11institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
22institutetext: IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 Avenue
Édouard Belin, 31400, Toulouse, France 33institutetext: STAR Institute,
Université de Liège, Allée du 6 Août 19, 4000 Liège, Belgium 44institutetext:
Department of Astronomy, School of Science, The University of Tokyo, 7–3–1
Hongo, Bunkyo-ku, Tokyo 113–0033, Japan

# First evidence of inertial modes in $\gamma$ Doradus stars

The core rotation revealed

R-M. Ouazzani 11 F. Lignières 22 M-A. Dupret 33 S.J.A.J. Salmon 33 J. Ballot
22 S. Christophe 11 and M. Takata 44

(Draft: ; Received xxx; accepted xxx)

###### Abstract

Context. The advent of space photometry with CoRoT and Kepler has allowed the
gathering of exquisite and long time series for a wealth of main sequence
stars, including $\gamma$ Doradus stars whose detailed seismology was
unachievable from the ground.

Aims. $\gamma$ Doradus stars present an incredibly rich pulsation spectra,
with gravito-inertial modes, in some cases supplemented with $\delta$ Scuti-
like pressure modes -for the hybrid stars- and in numerous cases with Rossby
modes. The present paper aims at showing that, in addition to these modes
established in the radiative envelope, pure inertial modes, trapped in the
convective core, can be detected in Kepler observations of $\gamma$ Doradus
stars, thanks to their resonance with the gravito-inertial modes.

Methods. We start by using a simplified model of perturbations in a full
sphere of uniform density. Under these conditions, the spectrum of pure
inertial modes is known from analytical solutions of the so-called Poincaré
equation. We then compute coupling factors which help select the pure inertial
modes which interact best with the surrounding dipolar gravito-inertial modes.
Using complete calculations of gravito-inertial modes in realistic models of
$\gamma$ Doradus stars, we are able to show that the pure inertial/gravito-
inertial resonances appear as dips in the gravito-inertial mode period spacing
series at spin parameters close to those predicted by the simple model. We
find the first evidence of such dips in the Kepler $\gamma$ Doradus star
KIC5608334. Finally, using complete calculations in isolated convective cores,
we find that the spin parameters of the pure inertial/gravito-inertial
resonances are also sensitive to the density stratification of the convective
core.

Results. In conclusion, we have discovered that certain dips in gravito-
inertial mode period spacings observed in some Kepler stars are in fact the
signatures of resonances with pure-inertial modes that are trapped in the
convective core.

Conclusions. This holds the promise to finally access the central conditions ,
i.e. rotation and density stratification, of intermediate-mass stars on the
main sequence.

###### Key Words.:

asteroseismology – stars: oscillations – stars: rotation – methods:

##  1 Introduction

Angular momentum transport in stellar radiative zones, and its evolution with
time is a major open issue in stellar physics. Observations of stars in
evolved stages provided by the Kepler mission (Beck et al. 2012; Deheuvels et
al. 2012; Mosser et al. 2012) have proved our then current models of transport
wrong (Eggenberger et al. 2012; Marques et al. 2013), and the various attempts
to invoke additional mechanisms (Cantiello et al. 2014; Fuller et al. 2014;
Belkacem et al. 2015; Pinçon et al. 2017; Mathis et al. 2018; Fuller et al.
2019) have not solved this issue, or are still matter of debate.

With the purpose of bringing a main-sequence counterpart to the seismic
constraint of rotation in evolved stars, $\gamma$ Doradus stars g-modes
monitored by Kepler have revealed near-core rotation rates in stars which are
progenitors of red giants (Van Reeth et al. 2016; Christophe et al. 2018; Li
et al. 2019). This has allowed the test of angular momentum transport models
against near-core rotation measurements in main-sequence intermediate-mass
stars. As a result, Ouazzani et al. (2019) have found that the rotational
transport as formalized by Zahn (1992) does not allow them to reproduce
observations of $\gamma$ Doradus stars. Hence, as for the more evolved stages,
another -yet to be found- mechanism seems to transport angular momentum from
the innermost part of the radiative zone outwards. Whether it leads to
rigidification of the rotation profile still remains unproved, but
measurements in a handful of stars seem to suggest so (Kurtz et al. 2014; Saio
et al. 2015; Murphy et al. 2016; Van Reeth et al. 2018; Li et al. 2019).

The constraint of the full internal rotation profile in intermediate-mass
stars would be extremely valuable, as it would answer this question and allow
further characterization of the transport mechanism of angular momentum at
stake. Once again, $\gamma$ Doradus stars could help bring such constraint. In
addition to the g-modes period spacing series, which interpretation allowed
near-core rotation measurements, these stars harbor pulsations in Rossby modes
(Saio et al. 2018; Li et al. 2019) which hopefully bring rotational
information in a different cavity than g-modes (Van Reeth et al. 2018,
Christophe et al. in prep). The present paper demonstrates the existence of a
signature of pure inertial modes trapped in the convective core within g-modes
period spacing series, holding the possibility to probe the rotation rate of
the convective core.

Inertial waves are caused by the Coriolis acceleration, their frequency in the
rotating frame ranging from $0$ to $2\Omega$. One usually refers to pure
inertial waves when other restoring such as buoyancy or pressure forces are
absent or negligible. This is relevant for slow motions in stellar convective
zones, where turbulent heat transport maintains an isentropic mean
stratification or equivalently a vanishing Brunt-Väisälä frequency $N$. We
thus expect convective zones and in particular the convective cores of massive
and intermediate-mass stars to harbor pure inertial waves. Pure inertial modes
in a full sphere have been studied for a fluid of constant density (Bryan
1889; Greenspan 1968; Rieutord 1991) and for isentropic polytropes (Papaloizou
& Pringle 1981; Lee et al. 1992; Dintrans & Ouyed 2001; Lockitch & Friedman
1999; Wu 2005). Pure inertial modes trapped in the solar convective zone have
been investigated by Guenther & Gilman (1985) and Dziembowski et al. (1987).
The coupling of pure inertial modes in stellar convective zones with gravito-
inertial modes in adjacent radiative zones has been studied in the case of
tidally excited modes in solar-type stars (Ogilvie & Lin 2004) and in massive
stars (Papaloizou & Savonije 1997). Lee & Saio (1987) computed oscillatory
convective modes present in a rotating convective core with a slightly
superadiabatic stratification coupled with gravito-inertial modes or Rossby
modes in the radiative enveloppe.

