# Damping of slow surface kink modes in solar photospheric waveguides modeled
by one-dimensional inhomogeneities

Shao-Xia Chen  Shandong Provincial Key Laboratory of Optical Astronomy and
Solar-Terrestrial Environment, Institute of Space Sciences, Shandong
University, Weihai 264209, China  Bo Li  Shandong Provincial Key Laboratory of
Optical Astronomy and Solar-Terrestrial Environment, Institute of Space
Sciences, Shandong University, Weihai 264209, China  Tom Van Doorsselaere
Centre for mathematical Plasma Astrophysics (CmPA), KU Leuven, Celestijnenlaan
200B bus 2400, B-3001 Leuven, Belgium  Marcel Goossens  Centre for
mathematical Plasma Astrophysics (CmPA), KU Leuven, Celestijnenlaan 200B bus
2400, B-3001 Leuven, Belgium  Hui Yu  Shandong Provincial Key Laboratory of
Optical Astronomy and Solar-Terrestrial Environment, Institute of Space
Sciences, Shandong University, Weihai 264209, China  Michaël Geeraerts  Centre
for mathematical Plasma Astrophysics (CmPA), KU Leuven, Celestijnenlaan 200B
bus 2400, B-3001 Leuven, Belgium

###### Abstract

Given the recent interest in magnetohydrodynamic (MHD) waves in pores and
sunspot umbrae, we examine the damping of slow surface kink modes (SSKMs) by
modeling solar photospheric waveguides with a cylindrical inhomogeneity
comprising a uniform interior, a uniform exterior, and a continuous transition
layer (TL) in between. Performing an eigen-mode analysis in linear, resistive,
gravity-free MHD, our approach is idealized in that, among other things, our
equilibrium is structured only in the radial direction. We can nonetheless
address two damping mechanisms simultaneously, one being the Ohmic
resistivity, and the other being the resonant absorption of SSKMs in the cusp
and Alfv$\acute{\rm e}$n continua. We find that the relative importance of the
two mechanisms depends sensitively on the magnetic Reynolds number ($R_{\rm
m}$). Resonant absorption is the sole damping mechanism for realistically
large values of $R_{\rm m}$, and the cusp resonance in general dominates the
Alfv$\acute{\rm e}$n one unless the axial wavenumbers are at the lower end of
the observationally relevant range. We also find that the thin-boundary
approximation holds only when the TL-width-to-radius ratios are much smaller
than nominally expected. The Ohmic resistivity is far more important for
realistically small $R_{\rm m}$. Even in this case, SSKMs are only marginally
damped, with damping-time-to-period-ratios reaching $\sim 10$ in the parameter
range we examine.

magnetohydrodynamics (MHD) — Sun: magnetic fields — Sun: photosphere — waves

††journal: ApJ

##  1 INTRODUCTION

Low-frequency waves and oscillations in the magnetohydrodynamic (MHD) regime
often prove key in the gas-magnetic field interactions throughout the solar
atmosphere (see e.g., the textbook by Roberts, 2019). When generated by, say,
sub-photospheric convective motions, these waves may be sufficiently energetic
to play an active role in atmospheric heating, thereby helping shape such
atmospheric structures as coronal loops (see e.g., Parnell & De Moortel, 2012;
Arregui, 2015; Cranmer & Winebarger, 2019, for recent reviews). Even when not
that energetic, these waves can passively encode some rich information on
their host magnetic structures in their observational signatures. Conversely,
this information can be deciphered with increasingly refined theories of MHD
waves in an inhomogeneous medium, constituting the field of “coronal
seismology” or more generally “solar atmospheric seismology” (SAS, Roberts et
al. 1984; see also recent reviews by e.g., Roberts 2000, Nakariakov &
Verwichte 2005, Nakariakov & Kolotkov 2020). Whichever the application, MHD
waves have been customarily classified with a scheme largely established for a
canonical configuration where a magnetic structure is seen as a field-aligned
plasma cylinder with circular cross-section, and the equilibrium quantities
are structured only in the radial direction (Wentzel 1979, Spruit 1982, Edwin
& Roberts 1983, hereafter ER83; see also Rosenberg 1970, Zajtsev & Stepanov
1975, Cally 1986). We restrict ourselves to collective waves, “collective” in
the sense that they involve coherent motions of the fluid parcels in the
structure itself and possibly its surroundings. Collective waves (“modes”
hereafter) may be classified as sausage ($m=0$), kink ($m=1$), and fluting
modes ($m\geq 2$) by their azimuthal wavenumber $m$. They may be independently
grouped into slow and fast modes according to their axial phase speeds.
Alternatively, the spatial dependence of the associated perturbations in the
surroundings enables the notions of trapped and leaky modes as a measure for
classification. Likewise, the notions of surface and body modes are enabled by
an examination of the spatial behavior within the magnetic structure. One
therefore ends up with an extensive list of combinations like fast kink body
modes and slow sausage surface modes 111 Some issue arises when one applies
this classification scheme to the abundantly observed kink modes that are
trapped in thin coronal loops and that possess axial phase speeds comparable
to the Alfv$\acute{\rm e}$n speed. Customarily called “fast kink body modes”,
they turn out to be essentially Alfv$\acute{\rm e}$nic surface waves with
mixed properties (for details, see Goossens et al., 2009, 2012). In
particular, they survive in incompressible MHD, making them difficult to be
classified as “fast”. We therefore deem it more appropriate to call them “the
radial fundamental kink mode”. . In the structured solar corona, there have
been ample observations for modes pertaining to a substantial fraction of
these combinations (see the reviews by e.g., Banerjee et al., 2007; Nakariakov
et al., 2016; Wang, 2016). In particular, the ubiquity of radial fundamental
kink modes has enabled one to seismologically construct the spatial
distributions of the magnetic field strength in a substantial portion of some
active region (AR, Anfinogentov & Nakariakov 2019) or even across several ARs
(Yang et al., 2020).

There is a long history of observational studies on MHD waves in the lower
solar atmosphere as well (see e.g., the review by Khomenko & Collados 2015 on
waves in sunspots, and that by Jess et al. 2015 on waves in the chromosphere).
However, it seems that the classification of these waves with the ER83 scheme
was made possible only relatively recently. Take the quiet chromosphere for
instance. Cyclic transverse displacements, the defining characteristic of
radial fundamental kink modes, have been abundantly observed in slender
fibrils imaged in both Ca II 3969 Å with SUNRISE/SuFI (Jafarzadeh et al.,
2017) and Hα with ROSA (Morton et al., 2012). Likewise, the latter study also
indicated the omnipresence of breathing-like motions of the Hα fibrils,
thereby suggesting the ubiquity of fast sausage modes. In both studies, the
width of chromospheric fibrils reaches down to $\sim 150$ km, which highlights
the importance of high spatial resolution in enabling such classifications as
kink or sausage modes. The same can be said for waves in chromospheres above
sunspot umbrae. For instance, the long-known spiral wave patterns (SWPs) with
periodicities of $\sim 3$ mins (e.g., Su et al., 2016, and references therein)
were recently shown to be compatible with the superposition of slow body modes
with $m=0$ and $m=1$ ($m=0$ and $m=2$) in the case of one-armed (two-armed)
SWPs (Kang et al., 2019). Likewise, clear signatures of slow kink body modes
were identified in the temporally and spatially filtered Hα image sequences
acquired with ROSA mounted on the Dunn Solar Telescope (DST, Jess et al.,
2017). Moving to the photosphere in either sunspot umbrae or pores, we note
that slow sausage modes tend to be ubiquitous (e.g., Dorotovič et al., 2008;
Fujimura & Tsuneta, 2009; Morton et al., 2011; Dorotovič et al., 2014; Moreels
et al., 2015a; Grant et al., 2015; Freij et al., 2016). In addition, surface
modes tend to be favored over body modes in pores with larger sizes and/or
stronger magnetic fields (Keys et al. 2018, Gilchrist-Millar et al. 2020; see
also Stangalini et al. 2018). The waves in the lower solar atmosphere, be them
kink or sausage modes, were suggested to reach chromospheric levels with a
Poynting flux of up to $\sim 27~{}{\rm kW}~{}{\rm m}^{-2}$ (e.g., Fujimura &
Tsuneta, 2009), a value sufficiently large to meet the chromospheric heating
requirement. Equally useful are their applications to SAS as initiated by
Fujimura & Tsuneta (2009) and Moreels et al. (2015a). Observations aside,
theoretical studies have proven crucial for establishing the phase-differences
between relevant perturbations and therefore for aiding mode identification
(Moreels & Van Doorsselaere, 2013; Moreels et al., 2013). Likewise,
expressions for the energy and energy flux densities are inevitable for
examining the energetics (Moreels et al., 2015b).

This study is intended to examine the damping of slow surface kink modes
(SSKMs) in photospheric waveguides representative of pores and sunspot umbrae,
thereby partially addressing the fate of SSKMs generated by, say, sub-
photospheric convective motions. Our motivation is twofold. Firstly, in
contrast to the apparent omnipresence of slow surface sausage modes (SSSMs),
photospheric kink modes were implicated only in the Hinode/SP measurements
reported by Fujimura & Tsuneta (2009) to our knowledge, and the kink mode
therein is more likely to be a fast surface one. We take this as encouraging
rather than discouraging for a theoretical study on SSKMs, the reason
primarily related to their theoretical properties and their likely excitation.
Theoretically, SSKMs tend to be indistinguishable from SSSMs in terms of
eigen-frequencies (see e.g., Figure 3 in ER83). While the azimuthal variations
of the two modes are distinctively different, the random broadband motions
that excite SSSMs are likely to excite SSKMs as well. Secondly, SSSMs in pores
were observationally demonstrated to be heavily damped over photospheric
heights (Grant et al., 2015; Gilchrist-Millar et al., 2020). So far this heavy
damping has only been partially addressed. Working in the thin-boundary (TB)
approximation, the largely analytical study by Yu et al. (2017a) indicated
that the resonant absorption of SSSMs in the cusp continuum may result in a
damping-time-to-period ratio ($\tau/P$) of $\sim 10$. This was taken further
by Sadeghi & Karami (2019) who incorporated magnetic twist in the equilibrium.
As such SSSMs further resonantly couple to the Alfvén continuum, thereby
leading to some stronger damping. Note that the Alfv$\acute{\rm e}$n and cusp
continua arise when the piece-wise constant model is replaced by a model in
which the equilibrium quantities vary in a continuous manner in a transitional
layer (TL) from their internal to their external values. By construction, the
TB results apply only when the TL width ($l$) is far below the cylinder radius
($R$). However, in the above-referenced TB studies, strong damping tends to
occur when $l/R$ is substantial, and one may question whether the TB damping
rates hold. Indeed, focusing on the damping of SSSMs due to the cusp
resonance, Chen et al. (2018, hereafter C18) showed with resistive MHD
computations that the numerically derived damping rates are in general well
below the TB values found by Yu et al. (2017a). C18 further showed that in
general Ohmic resistivity tends to be more important than resonant absorption
for damping SSSMs, a result that was further corroborated by the analytical
study in Geeraerts et al. (2020, G20 from here on). Compared with SSSMs, SSKMs
are theoretically appealing in that they can resonantly couple to the $m=1$
Alfvén waves even in the absence of magnetic twist (Yu et al., 2017b, Y17
hereafter). In Y17, the damping of SSKMs was solely due to resonant absorption
and the cusp resonance was shown to dominate for large axial wavenumbers,
resulting in values of down to $\sim 10$ for $\tau/P$. Now two questions
arise, given that Y17 adopted the TB approximation, and given the C18 results
regarding SSSMs. Firstly, how well do the Y17 results apply as far as the
damping due to resonant absorption is concerned? Secondly, what is the role of
Ohmic resistivity in the damping of SSKMs?

Some remarks prove necessary at this point. We will adopt resistive, gravity-
free, MHD as our theoretical framework, in accordance with which we will
additionally take our equilibrium configuration to be structured only
transversely. Both call into question how well our results apply to
photospheric structures in reality, the structuring of which is way more
complicated than assumed here. We leave the discussions on this aspect till
the end of this manuscript. It suffices to note here that by “photospheric
modes” we refer to the modes supported by some idealized magnetic structures,
for which the ordering of the internal and external characteristic speeds
nonetheless captures some realistic features of pores and sunspot umbrae.
Bearing the caveats in mind, we list two features that make our study
considerably new relative to available ones on photospheric modes. Firstly, so
far the damping of SSKMs was examined only by Y17 to our knowledge. Relative
to Y17, our study is new in that we will simultaneously account for the
effects of the Ohmic resistivity and resonant absorption, whereas Y17
considered only resonant absorption. Secondly, we will analyze the wave
damping from the perspective of energy balance, whereas all the afore-
referenced studies approached the wave damping exclusively from the
perspective of eigen-frequencies. As we will acknowledge where more
appropriate, this is not to say that wave damping has not been examined from
the energetics perspective. It is just that “photospheric modes” have not been
examined this way as far as we know.

This manuscript is organized as follows. Section 2 details the specification
of our equilibrium configuration, and how we formulate and numerically solve
the pertinent eigenvalue problem. The numerical results are then described in
Section 3. Section 4 summarizes the present study, ending with some concluding
remarks.

