# Topological modes in stellar oscillations

Armand Leclerc Univ Lyon, Univ Lyon1, ENS de Lyon, CNRS, Centre de Recherche
Astrophysique de Lyon
UMR5574, F-69230, Saint-Genis-Laval, France  Guillaume Laibe  Univ Lyon, Univ
Lyon1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon
UMR5574, F-69230, Saint-Genis-Laval, France  Institut Universitaire de France
Pierre Delplace  ENS de Lyon, CNRS, Laboratoire de Physique (UMR CNRS 5672),
F-69342 Lyon, France  Antoine Venaille  ENS de Lyon, CNRS, Laboratoire de
Physique (UMR CNRS 5672), F-69342 Lyon, France  Nicolas Perez  ENS de Lyon,
CNRS, Laboratoire de Physique (UMR CNRS 5672), F-69342 Lyon, France Guillaume
Laibe<EMAIL_ADDRESS>
###### Abstract

Stellar oscillations can be of topological origin. We reveal this deep and so-
far hidden property of stars by establishing a novel parallel between stars
and topological insulators. We construct an hermitian problem to derive the
expression of the stellar acoustic-buoyant frequency $S$ of non-radial
adiabatic pulsations. A topological analysis then connects the changes of sign
of the acoustic-buoyant frequency to the existence of Lamb-like waves within
the star. These topological modes cross the frequency gap and behave as
gravity modes at low harmonic degree $\ell$ and as pressure modes at high
$\ell$. $S$ is found to change sign at least once in the bulk of most stellar
objects, making topological modes ubiquitous across the Hertzsprung-Russel
diagram. Some topological modes are also expected to be trapped in regions
where the internal structure varies strongly locally.

Stellar oscillations – Internal waves – Theoretical techniques

##  1 Introduction

Stars are opaque. Fortunately, deformations of the stellar surface depend on
their interiors Cowling (1941); Gough (1993); Ledoux & Walraven (1958); Unno
et al. (1979); Christensen-Dalsgaard et al. (1996); Aerts et al. (2010) and as
such, asteroseismology is the Rosetta Stone to infer details of stellar
structures Christensen-Dalsgaard et al. (1996); Aerts et al. (2010). Stellar
spectra consist principally of low-frequencies gravity (g-) modes and high-
frequencies pressure (p-) modes, defining two bands separated by a finite
interval of frequencies, also referred as a gap. The stellar spectrum may also
be enriched by additional branches, such as surface wave modes confined in the
outer regions. In recent years, a novel type of waves propagating in
stratified compressible fluids has been discovered. This so-called Lamb-like
wave fills the gap between the p- and the g- band. Although this mode bears
similarities with the Lamb wave Lamb (1911); Iga (2001), it is confined around
peculiar values specific of the stratification profile, and not at the
boundaries. The key point is that these waves have been postulated using
arguments from topology Perrot et al. (2019). Modes in the original spatially
homogeneous system can be predicted from the analysis of the topological
invariant of a simpler dual wave problem with constant coefficients. Similar
topological approaches were developed in condensed matter since the eighties,
and flourished across all field of physics, including fluid dynamics and
plasma over the last few years Hasan & Kane (2010); Delplace et al. (2017);
Shankar et al. (2020); Parker (2021).

The Lamb wave has been detected in the atmosphere, but the Lamb-like wave is
hardly expected to propagate on Earth, neither in the atmosphere nor in
oceans. Stars were speculated to provide favourable conditions for it to
propagate Perrot et al. (2019). However, this study lacked the treatment of
self-gravity, spherical geometry and variations of sound speed, three critical
processes as we shall show. We therefore adapt tools that have been originally
developed by the topological insulator community to study the seminal case of
adiabatic perturbations of a non-rotating, non-magnetic, stably stratified
stellar fluid neglecting gravity perturbations (Cowling’s approximation
Cowling (1941)). The physical quantities are first rescaled to express the
evolution of linear perturbations under the form of a Schrödinger-like wave
equation

$i\partial_{t}{\mathbf{Y}}={\mathcal{H}}{\mathbf{Y}}$ (1)

where $\mathcal{H}=$
$i$ $\begin{pmatrix}0&0&0&0&-\frac{c_{\rm s}}{r}\partial_{\theta}\\\
0&0&0&0&-\frac{c_{\rm s}}{r\sin(\theta)}\partial_{\phi}\\\ 0&0&0&-N&S-c_{\rm
s}\partial_{r}-\frac{c_{\rm s}^{\prime}}{2}\\\ 0&0&N&0&0\\\ -\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\theta}(\sin(\theta)\cdot)&-\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\phi}&-S-c_{\rm s}\partial_{r}-\frac{c_{\rm
s}^{\prime}}{2}&0&0\end{pmatrix},$

and the perturbation vector contains rescaled velocities, density and pressure

$\mathbf{Y}=^{\top}\\!\\!\left(\tilde{u},\tilde{v},\tilde{w},\tilde{\Theta},\tilde{p}\right).$
(2)

See Appendix A for details.

As such, the $5\times 5$ wave operator $\mathcal{H}$ of the problem is
explicitely Hermitian. $\mathcal{H}$ depends on the sound speed
$c_{\mathrm{s}}$, the Brunt-Väisälä frequency $N$ and a characteristic
frequency further referred to as the acoustic-buoyant frequency $S$ that
emerges explicitly

$S\equiv\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-\frac{1}{2}\frac{{\rm
d}c_{\rm s}}{{\rm d}r}+\frac{c_{\mathrm{s}}}{r}.$ (3)

All three parameters vary with radius $r$. Usually, these equations are
combined into a single differential equation of high order. Instead,
preserving the vectorial structure of the problem is better suited for a
topological analysis.

##  2 Acoustic-buoyant frequency $S$

The acoustic-buoyant frequency $S$ is a coupling parameter for momentum
exchange between buoyant and acoustic oscillations, and was called
stratification parameter in Perrot et al. (2019). This role of mode coupling
is shown in details below. Two extra terms appear compared to the plan-
parallel case Perrot et al. (2019): $c_{\mathrm{s}}/r$, which accounts for
sphericity effects at small radii, and
$\frac{1}{2}\frac{\mathrm{d}c_{\mathrm{s}}}{\mathrm{d}r}$, which becomes
important when the internal structure of the object varies strongly. $S$
combines the four physical processes responsible for mirror-symmetry breaking
in the radial direction: gravity, density stratification, curvature, and
radial variations of sound speed. The profile $S(r)$ varies between stellar
objects ; however the sound speed is expected to go to 0 at the surface as a
positive power law of the density Chandrasekhar (1939); Horedt (1987). $S$ is
then $-\infty$ at the surface. At small radii, the curvature term guarantees
$S$ to reach $+\infty$. $S(r)$ being continuous, it must change sign in the
bulk of the star at least once. We confirm this analytically on a stellar
polytrope in Appendix B and numerically on models of typical stellar objects
computed with the MESA code Paxton et al. (2011) (Fig. 2).

