,z)+\omega
t]}\quantity(ik_{\perp}-\sin\alpha\partialderivative{z})f^{\prime},$
$\nabla_{||}f=e^{i[k_{\perp}(\cos\alpha\,y-\sin\alpha\,z)+\omega
t]}\cos\alpha\partialderivative{f^{\prime}}{z}.$

Therefore, Equations (14)-(18) can be simplified to

$\displaystyle i\omega u_{x}^{\prime}$ $\displaystyle=$ $\displaystyle
v_{A}^{2}(x,z)\quantity[\nabla_{||}^{\prime}\hat{b}_{x}^{\prime}-\partialderivative{\hat{b}_{||}^{\prime}}{x}],$
(72) $\displaystyle i\omega u_{\perp}^{\prime}$ $\displaystyle=$
$\displaystyle
v_{A}^{2}(x,z)\quantity[\nabla_{||}^{\prime}\hat{b}_{\perp}^{\prime}-\nabla_{\perp}^{\prime}\hat{b}_{||}^{\prime}],$
(73) $\displaystyle i\omega\hat{b}_{x}^{\prime}$ $\displaystyle=$
$\displaystyle\nabla_{||}^{\prime}u_{x}^{\prime},$ (74) $\displaystyle
i\omega\hat{b}_{\perp}^{\prime}$ $\displaystyle=$
$\displaystyle\nabla_{||}^{\prime}u_{\perp}^{\prime},$ (75) $\displaystyle
i\omega\hat{b}_{||}^{\prime}$ $\displaystyle=$
$\displaystyle-\quantity[\partialderivative{u_{x}^{\prime}}{x}+\nabla_{\perp}^{\prime}u_{\perp}^{\prime}],$
(76)

where

$\displaystyle\nabla_{\perp}^{\prime}$ $\displaystyle=$
$\displaystyle\quantity(ik_{\perp}-\sin\alpha\partialderivative{z}),$ (77)
$\displaystyle\nabla_{||}^{\prime}$ $\displaystyle=$
$\displaystyle\cos\alpha\partialderivative{z}.$ (78)

Eliminating $\hat{b}_{x}^{\prime}$ and $\hat{b}_{\perp}^{\prime}$ gives

$\displaystyle\partialderivative{u_{x}^{\prime}}{x}$ $\displaystyle=$
$\displaystyle-\quantity[i\omega\hat{b}_{||^{\prime}}+\nabla_{\perp}^{\prime}u_{\perp}^{\prime}],$
(79) $\displaystyle\partialderivative{\hat{b}_{||}^{\prime}}{x}$
$\displaystyle=$
$\displaystyle-\frac{i}{\omega}\mathcal{L}^{\prime}u_{x}^{\prime},$ (80)
$\displaystyle\mathcal{L}^{\prime}u_{\perp}^{\prime}$ $\displaystyle=$
$\displaystyle i\omega\nabla_{\perp}^{\prime}\hat{b}_{||}^{\prime},$ (81)

where

$\mathcal{L}^{\prime}(x,z)=\nabla_{||}^{\prime
2}+\frac{\omega^{2}}{v_{A}^{2}(x,z)}.$ (82)

###  5.2 Eigenfunctions and eigenfrequencies

The $z$-domain is given by $-L_{z}\leq z\leq L_{z}$. For mathematical
convenience and to help ensure the solutions to Equations (79)-(81) are unique
we impose periodic boundary conditions in $z$, i.e.

$f^{\prime}(x,z)=f^{\prime}(x,z+L_{z}).$ (83)

These boundary conditions model a loop going from the chromosphere to the
corona then back into the chromosphere and so on. We have a jump in density at
$z=-L_{z}$, $z=0$ and $z=L_{z}$. The region $0<z<L_{z}$ corresponds to the
corona and the region $-L_{z}<z<0$ corresponds to the chromosphere. To help
solve Equations (79)-(81), we assume the solutions are of the form

$f^{\prime}(x,z)=\sum_{n=0}^{\infty}f_{n}^{(1)}(x)\phi_{n}(z)+f_{n}^{(2)}(x)\varphi_{n}(z),$
(84)

where $\phi_{n}$, $\varphi_{n}$ are eigenfunctions, with eigenfrequencies,
$\omega_{n}$, $\varpi_{n}$, respectively. The eigenfunctions satisfy the
following Sturm-Liouville equations,

$\displaystyle\quantity[\nabla_{||}^{\prime
2}+\frac{\omega_{n}^{2}}{v_{A}^{2}(0,z)}]\phi_{n}(z)$ $\displaystyle=$
$\displaystyle 0,$ (85) $\displaystyle\quantity[\nabla_{||}^{\prime
2}+\frac{{\varpi}_{n}^{2}}{v_{A}^{2}(0,z)}]\varphi_{n}(z)$ $\displaystyle=$
$\displaystyle 0,$ (86)

as well as the periodic boundary conditions above. We normalise the
eigenfunctions such that

$\displaystyle\big{\langle}\phi_{n},\phi_{m}\big{\rangle}$ $\displaystyle=$
$\displaystyle\frac{v_{A+}^{2}}{L_{z}}\int_{-L_{z}}^{L_{z}}\frac{\phi_{n}\phi_{m}}{v_{A}^{2}(0,z)}dz=\delta_{nm},$
(87) $\displaystyle\big{\langle}\varphi_{n},\varphi_{m}\big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{v_{A+}^{2}}{L_{z}}\int_{-L_{z}}^{L_{z}}\frac{\varphi_{n}\varphi_{m}}{v_{A}^{2}(0,z)}dz=\delta_{nm},$
(88) $\displaystyle\big{\langle}\phi_{n},\varphi_{m}\big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{v_{A+}^{2}}{L_{z}}\int_{-L_{z}}^{L_{z}}\frac{\phi_{n}\varphi_{m}}{v_{A}^{2}(0,z)}dz=0,$
(89)

where $\delta_{nm}$ is the Kronecker delta function. By Sturm-Liouville
theory, the eigenfunctions form a basis for the infinite dimensional vector
space of twice-differentiable functions satisfying the boundary conditions,
this ensures that an expansion of the form (84) always exists. In Appendix C
we calculate expressions for $\phi_{n}$, $\varphi_{n}$, $\omega_{n}$ and
$\varpi_{n}$.

###  5.3 Analytic solution

In this section we aim to calculate the leading order singular/resonant
solution to Equations (79)-(81) analytically which we will then verify
numerically via a graphic approach in Section 5.4. We calculate the solution
by using the method of Frobenius with a similar approach to that used in
Thompson & Wright (1993); Wright & Thompson (1994); Soler et al. (2013). Note
that we only calculate the leading order singular solution here.

In Appendix D we show how $u_{x}^{\prime}$ and $\hat{b}_{||}^{\prime}$ can be
eliminated from Equations (79)-(81) to give

$\mathcal{L}^{\prime
2}\partialderivative[2]{u_{\perp}^{\prime}}{x}+^{\prime}\mathcal{L}_{x}^{\prime}\mathcal{L}^{\prime}\partialderivative{u_{\perp}^{\prime}}{x}+\mathcal{M}^{\prime}u_{\perp}^{\prime}=0,$
(D5)

where

$\mathcal{M}^{\prime}=\mathcal{L}^{\prime 3}+\mathcal{L}^{\prime
2}\nabla_{\perp}^{\prime
2}+\mathcal{L}_{xx}^{\prime}\mathcal{L}^{\prime}-\mathcal{L}_{x}^{\prime 2}.$
(90)

for $z\neq-L_{z}$, 0, $L_{z}$, where $\mathcal{L}^{\prime}_{x}(x,z)$ and
$a(x)$ are given by Equations (D3) and (D4). From the literature (e.g.
Thompson & Wright 1993; Wright & Allan 1996), we know that resonant absorption
can generate singularities in $x$. Equation (D5) shows that the coefficient of
the leading order $x$-derivative goes to zero if
$\mathcal{L}^{\prime}(x,z)=0$. Let

$x_{res,m}=x_{r,m}+ix_{i,m},$

be defined as a location which satisfies

$\mathcal{L}^{\prime}(x_{res,m},z)\phi_{m}(z)=0.$ (91)

We postulate that singularities can form at locations, $x_{res,m}$, which are
locations where a resonance can occur.