##  2 Model Description

###  2.1 Equilibrium Configuration

We adopt resistive, gravity-free MHD throughout, for which the primitive
variables are the mass density ($\rho$), velocity ($\bm{v}$), magnetic field
($\bm{B}$), and thermal pressure ($p$). We denote the equilibrium quantities
with the subscript $0$, and assume that no equilibrium flow is present
($\bm{v}_{0}=0$). Working in a cylindrical coordinate system $(r,\theta,z)$,
we assume that all equilibrium quantities depend only on $r$. The equilibrium
magnetic field ($\bm{B}_{0}$) is taken to be in the $z$-direction. As such
$B_{0}(r)$ and $p_{0}(r)$ are related by the transverse force balance
condition

$\displaystyle\displaystyle p_{0}(r)+\frac{B_{0}(r)^{2}}{2\mu_{0}}={\rm
const}~{},$ (1)

where $\mu_{0}$ is the magnetic permeability in free space. The adiabatic
sound speed $c_{\rm s}$, Alfv$\acute{\rm e}$n speed $v_{\rm A}$, and cusp
(tube) speed $c_{\rm T}$ are then defined by

$\displaystyle\displaystyle c_{\rm s}^{2}(r)=\frac{\gamma
p_{0}}{\rho_{0}}~{},~{}~{}~{}v_{\rm
A}^{2}(r)=\frac{B_{0}^{2}}{\mu_{0}\rho_{0}}~{},~{}~{}~{}c_{\rm
T}^{2}(r)=\frac{c_{\rm s}^{2}v_{\rm A}^{2}}{c_{\rm s}^{2}+v_{\rm A}^{2}}~{},$
(2)

with $\gamma=5/3$ being the adiabatic index.

We model a photospheric waveguide as a field-aligned cylinder with mean radius
$R$. Following C18, we realize this by assuming that $c_{\rm s}^{2}$ and
$c_{\rm T}^{2}$ take the following form

$\displaystyle\displaystyle{\cal E}(r)=\left\\{\begin{array}[]{lc}{\cal
E}_{\rm i}~{},&0<r<r_{\rm i}=R-l/2~{};\\\\[5.69046pt] \displaystyle\frac{{\cal
E}_{\rm i}+{\cal E}_{\rm e}}{2}-\frac{{\cal E}_{\rm i}-{\cal E}_{\rm
e}}{2}\sin\frac{\pi(r-R)}{l}~{},&r_{\rm i}\leq r\leq r_{\rm
e}=R+l/2~{};\\\\[5.69046pt] {\cal E}_{\rm e}~{},&r>r_{\rm
e}~{},\end{array}\right.$ (6)

where ${\cal E}$ represents $c_{\rm s}^{2}$ and $c_{\rm T}^{2}$. In addition,
the subscripts i and e refer to the equilibrium values at the cylinder axis
and in the ambient medium, respectively. As such the equilibrium configuration
comprises a uniform interior ($r<r_{\rm i}$), a uniform exterior ($r>r_{\rm
e}$) and a transition layer (TL) continuously connecting the two. This
equilibrium is fully specified by a set of dimensional parameters $[\rho_{\rm
i},v_{\rm Ai},R]$ together with a set of dimensionless parameters $[c_{\rm
si}/v_{\rm Ai},c_{\rm se}/v_{\rm Ai},v_{\rm Ae}/v_{\rm Ai},l/R]$. As in C18,
we fix $[c_{\rm si},c_{\rm se},v_{\rm Ae}]$ at $[0.5,0.75,0.25]v_{\rm Ai}$,
which is identical to ER83 and also in agreement with the generally accepted
values for sunspot umbrae and pores (e.g., Jess et al., 2017; Yu et al.,
2017b). The internal and external cusp speeds then read $[c_{\rm Ti},c_{\rm
Te}]=[0.4472,0.2372]~{}v_{\rm Ai}$. The ratio of the TL width to the mean
radius ($l/R$), on the other hand, is allowed to vary. The spatial
distributions for both the characteristic speeds and primitive variables are
displayed in Figure 1, where $l/R$ is taken to be $0.2$ for illustration
purposes. Among the dimensional parameters, $v_{\rm Ai}$ and $R$ are relevant
for our purposes. We take $v_{\rm Ai}=12~{}{\rm km}~{}{\rm s}^{-1}$ in view of
Figure 4 in Grant et al. (2015), who in turn constructed the values with the
“hot” umbral model from Maltby et al. (1986). With pores and sunspot umbrae in
mind, we assume that $R$ ranges from $500$ km (the lower limit for pores, see
Sobotka 2003) to $10^{4}$ km (for large sunspot umbrae, see Figure 1 in
Stangalini et al. 2018).

###  2.2 Formulation of the Eigenvalue Problem and Method of Solution

Let the subscript $1$ denote small-amplitude perturbations to the equilibrium.
The linearized resistive MHD equations then read

$\displaystyle\displaystyle\rho_{0}\frac{\partial\bm{v}_{1}}{\partial t}$
$\displaystyle=$ $\displaystyle-\nabla
p_{1}+\frac{(\nabla\times\bm{B}_{0})\times\bm{B}_{1}}{\mu_{0}}+\frac{(\nabla\times\bm{B}_{1})\times\bm{B}_{0}}{\mu_{0}}~{},$
(7) $\displaystyle\displaystyle\frac{\partial\bm{B}_{1}}{\partial t}$
$\displaystyle=$
$\displaystyle\nabla\times\left(\bm{v}_{1}\times\bm{B}_{0}-\frac{\eta}{\mu_{0}}\nabla\times\bm{B}_{1}\right)~{},$
(8) $\displaystyle\displaystyle\frac{\partial p_{1}}{\partial t}$
$\displaystyle=$ $\displaystyle-\bm{v}_{1}\cdot\nabla p_{0}-\gamma
p_{0}\nabla\cdot\bm{v}_{1}+\frac{2(\gamma-1)\eta}{\mu_{0}^{2}}\left(\nabla\times\bm{B}_{1}\right)\cdot\left(\nabla\times\bm{B}_{0}\right)~{}.$
(9)

Here $\eta$ is the Ohmic resistivity, taken to be constant for simplicity. We
adopt an eigen-value-problem (EVP) perspective by Fourier-analyzing any
perturbation as

$\displaystyle g_{1}(r,\theta,z;t)={\rm Re}\\{\tilde{g}(r)\exp[-i(\omega
t-m\theta-kz)]\\}~{},$ (10)

where $\omega$ is the angular frequency, and $k$ ($m$) represents the axial
(azimuthal) wavenumber. With tilde we denote the Fourier amplitude. We take
$k$ as real-valued, but allow $\omega$ to be complex-valued. If some quantity
is complex, then we denote its real and imaginary parts with subscripts R and
I, respectively. It follows that the period $P=2\pi/\omega_{\rm R}$, and the
damping time $\tau=1/|\omega_{\rm I}|$. Instabilities are not of interest,
hence $\omega_{\rm I}<0$. The axial phase speed is defined by $v_{\rm
ph}=\omega_{\rm R}/k$.

We proceed with normalizing Equations (7) to (9). Among the necessary
normalizing constants, we take $R$, $v_{\rm Ai}$, and $\rho_{\rm i}$ as
independent ones, following from which are the normalizing constants for time
($R/v_{\rm Ai}$), the magnetic field ($B_{\rm i}\equiv\sqrt{\mu_{0}\rho_{\rm
i}v_{\rm Ai}^{2}}$) and thermal pressure ($\rho_{\rm i}v_{\rm Ai}^{2}=B_{\rm
i}^{2}/\mu_{0}$). The Ohmic resistivity ends up in the magnetic Reynolds
number $R_{\rm m}=\mu_{0}Rv_{\rm Ai}/\eta$. In component form, the resulting
dimensionless equations then read

$\displaystyle\omega\tilde{v}_{r}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{B_{0}}{\rho_{0}}\left(k\tilde{B}_{r}+i\frac{{\rm
d}\tilde{B}_{z}}{{\rm d}r}\right)-\frac{i\tilde{B}_{z}}{\rho_{0}}\frac{{\rm
d}B_{0}}{{\rm d}r}-\frac{i}{\rho_{0}}\frac{{\rm d}\tilde{p}}{{\rm d}r}~{},$
(11) $\displaystyle\omega\tilde{v}_{\theta}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{B_{0}}{\rho_{0}}\left(k\tilde{B}_{\theta}-\frac{m\tilde{B}_{z}}{r}\right)+{}\frac{m\tilde{p}}{r\rho_{0}}~{},$
(12) $\displaystyle\omega\tilde{v}_{z}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{i\tilde{B}_{r}}{\rho_{0}}\frac{{\rm
d}B_{0}}{{\rm d}r}+\frac{k\tilde{p}}{\rho_{0}}~{},$ (13)
$\displaystyle\omega\tilde{B}_{r}$ $\displaystyle=$
$\displaystyle\displaystyle-kB_{0}\tilde{v}_{r}+\frac{1}{R_{\rm
m}}\left(i\frac{{\rm d^{2}}\tilde{B}_{r}}{{\rm
d}{r}^{2}}+\frac{i}{r}\frac{{\rm d}\tilde{B}_{r}}{{\rm
d}r}-\frac{im^{2}\tilde{B}_{r}}{{r}^{2}}-i{k}^{2}\tilde{B}_{r}+\frac{2m\tilde{B}_{\theta}}{r^{2}}-\frac{i\tilde{B}_{r}}{{r}^{2}}\right)~{},$
(14) $\displaystyle\omega\tilde{B}_{\theta}$ $\displaystyle=$
$\displaystyle\displaystyle-kB_{0}\tilde{v}_{\theta}+\frac{1}{R_{\rm
m}}\left(i\frac{{\rm d^{2}}\tilde{B}_{\theta}}{{\rm
d}{r}^{2}}+\frac{i}{r}\frac{{\rm d}\tilde{B}_{\theta}}{{\rm
d}r}-\frac{im^{2}\tilde{B}_{\theta}}{{r}^{2}}-i{k}^{2}\tilde{B}_{\theta}-\frac{2m\tilde{B}_{r}}{r^{2}}-\frac{i\tilde{B}_{\theta}}{{r}^{2}}\right)~{},$
(15) $\displaystyle\omega\tilde{B}_{z}$ $\displaystyle=$
$\displaystyle\displaystyle-i\tilde{v}_{r}\frac{{\rm d}B_{0}}{{\rm
d}r}-B_{0}\left(i\frac{{\rm d}\tilde{v}_{r}}{{\rm
d}r}+i\frac{\tilde{v}_{r}}{r}-\frac{m\tilde{v}_{\theta}}{r}\right)+\frac{1}{R_{\rm
m}}\left(i\frac{{\rm d^{2}}\tilde{B}_{z}}{{\rm
d}{r}^{2}}+\frac{i}{r}\frac{{\rm d}\tilde{B}_{z}}{{\rm
d}r}-\frac{im^{2}\tilde{B}_{z}}{{r}^{2}}-i{k}^{2}\tilde{B}_{z}\right)~{},$
(16) $\displaystyle\omega\tilde{p}$ $\displaystyle=$
$\displaystyle\displaystyle-c_{\rm s}^{2}\rho_{0}\left(i\frac{{\rm
d}\tilde{v}_{r}}{{\rm
d}r}+i\frac{\tilde{v}_{r}}{r}-\frac{m\tilde{v}_{\theta}}{r}-k\tilde{v}_{z}\right)-i\tilde{v}_{r}\frac{{\rm
d}p_{0}}{{\rm d}r}+\frac{2\left(\gamma-1\right)}{R_{\rm m}}\frac{{\rm
d}B_{0}}{{\rm d}r}\left(k\tilde{B}_{r}+i\frac{{\rm d}\tilde{B}_{z}}{{\rm
d}r}\right)~{}.$ (17)

We specialize to the case where $m=1$, given that SSKMs are of interest.
Equations (11) to (17) constitute a standard EVP when supplemented with
appropriate boundary conditions (BCs). The BCs at the cylinder axis ($r=0$)
are that ${\rm d}\tilde{v}_{r}/{\rm d}r={\rm d}\tilde{v}_{\theta}/{\rm
d}r={\rm d}\tilde{B}_{r}/{\rm d}r={\rm d}\tilde{B}_{\theta}/{\rm
d}r=\tilde{v}_{z}=\tilde{B}_{z}=\tilde{p}=0$. With trapped modes in mind, we
require that all variables vanish when $r$ approaches infinity. From here
onward, we see the physical variables as dimensional again for the ease of
description.