The present article holds the first unambiguous theoretical demonstration of
the existence of resonances between gravito-inertial modes in the radiative
zone, and pure inertial modes in the convective core in $\gamma$ Doradus
stars. By comparing with a $\gamma$ Doradus star observed by the Kepler
mission, we show the first observational signature of pure inertial modes. We
start by introducing inertial modes in the case of a sphere of uniform density
(Sect. 2.1) and then propose a simple model to describe their coupling with
gravito-inertial modes (Sect. 2.2). In Sect.3, we then explore the occurrence
of such gravito-inertial$/$pure inertial resonances in complete calculations.
Finally we investigate the dependency of these resonances on stellar
characteristics (Sect.4), before investigating the diagnosis potential of
these resonances regarding the measurement of rotation in the convective core.

##  2 Theoretical spectrum of pure inertial modes

Although inertial modes in a spherical shell with solid boundaries present
complex characteristics such as singular modes in the inviscid limit (Rieutord
& Valdettaro 2018), inertial modes in the full sphere are expected to be
smooth and, in the case of a uniform density sphere, are amenable to
analytical solutions (e.g. Greenspan 1968).

Following Wu (2005), it is straightforward to show that, in an isentropic
medium (where the Brunt-Vaissala frequency $N=0$), small-amplitude adiabatic
perturbations show a time and azimuthal (angle $\phi$) dependence
$\propto\exp(im\phi+i\omega t)$, and are governed by the wave equation :

$\nabla^{2}\Psi-s^{2}\frac{\partial^{2}\Psi}{\partial
z^{2}}=\frac{1}{H}\left(\frac{\partial\Psi}{\partial
r}-s^{2}\cos\theta\frac{\partial\Psi}{\partial
z}+\frac{ms}{r}\Psi\right)-(1-s^{2})\frac{\omega_{co}^{2}}{c_{s}^{2}}\Psi$ (1)

where $s=(2\Omega)/\omega_{co}$ is the spin parameter, that is the ratio of
twice the rotation rate $\Omega$ (which is taken uniform here) to the
frequency in the co-rotating frame $\omega_{co}$. $\Psi$ is related to the
Eulerian pressure perturbation $P^{\prime}$ and the equilibrium density
$\rho_{0}$ by $\Psi=P^{\prime}/(\varrho_{0}\omega_{co}^{2})$,
$H=-\varrho_{0}/(d\varrho_{0}/dr)$ is the density scale height, and
$c_{s}=\sqrt{\Gamma_{1}P_{0}/\rho_{0}}$ is the sound speed, here defined by
the help of the adiabatic exponent $\Gamma_{1}$ and the equilibrium pressure
$P_{0}$. Perturbations of the gravitational potential have been neglected
through the Cowling approximation. Away from the star surface, and as long as
the inertial frequencies are much lower than pressure wave frequencies, the
last term can also be neglected.

In the next two subsections, we first present properties of inertial modes
relevant for our purpose and then construct a simple model for the occurrence
of gravito-inertial/inertial mode resonance.

Figure 1:  Inertial modes in a uniform density sphere. Each of these modes
have a strong spatial matching with the dipolar gravito-inertial mode of the
same spin parameter. The dotted line specifies the critical latitude
$\theta_{c}=\arccos(\omega/(2\Omega))$.

###  2.1 Inertial modes in a full sphere of uniform density

In the absence of density stratification, $H=+\infty$, the wave equation
reduces to the so-called Poincaré equation
$\nabla^{2}\Psi-s^{2}\frac{\partial^{2}\Psi}{\partial z^{2}}=0$. This equation
is separable using the ellipsoidal coordinates $(x_{1},x_{2},\phi)$ defined in
Appendix A, and the solution, already obtained by Bryan (1889), reads

$\Psi(\@vec{x})=P_{\ell_{i}}^{m}(x_{1})P_{\ell_{i}}^{m}(x_{2})\exp(im\phi+i\omega
t)$ (2)

where $P_{\ell_{i}}^{m}$ is the associated Legendre polynomial of degree
$\ell_{i}$. For a given pair $(\ell_{i},m)$, the solution $\Psi(\@vec{x})$
also holds a frequency dependency through the endpoints of the meridional
coordinate intervals, $x_{1}\in[\mu,1]$ and $x_{2}\in[-\mu,\mu]$, where
$\mu=\omega_{co}/(2\Omega)=1/s$. In addition to regularity conditions at the
center and along the rotation axis, the boundary condition $\xi_{r}=0$,
applied at the sphere surface, determines the inertial mode frequencies as the
non-trivial positive roots of :

$\frac{dP_{\ell_{i}}^{m}}{d\mu}(\mu)=\frac{m}{1-\mu^{2}}P_{\ell_{i}}^{m}(\mu).$
(3)

The detail of the calculations together with a useful discussion on the
inertial mode properties can be found in Wu (2005).

Here we focus on the inertial modes which may interact resonantly with dipolar
gravito-inertial modes, that is, for symmetry reasons, anti-symmetric zonal
($m=0$) modes, symmetric prograde ($m=-1$) and retrograde ($m=1$) modes. For
each symmetry class, the set of inertial eigenfrequencies is dense in the
$[0,2\Omega]$ interval. However, we shall mostly select low degree $\ell_{i}$
modes. For each pair $(\ell_{i},m)$, there are
$[\ell_{i}-|m|-\epsilon(\ell_{i}-|m|)]/2$ eigenfrequencies, where
$\epsilon(\ell_{i}-|m|)=0$ for even modes and $\epsilon(\ell_{i}-|m|)=1$ for
odd modes. For example, for anti-symmetric zonal ($m=0$) modes, one mode is
associated with $\ell_{i}=3$, two modes with $\ell_{i}=5$, three modes with
$\ell_{i}=7$, and so forth. All modes of a given pair $(\ell_{i},m)$ have the
same number of nodes along a meridian line at the outer sphere, namely
$\ell_{i}-|m|$ nodes.

