# Standing Sausage Perturbations in Solar Coronal Slabs with Continuous
Transverse Density Profiles: cutoff wavenumbers, evanescent eigenmodes, and
oscillatory continuum

Zexing Wang  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  Shao-Xia Chen  Shandong
Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial
Environment, Institute of Space Sciences, Shandong University, Weihai 264209,
China  Mijie Shi  Shandong Provincial Key Laboratory of Optical Astronomy and
Solar-Terrestrial Environment, Institute of Space Sciences, Shandong
University, Weihai 264209, China

###### Abstract

The lack of observed sausage perturbations in solar active region loops is
customarily attributed to the relevance of cutoff axial wavenumbers and the
consequent absence of trapped modes (called “evanescent eigenmodes” here).
However, some recent eigenvalue problem studies yield that cutoff wavenumbers
may disappear for those equilibria where the external density varies
sufficiently slowly, thereby casting doubt on the rarity of candidate sausage
perturbations. We examine the responses of straight, transversely structured,
coronal slabs to small-amplitude sausage-type perturbations that excite axial
fundamentals by solving the pertinent initial value problem with
eigensolutions for a closed domain. The density variation in the slab exterior
is dictated by some steepness parameter $\mu$, and cutoff wavenumbers are
theoretically expected to be present (absent) when $\mu\geq 2$ ($\mu<2$).
However, our numerical results show no qualitative difference in the system
evolution when $\mu$ varies, despite the differences in the modal behavior.
Only oscillatory eigenmodes are permitted when $\mu\geq 2$. Our discrete
eigenspectrum becomes increasingly closely spaced when the domain broadens,
and an oscillatory continuum results for a truly open system. Oscillatory
eigenmodes remain allowed and dominate the system evolution when $\mu<2$. We
show that the irrelevance of cutoff wavenumbers does not mean that all fast
waves are evanescent. Rather, it means that an increasing number of evanescent
eigenmodes emerge when the domain size increases. We conclude that sausage
perturbations remain difficult to detect even for the waveguide formulated
here.

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

††journal: ApJ

##  1 INTRODUCTION

Solar coronal seismology (SCS) is an enterprise that unites theories and
observations, as was clear in the early studies where SCS was practiced
implicitly (e.g., Billings, 1959; Parks & Winckler, 1969; Rosenberg, 1970) or
advocated explicitly (e.g., Uchida, 1970; Roberts et al., 1984). Modern SCS
remains to rely on the wealth of measurements of low-frequency waves and
oscillations in the structured solar corona (see the reviews by e.g.,
Nakariakov & Verwichte 2005; De Moortel & Nakariakov 2012; Nakariakov &
Kolotkov 2020). Also key to SCS is the increasing refinement of
magnetohydrodynamic (MHD) theories for waves in structured media (see the
reviews by e.g., Roberts 2000; Goossens et al. 2011; Mathioudakis et al. 2013;
Nakariakov et al. 2016; also the textbooks by Goedbloed et al. 2019; Roberts
2019). Such terms as kink and sausage modes have been in routine use since
their introduction to the solar context for cylindrical (Edwin & Roberts 1983;
see also Zajtsev & Stepanov 1975; Spruit 1982; Cally 1986) and slab equilibria
(Edwin & Roberts 1982; see also Ionson 1978; Wentzel 1979). Observationally,
kink waves and oscillations have been frequently imaged in extreme ultraviolet
(EUV) in both magnetically open (see the review by Banerjee et al., 2021, and
references therein) and magnetically closed structures (see Nakariakov et al.
2021 for a recent review; see also Zimovets & Nakariakov 2015; Nechaeva et al.
2019 for catalogs). However, the reported instances of coronal fast sausage
waves are only sporadic and exclusively connected to quasi-periodic pulsations
(QPPs) in solar flares (see e.g., Van Doorsselaere et al., 2016; McLaughlin et
al., 2018; Li et al., 2020; Zimovets et al., 2021, for recent reviews). In
particular, they have not been reported or even implicated in oscillating
active region (AR) loops to our knowledge. This study is intended to address
the rarity of fast sausage waves in AR loops, emphasizing conceptual
understandings rather than digging into the seismological potential.

It proves necessary to detail the pertinent nomenclature. We adopt linear,
pressureless, ideal, MHD throughout, focusing on straight, field-aligned
equilibria that are structured only in one transverse direction. Only axial
standing sausage waves are of interest, with the axial wavenumber denoted by
$k$. The necessary terms are worded in a way that they apply to both slab and
cylindrical configurations, in which some interior (denoted by the subscript
${\rm i}$) is density-enhanced relative to its exterior (subscript ${\rm e}$).
Consider the cylindrical geometry for now. By $r$ and $R$ we denote the
transverse coordinate and the nominal cylinder radius, respectively. By
“modes” we broadly refer to the solutions to the pertinent eigenvalue problems
(EVPs) defined on the open domain $[0,\infty)$, with a mode jointly
characterized by some “mode frequency” and “mode function”. A mode is not
necessarily an eigenmode despite the broad term “eigenvalue problem”. We
reserve the term “eigenmodes” for those that qualify as such, deeming
“eigenfrequencies” and “eigenfunctions” as applicable only to eigenmodes 111In
our context, the distinction between modes and eigenmodes is necessary only
for EVPs defined on an open domain given the subtleties with the boundary
condition (BC) at infinity. The specification of BCs is straightforward if a
closed domain is adopted, all solutions qualifying as eigenmodes in terms of
orthogonality and completeness (e.g., Courant & Hilbert, 1989).. Eigenvalues
are necessarily real-valued in ideal MHD (see e.g., Andries & Goossens 2007;
Oliver et al. 2014, 2015 hereafter ; ; , respectively) 222 This does not
contradict the notion of ideal quasi-modes (see Chapter 10 in Goedbloed et al.
2019; also Section 3 in Goossens et al. 2014). Take resonantly damped kink
motions in a cylinder where the density varies continuously over some
transition layer (TL) around the boundary. For thin TLs, Soler & Terradas
(2015) showed that the coordinated kink motions can maintain their global
character, making their temporal variation describable by the complex
frequencies computed with either the dissipative eigenmode approach (e.g.,
Poedts & Kerner, 1991) or the ideal quasi-mode approach (e.g., Soler et al.,
2013). However, the meaning of quasi-modes becomes less and less clear when
the TL width increases.. Eigenfunctions can then be made real-valued, enabling
an eigenmode to be classified as either evanescent or oscillatory. By
“evanescent eigenmodes” we refer to those for which the eigenfunctions in the
exterior are evanescent somewhere, whereas by “oscillatory eigenmodes” we
refer to those whose eigenfunctions are oscillatory throughout the exterior.
Overall, two groups of EVPs have been extensively examined in the literature,
the difference being in the treatment of the nominal outer boundary
($r\to\infty$). No boundary condition (BC) is specified in one group (“EVP
open noBC” hereafter), whereas the BC for the other group (“EVP open noIC”) is
that no incoming waves are allowed.

Now consider the simplest case where the equilibrium density takes a step
profile, evaluating to $\rho_{\rm i}$ ($\rho_{\rm e}$) where $r<R$ ($r>R$).
Let $v_{\rm Ai}$ and $v_{\rm Ae}$ denote the Alfv$\acute{\rm e}$n speeds
evaluated with $\rho_{\rm i}$ and $\rho_{\rm e}$, respectively. A series of
cutoff axial wavenumbers $\\{k_{{\rm cutoff},m}\\}$ are relevant, satisfying
$k_{{\rm cutoff},1}<k_{{\rm cutoff},2}<\cdots$ (e.g., Roberts et al., 1984;
Vasheghani Farahani et al., 2014). By relevant we mean that a discrete set of
$m$ evanescent eigenmodes is allowed when $k>k_{{\rm cutoff},m}$, their real-
valued frequencies being consistently lower than the critical frequency
$\omega_{\rm crit}=kv_{\rm Ae}$. The focus of the classic study by Edwin &
Roberts (1983), this set is actually the evanescent part of the solutions to
both “EVP open noBC” (ORT15) and “EVP open noIC” (e.g., Cally, 1986; Chen et
al., 2015). Note that this evanescent set is empty when $k<k_{{\rm
cutoff},1}$. Note also that evanescent eigenmodes are customarily referred to
as “trapped modes” (e.g., Edwin & Roberts, 1983; Nakariakov & Verwichte, 2005;
Nakariakov & Kolotkov, 2020). The key difference between “EVP open noIC” and
“EVP open noBC” is then that, for arbitrary $k$, an infinity of discrete leaky
modes (DLMs) with complex-valued frequencies arise for the former (e.g.,
Meerson et al., 1978; Spruit, 1982; Cally, 1986), whereas a continuum of
oscillatory eigenmodes with real-valued frequencies arises for the latter
(ORT15; see also ; ). Now focus on the situation where $k<k_{{\rm cutoff},1}$,
which tends to be the only situation relevant for AR loops given their large
length-to-radius ratios and mild density contrasts (Aschwanden et al., 2004).
A multitude of initial value problem (IVP) studies have established that DLMs,
while not eigenmodes per se, may account for a substantial fraction of the
system evolution (e.g., Terradas et al., 2007; Nakariakov et al., 2012; Chen
et al., 2015; Guo et al., 2016) 333See, e.g., Figure 3 in Terradas et al.
(2007) for the temporal evolution of a sausage perturbation with $k<k_{\rm
cutoff,1}$. Figures 6 and 7 therein further suggest that the leaky phase
occurs only in a brief interval for the temporal evolution of kink
perturbations, which are beyond the scope of this study. . On the other hand,
it is well known that $P^{\rm DLM}\coloneqq 2\pi/\Re\omega^{\rm
DLM}<2.6R/v_{\rm Ai}$ and $\tau^{\rm DLM}/P^{\rm DLM}\coloneqq|\Re\omega^{\rm
DLM}/\Im\omega^{\rm DLM}|/(2\pi)\sim(\rho_{\rm i}/\rho_{\rm e})/\pi^{2}$, with
$P^{\rm DLM}$ and $\tau^{\rm DLM}$ representing the period and damping time,
respectively (e.g., Kopylova et al., 2007). Sausage perturbations are
therefore difficult to observe in AR loops given their short periodicities and
rapid attenuation.

Do sausage perturbations remain difficult to observe in AR loops for other
profiles of the equilibrium density $\rho_{0}$? This question is sensible
given the intricacies for quantifying the transverse density profiles with,
say, EUV measurements (see e.g., Aschwanden et al., 2003; Pascoe et al., 2017;
Goddard et al., 2017). We proceed with the choice labeled “outer $\mu$” in Yu
et al. (2017), where $\rho_{0}$ evaluates to $\rho_{\rm i}$ when $r<R$ but
otherwise reads $\rho_{\rm e}+(\rho_{\rm i}-\rho_{\rm e})(r/R)^{-\mu}$. Here
$R$ is some nominal cylinder radius, and $\mu\geq 1$ represents some steepness
parameter. Lopin & Nagorny (2015a, hereafter LN15a) were the first to examine
an EVP of the “EVP open noBC” type for this family of density profiles 444The
theoretical analyses in LN15a are not specific to the “outer $\mu$” family,
for which our discussions here follow directly from the more general results
therein., focusing on the behavior of the solutions at infinity to single out
evanescent eigenmodes. Two distinct regimes were shown to arise as dictated by
$\mu$. A series of cutoff axial wavenumbers $\\{k_{{\rm cutoff},m}\\}$ exist
when $\mu\geq 2$, satisfying $k_{{\rm cutoff},1}\leq k_{{\rm
cutoff},2}\leq\cdots$ (see also Li et al., 2018). The behavior of evanescent
eigenmodes is therefore qualitatively the same as in the step case. More
interesting is the regime where $\mu<2$, for which LN15a showed that cutoff
wavenumbers become irrelevant in that evanescent eigenmodes are permitted for
arbitrary $k$, regardless of $\mu$ and $\rho_{\rm i}/\rho_{\rm e}$.
Furthermore, the frequencies of evanescent eigenmodes tend to the critical
frequency $\omega_{\rm crit}=kv_{\rm Ae}$ when $kR\to 0$. Note that this
$\omega_{\rm crit}$ translates into a period $2L/v_{\rm Ae}$ for axial
fundamentals with $L$ being the loop length. LN15a (see also Lopin & Nagorny
2019) accepted the dominance of evanescent eigenmodes, arguing that sausage
perturbations may be invoked to account for flare QPPs with quasi-periods of
both $\mathcal{O}(R/v_{\rm Ai})$ and $\mathcal{O}(L/v_{\rm Ae})$. Now focus
again on AR loops. One deduces that sausage perturbations may be readily
captured in observations for AR loops with $\mu<2$, given the likely dominance
of evanescent eigenmodes and the long periodicities. However, this deduction
was not seen in the IVP study by Li et al. (2022, Paper I). For $\mu>2$ and
$\mu<2$ alike, the sausage perturbations are subject to rapid attenuation, and
their periodicities are consistently $\mathcal{O}(R/v_{\rm Ai})$ before the
signals become too weak to be of observational relevance.