####  5.3.1 Derive equation for $u_{\perp m}^{(1)}(x)$

Our goal now is to convert this PDE into an ODE for $u_{\perp m}^{(1)}(x)$
using the expansion given by Equation (84). Note that

$\displaystyle\mathcal{L}^{\prime}(x,z)\phi_{n}(z)$ $\displaystyle=$
$\displaystyle[\omega^{2}/\hat{v}_{A}^{2}(x)-\omega_{n}^{2}]\frac{\phi_{n}(z)}{v_{A}^{2}(0,z)},$
(92) $\displaystyle\mathcal{L}^{\prime}(x,z)\varphi_{n}(z)$ $\displaystyle=$
$\displaystyle[\omega^{2}/\hat{v}_{A}^{2}(x)-\varpi_{n}^{2}]\frac{\varphi_{n}(z)}{v_{A}^{2}(0,z)}.$
(93)

Multiplying Equation (D5) through by $\kappa_{m}^{2}(x,z)$ and taking the
inner product with $\phi_{m}(z)$ gives

$(x-x_{res,m})^{2}\derivative[2]{u_{\perp
m}^{(1)}}{x}+(x-x_{res,m})p_{m}(x)\derivative{u_{\perp
m}^{(1)}}{x}+q_{m}(x)u_{\perp m}^{(1)}(x)=f_{m}(x),$ (94)

where

$\displaystyle\kappa_{m}(x,z)$ $\displaystyle=$
$\displaystyle(x-x_{res,m})\frac{v_{A}^{2}(0,z)}{\omega^{2}/\hat{v}_{A}^{2}(x)-\omega_{n}^{2}},$
(95) $\displaystyle p_{m}(x)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{x}^{\prime}(x,z)\kappa_{m}(x,z),$ (96)
$\displaystyle q_{m}(x)$ $\displaystyle=$
$\displaystyle\left\langle\mathcal{M}^{\prime}\phi_{m}(z),\kappa_{m}^{2}(x,z)\phi_{m}(z)\right\rangle,$
(97) $\displaystyle f_{m}(x)$ $\displaystyle=$
$\displaystyle-\left\langle\mathcal{M}^{\prime}u_{\perp
m}^{(2)}\varphi_{m},\kappa_{m}^{2}\phi_{m}\right\rangle-\sum_{\begin{subarray}{c}n=0\\\
n\neq
m\end{subarray}}^{\infty}\left\langle\mathcal{M}^{\prime}\quantity[u_{\perp
n}^{(1)}\phi_{n}+u_{\perp
n}^{(2)}\varphi_{n}],\kappa_{m}^{2}\phi_{m}\right\rangle.$ (98)

####  5.3.2 Complementary function and particular integral

To solve Equation (94) we will calculate the complementary function first and
then the particular integral later. To calculate the complementary function we
solve the homogeneous version of Equation (94) where $f_{m}(x)=0$. This can be
solved using the method of Frobenius via an approach very similar to that used
by Soler et al. (2013) to solve Equation (15) in their paper. By assuming the
solutions are of the form

$u_{\perp
m}^{(1)}(x)=(x-x_{res,m})^{\sigma}\sum_{n=0}^{\infty}a_{n}(x-x_{res,m})^{n},$

we can convert the homogeneous version of Equation (94) into

$\sum_{n=0}^{\infty}a_{n}[\sigma(\sigma-1)+\sigma
p_{m}(x)+q_{m}(x)](x-x_{res})^{n+\sigma}.$

By considering the coefficient of the leading order term, we calculate the
indicial equation to be given by

$I(\sigma)=\sigma(\sigma-1)+p_{m}(x_{res,m})\sigma+q_{m}(x_{res,m}).$

For a non-trivial complementary function we require $I(\sigma)=0$. Since
$p_{m}(x_{res,m})=-q_{m}(x_{res,m})=1$ we know that $\sigma=\pm 1$. Note that
the allowed values of $\sigma$ differ by an integer. Therefore, the general
complementary solution for $u_{\perp m}^{(1)}(x)$ is given by a linear
combination of

$u_{\perp m1}^{(1)}(x)=\sum_{n=1}^{\infty}a_{n}(x-x_{res,m})^{n},$ (99)

and

$u_{\perp m2}^{(1)}(x)=Cu_{\perp
m1}^{(1)}(x)\ln(x-x_{res,m})+\sum_{n=-1}^{\infty}b_{n}(x-x_{res,m})^{n},$
(100)

where $a_{n}$, $b_{n}$, $C$ are constants.

If we know the power series in $x$ for $f_{m}(x)$ then we can calculate the
particular integral to Equation (94) by using the method of undetermined
coefficients. Since $f_{m}(x)$ is proportional to $u_{\perp n}^{(1)}(x)$,
$u_{\perp n}^{(2)}(x)$, we know that the leading order term can be at most
order $(x-x_{res,m})^{-1}$.

####  5.3.3 Singular solution

Our goal now is to calculate a leading order singular solution for
$u_{x}^{\prime}$, $u_{\perp}^{\prime}$ and $\hat{b}_{||}^{\prime}$ which we
can verify numerically. Assume that $x_{i,m}\ll R$ where $R$ is the radius of
convergence for the power series solution of $u_{\perp}^{\prime}(x,z)$ in $x$.
We choose the boundary conditions to ensure that to leading order $u_{\perp
n}^{(1)}=u_{\perp n}^{(2)}=u_{\perp m}^{(2)}=0$ for all $n\neq m$. This
ensures that the leading order solution for $u_{\perp m}(x)$ is given by the
leading term of the complementary solution. Let $u_{\perp}^{\prime}$ be given
by

$u_{\perp}^{\prime}(x,z)=u_{0}\quantity[\frac{\phi_{m}(z)}{X_{m}}+O(X_{m}\ln
X_{m},X_{m})],$ (101)

where we use the boundary conditions to ensure terms of order $X_{m}^{0}$
equal zero and $X_{m}$ is given by

$X_{m}=\frac{x-x_{res,m}}{x_{i,m}}.$ (102)

From Equation (D2) we know that

$u_{x}^{\prime}(x,z)=-u_{0}\Big{[}x_{i,m}\ln(X_{m})\nabla_{\perp}^{\prime}\phi_{,}(z)+O(X_{m},X_{m}^{2}\ln
X_{m})\Big{]},$ (103)

where we impose the boundary condition that terms of order $X_{m}^{0}$ equal
zero.

From Equation (81) we know that to leading order $\hat{b}_{||}^{\prime}$
satisfies

$\displaystyle\nabla_{\perp}^{\prime}\hat{b}_{||}^{\prime}(z)$
$\displaystyle=u_{0}\frac{\mathcal{L}_{x}^{\prime}(x_{r,m},z)x_{i,m}}{i\omega_{r}}\phi_{m}(z)+\frac{O(X_{m}\ln
X_{m},X_{m})}{L_{z}}$ (104)
$\displaystyle=2iu_{0}\frac{\omega_{r}x_{i,m}}{a(x_{r,m})v_{A}^{2}(x_{r,m},z)}\phi_{m}(z)+\frac{O(X_{m}\ln
X_{m},X_{m})}{L_{z}},$

assuming $\omega_{i}\ll\omega_{r}$. We need to solve Equation (104). For
$\sin\alpha=0$, $\hat{b}_{||}^{\prime}$ is given by

$\hat{b}_{||}^{\prime}(x,z)=\frac{2u_{0}\omega_{r}x_{i,m}}{a(x_{r,m})v_{A}^{2}(x_{r,m},z)k_{\perp}}\phi_{m}(z)+O(X_{m}\ln
X_{m},X_{m}).$ (105)