In table 1, the spin parameters $s$ of the first low $\ell_{i}-|m|$ eigenmodes
that can potentially couple with dipolar gravity modes are listed in ascending
$\ell_{i}-|m|$. The spatial distributions of the three $(\ell_{i}=3,5,7;m=0)$
modes of the first column in the table are displayed on Fig.1. As we shall see
below, these modes are the most likely to couple with a dipolar zonal mode
above the convective core. The remaining three axisymmetric modes of table 1,
one $(\ell_{i}=5;m=0)$ mode and two $(\ell_{i}=7;m=0)$ modes are shown on
Fig.8 of Appendix B. As apparent from these figures, and as mentionned before,
the different modes associated with a pair $(\ell_{i},m)$ have the same number
of nodes along the meridian line at the outer surface but they differ by the
number of nodes -on that outer surface meridian line- that are located between
the equator and the critical colatitude
$\theta_{c}=\arccos(\mu)=\arccos(1/s)$. For example, the three
$(\ell_{i},m)=(7,0)$ modes have either no node, one node or two nodes between
the equator and the critical colatitude $\theta_{c}$.

###  2.2 Coupling with gravito-inertial modes

Small-wavelength WKB analysis tells that gravity waves can propagate in
regions where $\omega<N$ as long as their frequency is sub-inertial, i.e.
$\omega<2\Omega$ (see for instance Unno et al. 1989). Low-frequency gravity
waves can thus propagate from a radiative zone down to a convective core
(Dintrans & Rieutord 2000; Prat et al. 2016). If conditions for positive
interferences are met, such waves can produce mixed gravito-inertial/pure
inertial modes.

Because the spectrum of pure inertial modes is dense in $[0,2\Omega]$, there
is always a possible frequency match between a sub-inertial gravity mode in
the radiative zone and a pure inertial mode in the core. However, a resonant
coupling is expected if there is matching between both their frequencies and
their spatial distributions at the convective core boundary. In the well-known
case of mixed gravity/pressure modes in slowly rotating evolved stars, the
spatial matching is ensured by considering modes with the same spherical
harmonic degree. There is no such simplification here as the latitudinal
distribution of both pure inertial modes and gravito-inertial modes in general
depends on the frequency.

Table 1: Inertial-mode spin parameters and coupling coefficients $\zeta_{I}^{GI}$ (in parenthesis) with the dipolar Hough function of the same symmetry class and spin parameter $\ell_{i}$ | $m=0$ |  |
---|---|---|---
3 | 2.2361 (0.62) |  |
5 | 3.5059 (0.51) | 1.3071 (0.014) |
7 | 4.7778 (0.44) | 1.6900 (3.3 10-3) | 1.1471 (1.6 10-4)
$\ell_{i}$ | $m=-1$ |  |
3 | 11.3245 (0.50) |  |
5 | 29.3302 (0.39) | 1.6900 (4.3 10-3) |
7 | 55.3317 (0.34) | 2.3182 (3.8 10-5) | 1.2926 (1.89 10-5)
$\ell_{i}$ | $m=1$ |  |
3 | 1.3246 (0.71) |  |
5 | 1.9128 (0.60) | 1.1074 (0.055) |
7 | 2.5309 (0.52) | 1.3570 (0.033) | 1.0546 (1.7 10-3)

Although a detailed mathematical model of the resonance conditions is beyond
the scope of the present paper, we propose here a simplified approach to
estimate the occurrences of strong resonant coupling between a pure inertial
mode and a gravito-inertial mode, respectively oscillating in the convective
core and above it in the radiative zone. We first define a coupling
coefficient $\zeta_{I}^{GI}$ as :

$\zeta_{I}^{GI}=\frac{\int_{0}^{\pi/2}\sin\theta\;{P^{\prime}}_{I}(r_{c},\theta){P^{\prime}}_{GI}(r_{c},\theta)d\theta}{\left(\int_{0}^{\pi/2}\sin\theta\;{P^{\prime}}_{I}^{2}d\theta\right)^{1/2}\left(\int_{0}^{\pi/2}\sin\theta\;{P^{\prime}}_{GI}^{2}d\theta\right)^{1/2}}$
(4)

where $r_{c}$ is the radius of the convective core boundary and,
${P^{\prime}}_{I}$ and ${P^{\prime}}_{GI}$ are the latitudinal pressure
profile of the inertial and gravito-inertial modes respectively.

To compute this coefficient we make the following assumptions : we assume that
the latitudinal variations of the gravito-inertial mode is given by the Hough
function, i.e. the expected form in the framework of the traditional
approximation. For the present work, Hough functions are numerically computed
by solving Laplace’s tidal equation projected on the basis of spherical
harmonics truncated to the degree $\ell_{\mathrm{max}}=500$. The linear system
to solve is then described in Unno et al. (1989). The latitudinal form of the
pure inertial modes results from two assumptions. The first one is to neglect
the effect of the density stratification. The second is to use the condition
$\xi_{r}=0$ at the convective core radius as a boundary condition to quantize
the inertial modes. The inertial modes then take the analytical form given in
the previous sub-section. In particular, the pressure perturbation profile at
the core radius ${P^{\prime}}_{I}$ is simply $\propto
P_{\ell_{i}}^{m}(\cos\theta)$.

Figure 2:  Period spacing as a function of the period for model 1z with
different value of the uniform rotation frequency, ranging from 7 $\mu$Hz to
23 $\mu$Hz, by increment of 2 $\mu$Hz, for zonal dipolar modes. Both the
period spacings and the periods are shown in the inertial frame of reference.
The filled circles are obtained with non-perturbative calculations, while the
grey curves are obtained with the traditional approximation of rotation. The
vertical lines give the location of the periods of pure inertial modes
associated with different rotation rates (only for $\nu_{rot}=13$ to
$23\mu$Hz), computed for an incompressible inviscid spherical box as described
in Sect.2.2.