This study aims to examine sausage perturbations in a slab configuration with
the “outer $\mu$” type of equilibrium density profiles. Our motivations are
threefold. Firstly, sausage perturbations in slab configurations prove to be
relevant for oscillatory signals observed in a broad range of passbands, some
examples being those in EUV (e.g., Verwichte et al., 2005) and radio (e.g.,
Roberts et al., 1983; Karlický et al., 2011; Jelínek & Karlický, 2012;
Kolotkov et al., 2021). Secondly, the mathematical treatments are less
involved for slab than for cylindrical configurations, making it easier to
concentrate on conceptual understandings. Thirdly, the EVP study by Lopin &
Nagorny (2015b, LN15b hereafter) demonstrated that the behavior of cutoff
wavenumbers in slab configurations is qualitatively identical to its
cylindrical counterpart 555By this assertion we mean that cutoff axial
wavenumbers exist only when $\mu\geq 2$, for slab and cylindrical geometries
alike (see Li et al., 2018, Figure 8). The mode functions, however, differ
quantitatively in their dependencies on the transverse coordinate. Consider
trapped sausage modes in coronal equilibria with step density profiles. The
transverse velocity in the exterior follows $K_{1}(\kappa_{\rm e}r)$ (${\rm
e}^{-\kappa_{\rm e}x}$) for cylindrical (slab) equilibria, with $\kappa_{\rm
e}=\sqrt{k^{2}-\omega^{2}/v_{\rm Ae}^{2}}$ and $K_{1}$ the modified Bessel
function of the second kind (e.g., Roberts, 2019). The former profile tends to
drop off more rapidly with distance. We note by passing that the same
difference in the drop-off rate takes place for trapped kink modes as well
(see e.g., Goossens et al. 1992; Arregui et al. 2007a; Hindman & Jain 2021; Yu
et al. 2021 for the connection between the two geometries). , thereby
warranting a slab version of the IVP study by Paper I. We nonetheless stress
that this study is new in the following aspects. One, we will consistently
solve the IVP in terms of eigensolutions to an EVP defined on a closed domain.
This “Modal Closed” approach has been employed only in the cylindrical studies
on sausage perturbations by Berghmans et al. (1996) and Paper I to our
knowledge. Two, we will revisit the IVP study for step density profiles by
AG07, offering a Fourier-integral-based formal solution rather than following
the Laplace transform approach therein. Key to this formal solution are the
eigensolutions to an EVP of the “EVP open noBC” type (see ; ; ). Our study
complements AG07 by further addressing the connection between continuum
eigenmodes and discrete leaky modes. We will additionally demonstrate how the
oscillatory continuum for open systems is recovered by the discrete
eigenspectrum in our “Modal Closed” approach. Three, we will work on the more
general situation with finite $\mu$, paying special attention to the influence
of $\mu$ on the connection between the discrete eigenspectrum and the
oscillatory continuum.

This manuscript is structured as follows. We formulate the IVP for a laterally
open system in Section 2, where we also detail the “Modal Closed” approach.
Section 3 then largely proceeds analytically to examine the IVP for a step
density profile. We move on to numerically examine the more general situation
where $\mu$ is finite in Section 4. Section 5 summarizes the present study,
ending with some concluding remarks.

##  2 Problem Formulation

###  2.1 Equilibrium and Overall Description

We work in the framework of pressureless, ideal, MHD throughout. Let $\rho$,
$\bm{v}$, and $\bm{B}$ denote the mass density, velocity, and magnetic field,
respectively. We denote the equilibrium quantities with a subscript $0$, and
assume that the equilibrium is static ($\bm{v}_{0}=0$). Working in a Cartesian
coordinate system $(x,y,z)$, we take the equilibrium magnetic field to be
uniform and $z$-directed ($\bm{B}_{0}=B_{0}\bm{e}_{z}$). We assume that the
equilibrium density ($\rho_{0}$) depends only on $x$ and is symmetric about
$x=0$. Specifically, $\rho_{0}(x)$ is chosen to be the “outer $\mu$” family in
Yu et al. (2017),

$\begin{split}&{\rho_{0}}(x)=\rho_{\rm e}+(\rho_{\rm i}-\rho_{\rm e})f(x),\\\
&f(x)=\left\\{\begin{array}[]{ll}1,&~{}~{}0\leq x\leq R,\\\\[2.84544pt]
\left(x/R\right)^{-\mu},&~{}~{}x\geq R.\end{array}\right.\end{split}$ (1)

Here $R$ represents some nominal slab half-width. With the subscript ${\rm i}$
(${\rm e}$) we denote the equilibrium quantities at $x=0$ ($x\to\infty$). The
Alfv$\acute{\rm e}$n speed is defined by $v_{\rm
A}^{2}=B_{0}^{2}/(\mu_{0}\rho_{0})$ with $\mu_{0}$ being the magnetic
permeability of free space. By $v_{\rm Ai}$ ($v_{\rm Ae}$) we then refer to
the Alfv$\acute{\rm e}$n speed at the slab axis (infinitely far). Evidently,
the steepness parameter $\mu$ is such that $\rho_{0}$ ($v_{\rm A}$) decreases
(increases) more rapidly with $x$ towards $\rho_{\rm e}$ ($v_{\rm Ae}$) when
$\mu$ increases.

Let the subscript $1$ denote small-amplitude perturbations. We neglect both
out-of-plane propagation ($\partial/\partial y=0$) and the out-of-plane
components of vectorial quantities. Consequently, only $v_{1x}$, $B_{1x}$, and
$B_{1z}$ survive in linearized, pressureless, ideal MHD. With axial standing
waves in mind, we take coronal slabs to be bounded by the planes $z=0$ and
$z=L$, and adopt the following ansatz

$\begin{split}v_{1x}(x,z,t)&=\hat{v}(x,t)\sin(kz),\\\
B_{1x}(x,z,t)&=\hat{B}_{x}(x,t)\cos(kz),\\\
B_{1z}(x,z,t)&=\hat{B}_{z}(x,t)\sin(kz),\end{split}$ (2)

where $k=n\pi/L$ is the quantized axial wavenumber ($n=1,2,\cdots$). One finds
that $\hat{B}_{x}$ and $\hat{B}_{z}$ are connected to $\hat{v}$ via

$\displaystyle\dfrac{\partial\hat{B}_{x}}{\partial t}$ $\displaystyle=$
$\displaystyle kB_{0}\hat{v},$ (3)
$\displaystyle\dfrac{\partial\hat{B}_{z}}{\partial t}$ $\displaystyle=$
$\displaystyle-B_{0}\dfrac{\partial\hat{v}}{\partial x}.$ (4)

We assume that

$\displaystyle\hat{B}_{x}(x,t=0)=\hat{B}_{z}(x,t=0)=0$ (5)

without loss of generality. The initial velocity perturbation is taken to be

$\displaystyle u(x)\coloneqq\hat{v}(x,t=0)=\left\\{\begin{array}[]{ll}v_{\rm
Ai}\sin^{3}(\pi x/\Lambda),&\quad 0\leq x\leq\Lambda,\\\\[2.84544pt] 0,&\quad
x\geq\Lambda,\end{array}\right.$ (8)

which is localized within $x=\Lambda$ and prescribed to be sufficiently smooth
with a magnitude arbitrarily set to be $v_{\rm Ai}$. Only sausage
perturbations arise given that $u(x=0)=0$ and that $\rho_{0}(x)$ is even. It
suffices to consider only the half-plane $x\geq 0$.

###  2.2 Initial Value Problem and Energetics

We retain only $\hat{v}$ to formulate our IVP for convenience.

###### IVP 1

Solutions are sought for the following equation

$\displaystyle\displaystyle\frac{\partial^{2}\hat{v}}{\partial t^{2}}$
$\displaystyle=$ $\displaystyle v_{\rm
A}^{2}(x)\left(\frac{\partial^{2}\hat{v}}{\partial
x^{2}}-k^{2}\hat{v}\right),$ (9)

subject to the initial conditions (ICs)

$\displaystyle\hat{v}(x,t=0)=u(x),\quad\frac{\partial\hat{v}}{\partial
t}(x,t=0)=0,$ (10)

together with the boundary condition (BC)

$\displaystyle\hat{v}(x=0,t)=0,$ (11)

on a domain spanning from $x=0$ to $\infty$.

Note that no BC is required at $x\to\infty$. Note further that the IC for
$\partial\hat{v}/\partial t$ is based on the IC (5), from which
$\hat{B}_{x}(x,t)$ and $\hat{B}_{z}(x,t)$ can be found by integrating
Equations (3) and (4) over time once $\hat{v}(x,t)$ is known.

It proves necessary to examine the associated energetics. Let $\mathcal{V}$
refer to a volume that is of unit length in the $y$-direction and spans the
rectangle $[0,x]\times[0,L]$ in the $x-z$ plane. An energy conservation law is
then readily derived, reading

$\displaystyle E_{\rm tot}(x,t)-E_{\rm tot}(x,t=0)=-F(x,t),$ (12)

where

$\displaystyle E_{\rm tot}(x,t)$ $\displaystyle=$
$\displaystyle\dfrac{L}{2}\int_{0}^{x}dx^{\prime}\left\\{\dfrac{1}{2}\rho_{0}(x^{\prime})\hat{v}^{2}(x^{\prime},t)+\dfrac{1}{2\mu_{0}}\left[\hat{B}_{x}^{2}(x^{\prime},t)+\hat{B}_{z}^{2}(x^{\prime},t)\right]\right\\}~{},$
(13) $\displaystyle F(x,t)$ $\displaystyle=$
$\displaystyle\dfrac{L}{2}\int_{0}^{t}dt^{\prime}\left[\hat{p}_{\rm
T}(x,t^{\prime})\hat{v}(x,t^{\prime})\right]~{}.$ (14)

Here $\hat{p}_{\rm T}=B_{0}\hat{B}_{z}/\mu_{0}$ represents the Eulerian
perturbation of total pressure, and the terms in the square parentheses in
Equation (14) stem from the lateral component of the Poynting vector.
Furthermore, the common factor $L/2$ is retained to ensure that $E_{\rm
tot}(x,t)$ represents the instantaneous total energy in $\mathcal{V}$, while
$F(x,t)$ represents the cumulative energy loss from $\mathcal{V}$.

###  2.3 Solution Methods

We choose to solve IVP 1 with two independent methods, one being a finite-
difference (FD) approach, and the other being a modal approach involving
eigensolutions to an EVP on a finite domain. The modal solutions turn out to
be more physically insightful. The FD approach, however, is orders-of-
magnitude less time-consuming. Two situations then arise where we present the
FD solutions. First, different approaches may need to be cross-validated.
Second, parametric studies are needed or the FD approach is more convenient to
work with.