For $\sin\alpha\neq 0$, we can calculate $b_{||}^{\prime}(x,z)$ by using an
integrating factor. However, if $k_{\perp}L_{z}/\sin\alpha=j\pi$ for
$j\in\mathds{Z}$ then $\hat{b}_{||}^{\prime}(x,z)$ is not uniquely defined by
the boundary conditions in $z$. To see this, assume $\hat{b}_{||}(x,z)$ is of
the form

$\hat{b}_{||}^{\prime}(x,z)=\sum_{n=-\infty}^{\infty}\hat{b}_{||n}(x)\exp(in\frac{\pi}{L_{z}}z),$

This ensures that $\hat{b}_{||}(x,z)$ is periodic in $z$ with period $2L_{z}$.
Substituting this into Equation (104), multiplying by $\exp(ij\pi
z/L_{z})/(2L_{z})$ and integrating from $-L_{z}$ to $L_{z}$ gives

$i\quantity(k_{\perp}-\sin\alpha\,j\frac{\pi}{L_{z}})\hat{b}_{||j}=\int_{-L_{z}}^{L_{z}}\frac{\hat{b}_{||}^{\prime}(x,z)}{2L_{z}}\exp(ij\frac{\pi}{L_{z}}z)dz.$

Hence, if $k_{\perp}L_{z}/\sin\alpha=j\pi$ for $j\in\mathds{Z}$ then
$\hat{b}_{||j}^{\prime}(x)$ and consequently $\hat{b}_{||}^{\prime}(x,z)$ is
not uniquely defined by the boundary conditions in $z$.

####  5.3.4 Poynting flux

Finally, we aim to calculate an expression for the Poynting flux because the
jump in Poynting flux across the resonant field line describes the rate at
which it absorbs energy. The Poynting flux, $\boldsymbol{S}$, is given by

$\boldsymbol{S}=\frac{1}{\mu}\big{[}(\boldsymbol{B}_{0}\dotproduct\real(\boldsymbol{b}))\real(\boldsymbol{u})-(\real(\boldsymbol{u})\dotproduct\real(\boldsymbol{b}))\boldsymbol{B}_{0}\big{]},$
(106)

where $\real$ denotes the real part operator. Hence, the Poynting flux
directed in the $x$-direction,
$S_{x}=\boldsymbol{S}\dotproduct\boldsymbol{\hat{x}}$ is given by

$S_{x}=\frac{B_{0}^{2}}{\mu}\real\quantity(u_{x})\real\quantity(\hat{b}_{||}).$
(107)

The spatial average in $y$ is given by

$\langle
S_{x}\rangle=\frac{B_{0}^{2}}{4\mu}\quantity(u_{x}^{\prime}\hat{b}_{||}^{\prime*}+u_{x}^{\prime*}\hat{b}_{||}^{\prime}),$
(108)

where an asterisk, $*$, is used to denote the complex conjugate. Substituting
Equation (103) gives

$\langle
S_{x}\rangle=-\frac{B_{0}^{2}}{4\mu}u_{0}\Big{[}x_{i,m}\Big{(}\ln(X_{m})\nabla_{\perp}^{\prime}\phi_{m}\hat{b}_{||}^{\prime*}+\ln(X_{m}^{*})\nabla_{\perp}^{\prime*}\phi_{m}\hat{b}_{||}^{\prime}\Big{)}+O(X_{m},X_{m}^{2}\ln
X_{m})\Big{]}.$ (109)

Note that

$\lim_{x\rightarrow\infty}\ln(X_{m})-\lim_{x\rightarrow-\infty}\ln(X_{m})=-i\pi\text{sign}(x_{i,m}),$
$\lim_{x\rightarrow\infty}\ln(X_{m}^{*})-\lim_{x\rightarrow-\infty}\ln(X_{m}^{*})=i\pi\text{sign}(x_{i,m}),$

where $\text{sign}(x_{i,m})=x_{i,m}/\absolutevalue{x_{i,m}}$. We define the
jump in Poynting flux, $\langle\Delta S_{x}\rangle$, across $x=x_{r,m}$ as

$\langle\Delta S_{x}\rangle=i\pi
u_{0}x_{i,m}\frac{B_{0}^{2}}{4\mu}\,\text{sign}(x_{i,m})\Big{(}\nabla_{\perp}^{\prime}\phi_{n}\hat{b}_{||}^{*}-\nabla_{\perp}^{\prime*}\phi_{n}\hat{b}_{||}\Big{)}.$
(110)

###  5.4 Numerical solution

In this section we aim to verify the Equations we derived in Section 5.3
numerically via a graphical approach. To solve the system numerically we use
the expansion given by (84) to convert Equations (79)-(81) from a set of PDEs
to a set of ODEs. We then solve the ODEs as an initial value problem in $x$
using solve_ivp in Virtanen et al. (2020).

For $z\neq 0$, $-L_{z}$, $L_{z}$, taking $\nabla_{\perp}$ of Equation (81)
gives

$\mathcal{L}^{\prime}\nabla_{\perp}^{\prime}u_{\perp}^{\prime}=i\omega\nabla_{\perp}^{\prime
2}\hat{b}_{||}^{\prime}.$ (111)

Let

$\nabla_{\perp}^{\prime}u_{\perp}^{\prime}=\Delta_{\perp
0}^{(1)}(x)\phi_{0}(z)+\sum_{n=1}^{\infty}\Delta_{\perp
n}^{(1)}(x)\phi_{n}(z)+\Delta_{\perp n}^{(2)}(x)\varphi_{n}(z).$ (112)

Using Equations (84), (112), (92) and (93), Equations (79), (80) and (111) can
be written as

$\displaystyle\partialderivative{u_{x}^{\prime}}{x}$ $\displaystyle=$
$\displaystyle-\sum_{n=0}^{\infty}i\omega[b_{||n}^{(1)}\phi_{n}(z)+b_{||n}^{(2)}\varphi_{n}]+\Delta_{\perp
n}^{(1)}\phi_{n}+\Delta_{\perp n}^{(2)}\varphi_{n},$ (113)
$\displaystyle\partialderivative{b_{||}^{\prime}}{x}$ $\displaystyle=$
$\displaystyle-\frac{i}{\omega}\sum_{n=0}^{\infty}u_{xn}^{(1)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\omega_{n}^{2}]\frac{\phi_{n}(z)}{v_{A}^{2}(0,z)}+u_{xn}^{(2)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\varpi_{n}^{2}]\frac{\varphi_{n}(z)}{v_{A}^{2}(0,z)},$
(114) $\displaystyle i\omega\nabla_{\perp}^{\prime 2}b_{||}^{\prime}$
$\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\Delta_{\perp
n}^{(1)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\omega_{n}^{2}]\frac{\phi_{n}(z)}{v_{A}^{2}(0,z)}+\Delta_{\perp
n}^{(2)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\varpi_{n}^{2}]\frac{\varphi_{n}(z)}{v_{A}^{2}(0,z)}.$
(115)

Taking the inner product of the above equations with $\phi_{m}$ and
$\varphi_{m}$ gives