Using Eq. 4, we compute the coupling coefficients $\zeta_{I}^{GI}$ between the
uniform-density low-order inertial modes listed in Tab. 1 and the dipolar
Hough functions corresponding to the same spin parameters. We find that the
coupling coefficients are either significant or very small. This property can
be readily understood from the geometry of the modes involved. Considering the
axisymmetric case, we know that the axisymmetric dipolar Hough functions are
single-signed between the equator and the critical colatitude $\theta_{c}$
corresponding to the spin parameter, and then rapidly goes to zero at higher
latitudes (Townsend 2003). Thus multiplying them by an inertial mode which
also have no node between the equator and $\theta_{c}$ and integrating over
latitude will give a significant normalized coefficient. By contrast, the
coupling with inertial modes that have nodes between the equator and
$\theta_{c}$ will be necessarily much smaller. This geometrical interpretation
explains why the inertial modes displayed in Fig. 1 have significant values of
$\zeta_{I}^{GI}$ with the axisymmetric dipolar gravito-inertial mode of
equivalent spin parameter and why those displayed in Fig. 8 have very low
$\zeta_{I}^{GI}$. Similar geometrical interpretations hold for the coupling
coefficient of the prograde and retrograde modes.

Thus, considering dipolar gravito-inertial modes scanning a given frequency
range, this simplified model predicts strong coupling with a pure inertial
mode at the spin parameters listed in Tab. 1. The assumptions of the present
model will be discussed later in the paper.

##  3 Occurrence of gravito-inertial resonances in complete calculations

Figure 3:  Period spacing as a function of the period for model 1z for a
uniform rotation frequency of 21 $\mu$Hz. Both the period spacings and the
periods are shown in the inertial frame of reference.

In this section, we compute oscillation spectra of realistic models of main-
sequence intermediate-mass stars. We wish to investigate further the behavior
of the mixed gravito-inertial/pure inertial modes, while avoiding the
assumptions made in the previous section.

###  3.1 Stellar and pulsation models

Stellar models were computed with the stellar evolution code cles (Scuflaire
et al. 2008b) for masses between 1.4 and 1.86 M⊙, and with initial helium mass
fraction $Y=0.27$. We adopted the AGSS09 metal mixture (Asplund et al. 2009)
and corresponding opacity tables obtained with OPAL opacities (Iglesias &
Rogers 1996), completed at low temperature ($logT<4.1$) with Ferguson et al.
(2005) opacity tables. We used the OPAL2001 equation of state (Rogers &
Nayfonov 2002) and the nuclear reaction rates from NACRE compilation (Angulo
et al. 1999), except for the ${}^{14}N(p,\gamma)^{15}O$ nuclear reaction, for
which we adopted the cross-section from Formicola et al. (2004). Surface
boundary conditions at $T=T_{\rm eff}$ were provided by ATLAS 9 model
atmospheres (Castelli & Kurucz 2003). Convection was treated using the mixing-
length theory (MLT) formalism (Böhm-Vitense 1958) with a parameter
$\alpha_{\textrm{MLT}}=1.70$, close to a solar calibration.

We considered models with turbulent diffusion. Since the cles code does not
include effects of rotation on transport of angular momentum or chemical
species, we instead introduced mixing by turbulent diffusion, following the
approach of Miglio et al. (2008). This reproduces an effect of rotationally-
induced mixing that is quite similar to overshooting, but in addition tends to
smooth chemical composition gradients inside the star. The coefficient of
turbulent diffusion was set to $D_{\textrm{t}}=700cm^{2}/s$ and kept constant
to this value during evolution and in every layer of the models. This value
was selected from a previous calibration to Geneva models with similar masses
(Miglio et al. 2008).

Figure 4:  Eigenfunction ($P^{\prime}/\sqrt{\rho}$, perturbed pressure over the equilibrium density) in the meridional plane for the three modes located at resonance in Fig.3: dipolar zonal modes computed with the acor code for model 1z, for a uniform rotation frequency of 21 $\mu$Hz. The left pannel corresponds to the first resonance, for the mode $n=-33$, the middle panel for the mode $n=-73$, and the right panel for the mode $n=-133$. The black solid line corresponds to the critical co-latitude $\theta_{c}=arccos(\omega_{co}/2\Omega)$, geometrical feature found in pure inertial modes as defined in Sect. 2.1. Table 2: Parameters of the models explored in detail in this study. They have been named as follows: the number 1, 2 3 corresponds to the mass (resp. 1.4, 1.6 and 1.86 M⊙), then the letter z stands for ZAMS, m for mid main sequence, and t for TAMS. These models were computed with turbulent diffusion (with Co = 700 cm2 s-1) and no overshooting.  Model name | 1z | 2m | 3t
---|---|---|---
$\rm M/M_{\odot}$ |  1.40 |  1.60 |  1.86
Teff | 6880 | 7190 | 6760
$\log L/L_{\odot}$ | 0.627 | 1.036 | 1.355
$\log$ g | 4.30 | 3.98 | 3.63
$\rm R/R_{\odot}$ | 1.38 | 2.13 | 3.48
Age (Myr) | 185 | 1830 | 1480
Xc | 0.68 | 0.35 | 0.06

The oscillation modes were computed as the adiabatic response of the structure
to small perturbations, i.e. of the density, pressure, gravitational
potential, and velocity field. The acor code was developed for this purpose
and is presented in Ouazzani et al. (2012) and Ouazzani et al. (2015). This
oscillation code solves the hydrodynamics equations perturbed by Eulerian
fluctuations, performing direct integration of the 2D problem. The numerical
method is based on a spectral multi-domain method which expands the angular
dependence of eigenfunctions onto spherical harmonics series, and whose radial
treatment is particularly well adapted to the behaviour of equilibrium
quantities in evolved models (at the interface of convective and radiative
regions, and at the stellar surface). The radial differentiation scheme is
made by means of a sophisticated finite difference method, which is accurate
up to the fifth order in terms of the radial resolution (developed by
Scuflaire et al. 2008a, for the losc adiabatic code).