####  2.3.1 The Finite-Difference Approach

Our FD scheme is second-order accurate in both space and time, and is
essentially identical to its cylindrical counterpart in Paper I. A uniform
grid with spacing $\Delta x=0.01~{}R$ is employed for simplicity. The timestep
is specified as $\Delta t=c\Delta x/v_{\rm Ae}$, where the Courant number $c$
is chosen to be $\sim 0.4$ to maintain numerical stability. We ensure that no
difference can be discerned when $\Delta x$ or $c$ varies. More importantly,
we make sure that our FD solutions remain the same when the outer boundary is
placed at larger distances.

####  2.3.2 The “Modal Closed” Approach

Our modal approach, labeled “Modal Closed” hereafter, relies on the following
EVP.

###### EVP 1

Nontrivial solutions are sought for the following equation

$\omega^{2}\breve{v}=-v_{\rm
A}^{2}(x)\left(\dfrac{d^{2}}{dx^{2}}\breve{v}-k^{2}\breve{v}\right)\equiv\mathcal{L}\breve{v},$
(15)

defined on a domain of $[0,d]$ and subject to the BCs

$\breve{v}(x=0)=\breve{v}(x=d)=0.$ (16)

Equation (15) is found by replacing $\hat{v}$ with
$\Re[\breve{v}(x)\exp(-i\omega t)]$ in Equation (9). One readily recognizes
that the operator $\mathcal{L}$ is Hermitian under the definition of the
scalar product

$\displaystyle\left<U|V\right>^{(d)}\equiv\int_{0}^{d}U^{*}(x)V(x)\rho_{0}(x)dx,$
(17)

where the asterisk represents complex conjugate. The superscript $(d)$ is
meant to emphasize that EVP 1 is defined on a closed domain. The following
general properties (GPs) then follow from general theory.

###### GP 1

The countable infinity of positive eigenvalues $\\{\omega^{2}_{l}\\}$ form a
monotonically increasing sequence with respect to the mode number
$l=1,2,\cdots$.

###### GP 2

The $l$-th eigenvalue for a domain never exceeds the $l$-th eigenvalue for its
subdomain.

###### GP 3

The $l$-th eigenfunction $\breve{v}_{l}(x)$ can be made real-valued,
possessing $l-1$ nodes inside the domain.

###### GP 4

The set $\\{\breve{v}_{l}(x)\\}$ is complete, satisfying the orthogonality
condition

$\displaystyle\left<\breve{v}_{l}|\breve{v}_{l^{\prime}}\right>^{(d)}=\left<\breve{v}_{l}|\breve{v}_{l}\right>^{(d)}\delta_{l,l^{\prime}},$
(18)

where $\delta_{l,l^{\prime}}$ denotes the Kronecker delta.

We specialize to our equilibrium density profile (1) from here onward. To
start, the eigenfrequency $\omega_{l}$ can be expressed formally as

$\dfrac{\omega_{l}R}{v_{\rm Ai}}={\mathcal{F}}_{l}(\rho_{\rm i}/\rho_{\rm
e},\mu;kR;d/R).$ (19)

We see $\omega_{l}$ as positive without loss of generality. It now proves
helpful to put Equation (15) in a Schrödinger form as

$\dfrac{d^{2}\breve{v}}{dx^{2}}+Q(x)\breve{v}=0,$ (20)

where the coefficient $Q(x)$ is defined through a potential $V(x)$ as

$V(x)=k^{2}v_{\rm A}^{2}(x),\quad Q(x)=\dfrac{\omega^{2}-V(x)}{v_{\rm
A}^{2}(x)}=\dfrac{\omega^{2}}{v_{\rm A}^{2}(x)}-k^{2}.$ (21)

The following specific properties (SPs) of EVP 1 then ensue, where by
“specific” we mean that the properties are more or less specific to our
equilibrium density profile. Nonetheless, they hold regardless of the specific
values of $[\rho_{\rm i}/\rho_{\rm e},\mu,kR,d/R]$.

###### SP 1

The eigenfrequency $\omega_{l}$ for any $l$ is bound to exceed $kv_{\rm Ai}$.

###### SP 2

High-frequency eigenmodes with $\omega_{l}^{2}\gg\omega_{\rm
crit}^{2}=(kv_{\rm Ae})^{2}$ are always permitted, following

$\omega_{l}\approx l\pi(v_{\rm Ae}/d),\quad\text{provided
that}~{}l^{2}\gg(kd/\pi)^{2}.$ (22)

###### SP 3

The eigenfunction $\breve{v}_{l}$ is oscillatory where $\omega_{l}^{2}>V(x)$
but evanescent otherwise.

We note that SP 1 can be justified by the heuristic argument that if
$\omega_{l}<kv_{\rm Ai}$, then $\breve{v}_{l}$ necessarily cusps somewhere in
the domain and therefore the continuity requirement for $d\breve{v}_{l}/dx$ is
violated. The same argument was first offered by Berghmans et al. (1996) in
cylindrical geometry. Likewise, a similar version of SP 2 holds in cylindrical
geometry as detailed in Appendix B in Paper I. Following the WKB treatment
therein yields Equation (22) for the Cartesian geometry, and the range of
validity turns out to be less restrictive. Important for this study is SP 3,
with which we classify the eigenmodes for a finite $d/R$ into two groups by
the spatial behavior of the corresponding eigenfunctions. An eigenmode,
labeled by $l$, is said to be evanescent if its eigenfunction is evanescent
somewhere in the domain. In contrast, an eigenmode is deemed oscillatory if
its eigenfunction is oscillatory throughout. Now that $v_{\rm A}(x)$ increases
monotonically toward $v_{\rm Ae}$ when $x$ increases, an eigenmode is
necessarily oscillatory for a given $d/R$ if $\omega_{l}>\omega_{\rm
crit}=kv_{\rm Ae}$ but evanescent when the opposite is true. Furthermore, if
an eigenmode transitions from an oscillatory to an evanescent one when $d/R$
exceeds some critical value, then property GP 2 dictates that it will remain
an evanescent eigenmode for larger still $d/R$.

We now connect EVP 1 to the general EVP analysis in LN15b. Put briefly, the
EVP therein belongs to the “EVP open noBC” category because the authors
analyzed the behavior of mode functions at infinity rather than specifying
some BC beforehand. The following deductions can be made by specializing to
our profile $f(x)$ (see Equation (1)). Consider the situation $\mu\geq 2$
first. Equations (12) to (16) in LN15b mean that there exist a series of
cutoff axial wavenumbers $k_{{\rm cutoff},m}$ with $m=1,2,\cdots$. Li et al.
(2018) further showed that $k_{{\rm cutoff},m}$ is expressible as

$k_{{\rm cutoff},m}R=\dfrac{g_{m}}{\sqrt{\rho_{\rm i}/\rho_{\rm
e}-1}}\quad(m=1,2,\cdots),$ (23)

where the $\rho_{\rm i}/\rho_{\rm e}$-independent coefficient $g_{m}\geq 1/2$.
One then deduces that no evanescent eigenmodes are permitted if $k<k_{{\rm
cutoff},1}$, with the inequality bound to hold when

$kR<\dfrac{1/2}{\sqrt{\rho_{\rm i}/\rho_{\rm e}-1}}.$ (24)

Now consider the situation where $\mu<2$, for which Equations (18) and (19) in
LN15b lead to the peculiar behavior for cutoff wavenumbers to disappear
altogether. However, what needs to be treated with caution is not this
peculiarity but the assertion that “all fast modes are trapped” for any $k$
and hence “such plasma slab configurations behave as perfect waveguides” (page
3, LN15b) in view of our EVP 1. Our point is, an infinity of oscillatory
eigenmodes are ensured by property SP 2 regardless of $d/R$, making them
difficult to disappear altogether in the limit $d/R\to\infty$. More
importantly, the solution to IVP 1 at any location can be expressed in terms
of the eigensolutions to EVP 1 provided that a timeframe of validity is
specified. Ideally, this confusion should be resolved with a theoretical
understanding of the spectrum associated with the EVP examined in LN15b.
However, we argue that there is no need to do so as long as the solutions to
IVP 1 can be understood with the aid of the textbook properties of EVP 1. In
particular, the “Modal Closed” approach allows to make clear what is meant by
the irrelevance of cutoff wavenumbers. We therefore restrict ourselves to the
situation where Inequality (24) holds.

Some remarks remain necessary on the role of $\mu$ for EVP 1 when $k<k_{{\rm
cutoff},1}$. To start, property GP 2 follows from Theorem 3 in Courant &
Hilbert (1989, Chapter VI), where it was noted that the $l$-th eigenvalue for
a domain is in fact smaller than the $l$-th eigenvalue for any of its
subdomains. When $\mu\geq 2$, one then directly deduces that EVP 1 allows only
oscillatory eigenmodes for any finite domain, because evanescent eigenmodes
are prohibited for a truly open domain. For $\mu<2$, on the other hand, some
concern may arise when we compute evanescent eigenmodes for EVP 1. Let $l$
refer to an evanescent eigenmode, and let $\omega_{l}(d)$ be its
eigenfrequency evaluated with a domain of size $d$. The stronger version of
property GP 2 then dictates that the eigenfrequency $\omega_{l}$ depends on
the domain size, meaning specifically that
$\omega_{l}(d^{\prime})<\omega_{l}(d)$ for any $d^{\prime}>d$. In practice,
however, we need to quote some domain-independent eigenfrequency for
definiteness (e.g., Figure 8). This discrepancy is nonetheless only apparent,
because what we mean by domain-independent is that $\omega_{l}(d^{\prime})$
and $\omega_{l}(d)$ are equal within the accuracy of our numerical solver when
$d$ is sufficiently large.

With the above preparations, the “Modal Closed” solution to IVP 1 writes

$\begin{split}&\hat{v}^{(d)}(x,t)=\sum\limits_{l=1}^{\infty}c_{l}\breve{v}_{l}(x)\cos(\omega_{l}t),\\\
&0\leq x\leq d,\quad t\leq\int_{\Lambda}^{d}\dfrac{dx}{v_{\rm
A}(x)}.\end{split}$ (25)

Here the coefficient $c_{l}$ measures the contribution from the $l$-th
eigenmode,

$\displaystyle
c_{l}=\dfrac{\left<u|\breve{v}_{l}\right>^{(d)}}{\left<\breve{v}_{l}|\breve{v}_{l}\right>^{(d)}}.$
(26)

Despite the superscript $(d)$, we stress that $\hat{v}^{(d)}(x,t)$ itself does
not depend on $d$ as long as the outermost edge of the perturbation has not
reached $x=d$. The relevant timeframe of validity is explicitly specified in
Equation (25).

Two steps are involved for numerically evaluating the “Modal Closed” solution.
Firstly, we follow Paper I to formulate and solve EVP 1 with the general-
purpose finite-element code PDE2D (Sewell, 1988), which was introduced into
the solar context by Terradas et al. (2005a) 666PDE2D has subsequently been
extensively employed in solar wave studies for addressing both IVPs (e.g.,
Terradas et al., 2006; Soler & Goossens, 2011; Oliver et al., 2016) and EVPs
(e.g., Arregui et al., 2007b, 2011; Chen et al., 2021). Recently, it found
application to the construction of magnetohydrostatic equilibria as well
(Terradas et al., 2022).. As output we have the sets $\\{\omega_{l}\\}$ and
$\\{\breve{v}_{l}\\}$ for a specified combination $[\rho_{\rm i}/\rho_{\rm
e},\mu;kR;d/R]$. Secondly, we evaluate the coefficient $c_{l}$ and then the
modal solution $\hat{v}^{(d)}$ with Equations (26) and (25), respectively. We
ensure that a sufficient number of eigenmodes are incorporated in the sum.