The EVP is formulated and solved with the general-purpose finite-element code
PDE2D (Sewell, 1988), which was first introduced to the solar context by
Terradas et al. (2006) to our knowledge. The computational domain spans from
$0$ to some outer boundary $r_{\rm M}$. A nonuniform grid is adopted, among
which a considerable number of grid points are deployed in the TL to resolve
the possible oscillatory behavior therein. The total number of grid points
typically ranges from $5\times 10^{4}$ up to $2.5\times 10^{5}$, among which
up to $50\%$ are deployed in the TL in some cases. Overall the strategy is
that deploying more grid points in the TL leads to no discernible difference
to our results, by which we mean both the eigen-frequencies and eigen-
functions. The outer boundary is placed at a sufficiently large distance of
$r_{\rm M}=50~{}R$ to ensure that further increasing $r_{\rm M}$ does not
introduce any appreciable change to our results. Our numerical results are
remarkably accurate, as will be seen from the energetics examinations. Taking
$[l/R,kR,R_{\rm m}]$ as free parameters, one may formally express $\omega$ as

$\displaystyle\frac{\omega R}{v_{\rm Ai}}={\cal G}\left(kR,\frac{l}{R},R_{\rm
m}\right)~{}.$ (18)

With $l/R$ difficult to constrain observationally, we allow $l/R$ to vary as
broadly as the code permits. Evaluating $R_{\rm m}$ necessarily involves the
electric resistivity, for which purpose we quote the values for electric
conductivity listed in Table 2 of Kovitya & Cram (1983) for a model sunspot
umbra in the temperature range of $3500$ to $\sim 6400$ K. Given the range of
$R$ we have discussed and the internal Alfv$\acute{\rm e}$n speed we have
fixed, $R_{\rm m}$ then ranges from $10^{4}$ to $6.6\times 10^{7}$.

The range of $kR$ is difficult to constrain, given that a concrete
identification of SSKMs has yet to appear. We nonetheless assume that
$kR\gtrsim 0.3$ with the limited knowledge on the axial wavelength
($\lambda=2\pi/k$) of photospheric modes. This adopted range of $kR$ is
inferred primarily from the measurements of photospheric slow sausage modes.
Gathering the measurements for pores and sunspot umbrae of various sizes, one
finds that the period tends to range between $\sim 1.5$ and $20$ mins
(Fujimura & Tsuneta, 2009; Morton et al., 2011; Moreels et al., 2015a; Grant
et al., 2015; Freij et al., 2016; Keys et al., 2018; Gilchrist-Millar et al.,
2020). Boldly assuming that this periodicity applies to SSKMs in our
equilibrium, we find that $\lambda$ ranges from $\sim 500$ to $\sim 6500$ km
by approximating the axial phase speed $v_{\rm ph}$ with $c_{\rm Ti}$. We
gather further information from the spectropolarimetric measurements with Fe I
6173 Å and Ca II 8542 Å for a large sunspot as reported by Stangalini et al.
(2018). The phase delay between the oscillatory signals in these two lines was
found to peak at $\delta\phi\approx 0.5$ rad (Figure 3 therein). As suggested
by the same study, the difference between the formation heights of the two
lines ($d$) is $\sim 800$ km. Equating $2\pi d/\lambda$ to $\delta\phi$ leads
to a $\lambda\approx 10^{4}$ km. We therefore deem $\lambda$ to range from
$500$ to $10^{4}$ km, from which we then deduce that $kR\gtrsim 0.3$. This
deduction is admittedly idealized. To name but one caveat, in general one
should equal $2\pi d/\lambda$ to $2n\pi+\delta\phi$ ($n=0,1,2,\cdots$).
Letting $n=1$, one finds a $\lambda$ of $\sim 740$ km, which is drastically
different from the case where $n=0$. This subtlety is representative of multi-
wavelength studies on waves in the lower solar atmosphere.

It is necessary to address some technical details regarding Equation (18), for
which purpose we recall that only the parameters in the parentheses are seen
as adjustable. The ER83 results were established in ideal MHD ($\eta\equiv 0$)
and for a step transverse distribution ($l=0$). In this case, an SSKM can be
identified without ambiguity for a given $kR$. However, care needs to be
exercised in resistive MHD ($\eta\neq 0$), in which case a dense set of
solutions arise (see e.g., Figure 2 in Geeraerts et al. 2020 even though SSSMs
were examined therein). We would like to find a unique solution out of this
dense set, “unique” in the sense that it falls back to a standard SSKM as in
ER83. As in C18, for a target combination $[kR,l/R,R_{\rm m}]$, this is done
by first fixing $[kR,l/R]$ and then looking for the solution lying on the
curve that becomes independent on $R_{\rm m}$ when $R_{\rm m}$ is sufficiently
large. This procedure relies on two facts, one being that the $R_{\rm
m}$-independent eigen-frequencies pertain to resonantly damped collective
modes, and the other being that resonantly damped modes are physically
connected to their ER83 counterparts (see the review by Goossens et al. 2011;
see also Poedts & Kerner 1991, and Van Doorsselaere et al. 2004).

###  2.3 Resonantly Damped SSKMs in the Thin-Boundary Limit

When $R_{\rm m}$ is irrelevant in Equation (18), the functional dependence
therein can be established with a largely analytical approach in the TB limit
($l/R\ll 1$). In this case, the eigen-frequencies for resonantly damped SSKMs
are governed by the dispersion relation (DR) derived by Y17, which reads

$\displaystyle\begin{array}[]{rcl}&&\displaystyle\rho_{\rm
i}(\omega^{2}-k^{2}v_{\rm Ai}^{2})-\rho_{\rm e}(\omega^{2}-k^{2}v_{\rm
Ae}^{2})\frac{\mu_{\rm i}}{\mu_{\rm e}}Q_{1}\\\\[5.69046pt]
&&\displaystyle+i\pi\rho_{\rm i}\rho_{\rm e}(\omega^{2}-k^{2}v_{\rm
Ai}^{2})(\omega^{2}-k^{2}v_{\rm Ae}^{2})\frac{G_{1}}{\mu_{\rm
e}}\left[\frac{k^{2}}{\rho_{\rm c}|\Delta_{\rm c}|}\left(\frac{c^{2}_{\rm
sc}}{c^{2}_{\rm sc}+v^{2}_{\rm Ac}}\right)^{2}+\frac{1}{\rho_{\rm
A}|\Delta_{\rm A}|r^{2}_{\rm A}}\right]=0~{}.\end{array}$ (21)

Out of the many symbols, relevant here are the definitions for $\Delta_{\rm
c}$ and $\Delta_{\rm A}$,

$\displaystyle\displaystyle\Delta_{\rm c}=\left.\frac{{\rm
d}(\omega^{2}-k^{2}c_{\rm T}^{2})}{{\rm d}r}\right|_{r=r_{\rm
c}}~{},~{}~{}\Delta_{\rm A}=\left.\frac{{\rm d}(\omega^{2}-k^{2}v_{\rm
A}^{2})}{{\rm d}r}\right|_{r=r_{\rm A}}~{},$ (22)

which were first introduced by Sakurai et al. (1991). Here the subscript c
pertains to the cusp resonance, which takes place at $r_{\rm c}$ where
$\omega_{\rm R}=kc_{\rm T}$. Likewise, by the subscript A we refer to the
Alfv$\acute{\rm e}$n resonance, which takes place at $r_{\rm A}$ where
$\omega_{\rm R}=kv_{\rm A}$. We note that the terms in the square parentheses
derive from the jumps in the radial Lagrangian displacement ($[\tilde{\xi}]$)
across the resonant locations, and these jumps are responsible for SSKMs to be
resonantly damped (Goossens et al., 2011). As such, the importance of the cusp
resonance relative to the Alfv$\acute{\rm e}$n one can be measured by the
ratio ($X$) between the two terms, which equals $[\tilde{\xi}]_{\rm
c}/[\tilde{\xi}]_{\rm A}$ in the TB limit (see Equations 19 and 20 in Y17).

Equation (21) is also necessary to solve given that we would like to compare
the damping rates in the TB limit with those found with the resistive
approach. We start with an initial guess for $\omega$, thereby finding
temporary values for $r_{\rm c}$ and $r_{\rm A}$. The terms in the square
parentheses are then evaluated, which enables us to solve Equation (21) to
update $\omega$. With this updated $\omega$ as a further initial guess, this
process is iterated until convergence is met, thereby yielding a unique eigen-
frequency that depends only on $[kR,l/R]$.

###  2.4 Energetics of Linear Waves in Resistive MHD

It turns out necessary to examine the wave damping from the perspective of
energy balance. For this purpose, we dot Equations (7) and (8) with
$\bm{v}_{1}$ and $\bm{B}_{1}/\mu_{0}$, respectively. Likewise, we multiply
Equation (9) by $p_{1}/(\gamma p_{0})$. Adding up the resulting equations then
leads to a conservation law

$\displaystyle\frac{\partial\epsilon}{\partial t}=-\nabla\cdot\bm{f}-s_{\rm
ideal}-s_{\rm res}~{},$ (23)

where

$\displaystyle\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{1}{2}\rho_{0}\bm{v}_{1}^{2}+\frac{\bm{B}_{1}^{2}}{2\mu_{0}}+\frac{p_{1}^{2}}{2\gamma
p_{0}}~{},$ (24) $\displaystyle\displaystyle\bm{f}$ $\displaystyle=$
$\displaystyle p_{\rm
tot}\bm{v}_{1}-\frac{1}{\mu_{0}}(\bm{v}_{1}\cdot\bm{B}_{1})\bm{B}_{0}+\frac{\eta}{\mu_{0}}\bm{j}_{1}\times\bm{B}_{1}$
(25) $\displaystyle\displaystyle s_{\rm ideal}$ $\displaystyle=$
$\displaystyle\bm{j}_{0}\cdot(\bm{v}_{1}\times\bm{B}_{1})+\frac{1}{\gamma
p_{0}}p_{1}\bm{v}_{1}\cdot\nabla p_{0}~{},$ (26) $\displaystyle s_{\rm res}$
$\displaystyle=$
$\displaystyle\eta\left[\bm{j}_{1}^{2}-\frac{2(\gamma-1)}{\gamma}\frac{p_{1}\bm{j}_{1}\cdot\bm{j}_{0}}{p_{0}}\right]~{}.$
(27)

Furthermore, $p_{\rm tot}$ is the Eulerian perturbation of total pressure, and
$\bm{j}_{0}$ ($\bm{j}_{1}$) is the equilibrium (perturbed) electric current
density,

$\displaystyle\displaystyle p_{\rm
tot}=p_{1}+\frac{\bm{B}_{0}\cdot\bm{B}_{1}}{\mu_{0}}~{},~{}~{}~{}\bm{j}_{0}=\frac{\nabla\times\bm{B}_{0}}{\mu_{0}}~{},~{}~{}~{}\bm{j}_{1}=\frac{\nabla\times\bm{B}_{1}}{\mu_{0}}~{}.$
(28)

Integrating Equation (23) over some fixed volume $V$, one finds that

$\displaystyle\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{V}\epsilon{{\rm
d}}V=-\oint_{\partial V}{{\rm d}}\bm{A}\cdot\bm{f}-\int_{V}\left(s_{\rm
ideal}+s_{\rm res}\right){{\rm d}}V~{},$ (29)

where $\partial V$ is the surface that encloses $V$.

Some remarks on Equation (23) are needed. Firstly, within the framework of
resistive MHD, Equation (23) governs the energetics of arbitrary small-
amplitude wave-like perturbations in an arbitrary nonuniform medium, provided
that an equilibrium flow ($\bm{v}_{0}$) is absent and hence the equilibrium is
in force balance as governed by

$\displaystyle-\nabla
p_{0}(\bm{x})+\bm{j}_{0}(\bm{x})\times\bm{B}_{0}(\bm{x})=0~{}.$ (30)

Evidently, Equation (30) simplifies to Equation (1) for a radially structured
equilibrium. Secondly, the ideal version ($\eta=0$) of Equation (23) was first
derived by Bazer & Hurley (1963) to our knowledge. Well known is that
$\epsilon$ and $\bm{f}$ represent the wave energy and energy flux densities,
respectively. Less well known is the $s_{\rm ideal}$ term, for which Leroy
(1985) offered a physical interpretation where both terms on the right-hand
side (RHS) of Equation (26) are connected to the rate of work done on the
perturbed flow field ($\bm{v}_{1}$) by some perturbations to the
Amp$\acute{\rm e}$re force density. The first term results directly from
$\bm{j}_{0}\times\bm{B}_{1}$, while the second term derives from the
perturbation to the mass density of a fluid parcel that experiences the
unperturbed Amp$\acute{\rm e}$re force ($\bm{j}_{0}\times\bm{B}_{0}$, see
Equation 30). Evidently, $s_{\rm ideal}$ is relevant, at least in principle,
for wave energetics unless $\bm{B}_{0}$ is potential. Thirdly, introducing a
finite $\eta$ results in both the $s_{\rm res}$ term and an additional Poyning
flux associated with the conductive electric field (the last term on the RHS
of Equation 25). The first term on the RHS of Equation (27) evidently
represents the Joule dissipation rate, while the second term can be loosely
interpreted as resulting from the interaction between the perturbed
($\bm{j}_{1}$) and unperturbed ($\bm{j}_{0}$) electric current densities.
While a conservation law in the full form of Equation (23) is not available as
far as we know, we tend not to claim to have derived it because introducing
$\eta$ is rather straightforward. On top of that, energy balance can be
expressed in multiple ways or possibly even in an infinite number of ways (see
e.g., Bogdan et al., 2003, Section 4). It suffices to note only one example,
where one dots Equations (7) and (8) with $\bm{v}_{1}$ and
$\bm{B}_{1}/\mu_{0}$, respectively. Combining the resulting equations and
integrating over $V$, one ends up with an alternative conservation law as
derived in Poedts et al. (1989, Equation 4.3).