According to Ballot et al. (2012) and Ouazzani et al. (2017), the 1D non-
perturbative approach – which presents the advantage of requiring less
numerical resources –, give satisfactory results compared to the full 2D
approach. Throughout this study we therefore work with one-dimensional
spherical stellar models. The pulsation modes have been computed using up to 5
spherical harmonics, i.e. for a given $m$, $\ell$=1 to 9 with odd value for
dipolar modes. We have made sure, by the mean of convergence tests, that the
frequencies do not vary significantly when adding a sixth spherical harmonic
in the series. The resulting modes have multiple $\ell$ characters, they are
then assigned an effective angular degree, taking the dominant contribution to
the kinetic energy in the series.

###  3.2 Gravito-inertial resonances in theoretical spectra

Table 3: Characteristics of the resonances for zonal dipolar modes for the three models 1z, 2m, 3t given in Tab.2 (radial order $n$, frequency of the mixed gravito-inertial/pure inertial mode $\nu_{in}$, and corresponding spin parameter $s$) and for three uniform rotation frequencies 15, 19 and 23$\mu$Hz.  $\nu_{rot}$ |  1z |  2m |  3t
---|---|---|---
15$\mu$Hz |  $n$ |  -43 |  -40 |  -45
$\nu_{in}$($\mu$Hz) |  13.30 |  13.89 |  13.93
$s$ |  2.256 |  2.160 |  2.153
19$\mu$Hz |  $n$ |  -34 |  -32 |  -35
$\nu_{in}$($\mu$Hz) |  16.81 |  17.52 |  17.72
$s$ |  2.261 |  2.169 |  2.145
23$\mu$Hz |  $n$ |  -29 |  -26 |  -29
$\nu_{in}$($\mu$Hz) |  20.04 |  21.29 |  21.15
$s$ |  2.296 |  2.161 |  2.149
Table 4: Same as Tab.3 for retrograde dipolar modes. $\nu_{rot}$ |  1z |  2m |  3t
---|---|---|---
15$\mu$Hz |  $n$ |  -37 |  -38 |  -42
$\nu_{in}$($\mu$Hz) |  7.35 |  7.24 |  7.33
$s$ |  1.342 |  1.349 |  1.344
19$\mu$Hz |  $n$ |  -29 |  -29 |  -33
$\nu_{in}$($\mu$Hz) |  9.35 |  9.38 |  9.27
$s$ |  1.340 |  1.339 |  1.344
23$\mu$Hz |  $n$ |  -25 |  -24 |  -28
$\nu_{in}$($\mu$Hz) |  10.74 |  11.23 |  10.49
$s$ |  1.363 |  1.344 |  1.374
Table 5: Same as Tab.3 for prograde dipolar modes, and a uniform rotation frequency of 25$\mu$Hz.  $\nu_{rot}$ |  1z |  2m |  3t
---|---|---|---
25$\mu$Hz |  $n$ |  -53 |  -44 |  -48
$\nu_{in}$($\mu$Hz) |  29.59 |  30.65 |  30.83
$s$ |  10.891 |  8.854 |  8.576

Using the set up described in Sect. 3.1 we have computed synthetic oscillation
spectra for dipolar gravito-inertial modes, assuming a set of uniform rotation
values ranging from 7$\mu$Hz to 25$\mu$Hz. In Fig. 2 we illustrate the period
spacing of such synthetic spectra for zonal modes with radial orders ranging
from -15 to -49. The pure inertial/gravito-inertial resonances appear clearly
in the period spacing series as dips, when a few modes show lower period
spacing than their neighbouring modes, for the spectra corresponding to
rotation frequencies of 13, 15, 17, 19, 21, and 23 $\mu$Hz. The vertical lines
in Fig.2 stand at the period of the pure inertial modes computed with the
method described in Sect. 2.1. Given the approximations used for their
calculations, the agreement with complete calculations is rather stunning, all
the more for the lowest rotation rates. The characteristics of the resonances
found in the range of excited modes in the complete synthetic spectra of
dipolar modes are given in Tab. 3, 4, 5, for different rotation rates. The
interpretation of the discrepancies between the analytical model presented in
Sect.2, and the complete ones is reported to Sect.4.

Concerning prograde modes, which are those that are more likely observed, the
resonances only appear at high rotation rates. For the three models explored
in this study, the value of the uniform rotation rate has to be increased up
to 25$\mu$Hz in order to find a resonance among the modes which are likely to
be excited (radial orders around -15 to -60). A particular attention should be
paid to this case, as the literature has mostly reported observations of
prograde modes in $\gamma$ Doradus and SPB stars. In particular, we believe
that the phenomenon of gravito-inertial resonance is responsible for the dips
in the period spacing series of dipolar prograde modes in KIC 5608334,
reported in Saio et al. (2018). This can be further confirmed by looking at
the value of the spin parameter at which this resonance occurs in KIC 5608334:
$s=9.01$, which is compatible with the values presented in Tab.5.

While relaxing the constraint of mode excitation, we seek for the next
resonances given in Tab.3, to confirm that they originate from the coupling
with pure inertial modes. We therefore have extended the complete calculations
to a broader range of radial orders, from -11 to -191. The spin parameters
relative to these eigenvalues are given in Fig. 3. As shown in Fig.3, three
resonances are found, and they correspond to the three eigenvalues of the
Poincaré problem (indicated in Fig.3 by the grey vertical lines), for which
$\zeta_{I}^{GI}$ is not negligible (see Tab. 1). In other words, for each
eigenvalue with a coupling coefficient of the order of 0.1 or higher, an
associated resonance has been found in the complete calculations. In Fig.4 are
given the 2D maps of the eigenfunctions of the three modes located in the dips
of the period spacing series Fig.3. These have to be compared to the three
panels giving the 2D maps of the pure inertial eigenmodes computed in Sect.2,
provided in Fig.1. From this comparison we see clearly that the modes trapped
in the convective cores in Fig.4 follow closely the morphology of the pure
inertial modes in a full sphere of uniform density computed in Sect. 2. The
mixed morphology of the complete modes clearly confirms their mixed nature:
pure inertial in the convective core, and gravito-inertial in the surrounding
radiative zone.