###  2.4 Parameter Specification

Let us start by recalling that the solution to IVP 1 is determined by the
combination $[\rho_{\rm i}/\rho_{\rm e},\mu;kR;\Lambda/R]$. We see the
steepness parameter $\mu$ as the primary adjustable, and occasionally adjust
the spatial extent $\Lambda$ as well. The combination $[\rho_{\rm i}/\rho_{\rm
e},kR]$ is fixed at $[2.25,\pi/15]$ throughout. We examine only axial
fundamentals, for which the adopted $kR$ translates into a value of $L/R=15$
for the ratio of the axial slab length to the nominal half-width. Note that
this pair of $[\rho_{\rm i}/\rho_{\rm e},L/R]$ is reasonable for AR loops,
although they somehow lie close to the lower ends of the observed ranges
deduced from EUV measurements (e.g., Aschwanden et al., 2004; Schrijver,
2007). Note further that this pair yields a $(kR)\sqrt{\rho_{\rm i}/\rho_{\rm
e}-1}$ of $0.23$, and hence ensures Inequality (24). Figure 1a offers a two-
dimensional (2D) representation of IVP 1 by presenting the $x-z$ distributions
of the equilibrium density (the filled contours) and the initial velocity
field (the arrows). The equilibrium density is further plotted against the
transverse coordinate in Figure 1b, where two values are examined for the
steepness parameter $\mu$ as labeled. In addition, the $x$-profile for the
initial perturbation is also shown, with $\Lambda$ chosen to be $4R$ for
illustrative purposes.

##  3 Step Density Profile ($\mu=\infty$)

This section examines IVP 1 for the simplest situation where the equilibrium
density takes a step profile. Our purposes are threefold. Firstly, the only
published analytical study on IVP 1 for the present case was due to AG07, who
adopted a Laplace transform approach. Our Section 3.1 will present a new
analytical solution by directly working with eigenmodes on an open domain
within the simpler Fourier framework 777Only real frequencies are involved in
the Fourier approach. The Laplace transform approach, on the other hand,
requires contour integrals in the complex frequency plane and hence the
identification of the branch cuts of the Green’s function. However, we follow
AG07 to stress that different expressions from different approaches are
necessarily equivalent because they describe the same temporal evolution..
Secondly, in Section 3.2 we will employ our Fourier solution to address the
connection between the oscillatory continuum modes and the much-studied
discrete leaky modes, which in some sense makes more apparent the notion
raised by AG07 that the initial perturbation is of critical importance.
Thirdly, in Section 3.3 we will address the connection between the oscillatory
continuum and the discrete spectrum associated with EVP 1.

###  3.1 Formal Solution to the IVP in Terms of Eigenmodes on an Open Domain

This subsection works out the formal solution to IVP 1 in terms of eigenmodes
on an open domain. The cylindrical version of this approach was presented in
Appendix A.1 of Paper I, which in turn was based on the Fourier approach
detailed by ORT14 and ORT15. We see the axial wavenumber $k$ as arbitrary in
this subsection. While derived for a given $k$, the resulting expressions can
find immediate applications if a continuous distribution of $k$ is involved
such as happens for wave trains impulsively excited by localized perturbations
(see ; ; also Li et al. 2023).

The EVP of interest in this subsection is of the “EVP open noBC” type, meaning
specifically that it is defined on $[0,\infty)$ and no BC is specified at
infinity. A series of cutoff wavenumbers exist and are given by

$\displaystyle k_{{\rm cutoff},m}R=\dfrac{(m-1/2)\pi}{\sqrt{\rho_{\rm
i}/\rho_{\rm e}-1}},$ (27)

where $m=1,2,\cdots$. Evidently, $k_{{\rm cutoff},1}<k_{{\rm
cutoff},2}<\cdots$. Defining

$\begin{split}&k_{\rm i}^{2}=\dfrac{\omega^{2}-k^{2}v_{\rm Ai}^{2}}{v_{\rm
Ai}^{2}},\\\ &k_{\rm e}^{2}=\dfrac{\omega^{2}-k^{2}v_{\rm Ae}^{2}}{v_{\rm
Ae}^{2}},\\\ &\kappa_{\rm e}^{2}=-\dfrac{\omega^{2}-k^{2}v_{\rm
Ae}^{2}}{v_{\rm Ae}^{2}}=-k_{\rm e}^{2},\end{split}$ (28)

we note that $k_{\rm i}^{2}>0$ because $\omega$ always exceeds $kv_{\rm Ai}$.
The eigensolutions are then discriminated by the sign of $k_{\rm e}^{2}$.
Proper eigenmodes ($k_{\rm e}^{2}<0$) arise when $k$ exceeds $k_{\rm
cutoff,1}$, characterized by a finite set of discrete frequencies. Let $j$
label a proper eigenmode, whose eigenfunction then writes

$\displaystyle\breve{v}_{j}(x)=\left\\{\begin{array}[]{ll}-\dfrac{v_{\rm
Ai}}{k_{\rm i}R}{\rm e}^{-\kappa_{\rm e}R}\sin(k_{\rm i}x),&\quad 0\leq x\leq
R,\\\\[2.84544pt] \dfrac{v_{\rm Ai}}{\kappa_{\rm e}R}\cos(k_{\rm i}R){\rm
e}^{-\kappa_{\rm e}x},&\quad x>R.\end{array}\right.$ (31)

Evidently, proper eigenmodes are necessarily evanescent. Equation (31) is
written to ensure the continuity of $d\breve{v}_{j}(x)/dx$. The continuity of
$\breve{v}_{j}(x)$, on the other hand, yields a dispersion relation (DR) that
dictates the eigenfrequency $\omega_{j}$,

$\displaystyle\kappa_{\rm e}\tan(k_{\rm i}R)+k_{\rm i}=0.$ (32)

Improper eigenmodes ($k_{\rm e}^{2}>0$) are relevant regardless of $k$. They
are characterized by a continuous frequency coverage in the range $(kv_{\rm
Ae},\infty)$, and hence are oscillatory. Note that the concept of dispersion
relation does not apply. We then derive the improper eigenfunction
$\breve{v}_{\omega}(x)$ by solving Equation (20) under the requirements that
both $\breve{v}_{\omega}$ and $d\breve{v}_{\omega}(x)/dx$ be continuous across
$x=R$. The result writes

$\displaystyle\breve{v}_{\omega}(x)=\left\\{\begin{array}[]{ll}-v_{\rm
Ai}\dfrac{k_{\rm e}}{k_{\rm i}}\sin(k_{\rm i}x),&\quad 0\leq x\leq
R,\\\\[2.84544pt] -v_{\rm Ai}\left[A_{\rm c}\cos(k_{\rm e}x)+A_{\rm
s}\sin(k_{\rm e}x)\right],&\quad x>R,\end{array}\right.$ (35)

where

$\begin{split}&A_{\rm c}=\dfrac{k_{\rm e}}{k_{\rm i}}\sin(k_{\rm
i}R)\cos(k_{\rm e}R)-\cos(k_{\rm i}R)\sin(k_{\rm e}R),\\\ &A_{\rm
s}=\dfrac{k_{\rm e}}{k_{\rm i}}\sin(k_{\rm i}R)\sin(k_{\rm e}R)+\cos(k_{\rm
i}R)\cos(k_{\rm e}R).\end{split}$ (36)

We scale $\breve{v}_{\omega}$ in such a way that it remains regular when
$k_{\rm e}$ approaches zero.

We solve IVP 1 by directly superposing the proper and improper eigensolutions.
This “Modal Open” approach is possible because the eigensolutions in the
discrete proper set and the improper continuum are complete, satisfying the
orthogonality condition

$\begin{split}&\left<\breve{v}_{j}|\breve{v}_{j^{\prime}}\right>^{(\infty)}=\left<\breve{v}_{j}|\breve{v}_{j}\right>^{(\infty)}\delta_{j,j^{\prime}},\\\
&\left<\breve{v}_{j}|\breve{v}_{\omega}\right>^{(\infty)}=0,\\\
&\left<\breve{v}_{\omega}|\breve{v}_{\omega^{\prime}}\right>^{(\infty)}=q(\omega)\delta(\omega-\omega^{\prime}).\end{split}$
(37)

Here the scalar product is defined as

$\left<U|V\right>^{(\infty)}\equiv\int_{0}^{\infty}U^{*}(x)V(x)\rho_{0}(x)dx,$
(38)

and the superscript $\infty$ emphasizes that the eigensolutions pertain to an
open domain. For the proper eigenmodes, it is straightforward to obtain that

$\left<\breve{v}_{j}|\breve{v}_{j}\right>^{(\infty)}=(\rho_{\rm i}v_{\rm
Ai}^{2}R)\dfrac{{\rm e}^{-2\kappa_{\rm e}R}}{2}\left[\dfrac{1}{k_{\rm
i}^{2}R^{2}}+\dfrac{\cos^{2}(k_{\rm i}R)}{\kappa_{\rm
e}R}\left(\dfrac{1}{k_{\rm i}^{2}R^{2}}+\dfrac{\rho_{\rm e}/\rho_{\rm
i}}{\kappa_{\rm e}^{2}R^{2}}\right)\right].$ (39)

For the improper eigenmodes, however, some algebra is necessary for evaluating
$q(\omega)$ in front of the generalized function
$\delta(\omega-\omega^{\prime})$ in Equation (37). The net result is that

$q(\omega)=(\rho_{\rm e}v_{\rm Ai}^{2})\dfrac{k_{\rm e}v_{\rm
Ae}^{2}}{\omega}\dfrac{\pi(A_{c}^{2}+A_{s}^{2})}{2}.$ (40)

The “Modal Open” solution to IVP 1 eventually writes

$\begin{split}&\hat{v}(x,t)=\sum\limits_{j=1}^{J}c_{j}\breve{v}_{j}(x)\cos(\omega_{j}t)+\int_{kv_{\rm
Ae}}^{\infty}S_{\omega}\breve{v}_{\omega}(x)\cos(\omega t)d\omega,\\\
&0<x<\infty,\quad 0<t<\infty,\end{split}$ (41)

where

$\begin{split}&c_{j}=\dfrac{\left<u|\breve{v}_{j}\right>^{(\infty)}}{\left<\breve{v}_{j}|\breve{v}_{j}\right>^{(\infty)}},\\\
&S_{\omega}=\dfrac{\left<u|\breve{v}_{\omega}\right>^{(\infty)}}{q(\omega)}=\dfrac{\omega\left<u|\breve{v}_{\omega}\right>^{(\infty)}}{(\rho_{\rm
e}v_{\rm Ai}^{2})(k_{\rm e}v_{\rm
Ae}^{2})(\pi/2)(A_{c}^{2}+A_{s}^{2})}.\end{split}$ (42)

We view $S_{\omega}\breve{v}_{\omega}(x)$ as some local spectral density.
Furthermore, $J$ counts all proper eigenmodes $j$ that satisfy $k>k_{{\rm
cutoff},j}$. Evidently, $J=0$ if $k<k_{{\rm cutoff},1}$.

Some remarks are necessary. To start, proper eigenmodes are the familiar
“trapped modes”, with Equations (27) to (32) well documented in, say, Chapter
5 of Roberts (2019). Somehow less familiar is the improper continuum $(kv_{\rm
Ae},\infty)$, which arises due to the absence of a BC at infinity. However,
the “Modal Open” approach itself is actually well established, with Equation
(40) resulting essentially from the techniques developed in the cylindrical
study of ORT14. We nonetheless note that the formal solution (41) has not been
derived this way to our knowledge. We also note that this solution is valid at
arbitrary $(x,t)$ for arbitrary $k$.

###  3.2 Discrete Leaky Modes vs. Improper Continuum Eigenmodes

This subsection aims at offering a better distinction between discrete leaky
modes (DLMs) and improper continuum eigenmodes.