Our way for examining the wave energetics differs from relevant previous ones
in three aspects. Firstly, we take Equation (23) (or equivalently Equation 29)
as our starting point. Compared with the formulation in, say, Poedts et al.
(1989), Equation (23) has the advantage that $\epsilon$ and $\bm{f}$ have
clearer physical interpretations. For instance, involving Equation (9) in the
derivation, we have a clearly defined internal energy term in $\epsilon$.
Likewise, $\bm{f}$ then involves explicitly the well-known acoustic flux
$p_{1}\bm{v}_{1}$. Secondly, our examination of the wave energetics is
intended to expand our understanding of the damping rates, which are found
simultaneously with the eigen-functions. This is different from driven
problems as examined in, say, Poedts et al. (1989), where waves with real
frequencies are driven into the system and the rate of energy absorption in
the Alfv$\acute{\rm e}$n continuum is evaluated by working with the resistive
eigen-functions (see also Goossens et al., 2011, for more on driven problems).
Thirdly, our way is also different from the limited number of available
studies where the philosophy is similar (Arregui et al. 2011, Soler et al.
2013; see also Wright & Thompson 1994). Basically the point is that these
studies were exclusively conducted by assuming a vanishing gas pressure
because the focus was on the resonant absorption of coronal kink modes in the
Alfv$\acute{\rm e}$n continuum. It follows from this assumption that the
equilibrium magnetic field $\bm{B}_{0}$ is potential and hence $s_{\rm ideal}$
is not involved. In our equilibrium setup, $\bm{B}_{0}$ is necessarily non-
potential and hence $s_{\rm ideal}$ needs to be evaluated. On top of that, we
will examine both resonantly and resistively damped modes.

##  3 Numerical Results

###  3.1 Overview of Different Damping Regimes

Now we are in a position to examine the damping of SSKMs. For this purpose we
start with Figure 2, which shows how the dispersive properties of SSKMs depend
on the magnetic Reynolds number ($R_{\rm m}$) for a number of combinations of
$[l/R,kR]$. Presented in Figure 2a is the damping-time-to-period ratio
$\tau/P$, which derives from the real ($\omega_{\rm R}$, Figure 2b) and
imaginary ($\omega_{\rm I}$, Figure 2c) parts of the eigen-frequency. The area
shaded yellow pertains to the values of $R_{\rm m}$ between $10^{4}$ and
$6.6\times 10^{7}$, a range likely to apply to pores and sunspot umbrae. Note
that Figure 2 is a pictorial representation of Equation (18), which in turn
represents the numerical solutions to the EVP in resistive MHD as laid out in
Equations (11) to (17). As such, the damping of SSKMs is expected to
experience the combined effects due to Ohmic resistivity and resonant
absorption. Indeed, three regimes show up for all pairs of $[l/R,kR]$. When
$R_{\rm m}$ is sufficiently large, all curves in Figure 2a level off, a
defining signature for resonant absorption to be solely responsible for
damping collective modes (Poedts & Kerner, 1991). We therefore label this
regime as the “resonant regime”. Moving away from this regime toward smaller
$R_{\rm m}$, one sees that $\tau/P$ tends to decrease until a knee is reached,
beyond which $\tau/P$ possesses a simple linear $R_{\rm m}$-dependence. We
call this last regime the “Ohmic regime” because the damping of SSKMs can be
solely attributed to Ohmic resistivity. To show this, we note that
$\tau/P\propto\omega_{\rm R}/|\omega_{\rm I}|$. We note further that the axial
phase speeds ($v_{\rm ph}=\omega_{\rm R}/k$, Figure 2b) of SSKMs differ little
from $c_{\rm Ti}$ for all combinations of $[l/R,kR,R_{\rm m}]$ examined in
this study, meaning in particular that $v_{\rm ph}$ and hence $\omega_{\rm R}$
are essentially constants for a given $[l/R,kR]$ in the present context. On
the other hand, for a given $[l/R,kR]$, one intuitively expects that
$|\omega_{\rm I}|\propto\eta\propto 1/R_{\rm m}$ for weak damping, which is
indeed the case as demonstrated by Figure 2c. The end result is then
$\tau/P\propto R_{\rm m}$. The range of $R_{\rm m}$ that lies between the two
regimes is to be named the “intermediate regime”, where Ohmic resistivity and
resonant absorption both play a role for damping SSKMs. This intermediate
regime is not as clearly separated from the resonant regime as it is from the
Ohmic one. For definitiveness, we nonetheless regard the resonant regime to
start at an $R_{\rm m}$ beyond which $\tau/P$ varies by no more than $1.5\%$
over a decade or so.

Overall, the point to draw from Figure 2 is that in general the importance of
Ohmic resistivity relative to resonant absorption for damping SSKMs needs to
be assessed on a case-by-case basis, given the broad coverage of the shaded
area. That said, the damping rates can be order- or even orders-of-magnitude
larger than the ones deriving from resonant absorption when Ohmic resistivity
dominates, as happens for waveguides with, say, small sizes and/or weak
magnetic fields (see the definition for $R_{\rm m}$). This is true despite
that $R_{\rm m}$ consistently exceeds $10^{4}$, a value that seems to be
sufficiently large for one to propose resonant absorption as the primary
damping mechanism by drawing experience from pertinent coronal studies.
Indeed, when coronal radial fundamental kink modes are of interest, the
Alfv$\acute{\rm e}$n resonance tends to become the sole damping mechanism when
$R_{\rm m}\gtrsim 3\times 10^{3}-10^{5}$ (see e.g., Figure 2 in Terradas et
al. 2006; Figure 9 in Guo et al. 2016). These values are many orders-of-
magnitude smaller than typically accepted coronal values for $R_{\rm m}$,
making resonant absorption a leading mechanism to account for the damping of
the abundantly measured coronal kink modes (Goossens et al., 1992; Ruderman &
Roberts, 2002; Goossens et al., 2002). What Figure 2 highlights is that in
general Ohmic resistivity needs to be incorporated in studies of collective
modes in photospheric waveguides. This point has already been raised by both
C18 and Geeraerts et al. (2020) who examined photospheric SSSMs, and derives
from the considerable difference between the physical conditions of
photospheric waveguides and the coronal ones. As an example, we note that the
waveguides in this study are under-dense ($\rho_{\rm i}/\rho_{\rm e}<1$,
Figure 1b), whereas bright coronal loops are consistently over-dense
($\rho_{\rm i}/\rho_{\rm e}>1$).

The $R_{\rm m}$-dependence of the behavior of SSKMs can also be examined from
the perspective of energy balance. We start by rewriting the Fourier ansatz
(Equation 10) as

$\displaystyle g_{1}(r,\theta,z;t)={\rm Re}\\{\tilde{g}(r)\exp[-i(\omega_{\rm
R}t-kz-\theta)]\\}\exp(\omega_{\rm I}t)~{},$

such that the oscillatory ($\omega_{\rm R}$) and damping ($\omega_{\rm I}$)
time-dependencies are explicitly separated. One then finds that the product of
any two first-order quantities $g_{1}$ and $h_{1}$, when averaged over one
axial wavelength ($\lambda=2\pi/k$), evaluates to

$\displaystyle\left<g_{1}(r,\theta,z;t)h_{1}(r,\theta,z;t)\right>\equiv\frac{1}{\lambda}\int_{0}^{\lambda}g_{1}(r,\theta,z;t)h_{1}(r,\theta,z;t){{\rm
d}}z=\overline{g_{1}h_{1}}\exp(2\omega_{\rm I}t)~{},$ (31)

where

$\displaystyle\overline{g_{1}h_{1}}=\frac{1}{2}{\rm
Re}\left\\{\tilde{g}^{*}(r)\tilde{h}(r)\right\\}=\frac{1}{2}{\rm
Re}\left\\{\tilde{g}(r)\tilde{h}^{*}(r)\right\\}~{}.$ (32)

The asterisks represent the complex conjugate, and $\overline{g_{1}h_{1}}$ is
seen to depend only on $r$. Now take a volume $V$ to be the space bounded by
two transverse planes that are an axial wavelength apart. One readily finds
that the net wave energy flux leaving this volume vanishes, given that only
trapped modes are of interest (see the surface integral in Equation 29). After
some further algebra, Equation (29) becomes

$\displaystyle-2\omega_{\rm I}E=S_{\rm ideal}+S_{\rm res}~{},$ (33)

where

$\displaystyle\displaystyle
E=\int_{0}^{\infty}\bar{\epsilon}(r^{\prime})r^{\prime}{{\rm
d}}r^{\prime}~{},~{}~{}S_{\rm ideal}=\int_{0}^{\infty}\bar{s}_{\rm
ideal}(r^{\prime})r^{\prime}{{\rm d}}r^{\prime}~{},~{}~{}S_{\rm
res}=\int_{0}^{\infty}\bar{s}_{\rm res}(r^{\prime})r^{\prime}{{\rm
d}}r^{\prime}~{}.$ (34)

The barred quantities in the integrand can be readily evaluated by applying
Equation (32) to the products of first-order quantities involved in Equations
(24) to (27). Physically speaking, Equation (33) means that the wave damping
is realized in two channels, one through the Ohmic resistivity and the other
through the mutual interaction between the waves and the equilibria.

Figure 3 examines the $R_{\rm m}$-dependence of some key terms in Equation
(33) for the same set of combinations $[l/R,kR]$ as in Figure 2. The shaded
area is also inherited from there. Let us start with Figure 3a, where $(S_{\rm
ideal}+S_{\rm res})/E$ is plotted in conjunction with $-2\omega_{\rm I}$ (the
black diamonds). Note that the latter is essentially a direct output from our
code, whereas the former needs to be evaluated with the eigen-functions. With
Equation (33), one expects that the diamonds should lie on the curves, which
is seen to be indeed the case. In fact, a close agreement between
$-2\omega_{\rm I}$ and $(S_{\rm ideal}+S_{\rm res})/E$ is found for all values
of $R_{\rm m}$ at a given $[l/R,kR]$, with the diamonds representing only a
small subset in order not to obscure the curves. On the one hand, this
agreement validates the remarkable numerical accuracy of our computations. On
the other hand, the curves leave out the explicit $k$-dependence and hence are
less crowded than in Figure 2c, making the three regimes of wave damping more
evident. Now move on to Figure 3b, where $S_{\rm ideal}/S_{\rm res}$ is
plotted to evaluate the relative importance of the two terms. This evaluation
is informative in that as far as we know, the $S_{\rm ideal}$ term has not
been explicitly examined, at least not in the context of photospheric modes.
To proceed, we note that the $S_{\rm res}$ term is always positive for all the
computations because it is by far dominated by the Joule dissipation rate (see
Equation 27). In contrast, the $S_{\rm ideal}$ term can be either positive or
negative, thereby serving as a sink or source in physical terms. One further
sees that $S_{\rm res}$ dominates $S_{\rm ideal}$ when $R_{\rm m}$ is either
sufficiently large or sufficiently small, or equivalently when the modes are
very deep in either the Ohmic or the resonant regime. When $S_{\rm
ideal}/S_{\rm res}$ is appreciable, one sees that its magnitude tends to be
stronger for smaller values of $l/R$ at a given $kR$ (see the red curves with
different linestyles). Likewise, this magnitude tends to be stronger for
smaller $kR$ when $l/R$ is given (see the solid curves with different colors).
These details aside, we conclude that $S_{\rm ideal}$ needs to be considered
for examining the energetics of SSKMs in our model equilibria, given that
$|S_{\rm ideal}/S_{\rm res}|$ may reach $\sim 28\%$ in the case where
$[l/R,kR]=[0.1,0.7]$.

Some further understanding of the different damping regimes can be gained from
Figure 4, where $\bar{s}_{\rm res}$ is plotted as a function of $r$ for a
fixed pair of $[l/R,kR]=[0.1,0.7]$. Here the shaded area represents the
transition layer (TL). A number of values for $R_{\rm m}$ are adopted, and the
corresponding $\bar{s}_{\rm res}$ is rescaled such that its maximum attains
unity. Above all, one sees that $\bar{s}_{\rm res}$ almost vanishes
identically in the exterior. The black curve in Figure 4a pertains to $\lg
R_{\rm m}=4.3$, and is representative of the Ohmic regime. Most notable in
this case is that $\bar{s}_{\rm res}$ is extended over both the interior and
the TL. On the contrary, pertaining to an $R_{\rm m}$ fairly deep in the
resonant regime, the red curve in Figure 4b is characterized by the
concentration of $\bar{s}_{\rm res}$ in two very narrow layers. Evidently, the
inner and outer layers correspond to the cusp and Alfv$\acute{\rm e}$n
resonances, respectively. Given that the axial phase speeds of resonantly
damped SSKMs are close to the internal tube speed ($c_{\rm Ti}$, see Figure
2b), one would have expected the locations of the resonances in view of the
radial profiles of $c_{\rm T}$ and $v_{\rm A}$ (Figure 1). Examining the rest
of the curves in Figure 4, one sees that $\bar{s}_{\rm res}$ becomes
increasingly concentrated around the resonances as $R_{\rm m}$ increases. This
behavior also offers a qualitative explanation for the rather involved
behavior of $S_{\rm ideal}/S_{\rm res}$ shown in Figure 3b. For this purpose
we note that by definition, $S_{\rm ideal}$ ($S_{\rm res}$) collects all
contributions from $s_{\rm ideal}$ ($s_{\rm res}$) over the entire range of
$r$. However, $s_{\rm ideal}$ does not vanish only in the TL. If then follows
that $S_{\rm ideal}/S_{\rm res}$ is determined by two factors, one being how
wide $s_{\rm ideal}$ is distributed relative to $s_{\rm res}$ and the other
being how $s_{\rm ideal}$ compares with $s_{\rm res}$ at specific locations.
It turns out that at sufficiently small $R_{\rm m}$, the former factor is more
important, leading to small values of $S_{\rm ideal}/S_{\rm res}$ despite that
$s_{\rm ideal}$ can be comparable to $s_{\rm res}$ locally. When $R_{\rm m}$
is sufficiently large, the latter factor is more important, once again
resulting in small values of $S_{\rm ideal}/S_{\rm res}$ even though $s_{\rm
ideal}$ can be distributed in a more extended manner.