$\displaystyle\derivative{u_{xm}^{(1)}}{x}$ $\displaystyle=$
$\displaystyle-[i\omega b_{||m}^{(1)}+\Delta_{\perp m}^{(1)}],$ (116)
$\displaystyle\derivative{u_{xm}^{(2)}}{x}$ $\displaystyle=$
$\displaystyle-[i\omega b_{||m}^{(2)}+\Delta_{\perp m}^{(2)}],$ (117)
$\displaystyle\derivative{b_{||m}^{(1)}}{x}$ $\displaystyle=$
$\displaystyle-\frac{i}{\omega}\sum_{n=0}^{\infty}u_{xn}^{(1)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\omega_{n}^{2}]I_{1},$
(118) $\displaystyle\derivative{b_{||m}^{(2)}}{x}$ $\displaystyle=$
$\displaystyle-\frac{i}{\omega}\sum_{n=0}^{\infty}u_{xn}^{(2)}[\omega^{2}/\hat{v}_{A}^{2}(x)-\varpi_{n}^{2}]I_{2},$
(119) $\displaystyle\Delta_{\perp m}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{i\omega}{\omega^{2}/\hat{v}_{A}^{2}-\omega_{m}^{2}}\sum_{n=0}^{\infty}b_{||n}^{(1)}I_{3}+b_{||n}^{(2)}I_{4},$
(120) $\displaystyle\Delta_{\perp m}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{i\omega}{\omega^{2}/\hat{v}_{A}^{2}-\varpi_{m}^{2}}\sum_{n=0}^{\infty}b_{||n}^{(1)}I_{5}+b_{||n}^{(2)}I_{6},$
(121)

where

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left\langle\frac{\phi_{n}(z)}{v_{A}^{2}(0,z)},\phi_{m}(z)\right\rangle,$
(122) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\left\langle\frac{\varphi_{n}(z)}{v_{A}^{2}(0,z)},\varphi_{m}(z)\right\rangle,$
(123) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\big{\langle}v_{A}^{2}(0,z)\nabla_{\perp}^{\prime
2}\phi_{n}(z),\phi_{m}(z)\big{\rangle},$ (124) $\displaystyle I_{4}$
$\displaystyle=$
$\displaystyle\big{\langle}v_{A}^{2}(0,z)\nabla_{\perp}^{\prime
2}\varphi_{n}(z),\phi_{m}(z)\big{\rangle},$ (125) $\displaystyle I_{5}$
$\displaystyle=$
$\displaystyle\big{\langle}v_{A}^{2}(0,z)\nabla_{\perp}^{\prime
2}\phi_{n}(z),\varphi_{m}(z)\big{\rangle},$ (126) $\displaystyle I_{6}$
$\displaystyle=$
$\displaystyle\big{\langle}v_{A}^{2}(0,z)\nabla_{\perp}^{\prime
2}\varphi_{n}(z),\varphi_{m}(z)\big{\rangle}.$ (127)

This is a set of ODEs which we can solve as an initial value problem in $x$
using solve_ivp in Virtanen et al. (2020). However, for numerical reasons,
only a finite number of terms can be calculated. Therefore, we truncate the
summations after a finite number of terms/harmonics, $N_{h}$.

The Alfvén speed $x$-dependence is given by

$\hat{v}_{A}(x)=1+\frac{x}{a_{0}},$ (128)

where $a_{0}=a(0)$ is a constant which controls the Alfvén speed length scale
in $x$. From Equation (91), we can calculate $x_{res,m}$ locations using

$\hat{v}_{A}(x_{res,m})=\frac{\omega}{\omega_{m}}.$ (129)

This implies that

$\displaystyle x_{r,m}$ $\displaystyle=$ $\displaystyle
a_{0}\quantity(\frac{\omega_{r}}{\omega_{m}}-1),$ (130) $\displaystyle
x_{i,m}$ $\displaystyle=$ $\displaystyle a_{0}\frac{\omega_{i}}{\omega_{m}}.$
(131)

We choose $v_{A-}=v_{A+}/21$ somewhat arbitrarily. A larger factor, such as
$10^{3}$, instead of 21 would be more typical in the solar atmosphere.
However, a factor of 21 ensures the wavelength in the $z<0$ region is long
enough for the shape of the solution to be clearly visible. 21 is also an odd
integer which is useful because we have an explicit analytic formula (instead
of the implicit formulas given by Equations C5 and C6) for some of the
eigenfrequencies including $\omega_{11}$. We set

$\omega_{r}=\omega_{11}.$ (132)

Hence, $x_{r,11}=0$ and $x_{i,11}=a_{0}\omega_{i}/\omega_{11}$.

The boundary conditions in $x$ are

$\displaystyle u_{x}^{\prime}(x_{min},z)$ $\displaystyle=$ $\displaystyle-
u_{0}x_{i,11}\ln(X_{11})\nabla_{\perp}^{\prime}\phi_{11}(z),$ (133)
$\displaystyle\nabla_{\perp}^{\prime}\hat{b}_{||}^{\prime}(x_{min},z)$
$\displaystyle=$ $\displaystyle
2iu_{0}\frac{\omega_{r}x_{i,11}}{a_{0}v_{A}^{2}(0,z)}\phi_{11}(z),$ (134)

where $k_{\perp}L_{z}/\sin\alpha\neq n\pi$ for $n\in\mathds{Z}$ to ensure
$\hat{b}_{||}^{\prime}(x_{min},z)$ is uniquely determined by Equation (134).
To calculate $u_{xn}^{(1)}(x_{min})$, $u_{xn}^{(2)}(x_{min})$,
$b_{||n}^{(1)}(x_{min})$, $b_{||n}^{(2)}(x_{min})$, for $n=0$, 1, 2, …,
$N_{h}$ we take the inner product of $u_{x}^{\prime}(x_{min},z)$ and
$\hat{b}_{||}^{\prime}(x_{min},z)$ with $\phi_{n}$ and $\varphi_{n}$.

Figure 4: This figure shows plots of the real and imaginary parts of
$u_{x}^{\prime}$ (top row), $u_{\perp}^{\prime}$ (middle row) and $\langle
S_{x}\rangle$ (bottom left) as a function of $z$ at $x=0$. The blue curves
were calculated numerically and the orange curves plot the analytic solutions
using Equations (101), (103) and (109). The velocity curves are normalised by,
$u_{0}$, which gives the $u_{\perp}^{\prime}$ amplitude at $x=0$. The Poynting
flux is normalised by $\absolutevalue{\langle\Delta S_{x}\rangle}$ at
$z=l_{z}$, see Equation (110). Figure 5: This figure is similar to Figure 4
except it plots the variables as function of $x$ at $z=l_{z}$. The bottom
right plot shows the relative error between the numerical and analytic
solutions for $u_{x}$ as a function of $\omega_{i}/\omega_{r}$. The solutions
are smooth because we use a complex frequency, $\omega$, which moves the
singularity/resonant location, $x_{res,11}$, off the real-axis and into the
complex plane.

To check that the numerical solutions agree with the analytic solutions given
by Equations (101), (103) and (109) and to help visualise the solutions, we
plot the solutions in Figure 4 and 5. Since $v_{A-}=v_{A+}/21$, the wavelength
of the waves in the chromosphere ($z<0$) are shorter by a factor 21 compared
with the waves in the corona ($z>0$). They show that the analytic and
numerical solutions agree provided $\omega_{i}/\omega_{r}$ is small enough.
The bottom right of the Figure 5 shows the maximum relative error between the
numerical and analytic solutions for $u_{x}$, which is given by

$\max_{\begin{subarray}{c}-8|x_{i,11}|\leq x\leq 8|x_{i,11}|\\\ -L_{z}\leq
z\leq
L_{z}\end{subarray}}\absolutevalue{\frac{u_{x,num}^{\prime}-u_{x,ana}^{\prime}}{u_{x,num}^{\prime}}},$

where $u_{x,num}$ denotes the $u_{x}$ solution which was calculated
numerically and $u_{x,ana}$ was calculated analytically using Equation (103).
The bottom right of the Figure 5 shows that decreasing $\omega_{i}/\omega_{r}$
causes the error to decrease. The plot appears to be converging to a non-zero
value, and this is because the numerical solution only uses a finite number of
harmonics $N_{h}$. Increasing $N_{h}$ acts to reduce the error further. Figure
4 shows that although the amplitude of $u_{\perp}$ may decrease significantly
in the chromosphere, this is not necessarily true for $u_{x}$. This was
predicted by Equation (A18), which showed that even if $v_{A-}\rightarrow 0$,
$u_{x}$ in the chromosphere will not go to zero provided the fast waves are
evanescent in the chromosphere. This suggests that imposing line-tied
($\boldsymbol{u}=0$) boundary conditions at the edge of the corona will lead
to the generation of unphysical structures. The usual justification for
imposing line-tied boundary conditions is that the chromosphere is
significantly denser than the corona and so it acts like a solid wall. Here,
$v_{A+}/v_{A-}=21$ and yet $u_{x}$ is showing no signs of going to zero in the
chromosphere. Although $u_{x}$ does not go to zero, the kinetic energy and
magnetic energy is dominated by $u_{\perp}$ and $b_{\perp}$. Therefore, the
absolute value of the velocity does decrease significantly in the
chromosphere.