The physical nature of the acoustic-buoyant frequency $S$ is disclosed by
considering the equivalent of Eq. (1) in the 2D plane-parallel $(y,z)$
geometry. After performing a Fourier transform in time and space in the
invariant direction $y$ and performing the rescaling $(u,w,\Theta,p)\mapsto
c_{\rm s}^{1/2}(u,w,\Theta,p)$, one obtains

$\displaystyle\partial_{t}u$ $\displaystyle=$ $\displaystyle
i{c_{\mathrm{s}}k_{y}}p,$ (4) $\displaystyle\partial_{t}\Theta$
$\displaystyle=$ $\displaystyle Nw,$ (5) $\displaystyle\partial_{t}w$
$\displaystyle=$ $\displaystyle-N\Theta-c_{\rm s}\partial_{z}p+Sp,$ (6)
$\displaystyle\partial_{t}p$ $\displaystyle=$ $\displaystyle
ic_{\mathrm{s}}k_{y}u-c_{\rm s}\partial_{z}w-Sw.$ (7)

Combining the equations gives

$\displaystyle(\partial_{tt}+N^{2})w=-\partial_{t}(c_{\rm
s}\partial_{z}p-Sp),$ (8)
$\displaystyle(\partial_{tt}+c_{\mathrm{s}}^{2}k_{y}^{2})p=-\partial_{t}(c_{\rm
s}\partial_{z}w+Sw),$ (9)

a system where acoustic and buoyant vibrations are explicitly coupled (no
Boussinesq or anelastic approximation is assumed). The first term of the
right-hand side of Eq. (8) consists of local pressure forces that competes
with buoyancy. The first term of the right-hand side of Eq. (9) comes from
fluid compression in the direction $z$ and is generic from 2D purely acoustic
waves. In the long wavelength limit in the stratification direction $z$, these
two terms become negligible and

$\displaystyle(\partial_{tt}+N^{2})w=S\partial_{t}p,$ (10)
$\displaystyle(\partial_{tt}+c_{\mathrm{s}}^{2}k_{y}^{2})p=-S\partial_{t}w,$
(11)

showing that $S$ is the frequency of periodic exchanges of momentum between
acoustic and buoyant vibrations. Non-Boussinesq contributions allow local
densities to be affected by acoustic compression, providing an effect that
compets with buoyancy when $S$ is large. Conversely, pressure increases not
only through compression, but also through advection in a differential
background. These two effects on coupling between g-modes and p-modes were
identified by Lighthill (1978). Multiplying Eq. (10) by $\partial_{t}p$ and
Eq. (11) by $\partial_{t}w$ shows that the power transmitted by one mode to
the other occurs without losses, as expected from the adiabatic assumption.
Such a coupling has been widely studied in polariton physics, and shown to
result in gap opening Lagoudakis (2013). The condition $S=0$ is therefore
associated to local mode decoupling (see Fig. 1).

Figure 1: Local dispersion relation of the problem, as modelled by Eqs.
(8)-(9). The p-mode and the g-mode (solid lines) result from the coupling of
acoustic and buoyant oscillations (dashed lines for $k_{z}=0$). Both $S$ and
$k_{z}$ pull the bands away. For any mode, including $k_{z}=0$, a gap exists
as soon as $S\neq 0$. Figure 2: Profiles of $S$ for four different typical
stellar objects. $N$ is plotted for comparison. Solid orange line indicates
the region where the topological mode is trapped, as measured by the trapping
length $\mathcal{L}$ defined by Eq. (25). Stellar interiors are computed with
Mesa. The High mass star is an $M=100M_{\odot}$ main-sequence star. The White
Dwarf mass is $0.6M_{\odot}$, during its cooling phase. The Jupiter model has
a solid core of 10 Earth masses. $S$ cancels always at least once, whether in
the radiative or convective region. Light grey area indicates the convective
zone.

##  3 Topological properties of the problem

Eigenvalues of $\mathcal{H}$ are constrained by topology when varying the
physical parameters. These constraints can be efficiently studied by
associating a simple matrix to $\mathcal{H}$ that retains the topological
constraints. The correspondence is established via a Wigner transform, which
allows us to define rigorously a wave that is locally plane without any
hypothesis of scale separation (Appendix C). Here, topological properties of
$\mathcal{H}$ can be characterised through the eigenvalue problem of the
matrix

$\displaystyle M$ $\displaystyle\equiv$
$\displaystyle\begin{pmatrix}N^{2}&-NS-iNc_{\mathrm{s}}k_{r}\\\
-NS+iNc_{\mathrm{s}}k_{r}&L_{\ell}^{2}+S^{2}+c_{\mathrm{s}}^{2}k_{r}^{2}\end{pmatrix},$
(12) $\displaystyle\omega\mathbf{X}$ $\displaystyle=$ $\displaystyle
M\mathbf{X},\ $ (13)

where the Lamb frequency is $L_{\ell}\equiv
c_{\mathrm{s}}\sqrt{\ell(\ell+1)}/r$. $M$ is Hermitian and parametrised by a
radial wavenumber $k_{r}$ and parameters $L_{\ell}$, $c_{\mathrm{s}}$, $N$ and
$S$ that are constant.