We start by summarizing some necessary properties of DLMs, which are solutions
to an EVP of the “EVP open noIC” type. We define $\mu_{\rm i}^{2}$ and
$\mu_{\rm e}^{2}$ in the same way as $k_{\rm i}^{2}$ and $k_{\rm e}^{2}$ in
Equation (28), seeing $\omega$ as possibly complex-valued and assuming
$-\pi/2<\arg\mu_{\rm i},\arg\mu_{\rm e}\leq\pi/2$ (e.g., Yu et al., 2015).
Note that the time-dependence of any perturbation is written as ${\rm
e}^{-i\omega t}$ in its Fourier decomposition. The mode function in the
exterior then writes ${\rm e}^{i\mu_{\rm e}x}$ to within a complex-valued
factor, because the other independent solution ${\rm e}^{-i\mu_{\rm e}x}$ to
Equation (20) is connected with incoming waves. All mode frequencies are
discrete, obeying the nominal DR (e.g., Terradas et al., 2005b; Yu et al.,
2015)

$i\mu_{\rm e}=\mu_{\rm i}\cot(\mu_{\rm i}R).$ (43)

Equation (43) is well known to allow solutions with purely imaginary $\mu_{\rm
e}$ when $k>k_{{\rm cutoff},1}$. These solutions coincide with the proper
eigensolutions to “EVP open noBC”, and are therefore not of further interest.
More interesting are the DLMs, which exist regardless of $k$. As first shown
by Terradas et al. (2005b), the complex-valued frequency of the $m$-th DLM may
be approximated by

$\begin{split}&\dfrac{R\Re\omega^{\rm DLM}_{m}}{v_{\rm
Ai}}\approx\left(m-\dfrac{1}{2}\right)\pi,\quad(m=1,2,\cdots)\\\
&\dfrac{R\Im\omega^{\rm DLM}_{m}}{v_{\rm
Ai}}\approx-\dfrac{1}{2}\ln\dfrac{1+\sqrt{\rho_{\rm e}/\rho_{\rm
i}}}{1-\sqrt{\rho_{\rm e}/\rho_{\rm i}}},\end{split}$ (44)

provided that $|\omega^{\rm DLM}_{m}|^{2}\gg k^{2}v_{\rm Ae}^{2}$. That DLMs
do not qualify as eigenmodes have been pointed out in the cylindrical study by
ORT15. The same holds for the Cartesian version, to corroborate which we add
that no scalar product can be properly defined for the relevant EVP. That
said, it is known that the periods and damping times predicted with Equation
(43) may describe a substantial portion of the time sequences of the relevant
perturbations (e.g., Terradas et al. 2005b; Hornsey et al. 2014; Yu et al.
2015). Whether this happens is solely determined by how the initial
perturbation $u(x)$ is prescribed, a notion highlighted by AG07. We will
visualize this notion by several specific computations, which in a sense makes
the notion more concrete.

Figure 2 presents the “Modal Open” solutions (Equation (41), the solid curves)
to the $\mu=\infty$ version of IVP 1. Only the improper continuum needs to be
considered, given that $[\rho_{\rm i}/\rho_{\rm e},kR]$ is fixed at
$[2.25,\pi/15]$. A number of values are examined for the spatial extent of the
initial perturbation ($\Lambda$) as discriminated by the different colors. The
lateral speeds at the nominal half-width $\hat{v}(R,t)$ as functions of time
are plotted in Figure 2a, where the finite difference (FD) solutions are shown
by the symbols for comparison. Note that the FD solutions are shown only for
$\Lambda/R=1$ and $\Lambda/R=2$ to avoid further crowding the curves.
Nonetheless, it holds in general that the FD solutions agree exactly with the
“Modal Open” ones, thereby validating both approaches. More importantly, one
sees that the temporal profile involves increasingly fine scales as $\Lambda$
decreases. This behavior is then quantified in Figure 2b, where the local
spectral densities $S_{\omega}\breve{v}_{\omega}(R)$ are shown against the
frequency $\omega$. The critical frequency $\omega_{\rm crit}=kv_{\rm Ae}$ is
given by the vertical dash-dotted line for reference. In agreement with Figure
2a, one sees that improper eigenmodes with increasingly high frequencies are
involved when $\Lambda$ decreases. Furthermore, one sees that the local
spectral densities tend to favor a set of discrete frequencies indicated by
the arrows, which represent the real parts of the mode frequencies
($\\{\Re\omega^{\rm DLM}_{m}\\}$) of the first several DLMs as found by
numerically solving Equation (43). Some further inspection indicates that the
values for the set $\\{\Re\omega^{\rm DLM}_{m}\\}$ can be safely approximated
by Equation (44), which is not surprising given that the inequality
$|\omega^{\rm DLM}_{m}|^{2}\gg k^{2}v_{\rm Ae}^{2}$ holds well. Equally
unsurprising is that these frequencies tend to stand out in
$S_{\omega}\breve{v}_{\omega}(R)$, for they tend to do so in $S_{\omega}$. We
capitalize on the expression for $S_{\omega}$ (see Equation (42)) to explain
this, and focus for now on the situation where $\omega^{2}\gg k^{2}v_{\rm
Ae}^{2}$. It then follows that $k_{\rm i}^{2}\approx\omega^{2}/v_{\rm Ai}^{2}$
and $k_{\rm e}^{2}\approx\omega^{2}/v_{\rm Ae}^{2}$. Consequently, the
$\omega$-dependence of $S_{\omega}$ is largely determined by the factor
$A_{\rm c}^{2}+A_{\rm s}^{2}$, which in turn approximates to

$A_{\rm c}^{2}+A_{\rm s}^{2}\approx\rho_{\rm e}/\rho_{\rm i}+(1-\rho_{\rm
e}/\rho_{\rm i})\cos^{2}(\omega R/v_{\rm Ai}).$ (45)

Evidently, the set $\\{\Re\omega^{\rm DLM}_{m}\approx(m-1/2)\pi v_{\rm
Ai}/R\\}$ minimizes $A_{\rm c}^{2}+A_{\rm s}^{2}$ and therefore tends to
maximize $|S_{\omega}|$. That $S_{\omega}\breve{v}_{\omega}(R)$ tends to
attain a local extreme at the set $\\{\Re\omega^{\rm DLM}_{m}\\}$ is further
strengthened by the fact that $\breve{v}_{\omega}(R)$ does so given that
$\breve{v}_{\omega}(R)\approx-v_{\rm Ai}\sqrt{\rho_{\rm e}/\rho_{\rm
i}}\sin(\omega R/v_{\rm Ai})$ (see Equation (35)). One therefore deduces with
Equation (42) that the deviation of the locations of local extrema from
$\\{\Re\omega^{\rm DLM}_{m}\\}$ when $\omega^{2}\gg k^{2}v_{\rm Ae}^{2}$ is
almost entirely due to the specification of the initial perturbation. This
deviation is reflected in two aspects. Firstly, the positions of some extrema
may be slightly different from $\\{\Re\omega^{\rm DLM}_{m}\\}$, with the first
extremum for $\Lambda/R=1$ or $\Lambda/R=2$ being an example. Secondly, and
more severely, some extrema may not be accounted for by $\\{\Re\omega^{\rm
DLM}_{m}\\}$. This is particularly true when $\Lambda/R$ is large, as
exemplified by the second (third) extremum for $\Lambda/R=4$ ($\Lambda/R=8$).
Now consider the situation where the inequality $\omega^{2}\gg k^{2}v_{\rm
Ae}^{2}$ does not hold. For large $\Lambda$, one sees some further extrema
that take place at frequencies below $\Re\omega^{\rm DLM}_{1}$. The first
extremum for $\Lambda/R=8$ serves as an example for this aspect, and it
therefore does not make much sense to further question whether the associated
damping timescale agrees with the expectation with Equation (43). Rather, with
Figure 2 we conclude that the details of the initial perturbation play an
important role in the temporal evolution of the system.

###  3.3 Connection Between the “Modal Closed” and “Modal Open” Solutions

This subsection discusses the connection between the “Modal Closed” and “Modal
Open” solutions, the aim being to further visualize the fact that improper
continuum modes qualify as eigenmodes. For simplicity, we focus on the
situation where $k<k_{{\rm cutoff},1}$ such that proper modes are irrelevant.

Let us start with some analytical progress for the eigensolutions to EVP 1,
which we recall is defined on a closed domain $[0,d]$. The eigenfunctions
remain expressible by Equation (35), except that the eigenfrequencies are
discrete as determined by the DR

$A_{\rm c}\cos(k_{\rm e}d)+A_{\rm s}\sin(k_{\rm e}d)=0.$ (46)

Evidently, this DR is nothing but the BC at the outer boundary. Consider now
low-frequency eigenmodes, by which we mean $\omega=kv_{\rm Ae}(1+\Delta)$
where $0<\Delta\ll 1$. One readily recognizes from Equation (28) that $k_{\rm
i}^{2}\approx k^{2}(\rho_{\rm i}/\rho_{\rm e}-1)$ and $k_{\rm e}^{2}\approx
k^{2}(2\Delta)$. Now suppose $k_{\rm e}R\ll 1$. Equation (36) then indicates
that $A_{\rm c}/A_{\rm s}\sim\mathcal{O}(k_{\rm e}R)$, meaning that the
leading order solution to Equation (46) is $k_{\rm e}d\approx l\pi$. Hence
$\Delta\approx(l\pi)^{2}/(2k^{2}d^{2})$. To sum up, the low-frequency
eigenmodes are characterized by

$\dfrac{\omega_{l}}{kv_{\rm Ae}}\approx
1+\dfrac{(l\pi)^{2}}{2(kd)^{2}},\quad\text{provided
that}\quad(kd)^{2}\gg(l\pi)^{2},\quad d/R\gg l\pi.\quad$ (47)

Now consider high-frequency eigenmodes, by which we mean $\omega^{2}\gg
k^{2}v_{\rm Ae}^{2}$. Equation (28) indicates that $k_{\rm i}\approx k_{\rm
e}\approx\omega/v_{\rm Ai}$. Some algebra then leads the DR (46) to become

$\displaystyle\sin\left[k_{\rm e}(d-R)+\phi_{\rm e}\right]\approx 0,$ (48)

where $\phi_{\rm e}$ is governed by

$\tan\phi_{\rm e}=\sqrt{\rho_{\rm e}/\rho_{\rm i}}\tan(\sqrt{\rho_{\rm
i}/\rho_{\rm e}}k_{\rm e}R).$ (49)

One readily sees that the leading order solution is given by Equation (22),
which is not surprising because this approximate expression applies to
arbitrary $\mu$. Nonetheless, the explicit DR (46) can be employed to yield
the next order correction, the net result being

$\begin{split}&\omega_{l}\approx(l\pi-\phi_{l})\dfrac{v_{\rm Ae}}{d-R},\\\
&\phi_{l}\approx\arctan\left[\sqrt{\rho_{\rm e}/\rho_{\rm
i}}\tan\left(\sqrt{\rho_{\rm i}/\rho_{\rm
e}}\dfrac{l\pi}{d/R-1}\right)\right]+\left\lfloor\sqrt{\rho_{\rm i}/\rho_{\rm
e}}\dfrac{l}{d/R-1}+\dfrac{1}{2}\right\rfloor\pi,\end{split}$ (50)

with $\left\lfloor\cdot\right\rfloor$ being the floor function. Equation (50)
is presented only for completeness, the derivation for which is the same as in
Appendix A.2 of Paper I.

More important to this study is that the discrete set of eigenfrequencies
${\omega_{l}}$ for EVP 1 converges to the improper continuum when the domain
size $d$ extends to infinity. Let

$\Delta\omega_{l}=\omega_{l+1}-\omega_{l},\quad l=1,2,\cdots$ (51)

define the frequency spacing. The continuum is then approached in such a way
that $\Delta\omega_{l}$ starts with a $1/d^{2}$ dependence for small $l$ (see
Equation (47)) and transitions to a $1/d$ dependence for large $l$ (see
Equation (22)). In view of Equation (25), we define

$\displaystyle S_{l}=\dfrac{c_{l}}{\omega_{l+1}-\omega_{l}}$ (52)

such that $S_{l}\breve{v}_{l}(x)$ represents some discrete local spectral
density. Let us demonstrate that the discrete set
$\\{S_{l}\breve{v}_{l}(x)\\}$ approaches the continuous
${S_{\omega}}\breve{v}_{\omega}(x)$ in Equation (41).