One may have noticed two peculiarities in Figure 2a, one being that some
curves do not extend to the largest $R_{\rm m}$ in the plot, the other being
that some curves are absent for some combinations of $[l/R,kR]$. Both result
from some limitations of our numerical method. Recall that, given a pair of
$[l/R,kR]$, we first look for the resonant regime and then vary $R_{\rm m}$
toward the target value such that a unique eigen-frequency can be found. Two
technical issues arise. When $R_{\rm m}$ is extremely large, sometimes the
computational resources required for resolving the extremely fine scales where
resonant absorption occurs may be too demanding. This explains the first
peculiarity. Another issue is that sometimes our code does not converge to a
unique solution when we reduce $R_{\rm m}$ from the resonant regime. This
happens when $kR$ is large for a given $l/R$ or when $l/R$ is large for a
given $kR$. Note that this is not to say that a proper resonant regime cannot
be found in these situations. It is just that for a given pair of $[l/R,kR]$,
the $\tau/P$ profile spans only a narrow range of $R_{\rm m}$ such that it is
not straightforward to address the importance of the Ohmic resistivity for
damping the SSKMs. These technical issues notwithstanding, we are allowed to
further examine resonant absorption by capitalizing on the computability of
the resonant regime for those pairs of $[l/R,kR]$ that cover a much broader
range than presented in Figure 2. This examination is important in its own
right, because resonant absorption is indeed the sole damping mechanism when
$R_{\rm m}$ is sufficiently large.

###  3.2 Resonantly Damped SSKMs

Figure 5 presents the $l/R$-dependence of (a) the damping-time-to-period-
ratios ($\tau/P$), (b) the axial phase speeds ($\omega_{\rm R}/k$), and (c)
the damping rates ($\omega_{\rm I}$) in units of $kv_{\rm Ai}$ for resonantly
damped SSKMs for a number of axial wavenumbers ($kR$). In addition to the
values computed with resistive MHD (the solid curves), Figure 5 also presents
the TB expectations (dashed) found by solving Equation (21). On top of that,
we have also plotted the results found by solving Equation (21) with the
second term in the square parentheses neglected (the dotted curves). For the
ease of description, let “full TB” and “cusp TB” refer to the dashed and
dotted curves, respectively. One may question why the “cusp TB” results are
shown together with the “full TB” ones. We will address this question shortly,
because it seems better to start with a comparison between the resistive and
“full TB” results. Examining Figure 5a, one sees that the smaller the value of
$l/R$, the smaller the difference between the solid and dashed curves, a
qualitative behavior implied in the TB approximation by construction. What
needs to be examined is how small an $l/R$ should be for the “full TB” results
to well agree with the resistive results for a given $kR$. We quantify this by
first defining $\epsilon\equiv|(\tau/P)^{\rm res}/(\tau/P)^{\rm TB}-1|$ to
measure the difference in the values for $\tau/P$ found with the two
approaches. We further define $(l/R)^{\rm TB}$ to be the critical value only
below which $\epsilon\leq 30\%$. One finds that in general $(l/R)^{\rm TB}$
decreases with $kR$, reading $0.074$ ($0.026$) for a $kR$ of $0.3$ ($4.3$).
This $k$-dependence of $(l/R)^{\rm TB}$ was also found for photospheric SSSMs
in C18, and can be explained by the same intuitive reasoning therein. After
all, with the axial wavelength ($\lambda=2\pi/k$) also a relevant lengthscale,
the TL width ($l$) should be much smaller than both $R$ and $\lambda$ for the
TB limit to hold. Somehow surprising is that $(l/R)^{\rm TB}$ is so small.
Indeed, $\epsilon$ evaluates to $2.6$ for an $l/R$ as small as $0.1$ when
$kR=4.3$. For this pair of $[l/R,kR]$, one sees that the substantial value for
$\epsilon$ results from the differences in the $l/R$-dependence of the
resistive and “full TB” profiles. While the $\tau/P$ profile in the TB limit
decreases monotonically with $l/R$ in the plotted range, its resistive
counterpart possesses a non-monotonic behavior as characterized by the
appearance of a minimal $\tau/P$. On the other hand, one sees from Figure 5b
that the axial phase speeds ($\omega_{\rm R}/k$) are consistently smaller than
$c_{\rm Ti}$ by only a small amount. It then follows that the behavior of
$\tau/P$, the existence of a minimal $\tau/P$ in particular, derives from the
$l/R$-dependence of the imaginary part of the eigen-frequencies ($\omega_{\rm
I}$, Figure 5c). We note that the existence of a minimal $\tau/P$ holds for
all the resistive computations we have conducted. We note further that this
non-monotonic $l/R$-dependence has been seen for the resonantly damped
photospheric SSSMs (C18) and coronal radial fundamental kink modes alike
(Soler et al., 2013, S13 hereafter). This S13 study approached the resonant
absorption of coronal kink modes in the Alfv$\acute{\rm e}$n continuum from
the ideal quasi-mode perspective with the Frobenius method. Furthermore, the
non-monotonic $l/R$-dependence of the damping rates for a particular
transverse profile (Figure 6 in S13) was interpreted from energetics
considerations by evaluating the energy flux injected into the resonance and
the integrated wave energy. In what follows, we analyze the $l/R$-dependence
of the damping rates of SSKMs from the energetics perspective as well.

We start by considering the entire volume bounded by two transverse planes
that are separated by an axial wavelength. We then exclude two cylindrical
shells, the inner shell bounded by $r_{\rm c}^{-}$ and $r_{\rm c}^{+}$, and
the outer one bounded by $r_{\rm A}^{-}$ and $r_{\rm A}^{+}$. Let $V$ denote
the resulting volume. The subscripts ${\rm c}$ and ${\rm A}$ are adopted
because these shells will be associated with the cusp and Alfv$\acute{\rm e}$n
resonances shortly. For now it suffices to assume that $r_{\rm c}^{-}<r_{\rm
c}^{+}<r_{\rm A}^{-}<r_{\rm A}^{+}$, and that the Ohmic resistivity is
negligible for determining the eigen-functions in $V$. Equation (29) then
becomes

$\displaystyle-2\omega_{\rm I}\hat{E}=\hat{F}_{\rm c}+\hat{F}_{\rm
A}+\hat{S}_{\rm ideal}~{},$ (35)

where

$\displaystyle\displaystyle\hat{E}$ $\displaystyle=$
$\displaystyle\left(\int_{0}^{r_{\rm c}^{-}}+\int_{r_{\rm c}^{+}}^{r_{\rm
A}^{-}}+\int_{r_{\rm
A}^{+}}^{\infty}\right)\bar{\epsilon}(r^{\prime})r^{\prime}{{\rm
d}}r^{\prime}~{},$ (36) $\displaystyle\hat{S}_{\rm ideal}$ $\displaystyle=$
$\displaystyle\left(\int_{0}^{r_{\rm c}^{-}}+\int_{r_{\rm c}^{+}}^{r_{\rm
A}^{-}}+\int_{r_{\rm A}^{+}}^{\infty}\right)\bar{s}_{\rm
ideal}(r^{\prime})r^{\prime}{{\rm d}}r^{\prime}~{},$ (37)
$\displaystyle\hat{F}_{\rm c}$ $\displaystyle=$ $\displaystyle r_{\rm
c}^{-}\bar{f}_{r}(r_{\rm c}^{-})-r_{\rm c}^{+}\bar{f}_{r}(r_{\rm
c}^{+})\equiv\left\\{r\bar{f}_{r}\right\\}_{\rm c}~{},$ (38)
$\displaystyle\hat{F}_{\rm A}$ $\displaystyle=$ $\displaystyle r_{\rm
A}^{-}\bar{f}_{r}(r_{\rm A}^{-})-r_{\rm A}^{+}\bar{f}_{r}(r_{\rm
A}^{+})\equiv\left\\{r\bar{f}_{r}\right\\}_{\rm A}~{}.$ (39)

Note that we have introduced the brace operator $\\{q\\}$ to evaluate the
difference of some quantity $q$ across a shell. In addition, $\bar{f}_{r}$ is
found by applying the bar operation (Equation 32) to the radial component of
the wave energy flux density ($\bm{f}$, Equation 25). In the absence of the
Ohmic resistivity, $f_{r}$ simplifies to $p_{\rm tot}v_{1r}$ and consequently
$\bar{f}_{r}={\rm Re}(\tilde{p}_{\rm tot}^{*}\tilde{v}_{r})/2$. Evidently,
Equation (35) means that the wave energy in the volume $V$ is lost via both
net fluxes into the shells ($\hat{F}_{\rm c}$ and $\hat{F}_{\rm A}$) and the
interactions with the equilibrium ($\hat{S}_{\rm ideal}$).

With Equation (35) we recall some features of the ideal MHD quasi-mode
approach such that some intricacies with our resistive MHD approach can be
appreciated. As far as resonantly damped modes are concerned, the ideal quasi-
mode approach has been exclusively conducted by assuming a vanishing gas
pressure (S13, and Soler 2017) 222Here by “ideal” we specifically refer to the
quasi-mode approach where dissipative effects are discarded from the outset.
To our knowledge, this approach has been realized only through the Frobenius
method formulated in Wright & Thompson (1994) and first practiced in the solar
context by S13. Some approaches can be regarded as nearly ideal, with the
formulation developed in Sakurai et al. (1991) as a much-employed prototype.
Restricting ourselves to Equation (21), we mean by “nearly ideal” that the
outcomes from these approaches do not explicitly involve dissipative effects
but dissipative MHD is nonetheless involved in the mathematical procedure. .
It then follows that only the Alfv$\acute{\rm e}$n resonance arises, and
$s_{\rm ideal}$ vanishes identically. On top of that, the Ohmic resistivity is
conceptually irrelevant in the first place, and resonant absorption can be
visualized as taking place in a resonant layer that is infinitely thin
($r_{\rm A}^{-}=r_{\rm A}^{+}$). In view of Equation (35), that $\omega_{\rm
I}$ does not vanish in S13 is then due to the discontinuity in $\bar{f}_{r}$
across the Alfv$\acute{\rm e}$n resonance, which in turn derives from the
discontinuities in $\tilde{p}_{\rm tot}$ and $\tilde{v}_{r}$ (see Figure 3
therein). Without concrete ideal MHD computations, we refrain from discussing
further how resonantly damped SSKMs behave when they are eventually found this
way. Rather, it suffices to note that the ideal eigen-frequencies are expected
to agree with their resistive counterparts but this is not true regarding the
eigen-functions. This latter aspect has some bearings on how we evaluate the
terms in Equation (35) with our resistive eigen-functions. The point is, if
neglecting the subscripts ${\rm c}$ and ${\rm A}$, then one finds that
Equation (35) holds for an arbitrary $V$ as long as resistivity can be
neglected therein. Evidently, we need to associate the two cylindrical shells
with the resonances for the evaluation to make physical sense. However, some
intricacies arise regarding how to determine the borders of the shells, known
as dissipative layers (DLs, e.g., Goossens et al., 2011). For the much-studied
Alfv$\acute{\rm e}$n resonance, the DL width ($\delta_{\rm A}$) is known to
take the larger one out of the resistive ($\delta_{\rm A,res}$) and non-
stationarity scales ($\delta_{\rm A,NS}$). Here $\delta_{\rm A,res}$ depends
on the Ohmic resistivity as $\eta^{1/3}$ (Sakurai et al., 1991), whereas
$\delta_{\rm A,NS}$ is independent of $\eta$ but proportional to $|\omega_{\rm
I}|$ (Ruderman et al., 1995; Tirry & Goossens, 1996). Regarding the DL width
pertaining to the cusp resonance ($\delta_{\rm c}$), a resistive spatial scale
($\delta_{\rm c,res}$) is also relevant and possesses the same
$\eta$-dependence (Sakurai et al., 1991). Likewise, the cusp version
($\delta_{\rm c,NS}$) of the non-stationarity scale was shown to be formally
the same as $\delta_{\rm A,NS}$ (Tirry et al., 1998, see the discussions in
connection to Equations 17 to 19 therein). It then follows that the DL widths
($\delta_{\rm c}$ and $\delta_{\rm A}$) in the present context will be $R_{\rm
m}$-independent when $R_{\rm m}$ becomes sufficiently large. In this case, no
$R_{\rm m}$-dependence will show up for the hatted terms in Equation (35),
making their evaluation more definitive. However, our code does not allow
$R_{\rm m}$ to increase indefinitely. This means in particular that
$\delta_{\rm A}$ never reaches the non-stationarity scale in our computations,
and the hatted terms in Equation (35) are in general dependent on $R_{\rm m}$.