As expected, the two eigenvalues of $M$ correspond to the square of the
frequencies of the local pressure and gravity modes. Interestingly, these two
bands intersect when $k_{r}=0$, $L_{\ell}=N$, $S=0$ for any value of
$c_{\mathrm{s}}$ and $N$, i.e. the two frequencies degenerate into a single
one (see Appendix C). Such a degeneracy point behaves like a topological
monopole in parameter space ($k_{r},L_{\ell},S$), which is characterised by an
integer called the Chern number Chern (1946). A non-zero Chern number
translates the topological obstruction to smoothly define the phase of the
eigenvectors – that describe the local polarization relations of $M$ – all
around the degeneracy point in parameter space. In that case, the eigenvectors
can only be defined smoothly over patches in parameter space, corresponding to
different gauge choices. The $U(1)$ gauge transformation that connects the
different patches is a phase whose winding is the Chern number. In our case,
we find the Chern numbers associated to the gravity and the pressure bands to
be $\mathcal{C}^{g}=+1$ and $\mathcal{C}^{p}=-1$ respectively (see Appendix D
for computations). These topological considerations can be back-connected to
the original problem : any change of sign of the acoustic-buoyant frequency
$S(r)$ is associated to the existence of a branch that transits from the
g-band towards the p-band as $\ell$ increases. Mathematically, this
correspondence is ensured by index theorems Atiyah & Singer (1963); Chern
(1946); Perrot et al. (2019); Faure (2019); Delplace (2022); Nakahara (1990);
Esposito (1997). The transiting branch flows from the upper-band to the lower-
band or vice-versa, depending on the sign of $S^{\prime}$ at the change of
sign of $S$. In stars, $S^{\prime}<0$ and the mode transits from the g- to the
p- band: this mode is the Lamb-like wave Perrot et al. (2019). Fig. 3 confirms
the deep relation between a change of sign of $S(r)$ and the existence of a
mode transiting from the g-band at small $\ell$ to the p-band at large $\ell$.
The physical validity of this mode is carefully verified in Appendix E.

Figure 3: A mode develops between the gravity band and the pressure band
(bottom) when the acoustic-buoyant frequency $S$ (top) of Eq. (3) changes
sign. From left to right: toy profile that cancels in the bulk, decaying
positive profile of $S$, polytrope with polytropic index $n=3$ and a Mesa
Solar-like profile. Physical values of the harmonic degree $\ell$ are integer,
and plotted with large points, from 0 to 20. Non-integer values are plotted
with small points for readability. Surface gravity waves are filtered out by
appropriate boundary conditions. The mode transiting between bands is
highlighted in black. These values are computed by solving Eq. (1) numerically
using Dedalus Burns et al. (2020).

By analogy with similar modes encountered in a variety of other physical
systems Hasan & Kane (2010); Delplace et al. (2017); Shankar et al. (2020);
Parker (2021), one may expect for the global stellar mode to have no node, and
to transit between the bands at a value of $\ell$ such that $L_{\ell}\sim N$.
One may also expect for the eigenfunctions to be located around the radius
$r_{0}$ where $S(r_{0})=0$. These properties of the topological mode can be
verified on a simple analytically solvable model presented in the next
section.

##  4 Topological mode in analytical model

We present a simple analytical model featuring a cancellation in $S$, and show
that the analytical solution of the wave equation include the topological
mode. Consider a fluid where all quantities but $S$ are constant in space:

$\displaystyle S(r)$ $\displaystyle=$ $\displaystyle-\alpha(r-r_{0}),$ (14)
$\displaystyle N(r)$ $\displaystyle=$ $\displaystyle N_{0},$ (15)
$\displaystyle c_{\mathrm{s}}(r)$ $\displaystyle=$ $\displaystyle
c_{\mathrm{s,0}},$ (16) $\displaystyle L_{\ell}^{2}(r)$ $\displaystyle=$
$\displaystyle L_{\ell,0}^{2}=c_{\mathrm{s,0}}\frac{\ell(\ell+1)}{r_{0}^{2}}.$
(17)

This parametrisation mimics a situation where variations of $S$ would be
infinitely more abrupt than the other quantities. In this minimal model, $S$
vary linearly and cancels in $r_{0}$. This model may thus be thought of as the
compressible-stratified analogue to the equatorial shallow water model solved
by Matsuno Matsuno (1966). Perform the transform $(u,v,w,\Theta,p)\mapsto
c_{\rm s}^{1/2}(u,v,w,\Theta,p)$ in Eq. (1), then apply a time-Fourier
Transform and project on spherical harmonics. The variables combine into a a
single ODE on $p$

$\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\left(\frac{S(r)}{c_{\mathrm{s,0}}}\right)^{2}-\left(\frac{S(r)}{c_{\mathrm{s,0}}}\right)^{\prime}+k_{r,0}^{2}\right)p=0,$
(18)

where we denote

$k_{r,0}^{2}\equiv\frac{(L_{\ell,0}^{2}-\omega^{2})(N_{0}^{2}-\omega^{2})}{c_{\mathrm{s,0}}^{2}\omega^{2}},$
(19)

and use the symbol ′ for derivatives with respect to $r$ for background
quantities. Eq. (18) holds for any $S(r)$, and can be seen as a Schrödinger
equation describing a particle of energy $k_{r,0}^{2}$ in the potential
$V=S^{2}/c_{\mathrm{s,0}}^{2}+S^{\prime}/c_{\mathrm{s,0}}$. For the model of
Eq. (14), this reduces to

$\left[\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-\left(\frac{1}{4}x^{2}-\frac{1}{2}(1+\frac{c_{\mathrm{s,0}}}{\alpha}k_{r,0}^{2})\right)\right]p=0,$
(20)

using the dimensionless quantity
$x\equiv\sqrt{2}\sqrt{\frac{\alpha}{c_{\mathrm{s,0}}}}(r-r_{0})$. The solution
is a Parabolic Cylinder Function $U$ Abramowitz & Stegun (1972)

$p=U\left(-\frac{1}{2}(1+\frac{c_{\mathrm{s,0}}}{\alpha}k_{r,0}^{2}),~{}x\right).$
(21)

Regularity at infinity imposes the first argument to be negative half-integer,
leading to the quantization

$\frac{c_{\mathrm{s,0}}}{\alpha}k_{r,0}^{2}=2n+1,$ (22)

for any $n\in\mathds{N}$.

Figure 4: Left: spectrum of the minimal model parametrized by Eq. (14). The
topological mode is the $n=0$ mode, and transits between the bands. Right:
Measure of the error a JWKB approximation of the solutions would make. The
error on the $n=0$ mode is not small.