###  3.3 Failure of the traditional approximation to reproduce the resonances

In rotating stars, the equation system for pulsations is not separable in the
radial and latitudinal coordinates. The TAR is an approximate treatment that
conserves the separability of the system. The first hypothesis is made on the
rotation profile by assuming solid-body rotation. The centrifugal distortion
is neglected and hence spherical symmetry is assumed. Furthermore, considering
the properties of low frequency high order g-modes, the TAR neglects the
Coriolis force associated with radial motion and radial component of the
Coriolis force associated with horizontal motion. Practically, it consists of
neglecting the horizontal component of the angular velocity so that
${\bf\Omega}=\left[\Omega\cos\theta,0,0\right]$ in the spherical polar
coordinates. Finally, the Cowling approximation is made (Cowling 1941), which
neglects the perturbation of the gravitational potential. As a result, the
motion equation for pulsations can be reduced to an equation for the radial
component, which is similar to the one without rotation, and a Laplace tidal
equation for the horizontal component, whose eigenfunctions are the Hough
functions. For a detailed derivation of these equation, we refer to Unno et
al. (1989) Sect. 34.3, or Lee & Saio (1987).

This approximation has been implemented in the losc adiabatic and mad non-
adiabatic oscillation codes, the details of which have been given by Bouabid
et al. (2013) and Salmon et al. (2014). We have computed a selection of
spectra with this method, for the sake of comparison with the non-perturbative
approach, for zonal modes in the model 1z, and for the whole rotation
frequency range from 7 to 23 $\mu$Hz. The results are plotted in Fig. 2, as
the grey solid lines. It appears from that figure that the TAR does not model
correctly the modes in the neighbourhood of the resonance. While it is in very
good agreement far from resonance, it completely misses the change of nature
of the mode which endorses a gravito-inertial/pure inertial mixed character.

This result is not surprising since the TAR only applies when $\Omega\ll N$.
Thus, in a convective zone where $N=0$, the use of the TAR is not justified.
Moreover, they cannot compute pure inertial modes which are not separable in
$r$ and $\theta$. Then, as long as the mode is confined in the radiative zone,
the TAR reproduces closely enough the complete calculations, but near the
resonances when the pure inertial character is significant in the convective
core, the TAR fails as expected. Finally, comparing, as in Fig. 2, TAR results
with complete calculations can be a way to identify resonances due to pure
inertial modes.

##  4 Impact of the stellar stratification and the rotation rate on the
resonances

Figure 5:  Density profile, scaled by the central value, for the three stellar
models given in Table 2. In dashed line are indicated the external boundary of
the convective core for each model.

The density stratification affects inertial waves through the first right hand
side term of equation Eq. (1) which is proportional to the inverse of the
density scale height and involves first order derivatives of $\Psi$. In the
short-wavelength limit, second order derivatives dominate over first order
ones. Moreover, $H$ does not vanish in stellar cores, thus the density
stratification term should have a negligible effect in this limit. However,
the short-wavelength limit is not expected to be accurate for the low
$\ell_{i}-|m|$ modes we are interested in, and some effects of the density
stratification on the inertial modes are to be expected.

###  4.1 Location of the resonances for different stellar models

We first investigate this effect by exploring the oscillation spectra of
stellar models with various density profiles. The three stellar models
presented in Tab.2 are investigated. Their stratifications are illustrated
through the density profiles plotted in Fig.5. Indeed the younger the stellar
model is, the closer to the homogeneous case. In order to find gravito-
inertial resonances, we computed synthetic oscillation spectra for dipolar
zonal and prograde modes at various uniform rotation rates.

Figure 6:  Period spacings as a function of the period for the three stellar
models given in Table 2. Top: for zonal dipolar modes with uniform rotation of
15 $\mu$Hz. Bottom: for prograde dipolar modes with uniform rotation of 25
$\mu$Hz. Both the period spacings and the periods are shown in the inertial
frame of reference.

For dipolar zonal modes, as well as dipolar retrograde modes, resonances
appear at relatively slow rotation, as illustrated Fig.6, and as given in
Tab.3 and Tab.4. Whereas for prograde modes, as mentioned above, one has to
push as high as 25$\mu$Hz to have the resonance occur in the range of
frequency where modes are excited. These values have to be compared to the
case of pure inertial mode frequencies computed in a homogeneous sphere given
in Tab.1: the strongest resonance, in that case occurs at a spin parameter of
$s=1.3246$ for the retrograde modes, $s=2.2361$ for the zonal modes, and
$s=11.3245$ for the prograde ones. For the three classes of modes, it appears
clearly, as expected, that the resonances are closer to the homogeneous case
for the youngest stellar model, than for the second youngest, than for the
oldest, ranging from a difference of less than 1% to 3% in the zonal case, 1%
to 2% in the retrograde case, and 4% to 24% in the prograde case.

The discrepancy between the homogeneous case and the stellar model case
remains relatively small for the retrograde and zonal modes whereas the
resonance of the prograde dipolar mode seems to be more sensitive to the
density stratification. As mentioned above, pure inertial modes with
wavelengths much shorter than the density scaleheight $H$ should not be
strongly affected by the density stratification. The wavelengths of the low
degree inertial modes considered here are of the order of a fraction of the
convective core radius $r_{cc}$. They are still shorter than the density
scaleheight because $r_{cc}\sim H$. This may explain the relatively small
effect of the density stratification for retrograde and zonal modes. However,
this argument does not hold for prograde inertial modes because their spatial
distribution is highly anisotropic with negligible variations in the vertical
direction, such behaviour being related to their high spin parameter. Indeed,
estimating the effect of the density stratification by comparing left hand
side and right hand terms in Eq. 1., we find that the ratio of the second left
hand side term to the second right hand side term is $\sim 1/(k_{z}H)$ where
$k_{z}$ the vertical component of the wavevector. This ratio and thus the
effect of the density stratification can not be neglected when $k_{z}$ nearly
vanishes, which occurs when the spin parameter is high. In our case, this
concerns the three prograde inertial modes with significant coupling listed in
Tab.1.

The behavior of these inertial/gravito-inertial resonances with rotation is
explored in more details next section.