So how do we evaluate the hatted terms, supposing a fixed pair of $[l/R,kR]$?
One possible way is to draw experience from the analytical study by Goossens
et al. (1995), who showed that

$\displaystyle r_{\rm A}^{\pm}\approx r_{\rm A}\pm 5\delta_{\rm A}~{},$ (40)

for the Alfv$\acute{\rm e}$n resonance, provided that $\delta_{\rm A}$ is
determined by $\delta_{\rm A,res}$. However, Equation (40) is no longer valid
when $\delta_{\rm A}$ is determined by $\delta_{\rm A,NS}$ (e.g., Tirry &
Goossens, 1996, Figure 1). We therefore choose an empirical procedure to
determine $r_{\rm A,c}^{\pm}$ for a given $R_{\rm m}$. The idea is simply that
by definition, resistivity is important only in the DLs, namely the two shells
in the present context. Basically, we compare the radial profile of some
eigen-function against that obtained at a substantially larger $R_{\rm
m}^{\prime}=\zeta_{R}R_{\rm m}$, and locate the shells by looking for where
the relative difference between the two profiles exceeds some critical value
$\zeta_{\rm cri}$. We consistently choose $|\tilde{v}_{r}|$ as the eigen-
function, and adopt a $\zeta_{R}=2$. In addition, we take $\zeta_{\rm cri}$ to
be $4\times 10^{-4}$ and $10^{-5}$ to determine the cusp and Alfv$\acute{\rm
e}$n DLs, respectively. Employing other reasonable values for $\zeta_{R}$ or
$\zeta_{\rm cri}$ leads only to insignificant differences to our results.
Figure 6 displays the radial profiles of ${\rm Re}\tilde{v}_{r}$ for a number
of $R_{\rm m}$ as labeled, with $[l/R,kR]$ fixed at $[0.2,2]$. Here two
different intervals are distinguished to show the details in the DLs
pertaining to the cusp (Figure 6a) and Alfv$\acute{\rm e}$n (Figure 6b)
resonances. Each profile is rescaled such that $\tilde{p}_{\rm tot}=1$ at
$r=1.25~{}R$, otherwise the comparison between different profiles will not
make sense. Note that the details for rescaling the eigen-functions are not
essential. Rather, with Figure 6a we show that the width of the cusp DL
($\delta_{\rm c}$) decreases when $R_{\rm m}$ increases, accompanied by the
appearance of more and more oscillations. This behavior is in line with what
was pointed out by Ruderman et al. (1995) and numerically demonstrated by S13
for coronal kink modes resonantly absorbed in the Alfv$\acute{\rm e}$n
continuum (see also Andries, 2003, Figure 2.3). If $R_{\rm m}$ further
increases, both studies showed that the width of the Alfv$\acute{\rm e}$n DL
($\delta_{\rm A}$) will eventually be determined by $\delta_{\rm A,NS}$. In
other words, eventually the Alfv$\acute{\rm e}$n DL width will become a
constant, and the effect of increasing $R_{\rm m}$ is to introduce an
increasing number of oscillations into the DL. We can discern some similar
signature for the cusp resonance here in that $\delta_{\rm c}$ is found to
depend on $R_{\rm m}$ only weakly when $\lg R_{\rm m}\gtrsim 9$. With Figure
6b as an illustration, we note that the Alfv$\acute{\rm e}$n DLs in all of our
computations consistently narrow when $R_{\rm m}$ increases, showing no
evidence that non-stationarity ultimately decides $\delta_{\rm A}$ for the
SSKMs in the $R_{\rm m}$-range that we examine.

Some remarks are necessary in this context. Our first set of remarks is
connected with Figure 6, from which one sees that the oscillations in the
DL(s) can be very strong when $R_{\rm m}$ is large. This does not invalidate
the examination of resonantly damped modes from the perspective of dissipative
eigen-modes. Rather, this means that if resonant absorption is examined from
the initial-value-problem (IVP) perspective, then the perturbation energy is
increasingly transferred from global coherent motions to localized motions
around resonant surfaces. For coronal radial fundamental kink modes, this was
established analytically by Ruderman & Roberts (2002) and has been shown by a
considerable number of numerical studies (e.g., Terradas et al., 2006; Soler &
Terradas, 2015; Ebrahimi et al., 2020; Guo et al., 2020, to name only a few).
To our knowledge, however, a similar study on photospheric modes from the IVP
perspective has yet to appear. Our second set of remarks pertains to Figure 2,
where the damping of SSKMs is seen to undergo a continuous transition from the
Ohmic to the resonant regime. We note that a similar problem was addressed for
surface Alfv$\acute{\rm e}$n waves in a one-dimensional slab equilibrium by
Ruderman & Goossens (1996), even though viscosity was adopted as the
dissipative factor therein. The transition was shown to be organized by a
dimensionless parameter ($R_{\rm g}$) that involves the relative variation of
the local Alfv$\acute{\rm e}$n speed, the ratio of the TL width to the axial
wavelength, and the viscous Reynolds number. Our Figure 2 plots the damping
rates as a function of $R_{\rm m}$ for a given pair of $[l/R,kR]$, showing
that different damping regimes can indeed be told apart. However, the range of
$R_{\rm m}$ pertaining to, say, the intermediate regime is different for
different combinations of $l/R$ and $kR$. If some $R_{\rm g}$ can be
established for photospheric modes in light of Ruderman & Goossens (1996),
then one expects a unified range of $R_{\rm g}$ for the intermediate regime
for different computations, leading to a substantially simpler way for
discriminating different damping regimes.

We are finally ready to examine the $l/R$-dependence of the damping rates with
the aid of Equation (35). It suffices to examine a fixed axial wavenumber of
$kR=2$. Figure 7 presents, with circles of different colors, the
$l/R$-dependencies of $\hat{F}_{\rm A}/\hat{E}$, $\hat{F}_{\rm c}/\hat{E}$,
and $\hat{S}_{\rm ideal}/\hat{E}$, as well as the sum of the three ratios for
a number of $R_{\rm m}$. Note that the range of $R_{\rm m}$ in general depends
on $l/R$. This is due to two reasons, one being that SSKMs at different $l/R$
in general enter into the resonant regime at different $R_{\rm m}$, and the
other being the technical difficulty for us to conduct computations when
$R_{\rm m}$ is too large. With the exception of the sum, the symbols
representing a given quantity for a given $R_{\rm m}$ are connected by a solid
line with the same color coding. At any examined $l/R$, one sees that the
circles of different colors representing the sum cannot be distinguished from
one another. This is understandable because they evaluate to $-2\omega_{\rm
I}$, which is $R_{\rm m}$-independent. In fact, replotting the pertinent set
of values for $\omega_{\rm I}$ with the dashed curve, we see the expected
behavior for the sums to agree with $-2\omega_{\rm I}$. On the other hand, one
sees that $\hat{F}_{\rm A}/\hat{E}$ increases monotonically with $l/R$ for all
the examined values of $R_{\rm m}$. Put another way, the wave damping due to
the Alfv$\acute{\rm e}$n resonance is stronger for a larger $l/R$. Two reasons
are therefore responsible for the apparently counter-intuitive behavior for
$|\omega_{\rm I}|$ to decrease with $l/R$ when $l/R\gtrsim 0.12$. Primarily,
it happens because the wave damping due to the cusp resonance behaves this
way. To a lesser but non-negligible extent, it happens because $\hat{S}_{\rm
ideal}/\hat{E}$ consistently weakens with $l/R$ in this range of $l/R$.

An important product of Figure 7 is that $\hat{F}_{\rm c}$ consistently
exceeds $\hat{F}_{\rm A}$, meaning that the cusp resonance is consistently
more important than the Alfv$\acute{\rm e}$n one for damping the SSKMs with
this particular $kR$. What about the relative importance for other values of
$kR$? Going back to Figure 5, one can most readily evaluate this for the TB
results, given the availability of an explicit dispersion relation (Equation
21). In this case, one is allowed to isolate, say, the cusp resonance by
keeping only the first term in the square parentheses, as we have done when
producing the “cusp TB” curves. Recall that the ratio of the first to the
second term is denoted by $X$. To proceed, let $\omega_{\rm I}^{\rm c}$ and
$\omega_{\rm I}^{\rm A}$ denote the damping rates due to the cusp and
Alfv$\acute{\rm e}$n resonances alone, respectively. Likewise, let
$\omega_{\rm I}^{\rm TB}$ denote the damping rate when both resonances are
accounted for. It then follows that $\omega_{\rm I}^{\rm TB}\approx\omega_{\rm
I}^{\rm c}+\omega_{\rm I}^{\rm A}$ and $\omega_{\rm I}^{\rm c}/\omega_{\rm
I}^{\rm A}\approx X$, when the damping is sufficiently weak. Now let $Y$
denote the ratio of the “cusp TB” value for $\tau/P$ to the “full TB” one. It
further follows that $Y\approx 1+1/X$ because $\tau/P\propto\omega_{\rm
R}/|\omega_{\rm I}|$. In other words, one expects that $Y<2$ when the cusp
resonance is stronger ($X>1$) and $Y\approx 1$ when it dominates ($X\gg 1$).
With this preparation, let us examine Figure 5a and focus on where the TB
limit holds ($l/R\leq(l/R)^{\rm TB}$). An inspection of the green curves
pertinent to $kR=0.7$ indicates that $Y$ happens to be $\approx 2$,
corresponding to the situation where the two resonances are equally important.
The cusp (Alfv$\acute{\rm e}$n) resonance becomes increasingly important when
$kR$ increases (decreases) from this critical value. Note that the transition
of the relative importance of the resonances consistently takes place at
$kR\approx 0.7$ when $l/R\leq(l/R)^{\rm TB}$. For now it suffices to consider
a given $l/R$. The existence of a critical $kR$ is not surprising in view of
Equation (21), which suggests that

$\displaystyle X=k^{2}\Lambda~{},$ (41)

where

$\displaystyle\displaystyle\Lambda=r_{\rm A}^{2}\frac{\rho_{\rm A}}{\rho_{\rm
c}}\frac{|\Delta_{\rm A}|}{|\Delta_{\rm c}|}\left(\frac{c^{2}_{\rm
sc}}{c^{2}_{\rm sc}+v^{2}_{\rm Ac}}\right)^{2}=r_{\rm A}^{2}\frac{\rho_{\rm
A}}{\rho_{\rm c}}\frac{|{\rm d}v_{\rm A}^{2}/{\rm d}r|_{r=r_{\rm A}}}{|{\rm
d}c_{\rm T}^{2}/{\rm d}r|_{r=r_{\rm c}}}\left(\frac{c^{2}_{\rm sc}}{c^{2}_{\rm
sc}+v^{2}_{\rm Ac}}\right)^{2}~{}.$ (42)

We arrive at the second equality by replacing $\Delta_{\rm A}$ and
$\Delta_{\rm c}$ with Equation (22). If one assumes that $r_{\rm A}\approx
r_{\rm c}\approx R$, then Equation (41) simplifies to Equation (14) in Soler
et al. (2009), where the effect of the cusp continuum was examined for damping
the transverse oscillations in prominence threads. Given the appearance of
$k^{2}$ in $X$, one would expect the cusp resonance to dominate for large
$kR$, an expectation that indeed holds. Less trivial is at which $kR$ the cusp
resonance starts to dominate. When $k$ varies, it turns out that the
$k$-dependence of $\Lambda$ derives essentially from the $k$-dependence of
$({\rm d}c_{\rm T}^{2}/{\rm d}r)_{r=r_{\rm c}}$. This subtlety results from
the extremely weak dispersion of SSKMs (see Figure 5b). With $\omega_{\rm
R}/k$ close to $c_{\rm Ti}$, neither $r_{\rm c}$ nor $r_{\rm A}$ is sensitive
to $k$, and hence the insensitivity of nearly all the terms involved in
$\Lambda$, with $({\rm d}c_{\rm T}^{2}/{\rm d}r)_{r=r_{\rm c}}$ being the only
exception. However weak its $k$-dependence is, $\omega_{\rm R}/k$ nonetheless
increases toward $c_{\rm Ti}$ with decreasing $k$. Therefore the cusp
resonance point ($r_{\rm c}$) moves toward the inner boundary of the
transition layer (see Figure 1a), thereby leading to some substantial decrease
in $|{\rm d}c_{\rm T}^{2}/{\rm d}r|_{r=r_{\rm c}}$ (see Equation 6) 333 It can
be readily shown that ${\rm d}c_{\rm T}^{2}/{\rm d}r\propto\delta
r\equiv(r-r_{\rm i})$ for a given $l/R$ when $0<|\delta r|\ll l$. Take
$l/R=0.03$ for instance. We find that $\delta r$ reads $[2.4,1.4,0.7]\times
10^{-3}~{}R$ for $kR=[2,0.7,0.3]$. These values of $\delta r$ are consistently
far below $l$, let alone $r_{\rm i}$. However, the decrease of $|{\rm d}c_{\rm
T}^{2}/{\rm d}r|_{r=r_{\rm c}}$ with decreasing $kR$ is substantial. . When it
comes to the $k$-dependence of $X$, the $k$-dependence of $({\rm d}c_{\rm
T}^{2}/{\rm d}r)_{r=r_{\rm c}}$ and hence that of $\Lambda$ somehow counteract
the $k^{2}$ term. While $X$ remains largely determined by $k^{2}$, this
subtlety means that the critical $kR\approx 0.7$ is specific to the
equilibrium configuration adopted in this study. The insensitivity of the
critical $kR$ to $l/R$, on the other hand, simply follows from the
$l/R$-insensitivity of $\omega_{\rm R}/k$ for a given $k$ (see Figure 5b).