##  6 Summary and conclusions

The first results presented in this paper showed how Alfvén waves propagating
from the corona to the chromosphere mode convert at the transition region to
form fast waves (see Section 3). If

$k_{x}\geq\real\quantity(\sqrt{k_{||+}^{2}-k_{y}^{2}}),$

then the fast waves are evanescent in the corona and will form boundary layers
at the transition region (see Figures 1 and 2). This phenomenon has been shown
in for example, Halberstadt & Goedbloed (1993, 1995); Arregui et al. (2003).
To simulate the conditions near a singularity in a resonant absorption
experiment, we calculated asymptotic expansions for
$k_{x}/k_{||+}\rightarrow\infty$ (see Equations A2-A16). They show that the
boundary layers do not form if the magnetic field is normal to the transition
region, i.e. if $\alpha=0$.

After that, we tested the validity of line-tied boundary conditions by
checking if these boundary layers still form if a piecewise constant
background Alfvén speed with continuity of $\boldsymbol{u}$ and
$\derivative*{\boldsymbol{u}}{z}$ is imposed instead (see Section 4). We
believe we are the first authors to test line-tied boundary conditions in this
way. The results (see Figures 1-3) suggest that line-tied boundary conditions
provide a good approximation provided the fast waves can propagate in the
chromosphere, i.e.

$k_{x}\leq\real\quantity(\sqrt{k_{||-}^{2}-k_{y}^{2}}).$

However, if the fast waves are evanescent in the chromosphere, i.e.

$k_{x}\geq\real\quantity(\sqrt{k_{||-}^{2}-k_{y}^{2}}).$

then line-tied boundary conditions will overestimate the size of the boundary
layers by a factor of the order $k_{x}/k_{||-}$. If $k_{x}\gg k_{||-}$ then
Equations (A18)-(A42) approximate the solution, however, if $k_{||+}\ll
k_{x}\ll k_{||-}$ then the line-tied solutions (Equations A2-A16) provide a
better approximation. We estimated that if the Alfvén waves are able to phase-
mix to the shortest length scales allowed by the resistivity and viscosity in
the corona then the $k_{x}\gg k_{||-}$ is usually most appropriate limit.
However, during the early stages, before the waves have phase-mixed, $k_{x}\ll
k_{||-}$ is the most valid limit.

The results from Section 4 suggest that the fast wave component of the
velocity should go to zero near singular locations in a resonant absorption
experiment since $k_{x}\rightarrow\infty$ approaching the singularity. In the
final section (Section 5), we tested this by calculating the normal mode
resonant absorption solution in a domain where the background Alfvén speed is
a function of $x$ and piecewise constant in $z$. We used periodic boundary
conditions in $z$ for convenience and because this simulates a loop which goes
from the chromosphere to the corona then back to the chromosphere and so on.
In Section 5.3 we calculated the leading order singular solutions near a
resonant location analytically. In Section 5.4 we verified the analytic
solutions by using eigenfunctions to convert the PDEs into a set of ODEs which
were solved numerically. In Figures 4 and 5 we plot the analytic and numerical
solutions together and show that they agree. We verified that to leading
order, $u_{x}$ and $u_{\perp}$ do not contain any boundary layers near the
singularity. This confirms that imposing line-tied boundary conditions can
cause the model to significantly overestimate the boundary layers’ amplitude.
However, in Section 3 we showed as $k_{x}\rightarrow\infty$ the amplitude of
the boundary layers does not increase. This suggests that the boundary layers
have a limited effect on the rate at which the resonant absorption occurs.
Moreover, since the boundary layers are evanescent, they cannot directly
transport energy away.

In conclusion, Alfvén waves will (in general) mode convert to form fast waves
at the transition region. The fast waves can be evanescent or can propagate
depending on the value of
$\boldsymbol{k}\dotproduct\boldsymbol{k}-\omega^{2}/v_{A}^{2}$. We have
demonstrated that line-tied boundary conditions can be more problematic than
many authors realise. If $k_{x}$ grows large enough such that it causes the
fast waves in the chromosome to be evanescent, then the line-tied model will
fail to reduce the boundary layer’s size accurately. We believe that few
authors are aware of this because it is quite counter-intuitive that $k_{x}$
(which gives the length scales in a direction tangential to the
chromosphere/corona interface) can be important in determining the validity of
line-tied boundary conditions. Increasing $k_{x}$ can result in the line-tied
model greatly overestimating the boundary layer’s amplitude by approximately a
factor $k_{x}/k_{||-}$. With that in mind, we believe authors should continue
to use line-tied boundary conditions for their simplicity and ability to
reduce computation time, provided that their flaws are understood. For
example, we have shown that line-tied boundary conditions provide a good
approximation (with an error of the order $v_{A-}/v_{A+}$) if the fast waves
can propagate in the chromosphere. If the fast waves are evanescent in the
chromosphere, then line-tied boundary conditions are less useful. However,
sometimes this won’t be important. For example, when studying energetics, as
the boundary layers are evanescent and cannot transport energy away. Moreover,
we showed in Section 3 that their energy does not grow as
$k_{x}\rightarrow\infty$.

We thank the referees for their constructive comments. This project has
received funding from the Science and Technology Facilities Council (U.K.)
through the consolidated grant ST/N000609/1.

##  Appendix A Asymptotic expansions

###  A.1 Uniform background Alfvén speed

In this section our goal is to present asymptotic expansions for the
coefficients, $u_{xn}$, $u_{\perp n}$, $b_{xn}$, $b_{\perp n}$, $b_{||n}$, in
the model where line-tied boundary conditions are imposed (see Section 3). To
simplify the notation, let

$\hat{k}_{y}=\frac{k_{y}}{k_{||+}}.$ (A1)

To assist with the algebra we use a symbolic computing environment called
Maple. In the supplementary material we provide the maple code as well as an
exported pdf to enable users without maple installed to view it. By using
Equations (34)-(41) as well the maple code,

$\text{maple\\_code\\_appendix\\_A1.mw},$

which can be accessed in the attached .tar.gz file (with a corresponding pdf
file to enable the user read the code without the need for a Maple
installation), We calculate that the leading order asymptotic expansions for
the $u_{x}$ coefficients are given by

$\displaystyle\frac{u_{x1}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}),$
(A2) $\displaystyle\frac{u_{x2}}{u_{0}}$ $\displaystyle=$
$\displaystyle\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}),$
(A3) $\displaystyle\frac{u_{x3}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\tan\alpha\,\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}).$
(A4)

For large $k_{x}$, the $u_{\perp}$ coefficients are given by

$\displaystyle\frac{u_{\perp 1}}{u_{0}}$ $\displaystyle=$ $\displaystyle 1,$
(A5) $\displaystyle\frac{u_{\perp 2}}{u_{0}}$ $\displaystyle=$
$\displaystyle-1+O\quantity(\frac{k_{||+}}{k_{x}}),$ (A6)
$\displaystyle\frac{u_{\perp 3}}{u_{0}}$ $\displaystyle=$ $\displaystyle
2i\frac{\sin^{2}\alpha}{\cos\alpha}\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}).$
(A7)

To leading order, the $\hat{b}_{x}$ coefficients are given by

$\displaystyle\frac{v_{A+}\hat{b}_{x1}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}),$
(A8) $\displaystyle\frac{v_{A+}\hat{b}_{x2}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\frac{k_{||+}}{k_{x}}+O\quantity(\frac{k_{||+}^{2}}{k_{x}^{2}}),$
(A9) $\displaystyle\frac{v_{A+}\hat{b}_{x3}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\sin\alpha+O\quantity(\frac{k_{||+}}{k_{x}}).$ (A10)

Next, the leading order asymptotic expansions for the $\hat{b}_{\perp}$
coefficients are given by