###  4.2 Pure inertial modes in a sphere with density stratification

Discrepancies between the homogeneous case and the real case can arise either
from the difference in stratification, or else from the very crude boundary
condition which has been applied at the surface of the sphere in the
homogeneous case (Sect. 2). In order to isolate the relevant physical effects,
a modified version of the acor code has been used, which allows the
computation of pure inertial modes in stellar models truncated above the
convective cores. The principle is that the convective core is extracted from
a stellar model, and the pulsations are then calculated in the complete non-
perturbative approach. We imposed a rigid condition at the surface of the
convective core which ensures that the radial displacement goes to zero:
$\xi_{r}(r=r_{cc})=0$. This condition allows the resolution of the core
problem without accounting for the radiative envelope.

This modification of the acor code allows us to compute pure inertial modes in
truncated convective cores taken from the models in Tab.2. The results are
given in Fig.7, where the eigenfrequencies computed for the truncated
convective cores, converted into spin parameters (in violet), are compared to
the values obtained for the resonances in the complete model (in pink), and to
the values obtained solving Eq.1 (also given in Tab.1, in grey). The first
striking observation, concerning the values of spin parameters obtained in the
convective core models, is their constant nature: they do not vary with
rotation. We will come back to this in the following subsection. Moreover, as
expected, the more evolved the model is –hence the more stratified– the
further away the spin parameters for the truncated core models are from the
uniform density case.

From Fig.7, it appears clearly that the resonances in the complete models
occur systematically much closer to the values corresponding to
eigenfrequencies of pure inertial modes determined in the convective core
models than in the homogeneous sphere. This does not come as a surprise, but
the differences here exposed show that the largest discrepancies between the
homogeneous case and the complete one come from the hypothesis of homogeneity
rather than from the crude outer boundary condition ($\xi_{r}=0$ at the
surface of the box), which seems to affect the results marginally.

###  4.3 Impact of the rotation rate

We have noted in the previous section that the spin parameter at which the
resonance between the pure inertial modes and the gravito-inertial modes does
not vary with rotation. This is a direct consequence of the fact that in
isentropic medium waves are governed by Eq. 1. Indeed, for frequencies much
smaller than the acoustic frequencies, the last right hand side term of Eq. 1
is negligible and the wave equation only depends on the spin parameter. Hence,
in Tab. 6 is given one single value of the spin parameter for each models and
for each type of modes.

This is not exactly the case for the complete models. The correct treatment at
a boundary separating two distinct propagation cavities is to ensure the
continuity of both pressure and radial displacement. Ogilvie & Lin (2004)
discussed this point at a convective/radiative interface. Arguing that in
their respective cavities radial displacements of inertial modes are much
larger than those of gravito-inertial modes limited by the Brunt-Väisälä
frequency, they deduced from the continuity condition that inertial waves see
a $\xi_{r}\approx 0$ condition at the cavity interface. This justification of
the boundary condition is only approximate though. In a more accurate model,
the conditions at the interface will also depend on the gravito-inertial mode
above, which means that the effective boundary condition for the inertial mode
and thus its eigenfrequency will depend on parameters such as the Brunt
Vaisala frequency. As a result, the resonance will not occur at a fixed spin
parameter but will also vary with, for example, the ratio of the rotation rate
to the Brunt-Vaisala frequency. In agreement with this discussion, Fig. 7
shows variables spin parameters in full calculations. These variations remain
relatively small for the stellar models used in our calculations and in the
rotation range considered.

Table 6: Eigenfrequencies of pure inertial modes in truncated models of convective cores. These are given for the three models 1z, 2m, 3t given in Tab.2, and for dipolar retrograde, zonal and prograde modes.  | 1z | 2m |  3t
---|---|---|---
retrograde | 1.33 | 1.34 |  1.35
zonal | 2.21 | 2.17 |  2.16
prograde | 10.43 | 9.18 |  8.83

Figure 7:  Values of spin parameters as a function of rotation frequency
determined through three different methods. In pink are plotted the values
computed with acor on the stellar model 2m (see Sect. 3.2), the violet curves
correspond to values computed in the truncated convective core of model 2m
(see Sect. 4.2), and the grey constant lines stand for the homogeneous
spherical box case (see Sect. 2.1). These are given for dipolar retrograde
modes (upper pannel), zonal (middle), and prograde ones (lower pannel).

##  5 Discussion and conclusions

We have studied the resonances which can occur between pure inertial modes
trapped in convective cores, and gravito-inertial modes in the surrounding
radiative envelopes. We have shown that these resonances leave a clear
signature in gravito-inertial modes period spacings which have been found in
$\gamma$ Doradus stars observed during the Kepler nominal mission, including
KIC 5608334 (Saio et al. 2018).

We have proposed a very simple model that provides the spin parameters at
which the resonances occur. Comparisons with full calculations show that the
model rightly predicts the occurrence of the resonances, while the values of
the spin parameters at resonance slightly differ. One of the assumption of the
model, namely the absence of density stratification, has been identified as
the main source of discrepancy on the resonance spin parameter. As compared to
uniform density model, taking into account density stratification does not
change the fact that the resonances occur at fixed spin parameters. The other
assumption, i.e. vanishing radial displacements at the core radius as a
boundary condition for inertial modes, should then account for the remaining
deviations with the full calculations.