Solutions reduce then to Hermite functions ${p={\rm e}^{-x^{2}/2}H_{n}(x)}$,
where $H_{n}$ denotes the $n$th Hermite polynomial. Fig. 4 shows the spectrum
associated to this problem. The values of $\omega$ can be inverted in Eq.
(22). For $n\geqslant 1$, each value of $k_{r,0}$ give two eigenfrequencies,
the $n$th g-mode and the $n$th p-mode. The expected topological mode
corresponds to $n=0$. One of the two eigenfrequencies associated with this
solution is unphysical, since the eigenfunctions diverges quickly at infinity.
The other verifies

$\omega=L_{\ell,0},$ (23)

which transits between the bands as $\ell$ increases, as shown on Fig. 4. This
property is associated to the fact that ${S^{\prime}(r_{0})=-\alpha<0}$ at the
cancellation point.
The topological mode has the profile

$p(r)=p_{0}\exp\left(-\frac{\alpha}{c_{\mathrm{s,0}}}(r-r_{0})^{2}\right),$
(24)

an expression that provides a definition of the length over which the mode has
significant amplitude

$\mathcal{L}\equiv\sqrt{\frac{c_{\mathrm{s,0}}}{\alpha}}=\sqrt{{c_{\mathrm{s,0}}}/{\Big{|}\frac{\mathrm{d}S}{\mathrm{d}r}\Big{|}_{S=0}}},$
(25)

which we call the trapping length. Denoting
${R(x)\equiv-\frac{1}{4}x^{2}+n+1/2}$ the second term of Eq. (20) that
corresponds to a solution for a given $n$, we find for JWKB approximation of
the solution to be valid when the condition
$|R^{-3/2}\frac{\mathrm{d}R}{\mathrm{d}x}|\ll 1$ is satisfied (Daghigh &
Green, 2012). Fig. 4 shows this quantity for the first modes. The topological
mode $n=0$ breaks strongly this validity condition. As expected, JWKB
techniques cannot capture the topological mode.

This analytical solution confirms that the topological mode is the mode with
zero node of the system, and that this mode is not accessible with scale
separation methods.

##  5 Discussion

Interestingly, the topological mode and the surface-gravity mode have both
zero node and similar dispersion relations. Numerical experiments show that
when they coexist, they hybridize to form a unique $n=0$ mode. A comprehensive
study including various boundary conditions is performed in Appendix F. We
interpret this hybridised mode as the f-mode of asteroseismology Gough (1993);
Rozelot & Neiner (2011), revealing its previously unexpected hybrid nature.
Finally, strong local gradients of thermodynamical quantities may give rise to
peaks of acoustic-buoyant frequency where $S$ changes sign twice over a short
scale, as in the White Dwarf model showed on Fig. 2. This results in two modes
of topological origin that may be used to probe fine details of the structure
of the stellar object. The white dwarf is the canonical object for application
of this study, as it is fully radiative. Its profile of $S$ cancels three
times, two of them resulting from a phase transition close to the surface. For
this model, we predict three topological modes, one for each cancellation. One
crossing the gap, with long trapping length $\mathcal{L}$, as the slope of $S$
where it changes sign is low at the first cancellation. Two more modes with
zero nodes are predicted close to the peak of $S$ just underneath the surface,
with much smaller trapping lengths $\mathcal{L}$, as the slope of $S$ is high
when $S$ changes sign. They potentially overlap each other, such that they
would hybridise. This hybridisation could serve as a measure of the peak in
$S$, meaning the modes could serve as probes for the associated phase
transition. This hybridization is illustrated on Fig. 5.

Figure 5: Peaks of $S$ through positive values imply the existence of two
topological modes in the spectrum. The sharper the peak, the more the modes
hybridize, and their branches avoid crossing in the spectrum.

The current study focuses on stably stratified stars, for which index theorems
on Hermitian systems apply. However the effect of a convective zone on the
Lamb-like wave remains to be investigated. Such a region, where $N^{2}$
vanishes, is indeed sustained by the convective circulation of the background.
Fig. 2 shows that in the Sun, $S$ cancels in the radiative zone, close to the
convective zone. The trapping length of the topological mode indicates
interactions with the convective zone, although convection is out of the scope
of this study. In High Mass stars, $S$ cancels within the convective core
where the topological mode is not guaranteed by this study, as no background
flow is considered in the wave equation Eq. (1). The same conclusion applies
to Jupiter, which is fully convective, and has interesting multiple
cancellations of $S$.

Lamb-like waves are neither Lamb waves, surface-gravity waves, nor mixed modes
Dziembowski et al. (2001); Dupret et al. (2009); Deheuvels & Michel (2010).
Mixed modes are linear combinations of g-modes and p-modes standing in
different cavities in the star, due to spatial variations of $N$ and
$L_{\ell}$ and can have a high number of nodes. The Lamb-like wave emanates as
a mode $n=0$ of a single cavity hosting both g-modes and p-modes.
We expect generic properties of stellar pulsations related to topology such as
ray tracing to be encoded in $S(r)$ Perez et al. (2021). Other discrete
symmetries can be broken in the presence of rotation Perez et al. (2021) and
magnetic fields Cally (2006); Parker et al. (2020), and one should expect the
emergence of new classes of topological waves when these additional
ingredients are taken into account, potentially at the stellar tachocline
where strong shear develops. The resilience of these topological modes on
unstable stratification when $N^{2}<0$, or with the inclusion of dissipative
effects, is a highly promising avenue of research in the currently flourishing
field of non-Hermitian topological waves Delplace et al. (2021); Gong et al.
(2018); Yao & Wang (2018); Bergholtz et al. (2021).

##  6 Conclusion

In this study, we revisit the old field of stellar pulsations under the bright
new prism of topology. By doing a novel parallel between stars and topological
insulators, we establish for the first time the existence of a wave of
topological origin in stars. We derive the expression of a novel key physical
parameter, the acoustic-buoyant frequency. We demonstrate in a comprehensive
analysis that topological modes are associated to zeros of this frequency, and
show the ubiquitous existence of at least one topological mode across the
entire spectrum of stellar object in the Universe. More importantly, we show
that local phase transitions, which are key for understanding the evolution of
stars within the cosmological context, may give rise to pairs of robust
topological modes. The hunt of these modes may therefore become a critical
target for future cutting-edge instruments such as the PLATO mission.