$\displaystyle\frac{v_{A+}\hat{b}_{\perp 1}}{u_{0}}$ $\displaystyle=$
$\displaystyle 1,$ (A11) $\displaystyle\frac{v_{A+}\hat{b}_{\perp 2}}{u_{0}}$
$\displaystyle=$ $\displaystyle 1+O\quantity(\frac{k_{||+}}{k_{x}}),$ (A12)
$\displaystyle\frac{v_{A+}\hat{b}_{\perp 3}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\sin^{2}\alpha+O\quantity(\frac{k_{||+}}{k_{x}}).$ (A13)

Finally, the $\hat{b}_{||}$ coefficients are given by

$\displaystyle\frac{v_{A+}\hat{b}_{||1}}{u_{0}}$ $\displaystyle=$
$\displaystyle 0,$ (A14) $\displaystyle\frac{v_{A+}\hat{b}_{||2}}{u_{0}}$
$\displaystyle=$ $\displaystyle 0,$ (A15)
$\displaystyle\frac{v_{A+}\hat{b}_{||3}}{u_{0}}$ $\displaystyle=$
$\displaystyle\sin(2\alpha)+O\quantity(\frac{k_{||+}}{k_{x}}).$ (A16)

###  A.2 Piecewise constant background Alfvén speed

In this section our goal is to present asymptotic expansions for the
coefficients, $u_{xn\pm}$, $u_{\perp n\pm}$, $b_{xn\pm}$, $b_{\perp n\pm}$,
$b_{||n\pm}$, in the model where a pieceiwse constant background Alfvén speed
is used (see Section 4). To simplify the notation, let

$r=\frac{k_{||+}}{k_{||-}}.$ (A17)

Assuming $k_{x}$ is large, and using Equations (53)-(60) as well as the maple
code,

$\text{maple\\_code\\_appendix\\_A2.mw},$

which can be be accessed in the attached .tar.gz file, we calculate the
leading order asymptotic expansions for the $u_{x}$ coefficients as

$\displaystyle\frac{u_{x1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2r\frac{r\hat{k}_{y}-\sin\alpha}{(1+r)\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A18) $\displaystyle\frac{u_{x4-}}{u_{0}}$ $\displaystyle=$ $\displaystyle-
ir(1-r)\frac{\sin\alpha}{\cos^{2}\alpha}\frac{k_{||-}^{2}}{k_{x}^{2}}+O\quantity(\frac{k_{||-}^{3}}{k_{x}^{3}}),$
(A19) $\displaystyle\frac{u_{x1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-r\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A20) $\displaystyle\frac{u_{x2+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
r\frac{1-r}{1+r}\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A21) $\displaystyle\frac{u_{x3+}}{u_{0}}$ $\displaystyle=$ $\displaystyle-
ir(1-r)\frac{\sin\alpha}{\cos^{2}\alpha}\frac{k_{||-}^{2}}{k_{x}^{2}}+O\quantity(\frac{k_{||-}^{3}}{k_{x}^{3}}).$
(A22)

The leading order asymptotic expansions for the $u_{\perp}$ coefficients are
given by

$\displaystyle\frac{u_{\perp 1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle\frac{2r}{r+1}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$ (A23)
$\displaystyle\frac{u_{\perp 4-}}{u_{0}}$ $\displaystyle=$ $\displaystyle
r(r-1)\tan^{2}\alpha\frac{k_{||-}^{2}}{k_{x}^{2}}+O\quantity(\frac{k_{||-}^{3}}{k_{x}^{3}}),$
(A24) $\displaystyle\frac{u_{\perp 1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle 1,$ (A25) $\displaystyle\frac{u_{\perp 2+}}{u_{0}}$
$\displaystyle=$
$\displaystyle-\frac{1-r}{1+r}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A26) $\displaystyle\frac{u_{\perp 3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-r(1-r)\tan^{2}\alpha\frac{k_{||-}^{2}}{k_{x}^{2}}+O\quantity(\frac{k_{||-}^{3}}{k_{x}^{3}}).$
(A27)

The leading order asymptotic expansions for the $b_{x}$ coefficients are given
by

$\displaystyle\frac{v_{A+}\hat{b}_{x1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\frac{r\hat{k}_{y}-\sin\alpha}{(1+r)\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A28) $\displaystyle\frac{v_{A+}\hat{b}_{x4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-(1-r)\tan\alpha\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}}{k_{x}^{2}}),$
(A29) $\displaystyle\frac{v_{A+}\hat{b}_{x1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-r\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A30) $\displaystyle\frac{v_{A+}\hat{b}_{x2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-r\frac{1-r}{1+r}\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A31) $\displaystyle\frac{v_{A+}\hat{b}_{x3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle(1-r)\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}).$
(A32)

The leading order asymptotic expansions for the $b_{\perp}$ coefficients are
given by

$\displaystyle\frac{v_{A+}\hat{b}_{\perp 1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle\frac{2}{1+r}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$ (A33)
$\displaystyle\frac{v_{A+}\hat{b}_{\perp 4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-i(1-r)\frac{\sin^{2}\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A34) $\displaystyle\frac{v_{A+}\hat{b}_{\perp 1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle 1,$ (A35) $\displaystyle\frac{v_{A+}\hat{b}_{\perp 2+}}{u_{0}}$
$\displaystyle=$
$\displaystyle\frac{1-r}{1+r}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A36) $\displaystyle\frac{v_{A+}\hat{b}_{\perp 3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-i(1-r)\frac{\sin^{2}\alpha}{\cos\alpha}\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}).$
(A37)

Finally, the leading order asymptotic expansions for the $\hat{b}_{||}$
coefficients are given by

$\displaystyle\hat{b}_{||1-}$ $\displaystyle=$ $\displaystyle 0,$ (A38)
$\displaystyle\frac{v_{A+}\hat{b}_{||4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle
ir(1-r)\sin\alpha\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}),$
(A39) $\displaystyle\hat{b}_{||1+}$ $\displaystyle=$ $\displaystyle 0,$ (A40)
$\displaystyle\hat{b}_{||2+}$ $\displaystyle=$ $\displaystyle 0,$ (A41)
$\displaystyle\frac{v_{A+}\hat{b}_{||3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle
ir(1-r)\sin\alpha\frac{k_{||-}}{k_{x}}+O\quantity(\frac{k_{||-}^{2}}{k_{x}^{2}}).$
(A42)

##  Appendix B Check recovery of line-tied equations

In this section, our goal is to check if we recover the equations we derived
in Appendix A.1 if $k_{x}\ll k_{||-}$. To simplify the notation, let

$\displaystyle\epsilon_{-}$ $\displaystyle=$
$\displaystyle\frac{k_{x}}{k_{||-}},$ (B1) $\displaystyle\epsilon_{+}$
$\displaystyle=$ $\displaystyle\frac{k_{||+}}{k_{x}}.$ (B2)

By using Equations (53)-(60) as well as the maple code,

$\text{maple\\_code\\_appendix\\_B.mw},$

which can be be accessed in the attached .tar.gz file, we calculate the
leading order multivariate Taylor expansions (assuming
$\epsilon_{-},\epsilon_{+}\ll 1$). The $u_{x}$ coefficients are given by

$\displaystyle\frac{u_{x1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\sin\alpha\,\epsilon_{-}\epsilon_{+}+O(\epsilon_{-}^{3},\epsilon_{-}^{2}\epsilon_{+},\epsilon_{-}\epsilon_{+}^{2},\epsilon_{+}^{3}),$
(B3) $\displaystyle\frac{u_{x4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\frac{\cos\alpha}{\sin\alpha}\epsilon_{-}^{2}\epsilon_{+}+O(\epsilon_{-}^{4},\epsilon_{-}^{3}\epsilon_{+},...\,,\epsilon_{+}^{4}),$
(B4) $\displaystyle\frac{u_{x1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B5) $\displaystyle\frac{u_{x2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B6) $\displaystyle\frac{u_{x3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\tan\alpha\,\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}).$
(B7)