When the TB approximation is not that satisfactory ($l/R>(l/R)^{\rm TB})$, one
needs to employ the resistive computations to examine the relative importance
of the resonances. While isolating one resonance is no longer possible, such
an examination remains possible because the importance of the cusp relative to
the Alfv$\acute{\rm e}$n resonance is quantified by $\hat{F}_{\rm
c}/\hat{F}_{\rm A}$ in view of Equation (35). At this point, we need to
clarify how $X$ in Equation (41) relates to $\hat{F}_{\rm c}/\hat{F}_{\rm A}$
to avoid possible confusion. For this purpose, we recall that the governing
equations on the primitive variables were sufficient for Sakurai et al. (1991)
to develop the mathematical prototype that enables the derivation of the TB
dispersion relation (21) where $X$ is involved. We recall further that
Equation (35) is valid regardless of the TB approximation. It is just that
$\hat{F}_{\rm c}/\hat{F}_{\rm A}$ can be explicitly expressed by Equation (41)
when the TB limit holds. To see this, we use Equations (38) and (39) to arrive
at

$\displaystyle\displaystyle\frac{\hat{F}_{\rm c}}{\hat{F}_{\rm
A}}=\frac{\left\\{r{\rm Re}(\tilde{p}_{\rm
tot}^{*}\tilde{v}_{r})\right\\}_{\rm c}}{\left\\{r{\rm Re}(\tilde{p}_{\rm
tot}^{*}\tilde{v}_{r})\right\\}_{\rm A}}~{},$ (43)

where the braces are recalled to evaluate the difference of some quantity
across a dissipation layer (DL) of finite width. In the TB limit, the DLs are
thin as well and $\tilde{p}_{\rm tot}$ is continuous across a DL (see Goossens
et al., 2011, for details). We can then assume that $r_{\rm c}^{-}\approx
r_{\rm c}^{+}\approx r_{\rm c}$, $r_{\rm A}^{-}\approx r_{\rm A}^{+}\approx
r_{\rm A}$, and replace the braces with a pair of square parentheses. Further
assuming that $r\tilde{p}_{\rm tot}$ is the same at the two resonances, one
finds that Equation (43) evaluates to $[\tilde{v}_{r}]_{r=r_{\rm
c}}/[\tilde{v}_{r}]_{r=r_{\rm A}}$ or equivalently
$[\tilde{\xi}_{r}]_{r=r_{\rm c}}/[\tilde{\xi}_{r}]_{r=r_{\rm A}}$ because
$\tilde{v}_{r}=-i\omega\tilde{\xi}$. The general expression (43) therefore
falls back to Equation (41). In other words, the relative importance
$\hat{F}_{\rm c}/\hat{F}_{\rm A}$ is fully determined by the jumps in the
radial speed in the TB limit. When the TB limit is not that good an
approximation, the relative importance can be roughly measured by how
$\\{\tilde{v}_{r}\\}_{\rm c}$ compares with $\\{\tilde{v}_{r}\\}_{\rm A}$ if
$\\{r\tilde{p}_{\rm tot}\\}_{\rm c}$ is not that different from
$\\{r\tilde{p}_{\rm tot}\\}_{\rm A}$.

Figure 8 presents the radial profiles of the Fourier amplitudes of some
relevant perturbations for an equilibrium profile with a TL-width-to-radius
ratio ($l/R$) of $0.2$. The vertical dash-dotted lines mark the boundaries of
the TL. The top row represents the Eulerian perturbation to total pressure
($\tilde{p}_{\rm tot}$), while the rest pertain to the radial
($\tilde{v}_{r}$), azimuthal ($\tilde{v}_{\theta}$), and axial
($\tilde{v}_{z}$) speeds. The eigen-functions are normalized such that
$\tilde{p}_{\rm tot}=1$ where $|\tilde{p}_{\rm tot}|$ attains its maximum. For
any eigen-function, we use the black and red curves to plot its real and
imaginary parts, respectively. Two axial wavenumbers ($kR$) are examined, one
being $0.3$ (the left column) and the other being $2$ (right). Note that the
resistive solutions for different values of $kR$ pertain to different magnetic
Reynolds numbers ($R_{\rm m}$). To be specific, for each $kR$, we choose an
$R_{\rm m}$ that is barely inside the resonant regime. The reason for doing
this is that when $R_{\rm m}$ is too large, $\tilde{v}_{r}$ in the DLs may be
too oscillatory for us to estimate the relative importance of the resonances.
Actually the oscillatory behavior can be seen in the lower two rows in Figure
8, from which the cusp (Alfv$\acute{\rm e}$n) resonance can be readily told by
the strong oscillations in $\tilde{v}_{z}$ ($\tilde{v}_{\theta}$). The so-
called $1/s$-singularities are also clearly visible. The presence of both
resonances is further seen in the plots for $\tilde{v}_{r}$, where both the
jumps across resonances (e.g., the black curves) and logarithmic singularities
(e.g., the red curves) are evident 444These singular profiles are only
apparent because the resistivity is nonetheless finite (see the review by
Goossens et al., 2011, for details).. In contrast, the curves for
$\tilde{p}_{\rm tot}$ are essentially continuous, as expected in the case of
weak damping. Comparing the two columns, one sees a change in the behavior of
the magnitude of $\\{\tilde{v}_{r}\\}_{\rm c}$ relative to
$\\{\tilde{v}_{r}\\}_{\rm A}$. While $\\{\tilde{v}_{r}\\}_{\rm c}$ is less
strong than $\\{\tilde{v}_{r}\\}_{\rm A}$ for $kR=0.3$, the opposite happens
for $kR=2$. One then expects that the cusp resonance is less important in the
former case, but is stronger in the latter.

Equation (35) can be used to perform a more quantitative evaluation of the
relative importance of the resonances as a function of the axial wavenumber
$kR$. For this purpose, we adopt a fixed $l/R=0.2$ and plot the ratios of the
relevant terms in Figure 9 in a format nearly identical to Figure 7. One sees
that the sum of the ratios, $(\hat{F}_{\rm c}+\hat{F}_{\rm A}+\hat{S}_{\rm
ideal})/\hat{E}$, agrees with the dashed curve representing the values of
$-2\omega_{\rm I}$, meaning that the computed eigen-functions are remarkably
accurate. More importantly, one sees that the cusp resonance surpasses the
Alfv$\acute{\rm e}$n one at $kR\sim 1$ in terms of the importance for damping
the SSKMs. That the cusp resonance dominates for large $kR$ is nothing
unexpected. What Figures 5 and 9 suggest is that the cusp resonance may not be
the dominant resonance throughout the entire $kR$ range that is likely to be
observationally relevant.

The relative importance of the two resonances aside, one may question how
efficient the resonant damping of SSKMs can be. For this purpose, we recall
that for each $kR$, there exists an $l/R$ that optimizes the resonant damping
(see Figure 5a; see also Yu & Van Doorsselaere 2019 for a relevant study on
coronal kink waves). Let $(l/R)_{\rm min}$ denote this $l/R$, and
$(\tau/P)_{\rm min}$ denote the value that the damping-time-to-period ratio
attains therein. Figure 10 presents both $(l/R)_{\rm min}$ (the red curve) and
$(\tau/P)_{\rm min}$ (green) as a function of $kR$. One sees that $(l/R)_{\rm
min}$ decreases with $kR$, reaching values below $0.02$ when $kR\gtrsim 16$.
This inspired us to examine the TB expectations at a given $(l/R)_{\rm min}$,
and the corresponding results are shown by the blue curve labeled
$(\tau/P)^{\rm TB}$. Despite that $(l/R)_{\rm min}$ eventually tends to be
small, the TB expectations are seen to consistently overestimate the resonant
damping rather considerably. In other words, at a given $kR$, the TB limit
holds only for $l/R$ that is even smaller than $(l/R)_{\rm min}$ (see Figure
5a). One further sees that when $kR$ increases, the resistive value for
$(\tau/P)_{\rm min}$ rapidly decreases at first but levels off when $kR$
exceeds, say, $10$. The asymptotic value that $(\tau/P)_{\rm min}$ attains at
$kR=30$ is merely $\sim 70$. From this we conclude that, in the
observationally relevant range of $kR$, resonant absorption does not play an
important role for damping the SSKMs in our equilibrium setup.

##  4 SUMMARY

Motivated by the considerable interest in solar photospheric collective modes,
we have examined the damping of slow surface kink modes (SSKMs) in
photospheric waveguides with equilibrium quantities representative of pores
and sunspot umbrae. We adopted a cylindrical equilibrium configuration
comprising a uniform interior, a uniform exterior, and a transition layer (TL)
that continuously connects the two. By formulating the relevant eigenvalue
problem in the framework of resistive, linear MHD, we were allowed to
simultaneously account for two mechanisms for damping SSKMs in a self-
consistent manner, one being the Ohmic resistivity, and the other being the
resonant absorption of SSKMs in the cusp and Alfv$\acute{\rm e}$n continua. We
find that the diversity of the geometric and physical parameters of
photospheric waveguides, wrapped up in the magnetic Reynolds number ($R_{\rm
m}$), means that the relative importance of the two mechanisms needs to be
assessed on a case-by-case basis. At realistically small values of $R_{\rm
m}$, the Ohmic resistivity plays a far more important role, resulting in
damping-time-to-period-ratios $\tau/P\sim 10$ in the parameter range we
examine. In contrast, resonant absorption is the only damping mechanism for
some realistically large $R_{\rm m}$. In this latter case, the cusp resonance
in general dominates the Alfv$\acute{\rm e}$n one except for very long axial
wavelengths. Regardless, in the observationally relevant wavelength range, the
resonant damping rate turns out to be negligible, leading to values for
$\tau/P$ to consistently exceed $\sim 70$.

When SSKMs are damped solely by resonant absorption, we have compared the
damping rates obtained by the resistive approach with those found by solving
the explicit dispersion relation valid in the thin-boundary (TB) limit. In
general we find that TB values agree with the resistive ones only when the
transition-layer-width-to-radius ratios ($l/R$) are much smaller than the
nominal values for the TB limit to apply. This is particularly true for short
axial wavelengths, which are more observationally relevant. From the practical
point, this means that when the resonant damping of SSKMs is of interest, one
needs to adopt approaches such as the resistive one, despite that the TB limit
proves much less computationally expensive.

Before closing, let us name a limited number of intricacies that are likely
inherent in theoretical studies on photospheric modes by contrasting our
findings with representative coronal studies on radial fundamental kink modes.
One, the range of $R_{\rm m}$ where resonant absorption is expected to
dominate for damping collective modes. Drawing experience from the coronal
studies by, say, Terradas et al. (2006) and Guo et al. (2016), one is inclined
to propose that photospheric SSKMs are damped primarily by resonant absorption
given that $R_{\rm m}$ tends to exceed $\sim 10^{4}$. However, this is not the
case. Two, the range of applicability of the TB limit regarding the resonant
damping of collective modes. While the TB limit is constructed to hold only
when $l/R$ is small, the range of its applicability is drastically different
in photospheric cases from that in coronal cases. In coronal studies, the TB
expectations for the resonant damping rates of kink modes tend to be a good
approximation well beyond the nominal range of applicability (e.g., Van
Doorsselaere et al., 2004; Soler et al., 2014). In particular, Figure 1 in the
latter study indicates that the true values for $\tau/P$ deviate from the TB
ones by no larger than $80\%$ even when $l/R$ reaches two, the maximum allowed
by the equilibrium configuration. And this deviation is $\lesssim 20\%$ when
$l/R\lesssim 0.2$. This is not the case for photospheric SSKMs, for which the
relative deviation can readily exceed unity even for $l/R$ as small as, say,
$0.1$. We take these two intricacies as strengthening rather than weakening
the need to further the theoretical examinations on photospheric modes. For
instance, the second intricacy means that other approaches like the one
outlined for coronal kink modes by Soler et al. (2013) can readily find
photospheric applications. Likewise, the first intricacy means that the
present study on the non-ideal mechanisms for damping SSKMs needs to be taken
further. Drawing experience from the study by Geeraerts et al. (2020) on the
damping of photospheric slow surface sausage modes (SSSMs) , we note that the
Ohmic resistivity can be so efficient that SSSMs are damped within a fraction
of the wave period even if $R_{\rm m}$ remains $\gtrsim 10^{4}$ (Figure 3
therein). While this was found for photospheric waveguides with piece-wise
constant transverse distributions of the equilibrium quantities, there seems
to be no reason to believe that photospheric waveguides cannot be transversely
structured this way.