Figure 3 is similar in format to Figure 2, the primary difference being that
the combination $[\rho_{\rm i}/\rho_{\rm e},kR,\Lambda/R]$ is now fixed at
$[2.25,\pi/15,4]$. The “Modal Open” solution is shown by the black curves. Two
domain sizes are employed for constructing the “Modal Closed” solutions, one
being $d/R=50$ (the curve and symbols in red), and the other being $d/R=100$
(blue). Examine Figure 3a first, from which one sees that the “Modal Open” and
“Modal Closed” solutions cannot be told apart. We therefore conclude with
Figure 3a and Figure 2a that the “FD”, “Modal Closed”, and “Modal Open”
approaches yield identical solutions to IVP 1. Now move on to Figure 3b, where
the symbols represent the values of $\\{S_{l}\breve{v}_{l}(R)\\}$ as functions
of the discrete eigenfrequencies $\\{\omega_{l}\\}$ associated with the “Modal
Closed” solutions. One sees that the set $\\{\omega_{l}\\}$ for $d/R=100$ is
finer than for $d/R=50$, a feature that holds in view of Equations (47) and
(22). More importantly, one sees that the sets $\\{S_{l}\breve{v}_{l}(R)\\}$
for both values of $d/R$ cannot be distinguished from the continuous
$S_{\omega}\breve{v}_{\omega}(R)$ associated with the improper continuum. We
therefore conclude that the continuous $S_{\omega}\breve{v}_{\omega}(R)$ can
be adequately resolved with the discrete eigenmodes for a domain size as
modest as $d/R=50$. This also visualizes the general behavior for
$\\{S_{l}\breve{v}_{l}(x)\\}$ to converge to $S_{\omega}\breve{v}_{\omega}(x)$
at arbitrary $x$.

###  3.4 Observational Implications

Some observational implications follow from our discussions on the connection
between DLMs and the improper continuum. For clarity, we restrict our
discussions to those observed signals that attenuate temporally. Our Figure 2a
suggests that the quasi-periods are consistently on the order of the
transverse Alfv$\acute{\rm e}$n time $R/v_{\rm Ai}$, provided that the spatial
extent of the initial perturbation differs not too much from the width of the
wave host. This supports the customary practice that candidate sausage
perturbations are identified primarily by their short quasi-periods (e.g.,
Nakariakov et al., 2003; Kolotkov et al., 2015; Carley et al., 2019).
Furthermore, the formal solution (Equation (41)) stresses that the temporal
damping derives from the destructive interference of the monochromatic
components in the improper continuum. It then follows that a rich variety of
damping envelopes may result, with the details of the initial perturbation
being critical (see the expression for $S_{\omega}$ in Equation (42)). This
damping envelope may be similar to an exponential one during some appropriate
interval, and hence agree with the expectations with DLMs (e.g., Figure 4 in
Terradas et al. 2005b). Conversely, nominal DRs of DLMs can be put to
seismological use when some exponential damping shows up, an example being the
analysis by Kopylova et al. (2007) of the flare pulsation originally reported
by McLean & Sheridan (1973). However, it is equally possible, if not more
common, that the damping envelope follows different patterns, given the
diversity of the initial perturbations (see Figure 2a). Two consequences then
arise. Firstly, there is no need to reject short-period signals as candidate
sausage perturbations simply because their damping envelopes deviate from an
exponential one. It is just that the meaning of the deduced damping time needs
to be clarified. In this sense, one may need to re-assess the seismology
performed by, say, Chen et al. (2015) for the pulsating flare emission
presented in Kolotkov et al. (2015). Our point is, the seismology therein
associated the damping signal with a DLM. The observed temporal damping,
however, does not show a particular pattern (Kolotkov et al., 2015, Figure 3).
Secondly, the details of an observed time-varying signal actually encapsulate
the information on not only the wave host but also the initial perturbation.
It is therefore possible, at least in principle, to decipher some appropriate
flare QPPs to deduce some parameters that characterize the energy-release
process. The same thinking applies to AR loops as well, if a candidate sausage
perturbation is eventually identified.

##  4 Continuous Density Profiles ($\mu<\infty$)

This section examines the situation where $\mu$ is finite. We largely rely on
the “Modal Closed” approach, and present the FD solutions only when necessary.

###  4.1 Comparison Between the Cases with $\mu<2$ and $\mu>2$

Figure 4 presents the FD solutions to IVP 1 for a fixed combination
$[\rho_{\rm i}/\rho_{\rm e},kR,\Lambda/R]=[2.25,\pi/15,4]$. Two steepness
parameters, $\mu=1.5$ and $\mu=5$, are examined as discriminated by the
different colors. Note that FD solutions are shown here to avoid discussing
the outer boundary, which is inherent to the “Modal Closed” approach. Figure 4
offers some gross properties of the solutions, where by gross we choose to
examine the energetics in the volume $\mathcal{V}$ within which the initial
perturbation is implemented. Specifically, the total energy $E_{\rm tot}$ in
$\mathcal{V}$ and the cumulative energy loss $F$ from $\mathcal{V}$ are
evaluated with Equations (13) and (14) by letting $x=\Lambda$ therein. One
sees no qualitative difference between the solutions for different values of
$\mu$, despite that these values lie astride the nominally critical value of
two. In both cases, $E_{\rm tot}$ features some rather rapid decrease with
time, accompanied by some steady increase in $F$. The sum of the two
quantities ($E_{\rm tot}+F$) remains constant throughout, indicating that
energy is conserved remarkably well (see Equation (12)). In this regard, the
plateaus in $E_{\rm tot}$ or $F$ arise simply because the transverse component
of the Poynting vector is small during the corresponding intervals (see
Equation (14)). More importantly, $E_{\rm tot}$ drops to extremely small
values when, say, $t\gtrsim 10R/v_{\rm Ai}$ for both $\mu$. One then expects
that the signals in the entire interval $x\leq\Lambda$ behave in a similar
manner. This is indeed true, as evidenced by Figure 4b where we arbitrarily
single out the transverse speed at the nominal slab half-width $\hat{v}(R,t)$.
One then sees again the rapid attenuation for both steepness parameters, with
only two (four) extrema discernible when $\mu=1.5$ ($\mu=5$). We remark that
the symbols represent the “Modal Closed” solutions constructed with Equation
(25) with the domain size chosen to be $d/R=50$. That the FD and “Modal
Closed” solutions agree exactly means that this modest domain size is adequate
for describing the behavior of $\hat{v}(R,t)$ in the examined timeframe.

Figure 5 capitalizes on Equation (25) to examine the contributions from
individual eigenmodes. For this purpose, the discrete sets of position-
independent coefficients $\\{c_{l}\\}$ and the specific contributions at the
half-width $\\{c_{l}\breve{v}_{l}(R)\\}$ are grabbed from the “Modal Closed”
solutions from Figure 4. They are then plotted against the sets of
eigenfrequencies $\\{\omega_{l}\\}$ in Figures 5a and 5b, with the two
different steepness parameters again discriminated by the different colors.
The critical frequency $\omega_{\rm crit}=kv_{\rm Ae}$ is given by the
vertical dash-dotted lines for comparison. Examine Figure 5a first. One sees
that the frequency-dependency of $|c_{l}|$ for $\mu=1.5$ is qualitatively the
same as that for $\mu=5$, both featuring a series of peaks at frequencies
substantially higher than $\omega_{\rm crit}$. Now examining Figure 5b, one
sees the same set of peak frequencies in $\\{c_{l}\breve{v}_{l}(R)\\}$ for
each examined $\mu$. The values of these peak frequencies then explain why the
transverse Alfv$\acute{\rm e}$n time $R/v_{\rm Ai}$ characterizes the
periodicities in the signals themselves ($\hat{v}(R,t)$) shown in Figure 4b.
It is just that the term “periodicity” applies only marginally to
$\hat{v}(R,t)$ for $\mu=1.5$ to say the best, given the extremely rapid
attenuation of the signal. Regardless, more important is that all
eigenfrequencies are above $\omega_{\rm crit}$, meaning that all eigenmodes
are oscillatory. This is true not just for $\mu=5$ but for $\mu=1.5$ at least
at the chosen domain size for EVP 1, despite that evanescent eigenmodes are
ensured for an infinite domain when $\mu<2$.

###  4.2 Dependence of Modal Structure on Domain Size

This subsection examines how the modal structure depends on the domain size
$d$ for EVP 1, where by “modal structure” we refer to the $l$-dependence of
the eigenfrequency $\omega_{l}$. Our purposes are twofold. Firstly, we will
explore how the solutions to EVP 1 connect to the theoretical results on
cutoff wavenumbers for an open system by LN15b. In particular, we will
visualize what is meant by the irrelevance of cutoff wavenumbers when $\mu<2$,
paying special attention to how the number of evanescent eigenmodes varies
with $d$ to evaluate the assertion that “all fast modes are trapped”.
Secondly, in physical terms we will demonstrate when evanescent eigenmodes are
guaranteed to be negligible as far as the time-dependent solutions to IVP 1
are concerned.

Defining some fractional difference $\delta_{l}=\omega_{l}/\omega_{\rm
crit}-1$, Figure 6 examines how its modulus varies in response to the
variation of the dimensionless domain size $d/R$. The combination $[\rho_{\rm
i}/\rho_{\rm e},kR]$ is fixed at $[2.25,\pi/15]$, whereas two different values
are examined for the steepness parameter, one being $\mu=1.5$ (Figure 6a) and
the other being $\mu=5$ (Figure 6b). For each $\mu$, we choose to always
present the first five eigenmodes but sample other eigenmodes with a uniform
step of five in $l$ when the mode number $l$ ranges from $10$ to $50$. The
values for $\delta_{l}$ for any given $l$ are connected by a solid (dashed)
curve when $\delta_{l}$ is positive (negative). The dash-dotted curves, on the
other hand, represent a $1/d^{2}$ dependence as inspired by the analytical
result for step density profiles (see Equation (47)). Let us start with the
case $\mu=5$ as presented in Figure 6b, where one sees the unsurprising result
that $\delta_{l}$ is consistently positive. All eigenmodes are therefore
oscillatory, a behavior that can be better visualized by the spatial profiles
of the first three eigenfunctions shown in the right column of Figure 7. It is
also unsurprising to see that $\delta_{l}$ decreases monotonically with the
domain size. What Figure 6b demonstrates for sufficiently large $d/R$ is that
the frequency spacing $\Delta\omega_{l}$ starts with a $1/d^{2}$ dependence
for small $l$ and gradually transitions to a $1/d$ dependence for large $l$,
with the latter dependence following from Equation (22). This behavior is
qualitatively the same as happens for the step density profiles (see Section
3.3). We therefore conclude that a genuine continuum $(\omega_{\rm
crit},\infty)$ can be properly defined for an open system, provided that the
limit $d/R\to\infty$ is taken. However, the modal behavior for $\mu=1.5$ is
qualitatively different. One sees from Figure 6a that evanescent eigenmodes
indeed show up eventually for large enough $d/R$, as indicated by the changes
of sign of $\delta_{l}$. For instance, the first eigenmode transitions from an
oscillatory to an evanescent one at $d/R\approx 900$, and so does the second
(third) eigenmode at $d/R\approx 6600$ ($d/R\approx 29000$). Once again, this
behavior is made clearer by the spatial dependencies of the first several
eigenfunctions (see the left column of Figure 7). One then deduces that an
increasing number of evanescent eigenmodes emerge from the sea of oscillatory
eigenmodes as the domain size increases, which we take as a visualization of
the notion that cutoff wavenumbers are indeed irrelevant when $\mu<2$.
However, this notion needs to be understood as meaning that evanescent
eigenmodes are always permitted for sufficiently large domains, regardless of
the combination $[\rho_{\rm i}/\rho_{\rm e},kR]$. On the other hand, with the
exception of the lowest subset, the frequencies of oscillatory eigenmodes are
such that the frequency spacing $\Delta\omega_{l}$ also follows a $1/d^{2}$
dependence at small $l$ and gradually gives way to a $1/d$ dependence at large
$l$. Despite this $d$-dependence, we conclude that a genuine continuum
$(\omega_{\rm crit},\infty)$ cannot be defined, at least not by the
straightforward limiting procedure $d/R\to\infty$ because the critical
frequency is continuously crossed when $d/R$ increases continuously. We also
argue that it does not much sense to explore the spectrum for a truly open
system per se, because all necessary physics can be addressed by the “Modal
Closed” approach, which is simple both technically and conceptually. In
particular, one safely concludes that evanescent eigenmodes makes no
contribution to the solution to IVP 1 in Figure 4 as far as the timeframe
therein is of interest, despite that a large number of evanescent eigenmodes
are permitted for extraordinarily large domains. This conclusion holds because
the solution in the pertinent timeframe is entirely contributed by oscillatory
eigenmodes by any practical standard, and evanescent eigenmodes are incomplete
no matter how dense the set of their frequencies may be.