The $u_{\perp}$ coefficients are given by

$\displaystyle\frac{u_{\perp 1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\cos\alpha\,\epsilon_{-}^{2}\epsilon_{+}+O(\tiny{\epsilon_{-}^{4},\epsilon_{-}^{3}\epsilon_{+},...\,,\epsilon_{+}^{4}})$
(B8) $\displaystyle\frac{u_{\perp 4-}}{u_{0}}$ $\displaystyle=$ $\displaystyle
2\cos\alpha\,\epsilon_{-}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B9) $\displaystyle\frac{u_{\perp 1+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
1,$ (B10) $\displaystyle\frac{u_{\perp 2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-1+O(\epsilon_{-},\epsilon_{+}),$ (B11)
$\displaystyle\frac{u_{\perp 3+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
2i\frac{\sin^{2}\alpha}{\cos\alpha}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}).$
(B12)

The $b_{x}$ coefficients are given by

$\displaystyle\frac{v_{A+}\hat{b}_{x1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\sin\alpha+O(\epsilon_{-},\epsilon_{+}),$ (B13)
$\displaystyle\frac{v_{A+}\hat{b}_{x4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\frac{\cos^{2}\alpha}{\sin\alpha}\epsilon_{-}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B14) $\displaystyle\frac{v_{A+}\hat{b}_{x1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}-\sin\alpha}{\cos\alpha}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B15) $\displaystyle\frac{v_{A+}\hat{b}_{x2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\hat{k}_{y}+\sin\alpha}{\cos\alpha}\epsilon_{+}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B16) $\displaystyle\frac{v_{A+}\hat{b}_{x3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\sin\alpha+O(\epsilon_{-},\epsilon_{+}).$ (B17)

The $b_{\perp}$ coefficients are given by

$\displaystyle\frac{v_{A+}b_{\perp 1-}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2i\cos\alpha\,\epsilon_{-}+O(\epsilon_{-}^{2},\epsilon_{-}\epsilon_{+},\epsilon_{+}^{2}),$
(B18) $\displaystyle\frac{v_{A+}b_{\perp 4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle 2\cos^{2}\alpha+O(\epsilon_{-},\epsilon_{+}),$ (B19)
$\displaystyle\frac{v_{A+}b_{\perp 1+}}{u_{0}}$ $\displaystyle=$
$\displaystyle 1,$ (B20) $\displaystyle\frac{v_{A+}b_{\perp 2+}}{u_{0}}$
$\displaystyle=$ $\displaystyle 1+O(\epsilon_{-},\epsilon_{+}),$ (B21)
$\displaystyle\frac{v_{A+}b_{\perp 3+}}{u_{0}}$ $\displaystyle=$
$\displaystyle-2\sin^{2}\alpha+O(\epsilon_{-},\epsilon_{+}).$ (B22)

Finally, the $\hat{b}_{||}$ coefficients are given by

$\displaystyle\frac{v_{A+}b_{||1-}}{u_{0}}$ $\displaystyle=$ $\displaystyle
0,$ (B23) $\displaystyle\frac{v_{A+}b_{||4-}}{u_{0}}$ $\displaystyle=$
$\displaystyle\sin(2\alpha)+O(\epsilon_{-},\epsilon_{+}),$ (B24)
$\displaystyle\frac{v_{A+}b_{||1+}}{u_{0}}$ $\displaystyle=$ $\displaystyle
0,$ (B25) $\displaystyle\frac{v_{A+}b_{||2+}}{u_{0}}$ $\displaystyle=$
$\displaystyle 0,$ (B26) $\displaystyle\frac{v_{A+}b_{||3+}}{u_{0}}$
$\displaystyle=$ $\displaystyle\sin(2\alpha)+O(\epsilon_{-},\epsilon_{+}).$
(B27)

Taking the limit $\epsilon_{-}\rightarrow 0$ we do indeed recover the
equations derived in Appendix A.1 for $z>0$.

##  Appendix C Eigenfunctions and eigenfrequencies

In this section our goal is to calculate expressions for the eigenfunctions
and corresponding eigenfrequencies, $\phi_{n}(z)$, $\varphi_{n}(z)$,
$\omega_{n}$, $\varpi_{n}$, which were discussed in Section 5.2. By symmetry,
we assume the eigenfunctions are of the form

$\displaystyle\phi_{n}(z)$ $\displaystyle=$ $\displaystyle
A_{n}\begin{cases}a_{n}\cos[k_{zn-}(z+l_{z})],&z<0,\\\
c_{n}\cos[k_{zn+}(z-l_{z})],&z\geq 0,\\\ \end{cases}$ (C1)
$\displaystyle\varphi_{n}(z)$ $\displaystyle=$ $\displaystyle
B_{n}\begin{cases}b_{n}\sin[\bar{k}_{zn-}(z+l_{z})],&z<0,\\\
d_{n}\sin[\bar{k}_{zn+}(z-l_{z})],&z\geq 0,\\\ \end{cases}$ (C2)

where $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$ are constants, $L_{z}=2l_{z}$ and

$\displaystyle k_{zn\pm}$ $\displaystyle=$
$\displaystyle\frac{\omega_{n}}{v_{A\pm}\cos\alpha},$ (C3)
$\displaystyle\bar{k}_{zn\pm}$ $\displaystyle=$
$\displaystyle\frac{\varpi_{n}}{v_{A\pm}\cos\alpha}.$ (C4)

We require continuity of $\phi_{n}$, $\varphi_{n}$,
$\derivative*{\phi_{n}}{z}$, $\derivative*{\varphi_{n}}{z}$ at $z=0$ and
$z=\pm L_{z}$, this gives the following equations

$\displaystyle\begin{pmatrix}\cos(k_{zn-}l_{z})&-\cos(k_{zn+}l_{z})\\\
k_{zn-}\sin(k_{zn-}l_{z})&k_{zn+}\sin(k_{zn+}l_{z})\end{pmatrix}\begin{pmatrix}a_{n}\\\
c_{n}\end{pmatrix}$ $\displaystyle=0,$
$\displaystyle\begin{pmatrix}\sin(\bar{k}_{zn-}l_{z})&\sin(\bar{k}_{zn+}l_{z})\\\
\bar{k}_{zn-}\cos(\bar{k}_{zn-}l_{z})&-\bar{k}_{zn+}\cos(\bar{k}_{zn+}l_{z})\end{pmatrix}\begin{pmatrix}b_{n}\\\
d_{n}\end{pmatrix}$ $\displaystyle=0.$

For non-trivial solutions to exist, $\omega_{n}$, $\varpi_{n}$ must satisfy

$\displaystyle
k_{zn+}\sin(k_{zn+}l_{z})\cos(k_{zn-}l_{z})+k_{zn-}\sin(k_{zn-}l_{z})\cos(k_{zn+}l_{z})$
$\displaystyle=$ $\displaystyle 0,$ (C5)
$\displaystyle\bar{k}_{zn+}\sin(\bar{k}_{zn-}l_{z})\cos(\bar{k}_{zn+}l_{z})+\bar{k}_{zn-}\sin(\bar{k}_{zn+}l_{z})\cos(\bar{k}_{zn-}l_{z})$
$\displaystyle=$ $\displaystyle 0.$ (C6)

These equations define, $\omega_{n}$, $\varpi_{n}$ and we order them such that

$\omega_{0}<\omega_{1}<\omega_{2}...,\quad\varpi_{0}<\varpi_{1}<\varpi_{2}...\,,$
(C7)

where $\omega_{0}=\varpi_{0}=0$.