## Acknowledgments

A.L. lead the derivation of the acoustic-buoyant frequency and performed
numerical simulations. G.L. lead the astrophysical analysis and the writing of
the manuscript. P.D. and A.V. lead the topological analysis and the analogy
with the plane-parallel case. N.P. performed the numerical experiments on
surface-gravity wave and topological mode hybridisation.
G.L. acknowledges funding from the ERC CoG project PODCAST No 864965. This
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
823823. This project was partly supported by the IDEXLyon project (contract
nANR-16-IDEX-0005) under the auspices University of Lyon. We acknowledge
financial support from the national programs (PNP, PNPS, PCMI) of CNRS/INSU,
CEA, and CNES, France. AV and PD were supported by the national grant
ANR-18-CE30-0002-01 and Idex Tore. N.P. was funded by a PhD grant allocation
Contrat doctoral Normalien. We thank S. Deheuvels, I. Baraffe, G. Chabrier, E.
Jaupart, J. Fensch and E. Lynch for useful comments and discussions. The
authors are grateful to the anonymous referee, whose thorough comments helped
to improved the quality of this article significantly.

##  Appendix A Wave equation as Schrödinger-like

We study the evolution of a perturbation in velocity, pressure and density of
a stable equilibrium of a star. $(w^{\prime},u^{\prime},v^{\prime})$ is the
perturbation’s velocity in spherical coordinates, $\rho^{\prime}$ and
$p^{\prime}$ are the perturbations in density and pressure. The system of
equations is obtained by linearizing the equations of mass and momentum
conservation assuming adiabatic evolution. As a first step, the hermiticity of
the linear system is made explicit by the mean of the following transformation

$\displaystyle(w^{\prime},u^{\prime},v^{\prime})$
$\displaystyle\mapsto(\tilde{w},\tilde{u},\tilde{v})=\rho_{0}^{1/2}r(w^{\prime},u^{\prime},v^{\prime}),$
(A1) $\displaystyle p^{\prime}$
$\displaystyle\mapsto\tilde{p}=\rho_{0}^{-1/2}c_{\mathrm{s}}^{-1}rp^{\prime},$
$\displaystyle\rho^{\prime}$
$\displaystyle\mapsto\tilde{\Theta}=\rho_{0}^{-1/2}r\frac{g}{N}(\rho^{\prime}-\frac{1}{c_{\mathrm{s}}^{2}}p^{\prime}),$

where $\tilde{\Theta}$ is the potential density of the fluid. The evolution of
the perturbation is then

$i\partial_{t}{\mathbf{Y}}={\mathcal{H}}{\mathbf{Y}},$ (A2)

where $\mathcal{H}$ is the differential operator

$i\begin{pmatrix}0&0&0&0&-\frac{c_{\rm s}}{r}\partial_{\theta}\\\
0&0&0&0&-\frac{c_{\rm s}}{r\sin(\theta)}\partial_{\phi}\\\ 0&0&0&-N&S-c_{\rm
s}\partial_{r}-\frac{c_{\rm s}^{\prime}}{2}\\\ 0&0&N&0&0\\\ -\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\theta}(\sin(\theta)\cdot)&-\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\phi}&-S-c_{\rm s}\partial_{r}-\frac{c_{\rm
s}^{\prime}}{2}&0&0\end{pmatrix},$

the perturbation vector is

$\mathbf{Y}=^{\top}\\!\\!\left(\tilde{u},\tilde{v},\tilde{w},\tilde{\Theta},\tilde{p}\right).$
(A3)

and
$c_{\mathrm{s}}^{\prime}\equiv\frac{\mathrm{d}c_{\mathrm{s}}}{\mathrm{d}r}$.
This rescaled system of equations reveals that three functions govern the
perturbations: $c_{\mathrm{s}}(r)$, $N(r)$ and the acoustic-buoyant frequency

$S(r)\equiv\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-\frac{1}{2}\frac{{\rm
d}c_{\rm s}}{{\rm d}r}+\frac{c_{\mathrm{s}}}{r}.$ (A4)

.

##  Appendix B Polytropic stars

We derive the expressions for the parameters
$c_{\mathrm{s}}(r),~{}N(r),~{}S(r)$ for polytropic stars that verify the
equation of state $P=k\rho^{1+1/n}$. Static equilibrium satisfies the
continuity and Poisson equations, and is given by a seminal solution in terms
of the Lane-Emden equation

$\displaystyle\frac{1}{x^{2}}\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\frac{\mathrm{d}f}{\mathrm{d}x})+f^{n}=0,$
(B1) $\displaystyle f(0)=1,$ (B2) $\displaystyle f^{\prime}(0)=0,$ (B3)

where $\rho(r)=\rho_{c}f^{n}(x=r/a)$, and
$a^{2}=(n+1)\frac{P_{c}}{4\pi\mathcal{G}\rho_{c}^{2}}$. We adopt length and
time units such that $a=1$ and $k\rho_{c}^{1/n}=P_{c}/\rho_{c}=1$ and assume
the fluid to be a monoatomic perfect gas ($\Gamma_{1}$ = 5/3). We then obtain

$\displaystyle c_{\mathrm{s}}^{2}$ $\displaystyle=$
$\displaystyle\frac{5}{3}f,$ (B4) $\displaystyle S$ $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{3}}f^{1/2}\left(\frac{1}{r}+\alpha(n)\frac{f^{\prime}}{f}\right),$
(B5) $\displaystyle N^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{3}(n+1)\left(n-\frac{3}{2}\right)\frac{f^{\prime
2}}{f},$ (B6)

where $\alpha(n)=(n+7/3)/10$. Since $f(r=R)=0$ and $f^{\prime}(r=R)<0$, a
polytropic star verifies $S(r=R)=-\infty$.

##  Appendix C Local properties of $\mathcal{H}$: Wigner transform

Symbolic calculus gives a way to associate to differential operators acting on
functions other functions called symbols acting on a phase space. The symbol
$\text{Symb}\left[\hat{f}\right]$ of an operator $\hat{f}$ (e.g. a
differential operator) is obtained by a Wigner transform, defined as

$\text{Symb}[\hat{f}]\equiv
f(x,k)=\int\mathrm{d}yK_{\hat{f}}\left(x+\frac{y}{2},x-\frac{y}{2}\right)\mathrm{e}^{-iky},$
(C1)

where $K_{\hat{f}}$ is the integral kernel of the operator $\hat{f}$:
$(\hat{f}\Psi)(x)\equiv\frac{1}{2\pi}\int\mathrm{d}yK_{\hat{f}}\left(x,y\right)\Psi(y)$.
The inverse correspondence is the Weyl quantification of the symbol