We further exploit the “Modal Closed” approach to examine the condition for
evanescent eigenmodes to be negligible in terms of their contributions to the
system evolution. This discussion turns out to be not that straightforward. We
therefore decide to employ a fixed combination $[\rho_{\rm i}/\rho_{\rm
e},kR]=[2.25,\pi/15]$ and additionally assume $\Lambda\geq R$. Only the
situation $\mu<2$ is relevant now. Consider some extremely large domain size
$d$ such that a substantial number of evanescent eigenmodes solve EVP 1. Let
the $l$-th evanescent eigenmode be characterized by some spatial scale
$D_{l}$, which can be unambiguously identified as the position beyond which
the $l$-th eigenfunction is evanescent. It follows from Equation (21) that

$\dfrac{D_{l}}{R}=\left[\dfrac{(1-\alpha_{l})(\rho_{\rm i}/\rho_{\rm
e}-1)}{\alpha_{l}}\right]^{1/\mu},$ (53)

where $\alpha_{l}=1-\omega_{l}^{2}/\omega_{\rm crit}^{2}$. Evidently,
$D_{1}<D_{2}<\cdots$ for a given $\mu$ because $\omega_{1}<\omega_{2}<\cdots$.
For definiteness, we evaluate $\omega_{1}$ at some $d$ beyond which
$\omega_{1}$ does not vary with the domain size within the accuracy of our EVP
solver (see Figure 6a). Figure 8a then displays $1-\omega_{1}/\omega_{\rm
crit}$ and $D_{1}$ against $\mu$. One sees that $1-\omega_{1}/\omega_{\rm
crit}$ decreases and consequently $D_{1}$ increases extremely rapidly with
$\mu$, the reason being that $\omega_{1}$ becomes increasingly close to
$\omega_{\rm crit}$ from below when $\mu$ approaches two.

Our following discussions center around $D_{1}$. Immediately ensuing is a
timescale $\tau_{D1}$, namely the time at which the location $x=D_{1}$ is
disturbed. Evidently,

$\tau_{D1}=\int_{\Lambda}^{D_{1}}\dfrac{dx}{v_{\rm A}(x)}.$ (54)

Evanescent eigenmodes are guaranteed to play no role in the timeframe
$t<\tau_{D1}$. Figure 8b offers an example where we survey the time-dependent
solutions for a considerable number of combinations of $\mu$ and $\Lambda$.
Note that the FD approach is adopted for convenience. For each
$[\mu,\Lambda]$, we determine a further timescale $\tau_{\rm ener}$ as the
time at which $E_{\rm tot}(\Lambda,t)/E_{\rm tot}(\Lambda,t=0)$ decreases to
some threshold $\delta_{\rm ener}$ with $E_{\rm tot}(\Lambda,t)$ being the
instantaneous total energy in the volume where the initial perturbation is
applied (see Figure 4). The solid curves represent the $\mu$-dependencies of
$\tau_{\rm ener}$ determined with $\delta_{\rm ener}={\rm e}^{-4}\approx 1/55$
for a number of $\Lambda$ as discriminated by the different colors. One sees
that $\tau_{\rm ener}$ thus determined for a given $\Lambda$ is largely
insensitive to $\mu$, the only exception being for $\Lambda/R=8$ in the
portion $\mu\lesssim 1.1$. There is actually some subtlety associated with
this exception, as will be discussed shortly. It suffices for now to notice
the overall behavior that $\tau_{\rm ener}$ increases with $\Lambda$ at a
given $\mu$. This behavior is understandable given that, when $\Lambda$
increases, the eigenmodes with increasingly low frequencies make more
contributions to the system evolution, thereby increasing the duration it
takes for the eigenmodes to interfere to weaken $E_{\rm tot}(\Lambda,t)$ to
some threshold. Regardless, one sees that $\tau_{\rm ener}$ at any examined
$[\mu,\Lambda]$ is lower than the corresponding $\tau_{D1}$, which is
represented by the dash-dotted curves and color-coded by $\Lambda$ in the same
way as for $\tau_{\rm ener}$. This is not surprising because $\tau_{\rm
ener}\lesssim 20.8R/v_{\rm Ai}$ with the maximum reached at
$[\mu,\Lambda/R]=[1,8]$, while Equation (54) indicates that
$\tau_{D1}>(D_{1}-\Lambda)/v_{\rm Ae}$ and the latter attains a minimum of
$\sim 44.2~{}R/v_{\rm Ai}$ at the same $[\mu,\Lambda/R]$. The comparison
between $\tau_{\rm ener}$ and $\tau_{D1}$ therefore validates the negligible
role of evanescent eigenmodes, which was explicitly shown by the specific
“Modal Closed” solution for $[\mu,\Lambda/R]=[1.5,4]$ (see Figures 4 and 5).

We now come back to the behavior of $\tau_{\rm ener}$ for $\Lambda/R=8$ as
shown in Figure 8b. By subtlety we refer to the apparent discontinuity at
$\mu\approx 1.1$, and refer further to the behavior of signals for
$x\leq\Lambda$ in the interval $t<\tau_{D1}$. We restrict ourselves to the
parametric survey presented in Figure 8 for the ease of description. The
signals turn out to always evolve in three stages, to be labeled I to III
hereafter. Stage I is characterized by some rapid attenuation, and the
periodicity tends to be substantially shorter than $P_{\rm
longi}=2\pi/\omega_{\rm crit}$. Here the subscript “longi” is meant to
indicate that this periodicity evaluates to the longitudinal Alfv$\acute{\rm
e}$n time $2L/v_{\rm Ae}$ for axial fundamentals. In addition, $E_{\rm
tot}(\Lambda,t)$ hardly shows any increase with time. Stage III, on the other
hand, is characterized by a nearly monochromatic variation with the
periodicity being very close to $P_{\rm longi}$. The signals still weaken with
time but in a fashion different from Stage I, meaning in particular that
$E_{\rm tot}(\Lambda,t)$ is wavy and its envelope declines at a much lower
rate. Stage II corresponds to the transition from Stage I to Stage III, and
tends to be short in duration. The appearance of the three stages is actually
a natural result of Equation (25). Stage I tends to involve eigenmodes with
frequencies $\omega_{l}$ substantially higher than $\omega_{\rm crit}$, and
the signals therefore tend to attenuate rapidly because the involved
monochromatic components can readily interfere to largely cancel out their
contributions. Consequently, modes with frequencies very close to $\omega_{\rm
crit}$ are allowed to stand out in stage III, and the associated monochromatic
components require longer time to interfere to yield some appreciable
attenuation. Note that the physics discussed herein only requires the
negligible role of evanescent eigenmodes, and hence applies to the case for
$\mu>2$ when $k<k_{{\rm cutoff},1}$. It is just that evanescent eigenmodes and
$\tau_{D1}$ become irrelevant altogether in that case, and there is no need to
consider any complication beyond $t=\tau_{D1}$. Now focus once again on Figure
8b. The details of Stage II, in particular where it is positioned in time,
turn out to possess a rather intricate dependence on $\mu$ and $\Lambda/R$.
One end result is that $\tau_{\rm ener}$ always appears in Stage I for all
examined pairs $[\mu,\Lambda/R]$ with the exception being for $\mu\lesssim
1.1$ when $\Lambda/R=8$. In this case $\tau_{\rm ener}$ appears in Stage II,
and the somehow wavy behavior in $E_{\rm tot}(\Lambda,t)/E_{\rm
tot}(\Lambda,t=0)$ means that the threshold $\delta_{\rm ener}$ is reached on
the opposite side of a ripple to what happens for larger $\mu$. Hence the
discontinuity in the $\mu$-dependence of $\tau_{\rm ener}$.

###  4.3 Further Remarks

The results in this section can be further digested as follows. We
consistently see the interval $t\leq\tau_{\rm ener}$ as the duration of
interest, and restrict ourselves to Figures 4 and 8. Let us start with the FD
solutions in Figure 4 and recall that no outer boundary is involved in IVP 1.
The drop of $E_{\rm tot}(t)/E_{\rm tot}(t=0)$ to the designated threshold
$\delta_{\rm ener}$ can then be explained by that almost all of the wave
energy in the volume $x\leq\Lambda$ is carried away by the first two outgoing
wave fronts. The passage of either front through $x=\Lambda$ is readily
identified by a rapid drop in $E_{\rm tot}(t)$ in front of a plateau-like
feature. This behavior happens for both $\mu=1.5$ and $\mu=5$, and holds also
for the cylindrical computations (see Figure 2 in Paper I). Now consider the
“Modal Closed” solutions, and let $D_{\rm ener}$ denote the distance that the
outermost perturbation reaches when $t=\tau_{\rm ener}$. Furthermore, let
$D_{\rm ener}$ be chosen as the domain size for EVP 1. By construction, the FD
solution for $t\leq\tau_{\rm ener}$ is identical to the “Modal Closed”
solution (25) constructed with eigensolutions to EVP 1. Consequently, the
outgoing fronts in the FD solution for $\mu=5$ are in fact a manifestation of
oscillatory eigensolutions, which are the only solutions to EVP 1 regardless
of the domain size. Let us move on to $\mu=1.5$, for which Figure 8b shows
that $\tau_{D1}\gg\tau_{\rm ener}$. Hence $D_{1}\gg D_{\rm ener}$. The
outgoing fronts in the FD solution for $\mu=1.5$ then also derive from the
oscillatory eigensolutions to EVP 1, given that only eigensolutions of this
type are allowed for any $d<D_{1}$. All in all, only oscillatory
eigensolutions are allowed for both $\mu=1.5$ and $\mu=5$ if $d=D_{\rm ener}$.
This then explains the qualitative similarity in the behavior of $E_{\rm tot}$
in the two FD solutions when $t<\tau_{\rm ener}$.

Some additional implications follow. Our key message is that IVP studies in
general offer a fuller picture for the wave dynamics than EVP analyses. Take
the case $\mu=1.5$ for instance. Let $d^{\prime}\gg D_{1}$ denote another
domain size for EVP 1. A considerable number of evanescent eigensolutions may
be allowed together with an extremely closely spaced set of oscillatory ones.
When $t<\tau_{\rm ener}$, however, the superposition of all eigensolutions
necessarily yields the same temporal behavior as computed with eigensolutions
to EVP 1 on the domain of size $d=D_{\rm ener}$. In other words, evanescent
eigenmodes can be safely discarded for both the gross energetics and the
signals sampled at individual locations (see Figure 4). This then calls into
question the seismological applications of the EVP analyses for $\mu<2$ by,
say, Lopin & Nagorny (2015a). Of interest therein was a pulsating flare event
observed by the Nobeyama Radioheliograph (NoRH) and originally reported in
Kupriyanova et al. (2013). Three periodicities were found, and the spatial
distributions of individual power spectral densities favor their
identification as the first three axial harmonics. The interpretation of these
signals as sausage perturbations was rejected by Kupriyanova et al. (2013),
for the measured periods are difficult to reconcile with sausage modes in
flare loops with step transverse profiles. However, Lopin & Nagorny (2015a)
argued that this difficulty may be circumvented if one invokes transverse
profiles similar to our “outer $\mu$” formulation with $\mu<2$. We remark that
evanescent eigensolutions were implied to dominate the system behavior for
this argument to apply. Our results in this section indicate otherwise,
stressing that the physical relevance of a mode needs to be established by IVP
studies before it is invoked for seismology.