If $\omega_{n}$, $\varpi_{n}$ satisfy Equations (C5) and (C6), then the
boundary conditions and continuity conditions are ensured if

$\displaystyle a_{n}$ $\displaystyle=$
$\displaystyle\begin{cases}\cos(k_{zn+}l_{z}),&\cos(k_{zn+}l_{z})\neq 0,\\\
k_{zn+}\sin(k_{zn+}l_{z}),&\cos(k_{zn+}l_{z})=0,\\\ \end{cases}$ (C8)
$\displaystyle c_{n}$ $\displaystyle=$
$\displaystyle\begin{cases}\cos(k_{zn-}l_{z}),&\cos(k_{zn-}l_{z})\neq 0,\\\
-k_{zn-}\sin(k_{zn-}l_{z}),&\cos(k_{zn-}l_{z})=0,\\\ \end{cases}$ (C9)
$\displaystyle b_{n}$ $\displaystyle=$
$\displaystyle\begin{cases}\sin(\bar{k}_{zn+}l_{z}),&\sin(\bar{k}_{zn+}l_{z})\neq
0,\\\ \bar{k}_{zn+}\cos(\bar{k}_{zn+}l_{z}),&\sin(\bar{k}_{zn+}l_{z})=0,\\\
\end{cases}$ (C10) $\displaystyle d_{n}$ $\displaystyle=$
$\displaystyle\begin{cases}-\sin(\bar{k}_{zn-}l_{z}),&\sin(\bar{k}_{zn-}l_{z})\neq
0,\\\ \bar{k}_{zn-}\cos(\bar{k}_{zn-}l_{z}),&\sin(\bar{k}_{zn-}l_{z})=0.\\\
\end{cases}$ (C11)

We normalise such that

$\Big{\langle}\phi_{n}(z),\phi_{m}(z)\Big{\rangle}=\Big{\langle}\varphi_{n}(z),\varphi_{m}(z)\Big{\rangle}=\delta_{nm},$
(C12)

where $\delta_{nm}$ is the Kronecker delta. Note that

$\displaystyle\langle\phi_{n},\phi_{n}\rangle$ $\displaystyle=$
$\displaystyle\frac{v_{A+}^{2}A_{n}^{2}}{L_{z}}\quantity{\frac{a_{n}^{2}}{2v_{A-}^{2}k_{zn-}}\Big{[}2k_{zn-}l_{z}+\sin(2k_{zn-}l_{z})\Big{]}+\frac{c_{n}^{2}}{2v_{A+}^{2}k_{zn+}}\Big{[}2k_{zn+}l_{z}+\sin(2k_{zn+}l_{z})\Big{]}},$
(C13) $\displaystyle\langle\varphi_{n},\varphi_{n}\rangle$ $\displaystyle=$
$\displaystyle\frac{v_{A+}^{2}B_{n}^{2}}{L_{z}}\quantity{\frac{b_{n}^{2}}{2v_{A-}^{2}\bar{k}_{zn-}}\Big{[}2\bar{k}_{zn-}l_{z}-\sin(2\bar{k}_{zn-}l_{z})\Big{]}+\frac{d_{n}^{2}}{2v_{A+}^{2}\bar{k}_{zn-}}\Big{[}2\bar{k}_{zn-}l_{z}-\sin(2\bar{k}_{zn-}l_{z})\Big{]}}.$
(C14)

Hence,

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\quantity(\frac{v_{A+}^{2}}{L_{z}}\quantity{\frac{a_{n}^{2}}{2v_{A-}^{2}k_{zn-}}\Big{[}2k_{zn-}l_{z}+\sin(2k_{zn-}l_{z})\Big{]}+\frac{c_{n}^{2}}{2v_{A+}^{2}k_{zn+}}\Big{[}2k_{zn+}l_{z}+\sin(2k_{zn+}l_{z})\Big{]}})^{-1/2},$
(C15) $\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle\quantity(\frac{v_{A+}^{2}}{L_{z}}\quantity{\frac{b_{n}^{2}}{2v_{A-}^{2}\bar{k}_{zn-}}\Big{[}2\bar{k}_{zn-}l_{z}-\sin(2\bar{k}_{zn-}l_{z})\Big{]}+\frac{d_{n}^{2}}{2v_{A+}^{2}\bar{k}_{zn-}}\Big{[}2\bar{k}_{zn-}l_{z}-\sin(2\bar{k}_{zn-}l_{z})\Big{]}})^{-1/2}.$
(C16)

##  Appendix D Deriving equation for $u_{\perp}^{\prime}$

Eliminating $\hat{b}_{||}^{\prime}$ from Equations (79)-(81) gives

$\displaystyle\quantity[\mathcal{L}^{\prime}+\partialderivative[2]{x}]u_{x}^{\prime}$
$\displaystyle=$
$\displaystyle-\partialderivative{x}\nabla_{\perp}^{\prime}u_{\perp}^{\prime},$
(D1)
$\displaystyle[\mathcal{L}^{\prime}+\nabla_{\perp}^{2}]u_{\perp}^{\prime}$
$\displaystyle=$
$\displaystyle-\partialderivative{x}\nabla_{\perp}^{\prime}u_{x}^{\prime}$
(D2)

We can eliminate $u_{x}^{\prime}$ with the following procedure. Take the
$x$-derivative of the Equation (D1)

$\mathcal{L}_{x}^{\prime}u_{x}^{\prime}+\quantity[\mathcal{L}^{\prime}+\partialderivative[2]{x}]\partialderivative{u_{x}^{\prime}}{x}=-\nabla_{\perp}\partialderivative[2]{u_{\perp}^{\prime}}{x},$
$\implies
u_{x}^{\prime}=-\frac{1}{\mathcal{L}_{x}^{\prime}}\quantity{\quantity[\mathcal{L^{\prime}}+\partialderivative[2]{x}]\partialderivative{u_{x}^{\prime}}{x}+\nabla_{\perp}^{\prime}\partialderivative[2]{u_{\perp}^{\prime}}{x}}.$

where

$\displaystyle\mathcal{L}_{x}^{\prime}$ $\displaystyle=$
$\displaystyle\partialderivative{\mathcal{L}^{\prime}}{x}=\frac{-2\omega^{2}}{a(x)v_{A}^{2}(x,z)},$
(D3) $\displaystyle a(x)$ $\displaystyle=$
$\displaystyle\frac{\hat{v}_{A}(x)}{\derivative*{\hat{v}_{A}}{x}}.$ (D4)

Substitute this into Equation (14) to give

$-\frac{\mathcal{L}^{\prime}}{\mathcal{L}_{x}^{\prime}}\quantity{\quantity[\mathcal{L}^{\prime}+\partialderivative[2]{x}]\partialderivative{u_{x}^{\prime}}{x}+\nabla_{\perp}^{\prime}\partialderivative[2]{u_{\perp}^{\prime}}{x}}+\partialderivative[2]{u_{x}^{\prime}}{x}=-\nabla_{\perp}^{\prime}\partialderivative{u_{\perp}^{\prime}}{x},$

Applying $-\nabla_{\perp}^{\prime}$ to both sides, substituting for
$u_{x}^{\prime}$ and assuming $z\neq 0,-L_{z},L_{z}$ gives

$-\frac{\mathcal{L}^{\prime}}{\mathcal{L}_{x}^{\prime}}\quantity{\quantity[\mathcal{L}^{\prime}+\partialderivative[2]{x}][\mathcal{L}^{\prime}+\nabla_{\perp}^{\prime
2}]u_{\perp}^{\prime}-\nabla_{\perp}^{\prime
2}\partialderivative[2]{u_{\perp}^{\prime}}{x}}+\partialderivative{x}[\mathcal{L}^{\prime}+\nabla_{\perp}^{\prime
2}]u_{\perp}^{\prime}=\nabla_{\perp}^{\prime
2}\partialderivative{u_{\perp}^{\prime}}{x}.$

This can be simplified to give

$\mathcal{L}^{\prime
2}\partialderivative[2]{u_{\perp}^{\prime}}{x}+^{\prime}\mathcal{L}_{x}^{\prime}\mathcal{L}^{\prime}\partialderivative{u_{\perp}^{\prime}}{x}+(\mathcal{L}^{\prime
3}+\mathcal{L}^{\prime 2}\nabla_{\perp}^{\prime
2}+\mathcal{L}_{xx}^{\prime}\mathcal{L}^{\prime}-\mathcal{L}_{x}^{\prime
2})u_{\perp}^{\prime}=0.$ (D5)

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