$\displaystyle\hat{f}=\mathrm{Op}\left[f(x,k)\right]$
$\displaystyle\equiv\frac{1}{(2\pi)^{2}}\int\mathrm{d}x\mathrm{d}k\,\mathrm{d}\xi\mathrm{d}\eta
f(x,k)\mathrm{e}^{i\left[\xi(x-\hat{x})+\eta(k-\hat{k})\right]},$ (C2)

such that $\mathrm{Symb}\left[\mathrm{Op}\left[f(x,k)\right]\right]=f(x,k)$.
Eq. C2 often gives a convenient way to relate an operator to a given
functional form of its symbol. For example, the Wigner symbol of the
differential operator $\partial_{x}$ is
Symb$\left[\partial_{x}\right]=-ik_{x}$, a similar expression as the Fourier
transform in this case. Weyl quantification involves commutators
$\left[x,\partial_{x}\right]=-1\neq 0$, that provide a rigorous framework for
making a correspondence of differential operators with varying coefficients to
a phase space and this, without assumptions on the wavelengths of its
eigenfunctions contrary to JWKB approaches Faure (2019); Onuki (2020);
Venaille & Delplace (2021). Hence the following relation:

$\displaystyle\mathrm{Op}\left[f(x)k\right]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(2\pi)^{2}}\int\\!\\!\\!\int\\!\\!\\!\int\\!\\!\\!\int\mathrm{d}x\mathrm{d}k\,\mathrm{d}\xi\mathrm{d}\eta\>f(x)k\>\mathrm{e}^{i\left[\xi(x-\hat{x})+\eta(k-\hat{k})\right]},$
(C3) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int\\!\\!\\!\int\\!\\!\\!\int\\!\\!\\!\int\mathrm{d}x\mathrm{d}k\,\mathrm{d}\xi\mathrm{d}\eta\>f(x)k\>\mathrm{e}^{i\xi(x+\eta/2)+i\eta
k}\mathrm{e}^{-i\xi\hat{x}}\mathrm{e}^{-i\eta\hat{k}},$ (C4) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int\\!\\!\\!\int\\!\\!\\!\int\\!\\!\\!\int\mathrm{d}x\mathrm{d}k\>\,\mathrm{d}\xi\mathrm{d}\eta
f(x-\frac{\eta}{2})k\mathrm{e}^{i\xi x+i\eta
k}\mathrm{e}^{-i\xi\hat{x}}\mathrm{e}^{-i\eta\hat{k}},$ (C5) $\displaystyle=$
$\displaystyle\int\\!\\!\\!\int\mathrm{d}x\mathrm{d}k\>f(x)k\>\delta(x-\hat{x})\delta(k-\hat{k})\;$
(C7)
$\displaystyle\mspace{70.0mu}+\;\int\\!\\!\\!\int\mathrm{d}x\mathrm{d}k\>\frac{i}{2}f^{\prime}(x)k\>\delta(x-\hat{x})\delta^{\prime}(k-\hat{k}),$
$\displaystyle=$ $\displaystyle
f(\hat{x})\hat{k}-\frac{i}{2}f^{\prime}(\hat{x}).$ (C8)

Microlocal analysis connects topological properties of the eigenvectors of the
Wigner symbol to spectral properties of the operator $\mathcal{H}$. The
correspondence relies on index theorems Atiyah & Singer (1963), and provide a
powerful tool to identify spectral properties of an operator from a much
simpler scalar dual problem. In particular, this procedure allows analysis at
long wavelengths that are filtered out by JWKB approximation.
The operator $\mathcal{H}$ depends on parameters that vary with radius $r$.
Key manipulation concerns the term

$\displaystyle\mathrm{Symb}\left[\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-c_{\mathrm{s}}^{\prime}-c_{\mathrm{s}}\partial_{r}\right]$
$\displaystyle=$
$\displaystyle\mathrm{Symb}\left[\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-\frac{c_{\mathrm{s}}^{\prime}}{2}-c_{\mathrm{s}}\partial_{r}-\frac{c_{\mathrm{s}}^{\prime}}{2}\right]$
(C9) $\displaystyle=$
$\displaystyle\mathrm{Symb}\left[\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-\frac{c_{\mathrm{s}}^{\prime}}{2}\right]-\mathrm{Symb}\left[c_{\mathrm{s}}\partial_{r}+\frac{c_{\mathrm{s}}^{\prime}}{2}\right]$
(C10) $\displaystyle=$
$\displaystyle\frac{c_{\mathrm{s}}}{2g}\left(N^{2}-\frac{g^{2}}{c_{\mathrm{s}}^{2}}\right)-\frac{c_{\mathrm{s}}^{\prime}}{2}-ic_{\mathrm{s}}k_{r}$
(C11) $\displaystyle=$ $\displaystyle S-ic_{\mathrm{s}}k_{r},$ (C12)

applying identity Eq. (C8) with $f=c_{\mathrm{s}}$. The Wigner symbol of
$\mathcal{H}$ in the radial direction is then

$H\equiv i\begin{pmatrix}0&0&0&0&-\frac{c_{\rm s}}{r}\partial_{\theta}\\\
0&0&0&0&-\frac{c_{\rm s}}{r\sin(\theta)}\partial_{\phi}\\\
0&0&0&-N&S+ic_{\mathrm{s}}k_{r}\\\ 0&0&N&0&0\\\ -\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\theta}(\sin(\theta)\cdot)&-\frac{c_{\rm
s}}{r\sin(\theta)}\partial_{\phi}&-S+ic_{\mathrm{s}}k_{r}&0&0\end{pmatrix}.$
(C13)

The object $H$ is a function with respect to $r$, and an operator over the
angles $(\theta,\phi)$. Performing a Fourier transform with respect to time,
one obtains

$-\omega\Psi=H\Psi,$

which gives

$\omega^{2}\Psi=H^{2}\Psi.$

The operator $H^{2}$ is block diagonal

$H^{2}=\begin{pmatrix}A&0_{3,2}\\\ 0_{2,3}&\bar{M}\\\ \end{pmatrix},$ (C14)

where $0_{2,3}$ and $0_{3,2}$ denote null matrices of dimensions $2\times 3$
and $3\times 2$ respectively. The eigenvalues $\omega^{2}$ of $H^{2}$ consist
generically of the union of both the eigenvalues of $A$ and $\bar{M}$. Here,
the eigenvalues of $A$ and $\bar{M}$ are the same. Indeed, a nonzero
eigenvector
$\Psi=^{\top}\\!\\!\left(\tilde{u},\tilde{v},\tilde{w},\tilde{\Theta},\tilde{p}\right)$
cannot have $\tilde{\Theta}=\tilde{p}=0$ or $\tilde{u}=\tilde{v}=\tilde{w}=0$,
as a perturbation cannot be made of only velocity with no pressure/density or
pressure/density with no velocity. Hence, no eigenvector of $H^{2}$ can be of
the form ${}^{\top}(u,v,w,0,0)$ or of the form ${}^{\top}(0,0,0,\Theta,p)$,
implying that $A$ and $\bar{M}$ cannot have different eigenvalues. The
eigenvalues of $H^{2}$ are therefore the eigenvalues of $\bar{M}$, which is
the $2\times 2$ matrix