##  5 Summary

This study was motivated by some recent theoretical progresses that may cast
doubt on the well-known lack of candidate sausage perturbations in solar
active region (AR) loops. We adopted linear, pressureless, ideal MHD to
examine the responses of straight, field-aligned, axially uniform, coronal
slabs to localized initial velocity perturbations that excite axial
fundamentals. The equilibrium density ($\rho_{0}$) was taken to be structured
only in one lateral direction ($x$), following an “outer $\mu$” prescription
where a steepness parameter $\mu$ pertains to the exterior (see Equation (1)).
An initial value problem (IVP, see IVP 1) was formulated on the open domain
$[0,\infty)$, to which two types of eigenvalue problems (EVPs) are relevant.
No boundary condition (BC) is specified at infinity in the “EVP open noBC”
type, whereas the BC at infinity for the “EVP open noIC” type is that no
incoming waves are permitted. While working on an EVP of the “EVP open noBC”
type, Lopin & Nagorny (2015b, LN15b) showed that the steepness of $\rho_{0}$
in the exterior is critical to the existence of cutoff axial wavenumbers. An
application of the LN15b analysis to our “outer $\mu$” profile yields that
cutoff wavenumbers arise only when $\mu\geq 2$. The lowest cutoff exceeds
axial wavenumbers expected for typical AR loops (Li et al., 2018). Evanescent
eigenmodes (or equivalently trapped modes) are therefore prohibited,
supporting the customary argument for sausage perturbations to be difficult to
detect. When $\mu<2$, however, cutoff wavenumbers are irrelevant in that
evanescent eigenmodes are always allowed regardless of the axial wavenumber
($k$) or the density contrast ($\rho_{\rm i}/\rho_{\rm e}$). This led LN15b to
further argue that all fast waves are evanescent. By implication, sausage
perturbations may then have been overlooked in AR loops with $\mu<2$. We
focused on a value of $k$ that ensures the absence of evanescent eigenmodes
for $\mu\geq 2$, and solved IVP 1 by superposing the eigenmodes on a closed
domain (EVP 1; also Equation (25)). Our study can be summarized as follows.

While revisiting IVP 1 for step density profiles ($\mu\to\infty$), we offered
an essentially analytical solution in terms of eigensolutions to the EVP of
the “EVP open noBC” type. The importance of initial perturbations was
demonstrated. In particular, their specification impacts whether the system
evolution can be understood with discrete leaky modes, which arise for “EVP
open noIC”. Our “Modal Closed” solutions then complemented previous studies of
Andries & Goossens (2007) and Oliver et al. (2015) by visualizing how the
oscillatory continuum in “EVP open noBC” is approached by the discrete
eigenspectrum. We further applied the “Modal Closed” approach to the case
where $\mu$ is finite, and showed that the situation with $\mu\geq 2$ is
qualitatively the same as in the step case. Evanescent eigenmodes are
prohibited by the relevance of cutoff axial wavenumbers, and an oscillatory
continuum can result for open systems. When $\mu<2$, we showed that cutoff
wavenumbers are indeed irrelevant. However, we stressed that this irrelevance
does not mean that all fast waves are evanescent for truly open systems.
Rather, this means that an increasing number of evanescent eigenmodes result
when the domain size increases for EVP 1. Oscillatory eigenmodes are always
permitted, although a genuine continuum cannot be obtained simply by letting
the domain size approach infinity. We quantified the condition for oscillatory
eigenmodes to dominate the solutions to IVP 1, justifying the absence of any
qualitative change of the system behavior when $\mu$ crosses the nominal
critical value of two.

Before closing, we emphasize again the importance of the IVP approach for
examining the wave dynamics. The physical relevance of a mode found in some
EVP analysis needs to be established with an IVP examination before it is
employed for seismology. Likewise, the detailed temporal behavior of an
observed oscillatory signal actually carries some rich information on both the
wave host and the initial perturbation. Furthermore, we remark that our
numerical findings can be summarized in a simpler way, provided that one is
interested in the solutions to IVP 1 rather than any conceptual difficulty
associated with the domain being open for some relevant EVPs. First of all,
the mathematical properties of EVP 1 ensure that the associated eigenmodes can
be used to solve IVP 1. That EVP 1 is defined on a closed domain merely means
that the “Modal Closed” solution (Equation (25)) holds only in some
appropriate timeframe. For our chosen parameters, the key difference in the
modal behavior between $\mu\geq 2$ and $\mu<2$ is that only oscillatory
eigenmodes are allowed in the former, whereas evanescent eigenmodes eventually
show up for sufficiently large domains in the latter. The system evolution,
however, necessarily shows no qualitative difference for different $\mu$ in
those timeframes for which the required domain sizes of EVP 1 are not large
enough to accommodate evanescent eigenmodes. Only oscillatory eigenmodes are
therefore involved in Equation (25). The signal at a given location then
weakens with time, for the monochromatic components will interfere to cancel
out their contributions. On this last aspect, we note that a close analogy can
be drawn for kink waves resonantly absorbed in the Alfv$\acute{\rm e}$n
continuum (Soler & Terradas 2015; see also Cally 1991 and Andries & Goossens
2007).

We thank the referee for thorough and constructive comments. This research was
supported by the National Natural Science Foundation of China (41974200,
41904150, and 11761141002). We gratefully acknowledge ISSI-BJ for supporting
the international team “Magnetohydrodynamic wavetrains as a tool for probing
the solar corona”.

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Figure 1:  (a) Two-dimensional (2D) representation of the 1D initial value
problem (IVP 1). Plotted are the $x-z$ distributions of the equilibrium
density (the filled contours) and the initial velocity field (the arrows). The
examined slabs possess a nominal half-width $R$. Axial fundamentals are
ensured by the $z$-dependence of the initial perturbation. (b) Illustrative
transverse profiles of the initial perturbation ($\hat{v}$, the blue dashed
curve) and the equilibrium density ($\rho_{0}$, the solid curves), both
involved in IVP 1. The density profile follows the outer $\mu$ prescription in
Equation (1), with the density contrast $\rho_{\rm i}/\rho_{\rm e}$ chosen to
be $2.25$. Two steepness parameters are plotted, namely $\mu=1.5$ (the black
curve) and $\mu=5$ (red). The initial perturbation is described by Equation
(10), with the spatial extent $\Lambda$ chosen to be $4R$ here.

Figure 2:  “Modal Open” solutions to IVP 1 for a coronal slab with step
density profile ($\mu=\infty$) and a density contrast of $\rho_{\rm
i}/\rho_{\rm e}=2.25$. The dimensionless axial wavenumber is fixed at
$kR=\pi/15$. By “Modal Open” we mean that the solution is constructed by
superposing the eigenmodes for an open system. A number of values are examined
for the spatial extent ($\Lambda$) of the initial perturbation as
discriminated by the different colors. Plotted in (a) are the temporal
profiles of the transverse speed at the nominal slab half-width, namely
$\hat{v}(R,t)$. In addition to the “Modal Open” solutions (the solid curves),
the finite-difference solutions are also presented for two values of $\Lambda$
for comparison (the black and red symbols). Presented in (b) are the frequency
dependencies of the local spectral densities at the nominal slab half-width,
namely $S_{\omega}\breve{v}_{\omega}(R)$. The vertical dash-dotted line
represents the critical frequency $\omega_{\rm crit}=kv_{\rm Ae}$. The real
parts of the mode frequencies for the first several discrete leaky modes
(DLMs) are indicated by the magenta arrows. These DLMs are computed with
Equation (43), the dispersion relation (DR) pertaining to an eigenvalue
problem (EVP) that does not allow incoming waves at infity. The continuous
curves, however, derive from the improper continuum that results from an EVP
for which no boundary condition is specified at infinity. See text for more
details.

Figure 3:  Similar to Figure 2 but for a fixed combination $[\rho_{\rm
i}/\rho_{\rm e},kR,\Lambda/R]=[2.25,\pi/15,4]$. The “Modal Open” solution is
represented by the black curves in both (a) and (b). Additional “Modal Closed”
solutions are presented as found by superposing the eigenmodes for the EVP
defined on closed domains (EVP 1). Two domain sizes are discriminated by the
different colors, one being $d/R=50$ and the other being $d/R=100$. The
eigenfrequencies for EVP 1 are discrete by construction. See text for more
details.

Figure 4:  FD solutions to IVP 1 for two steepness parameters, one being
$\mu=1.5$ (the black curves) and the other being $\mu=5$ (red), given a fixed
combination $[\rho_{\rm i}/\rho_{\rm e},kR,\Lambda/R]=[2.25,\pi/15,4]$.
Plotted in (a) are the terms characterizing the energy content in the volume
$\mathcal{V}$ where the initial perturbation is applied, with $E_{\rm tot}$
and $F$ representing the total wave energy in and the cumulative energy loss
from $\mathcal{V}$ (see Equations (12) to (14)). The transverse speeds at the
nominal half-width $\hat{v}(R,t)$ are shown against time in (b), where the
“Modal Closed” solutions are presented by the symbols as labeled. These
solutions are obtained with Equation (25), with the domain size chosen to be
$d=50~{}R$.

Figure 5:  Frequency-dependencies of the contributions of individual modes for
a fixed combination $[\rho_{\rm i}/\rho_{\rm
e},kR,\Lambda/R,d/R]=[2.25,\pi/15,4,50]$. Two steepness parameters, $\mu=1.5$
and $\mu=5$, are examined as labeled. The sets of position-independent
coefficients $\\{c_{l}\\}$ are plotted in (a), while the specific
contributions at the half-width $c_{l}\breve{v}_{l}(R)$ are presented in (b).
The critical frequency $\omega_{\rm crit}=kv_{\rm Ae}$ is represented by the
vertical dash-dotted lines for reference.

Figure 6:  Dependencies of the modulus of the fractional difference
$\delta_{l}=\omega_{l}/\omega_{\rm crit}-1$ on the dimensionless domain size
($d/R$) for two steepness parameters, (a) $\mu=1.5$ and (b) $\mu=5$, given a
combination $[\rho_{\rm i}/\rho_{\rm e},kR]=[2.25,\pi/15]$. For each pair
$[\mu,d/R]$, the first five eigenmodes are always presented, while the rest
are uniformly sampled in $l$ with a step of five when $l$ lies in the range
between $10$ and $50$. A solid (dashed) curve is employed to connect
$|\delta_{l}|$ for a given $l$ when $\delta_{l}$ is positive (negative). The
blue dash-dotted lines represent the $1/d^{2}$-dependence for comparison. See
text for more details.

Figure 7:  First three eigenfunctions for EVP 1 on a number of domains
differentiated by the linestyles, given a fixed combination $[\rho_{\rm
i}/\rho_{\rm e},kR]=[2.25,\pi/15]$. Two steepness parameters are examined, one
being $\mu=1.5$ (the left column) and the other being $\mu=5$ (right). The
eigenfunctions are arbitrarily scaled such that their dependencies on the
domain size can be better visualized. See text for more details.

Figure 8:  Some scales characterizing the solutions to EVP 1 and IVP 1 for a
fixed pair $[\rho_{\rm i}/\rho_{\rm e},kR]=[2.25,\pi/15]$. Plotted in (a) is
the $\mu$-dependence of $D_{1}$, the distance beyond which the first
eigenfunction is evanescent. The fractional difference
$1-\omega_{1}/\omega_{\rm crit}$ is also displayed for reference. Panel (b)
presents, by the solid curves, the variations of some timescale $\tau_{\rm
ener}$ against $\mu$ for a number of $\Lambda$ as determined from the FD
solutions to IVP 1. Here $\tau_{\rm ener}$ denotes the time at which the total
energy drops by a factor of ${\rm e}^{4}\approx 55$ relative to its initial
value in the volume where the initial perturbation is applied. The dash-dotted
curves represent the $\mu$-dependencies of another $\Lambda$-related timescale
$\tau_{D1}$, the time when the position $x=D_{1}$ is disturbed. The curves for
both $\tau_{\rm ener}$ and $\tau_{D1}$ are color-coded by the values of
$\Lambda$. See text for more details.