Some remarks on the caveats of our approach are also necessary, for which
purpose we start by noting that in reality the solar photosphere is structured
in a way much more complicated than modeled here. In fact, this complication
is to such an extent that we have to restrict the following discussions to
only one additional factor that is expected to be important in the wave
dynamics. This factor, the effect of gravity, was neglected in our study from
the outset. As a consequence, essentially only three spatial scales were
relevant, namely the waveguide radius ($R$), the TL width ($l$), and the axial
wavelength ($\lambda=2\pi/k$). Introducing gravity considerably complicates
the problem, some aspects of which were summarized by Roberts (2019, Chapters
9 to 11 and references therein). For our purpose, it suffices to assume that
the gravitational acceleration is aligned with the waveguide axis. A number of
additional spatial scales become relevant even if we assume that neither the
equilibrium temperature nor the mean molecular weight varies much over
photospheric heights. These scales characterize how rapid the equilibrium
pressure and magnetic field strength vary. Let $H$ denote the shortest one
among these scales. Given the axial stratification, the axial wavelength
($\lambda$) cannot be unambiguously defined, but nonetheless can be understood
as the spatial scale characterizing the axial variation of the perturbations.
For our results to apply, the condition $\lambda\ll H$ should be satisfied but
proves quite demanding for the majority of our computations in view of the
small values of $H$ at photospheric temperatures. It then follows that, in
principle our results apply only to ER83-like photospheric waveguides, by
which we mean the idealized configurations where the equilibrium quantities
are structured only transversely and the involved characteristic speeds follow
some canonical ordering. Note that here we distinguish between the ER83-like
and ER83 equilibria by one key difference, namely the transverse structuring
is allowed to be continuous in the former as realized by a non-vanishing $l$
in this study. With this distinction we note that all the referenced
observational studies on photospheric modes have adopted the ER83 framework to
aid the mode identification and/or the evaluation of energetics. Focusing on
wave damping, we believe that our computations contributed to the better
understanding of photospheric collective modes, “photospheric modes” in the
sense as practiced in available observational and theoretical efforts. In
addition, our results can be of help for further theoretical/numerical studies
that address additional realistic effects on photospheric modes. For instance,
if one adopts a theoretical framework that extends ours by accounting for
gravity, then our results can be used to validate the pertinent computations
by specializing to the cases where $\lambda\ll H$. We believe that such a
validation is warranted, given that such further theoretical developments are
necessarily more involved than presented here. That said, we concede that it
is difficult to assess how well our results carry over to the extremely
complicated photospheric structures (see also Roberts, 2019, the introduction
sections in Chapters 9 and 10).

With the above-mentioned caveats in mind, let us discuss whether our results
can help constrain the likely damping mechanism(s) for an identified SSKM. We
start by noting that, in the case of weak damping, the damping-time-to-period
ratios $\tau/P$ for temporally damped modes are nearly identical to the
damping-length-to-wavelength ratios $L_{\rm D}/\lambda$ for spatially damped
ones if the wave dispersion is weak (see e.g., Terradas et al., 2010, Equation
40). In view of the extremely weak dispersion of SSKMs, this means that the
following discussions apply to SSKMs that experience either temporal or
spatial damping. For the ease of discussion, let us slightly generalize
Equation (18) by counting all possible physical quantities that may influence
the dispersive properties of SSKMs in our framework. Some straightforward
dimensional analysis yields that the complex-valued frequency $\omega$ can be
expressed as

$\displaystyle\displaystyle\frac{\omega R}{v_{\rm Ai}}={\cal
H}\left(\frac{c_{\rm si}}{v_{\rm Ai}},\frac{c_{\rm se}}{v_{\rm
Ai}},\frac{v_{\rm Ae}}{v_{\rm Ai}},\frac{l}{R},R_{\rm m};kR\right)~{},$ (44)

where the function ${\cal H}$ is dictated by the eigen-value problem. Note
that the arguments on the RHS of Equation (44) are all dimensionless, and are
independent from one another. One may further formally express $\tau/P$ (or
equivalently $L_{\rm D}/\lambda$) as

$\displaystyle\displaystyle\frac{\tau}{P}={\cal L}\left(\frac{c_{\rm
si}}{v_{\rm Ai}},\frac{c_{\rm se}}{v_{\rm Ai}},\frac{v_{\rm Ae}}{v_{\rm
Ai}},\frac{l}{R},R_{\rm m};kR\right)~{}.$ (45)

To proceed, let us assume that the set of $[c_{\rm si}/v_{\rm Ai},c_{\rm
se}/v_{\rm Ai},v_{\rm Ae}/v_{\rm Ai}]$ is measurable. Our Figure 10 then
indicates that $\tau/P$ cannot be lower than the asymptotic value
$(\tau/P)_{\rm min,asymp}$ that the green curve attains at large $kR$,
provided that resonant absorption is the sole damping mechanism. Conversely,
the wave damping cannot be attributed to resonant absorption if $\tau/P$ is
observed to be smaller than $(\tau/P)_{\rm min,asymp}$, which depends only on
$[c_{\rm si}/v_{\rm Ai},c_{\rm se}/v_{\rm Ai},v_{\rm Ae}/v_{\rm Ai}]$. In this
case, one is allowed to rule out resonant absorption even without knowing any
quantity in the set $[l/R,R_{\rm m},kR]$. The situation is much more involved
when the observed $\tau/P$ exceeds $(\tau/P)_{\rm min,asymp}$, and one needs
to survey the combinations of $[l/R,R_{\rm m},kR]$ (see Figure 2). However, it
remains possible to say a few words on the likely damping mechanism(s). For
this purpose, we further assume that $l/R$ and $kR$ are known. Note that it is
non-trivial to evaluate the electric resistivity ($\eta$) in a photospheric
medium (e.g., Wang, 1993; Khomenko & Collados, 2012; Ni et al., 2016).
Nonetheless, one can rather safely assume that the resulting $R_{\rm m}$ lies
in the range pertaining to the shaded area in Figure 2, given that this range
encompasses the rather extreme values of the waveguide radius ($R$). With a
known set of $[c_{\rm si}/v_{\rm Ai},c_{\rm se}/v_{\rm Ai},v_{\rm Ae}/v_{\rm
Ai},l/R;kR]$, one may construct the $R_{\rm m}$-dependence of $\tau/P$,
thereby finding the range of $\tau/P$ allowed by the range of $R_{\rm m}$. Now
two situations arise. If the observed $\tau/P$ lies in this range, then one is
allowed to constrain the possible $R_{\rm m}$. If the measured $\tau/P$ is
outside this range, then the wave damping can be attributed to neither the
Ohmic resistivity nor the resonant absorption. To name but one additional
mechanism, we note that we have adopted the values for the electric
conductivity from Kovitya & Cram (1983). As such, only the Ohmic resistivity
due to electron-neutral and electron-ion collisions was addressed. In reality,
the Cowling resistivity due to ion-neutral collisions may be substantially
stronger (e.g., Wang, 1993, Figure 1). A detailed study on the effects of the
Cowling resistivity on photospheric modes is certainly warranted, but is
nonetheless left for a future work.

We thank both referees for their comments, which helped improve the manuscript
substantially. This research was supported by the National Natural Science
Foundation of China (BL: 41674172, 11761141002, 41974200; SXC:41604145,
HY:41704165). TVD was supported by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 724326) and the C1 grant TRACEspace of Internal Funds KU Leuven.
This research benefited greatly from the discussions at ISSI-BJ. This research
was also supported by Shandong University via grant No 2017JQ07.

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Figure 1: Transverse profiles for (a) the characteristic speeds and (b)
primitive variables of a photospheric waveguide representative of pores and
sunspot umbrae. These profiles are generated by specifying the transverse
distributions of the adiabatic sound ($c_{\rm s}$) and cusp ($c_{\rm T}$)
speeds, for which a continuous transition layer (TL) connects a uniform
interior to a uniform exterior (see Equation 6). This TL is located between
$r_{\rm i}=R-l/2$ and $r_{\rm e}=R+l/2$, where $R$ is the mean waveguide
radius and $l$ the TL width. For illustration purposes, here $l/R$ is
arbitrarily chosen to be $0.2$.

Figure 2:  Dependence on the magnetic Reynolds number ($R_{\rm m}$) of the
dispersive properties of slow surface kink modes in a photospheric waveguide.
Plotted are (a) the damping-time-to-period ratio $\tau/P$, (b) the axial phase
speed $\omega_{\rm R}/k$, and (c) the damping rate $\omega_{\rm I}$ in units
of $kv_{\rm Ai}$. A number of combinations of the transition layer width
($l/R$) and axial wavenumber ($kR$) are examined as labeled. The area shaded
yellow corresponds to the range of $R_{\rm m}$ that is likely to be relevant
for pores and sunspot umbrae. See the text for details.

Figure 3:  Dependence on the magnetic Reynolds number ($R_{\rm m}$) of the
terms involved in the energy balance (see Equation 33 and the associated
definitions). A number of combinations of the transition layer width ($l/R$)
and axial wavenumber ($kR$) are examined as labeled. The area shaded yellow
corresponds to the range of $R_{\rm m}$ that is likely to be relevant for
pores and sunspot umbrae. The diamonds represent $-2\omega_{\rm I}$, taken
from a small subset of the values presented in Figure 2c. See the text for
details.

Figure 4:  Radial distributions of the $\bar{s}_{\rm res}$ term for a number
of representative values of the magnetic Reynolds number ($R_{\rm m}$) at a
given pair of $[l/R,kR]=[0.1,0.7]$. Here $\bar{s}_{\rm res}$ is evaluated by
applying the bar operation (Equation 32) to the definition of $s_{\rm res}$
(Equation 27). The shaded area represents where the equilibrium quantities are
nonuniform. See the text for details.

Figure 5:  Dispersive properties of resonantly damped slow surface kink modes,
shown by the dependence on the dimensionless layer width ($l/R$) of (a) the
damping-time-to-period ratios ($\tau/P$), (b) the axial phase speeds
($\omega_{\rm R}/k$), and (c) the damping rates ($\omega_{\rm I}$) in units of
$kv_{\rm Ai}$. A number of axial wavenumbers ($kR$) are examined as
represented by the different colors. The results from self-consistent
resistive computations are given by the solid curves. For each pair of
$[l/R,kR]$, the explicit dispersion relation (Equation 21) in the thin-
boundary limit is also solved. The dashed curves represent the computations
where both the cusp and Alfv$\acute{\rm e}$n resonances are accounted for,
whereas the dotted curves take into account only the cusp resonance.

Figure 6:  Radial distributions of the real part of the Fourier amplitude of
the radial flow speed (${\rm Re}\tilde{v}_{r}$) in the intervals embracing (a)
the cusp, and (b) the Alfv$\acute{\rm e}$n dissipative layers. A number of
magnetic Reynolds numbers ($R_{\rm m}$) are examined as labeled, whereas
$[l/R,kR]$ is fixed at $[0.2,2]$. The eigenfunctions are rescaled such that
$\tilde{p}_{\rm tot}=1$ at $r=1.25~{}R$.

Figure 7:  Dependence on the transition layer width ($l/R$) of the ratios of
the hatted terms in Equation (35). The axial wavenumber is fixed at $kR=2$.
For a given $l/R$, a number of magnetic Reynolds numbers ($R_{\rm m}$) are
examined to evaluate the hatted terms as represented by the symbols of
different colors. With solid curves we connect the symbols representing
$\hat{F}_{\rm c}$, $\hat{F}_{\rm A}$, and $\hat{S}_{\rm ideal}$ for the same
$R_{\rm m}$ but different $l/R$. The dashed curve represents $-2\omega_{\rm
I}$, the values of which are direct outputs of our code. See the text for
details.

Figure 8:  Radial profiles of some relevant eigen-functions found in self-
consistent resistive computations for resonantly damped slow surface kink
modes. Two different values for the axial wavenumber are examined, one
corresponding to $kR=0.3$ (the left column) and the other to $kR=2$ (right).
The vertical dash-dotted lines mark the boundaries of the transition layer,
which corresponds to a layer-width-to-radius-ratio of $l/R=0.2$. The black
(red) curves correspond to the real (imaginary) parts. The top row pertains to
the Eulerian perturbation to total pressure ($\tilde{p}_{\rm tot}$), while the
rest correspond to the radial ($\tilde{v}_{r}$), azimuthal
($\tilde{v}_{\theta}$), and axial ($\tilde{v}_{z}$) speeds. The eigen-
functions are normalized such that $\tilde{p}_{\rm tot}=1$ where its modulus
attains maximum. Different values for the magnetic Reynolds number ($R_{\rm
m}$) are adopted in the two columns, see the text for details.

Figure 9:  Similar to Figure 7 except that the dependence on the axial
wavenumber ($kR$) is examined. Here the dimensionless transition layer width
is fixed at $l/R=0.2$.

Figure 10:  Dependence on the axial wavenumber ($k$) of the optimal damping-
time-to-period ratio $(\tau/P)_{\min}$ and the transition-layer-width-to-
radius ratio $(l/R)_{\min}$ where the optimal damping is reached. The values
for $(\tau/P)_{\min}$ and $(l/R)_{\min}$ are derived from self-consistent
resistive computations. At an $(l/R)_{\min}$ thus derived, the explicit
dispersion relation in the TB limit (Equation 21) is also solved, yielding
$(\tau/P)^{\rm TB}$ for comparison. See the text for details.