$\bar{M}\equiv\begin{pmatrix}N^{2}&-NS-iNc_{\mathrm{s}}k_{r}\\\
-NS+iNc_{\mathrm{s}}k_{r}&-\frac{c_{\mathrm{s}}^{2}}{r^{2}}\mathcal{L}+S^{2}+c_{\mathrm{s}}^{2}k_{r}^{2}\end{pmatrix},$
(C15)

where
$\mathcal{L}\equiv\frac{1}{\sin(\theta)}\partial_{\theta}(\sin(\theta)\partial_{\theta}\cdot)+\frac{1}{\sin(\theta)^{2}}\partial_{\phi\phi}$.
After projecting onto spherical harmonics $Y_{\ell}^{m}$, one obtains the
matrix $M$

$M\equiv\begin{pmatrix}N^{2}&-NS-iNc_{\mathrm{s}}k_{r}\\\
-NS+iNc_{\mathrm{s}}k_{r}&L_{\ell}^{2}+S^{2}+c_{\mathrm{s}}^{2}k_{r}^{2}\end{pmatrix},$
(C16)

where $L_{\ell}$ is the Lamb frequency as presented in the main text.

The matrix $M$ is hermitian and as such is diagonalisable. Its two eigenvalues
are degenerate and both take the value $N^{2}$ when

$\begin{pmatrix}k_{r}\\\ L_{\ell}\\\ S\end{pmatrix}=\begin{pmatrix}0\\\ N\\\
0\end{pmatrix}\equiv\mathbf{p_{0}}.$ (C17)

This degenerescence is identical as the one found in Perrot et al. (2019). An
other degenerescence of the eigenvalues occur at ${L_{\ell}=-N}$, which we
will ignore as it corresponds to negative values of $L_{\ell}$.

##  Appendix D Chern numbers

The first Chern number $\mathcal{C}^{\left(n\right)}$ of the $n-$th band is
the topological charge associated to the flux of the Berry curvature
$\mathbf{F}^{\left(n\right)}$ over a close surface $\Sigma$ of the parameter
space of the matrix $M$ Chern (1946). Its expression is

$\mathcal{C}^{\left(n\right)}~{}=~{}\frac{1}{2\pi}\int_{\Sigma}~{}\mathbf{F}^{\left(n\right)}\cdot\mathrm{d}\mathbf{\Sigma}.$
(D1)

where we denote $n=1$ for the g-band and $n=2$ for the p-band. The relevant
parameter space for our study is $\\{k_{r},L_{\ell},S\\}$, such that the Berry
curvature is a vector with three components denoted
$\left(F^{(n)}_{k_{r},L_{\ell}},\;F^{(n)}_{L_{\ell},S},\;F^{(n)}_{S,k_{r}}\right)$.
A degeneracy point for the eigenvalues of the symbol matrix $M$ is
topologically non-trivial if the Chern numbers $\mathcal{C}^{\left(n\right)}$
take non-zero values at this point. This topological property is reflected in
the spectrum of the original operator problem $\mathcal{M}$ by
$\absolutevalue{\mathcal{C}^{\left(n\right)}}$ modes that transit from one
band to another.
To calculate the value of the Chern numbers of $M$ at the degeneracy point
$\mathbf{p_{0}}$, we start from the definition

$F_{p,p^{\prime}}=i\left(\partial_{p}\Psi_{j}^{*}\partial_{p^{\prime}}\Psi_{j}-\partial_{p^{\prime}}\Psi_{j}^{*}\partial_{p}\Psi_{j}\right),$
(D2)

where the summation on $j$ is implied, and $\mathbf{\Psi}=(\Psi_{j})_{j=1,2}$
is the normalized eigenvector of $M$ corresponding to the p-mode, and $p$ and
$p^{\prime}$ are directions in parameter space $\\{k_{r},L_{\ell},S\\}$ (we
ignore $c_{\mathrm{s}}$ and $N$ as they are not involved in the degeneracy
$\mathbf{p_{0}}$). We decompose $M$ on the Pauli matrices as

$M=\frac{c_{\mathrm{s}}^{2}k_{r}^{2}+L_{\ell}^{2}+N^{2}+S^{2}}{2}I_{2}+\mathbf{g}\cdot\bm{\sigma},$
(D3)

where $\bm{\sigma}$ is the 3-vector of Pauli matrices, and

$\mathbf{g}(\mathbf{p})=\begin{pmatrix}-c_{\mathrm{s}}k_{r}N\\\ -NS\\\
\frac{N^{2}-c_{\mathrm{s}}^{2}k_{r}^{2}-L_{\ell}^{2}-S^{2}}{2}\end{pmatrix},$
(D4)

with $\mathbf{p}={}^{\top}(k_{r},L_{\ell},S)$. Following classical
computations found in Bernevig (2013), one shows that for the p-band

$\frac{1}{2\pi}{F}_{p,p^{\prime}}=(-1)^{2}\frac{1}{4\pi}\frac{\mathbf{g}}{\left\lVert\mathbf{g}\right\rVert^{3}}\cdot(\partial_{p}\mathbf{g}\times\partial_{p^{\prime}}\mathbf{g}),$
(D5)

which gives a simple expression for the Chern number

$\mathcal{C}^{p}=-\mathrm{sign}\left(\det(\nabla_{\mathbf{p}}\mathbf{g})\big{|}_{\mathbf{p_{0}}}\right).$
(D6)

Finally, one has

$\displaystyle\mathcal{C}^{p}$ $\displaystyle=$
$\displaystyle-\mathrm{sign}\begin{vmatrix}-c_{\mathrm{s}}N&0&0\\\ 0&0&-N\\\
0&-N&0\end{vmatrix}$ (D7) $\displaystyle=$
$\displaystyle-\mathrm{sign}(c_{\mathrm{s}}N^{3})$ (D8) $\displaystyle=$
$\displaystyle-1.$ (D9)

Since the sum of the Chern numbers over the different bands is zero, one
obtains directly $\mathcal{C}^{g}=+1$. The theorem of spectral flow ensures
that the number of modes in each band varies by
$\left|\mathcal{C}^{g}\right|=\left|\mathcal{C}^{p}\right|=1$ when $S$ changes
sign. Faure (2019). More precisely, when $S(r)$ changes sign from negative to
positive values, the p-band of the operator $\mathcal{M}$ loses one mode to
the g-band. When $S(r)$ changes sign from positive to negative values, the
topological mode transits from the g- towards the p-band Perrot et al. (2019).

For clarity, detailed expressions of the Berry curvature $\mathbf{F}^{(n)}$
are provided below for both bands, together with representations of the
vectors fields in the parameter space (Fig. 6). One has

$\displaystyle\scriptstyle\mathbf{F}^{(1)}$
$\displaystyle\scriptstyle=\left(\frac{-4kL}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}},\frac{2\left(k^{2}-L^{2}+S^{2}+1\right)}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}},\frac{-4LS}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}}\right),$
(D10) $\displaystyle\scriptstyle\mathbf{F}^{(2)}$
$\displaystyle\scriptstyle=\left(\frac{4kL}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}},\frac{-2\left(k^{2}-L^{2}+S^{2}+1\right)}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}},\frac{4LS}{\left(\left(k^{2}+(L-1)^{2}+S^{2}\right)\left(k^{2}+(L+1)^{2}+S^{2}\right)\right)^{3/2}}\right).$
(D11)

Poles of the curvature are found to be at the two points $k=S=L_{\ell}\pm
N=0$, as shown on Fig. 6.

Figure 6: Vector field of the Berry curvature in parameter space
$\\{k_{r},L_{\ell},S\\}$. Left: $\mathbf{F}^{(1)}$, associated to the g-band
(source). Right: $\mathbf{F}^{(2)}$, associated to the p-band (sink). The red
dot is the degenerated point at $(0,N,0)$.

##  Appendix E Regularity at the center

Let us verify that the change of variables Eq. (A1) used does not include
diverging modes in the spectrum. In the vicinity of the center, one has

$\displaystyle c_{\mathrm{s}}$ $\displaystyle\rightarrow$ $\displaystyle
c_{\mathrm{s,0}},$ (E1) $\displaystyle N$ $\displaystyle\rightarrow$
$\displaystyle 0,$ (E2) $\displaystyle S/c_{\mathrm{s}}$ $\displaystyle\sim$
$\displaystyle r^{-1},$ (E3) $\displaystyle k_{r,0}^{2}$ $\displaystyle\sim$
$\displaystyle-L_{\ell}^{2}/c_{\mathrm{s,0}}^{2},$ (E4) $\displaystyle
L_{\ell}^{2}/c_{\mathrm{s}}^{2}$ $\displaystyle\sim$
$\displaystyle\ell(\ell+1)r^{-2},$ (E5)

such that Eq. (18) becomes

$\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{\ell(\ell+1)}{r^{2}}\right)p=0.$
(E6)

This equation has two solutions, only one of which is regular, which is

$p\propto r^{\ell}.$ (E7)

We inverse the transform Eq. (A1) to obtain the behavior of physical
quantities of the perturbation, which are

$\displaystyle p^{\prime}$ $\displaystyle\propto$ $\displaystyle r^{\ell-1},$
(E8) $\displaystyle w^{\prime},v^{\prime},u^{\prime}$ $\displaystyle\propto$
$\displaystyle r^{\ell-2}.$ (E9)

As a consequence the radial flux as well as kinetic energy remain finite at
the center:

$\displaystyle F(r\rightarrow 0)$ $\displaystyle\sim$ $\displaystyle 4\pi
r^{2}w^{\prime}\propto r^{\ell},$ (E10) $\displaystyle E_{\mathrm{kin}}$
$\displaystyle\equiv$ $\displaystyle
4\pi\int_{0}^{R}\rho\bm{V}^{2}r^{2}\mathrm{d}r\propto\int_{0}^{R}r^{2(\ell-1)},$
(E11)

which is finite for $\ell>1$. The radial pulsations case $\ell=0$ is left
aside, as the topological has zero frequency in this case. The behavior at the
other boundary $r=R$ is depend on the given model.

##  Appendix F Lamb-like and f-mode

The f-mode is defined by Cowling as the stellar mode with zero node in the
radial direction Cowling (1941). Since the topological mode as well as the
surface-gravity wave has zero node, we lead numerical experiments to study
their coexistence. The Lamb-like wave is present in the spectrum when $S$
changes sign somewhere in the bulk. The surface-gravity wave is present in the
spectrum when peculiar boundary conditions are enforced at the surface, namely
Poisson’s boundary conditions

$\partial_{t}p=\frac{g}{c_{\mathrm{s}}}w.$ (F1)

We lead numerical experiments in plane-parallel geometry, with $z$-direction
being stratified, and $x$-direction being invariant by translation. We note
$z_{2}$ the top of the medium, and $z_{1}$ the bottom. The average
localisation of a normalized mode $\Psi=(v,w,\Theta,p)$ is the average
position of its energy

$\bar{z}_{\Psi}\equiv\int_{z_{1}}^{z_{2}}\mathrm{d}z\>z\>\Psi\cdot\Psi^{*},$

since the sum of kinetic and potential energy of the mode is
$\Psi\cdot\Psi^{*}$.

In Fig. 7, we show that if the Lamb-like is trapped sufficiently far away from
the top-surface, it does not hybrid with the surface-gravity wave. In Fig. 8,
we show that if they overlap, they hybrid into a single zero-node mode. We
also show the eigenfunctions of the density of the perturbation of a few
modes.

Figure 7: Spectrum of the waves in a stratified medium with 2D plane-parallel
geometry. The buoyancy frequency $N$ is set as a constant. $S$ is set
according to the profile displayed on the right panel of the figure. The Lamb-
like wave develops around the cancellation point of $S$ (pink interval). The
surface-gravity wave is trapped at the top surface (green interval). The two
waves do not overlap significantly: the system have two modes with zero nodes.
Figure 8: Spectrum of the same problem as presented in Fig. 7, for a profile
of $S$ whose zero is closer to the surface $z=z_{2}$. TheLamb-like and
surface-gravity waves overlap and hybrid. This results in one $n=0$ mode: the
f-mode.

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