In this paper we assumed that the convective core is non-magnetic and
uniformly rotating. One could question the relevance of such hypotheses:
indeed, convective cores may be differentially rotating, and numerical
simulations of A-type star convective cores (see Brun et al. 2005; Augustson
et al. 2016, for B stars) show that strong magnetic fields can be generated by
a dynamo process. However, in the low Rossby number regime characterizing
these convective motions, (i) the maxwell stresses are found to largely
suppress the differential rotation (no more than a few percents in Brun et al.
2005, based on simulations of a 7-days period rotating star) and (ii) the
ratio of the magnetic energy to the kinetic energy of the rotational motions
is much smaller than one ($B_{rms}^{2}/(4\pi\rho r_{c}^{2}\Omega^{2})\sim
2.10^{-4}$ half way through the convective core for the same simulation). The
latter property suggests that the magnetic fields generated in the $\gamma$
Dor convective cores have either a very small or a negligible effect on the
inertial modes. Indeed local and global analysis (e.g. Finlay 2008; Canet et
al. 2014) show that the relative deviation of the frequency of an inertial
mode provoked by a magnetic field is proportional to this ratio (when this
ratio is small). Contrary to magnetic fields, even a small differential
rotation would matter at least because it will impact the definition of the
spin parameter. Baruteau & Rieutord (2013) studied the effect of a
differential rotation in the case of an uniform density fluid, and showed that
it can strongly modifiy some inertial modes by reducing the volume of their
resonant cavity. This effect occurs in specific frequency domains determined
by the distribution $\Omega(r,\theta)$. As an additional effect, the non-
axisymmetric inertial modes may be affected by the presence of critical layers
when their phase speed equals the azimuthal velocity somewhere in their
resonant cavity. Assuming a weak cylindrical differential rotation similar to
the one found by Brun et al. (2005) for a 7-days rotating A-star,
$\Omega/\Omega_{0}=1+\eta(r\sin\theta/R)^{2}$ with $\eta=-0.05$, and applying
the criteria given in Baruteau & Rieutord (2013), we find that the inertial
modes producing the dips in the dipolar gravity mode period spacings (see
Table 1) are not concerned by these two strong effects of the differential
rotation. In order to explore further the influence of differential rotation
on the exact spin parameter of the dips, a dedicated study in a stellar
context will be necessary.

In future works, it would also be worth investigating whether the dip spin
parameters are sensitive to stellar modelling options especially those that
modify the structure near the convective core. Furthermore, beyond the central
position of the dip, another interesting development concerns its shape and
how it depends on the star properties.

In time, one expects to be able to obtain, given one single series of gravito-
inertial modes holding the signature of resonance with a pure inertial mode,
the rotation rate both in the convective core and in the surrounding radiative
near-core region at once. The diagnostic potential of these resonances hold
the promise to finally probe the central layers of stars on the main sequence.

In this paper, we focused our attention on mixed pure inertial/dipolar
gravito-inertial modes because mostly dipolar-inertial modes have been
identified so far. However, resonances will also occur with the other gravito-
inertial modes. They can be investigated theoretically as in the present work,
using the simplified model and the full calculations.

###### Acknowledgements.

This work was supported by the ”Programme National de Physique Stellaire”
(PNPS) of CNRS/INSU co-funded by CEA and CNES. We thank ISSI (“International
Space SciencevInstitute”) through the SoFAR (“Seismology of Fast Rotating
Stars”) program for their support. The authors are very grateful to Hideyuki
Saio for his great patience. F.L. and J.B. also would like to thank P.-M.
Culpin for his significative contribution at early stages of this work during
his Master internship.

## References

  * Angulo et al. (1999) Angulo, C., Arnould, M., Rayet, M., et al. 1999, Nuclear Physics A, 656, 3
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Augustson et al. (2016) Augustson, K. C., Brun, A. S., & Toomre, J. 2016, ApJ, 829, 92
  * Ballot et al. (2012) Ballot, J., Lignières, F., Prat, V., Reese, D. R., & Rieutord, M. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 462, Progress in Solar/Stellar Physics with Helio- and Asteroseismology, ed. H. Shibahashi, M. Takata, & A. E. Lynas-Gray, 389
  * Baruteau & Rieutord (2013) Baruteau, C. & Rieutord, M. 2013, Journal of Fluid Mechanics, 719, 47
  * Beck et al. (2012) Beck, P. G., Montalban, J., Kallinger, T., et al. 2012, Nature, 481, 55
  * Belkacem et al. (2015) Belkacem, K., Marques, J. P., Goupil, M. J., et al. 2015, A&A, 579, A31
  * Böhm-Vitense (1958) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * Bouabid et al. (2013) Bouabid, M.-P., Dupret, M.-A., Salmon, S., et al. 2013, MNRAS, 429, 2500
  * Brun et al. (2005) Brun, A. S., Browning, M. K., & Toomre, J. 2005, ApJ, 629, 461
  * Bryan (1889) Bryan, G. H. 1889, Philosophical Transactions of the Royal Society of London Series A, 180, 187
  * Canet et al. (2014) Canet, E., Finlay, C. C., & Fournier, A. 2014, Physics of the Earth and Planetary Interiors, 229, 1
  * Cantiello et al. (2014) Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., & Paxton, B. 2014, ApJ, 788, 93
  * Castelli & Kurucz (2003) Castelli, F. & Kurucz, R. L. 2003, in IAU Symposium, Vol. 210, Modelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Weiss, & D. F. Gray, A20
  * Christophe et al. (2018) Christophe, S., Ballot, J., Ouazzani, R.-M., Antoci, V., & Salmon, S. J. A. J. 2018, A&A, 618, A47
  * Cowling (1941) Cowling, T. G. 1941, MNRAS, 101, 367
  * Deheuvels et al. (2012) Deheuvels, S., García, R. A., Chaplin, W. J., et al. 2012, ApJ, 756, 19
  * Dintrans & Ouyed (2001) Dintrans, B. & Ouyed, R. 2001, A&A, 375, L47
  * Dintrans & Rieutord (2000) Dintrans, B. & Rieutord, M. 2000, A&A, 354, 86
  * Dziembowski et al. (1987) Dziembowski, W., Kosovichev, A., & Kozlowski, M. 1987, Acta Astron., 37, 331
  * Eggenberger et al. (2012) Eggenberger, P., Montalbán, J., & Miglio, A. 2012, A&A, 544, L4
  * Ferguson et al. (2005) Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585
  * Finlay (2008) Finlay, C. C. 2008, Physics of the Earth and Planetary Interiors, 170, 1
  * Formicola et al. (2004) Formicola, A., Imbriani, G., Costantini, H., et al. 2004, Physics Letters B, 591, 61
  * Fuller et al. (2014) Fuller, J., Lecoanet, D., Cantiello, M., & Brown, B. 2014, ApJ, 796, 17
  * Fuller et al. (2019) Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485, 3661
  * Greenspan (1968) Greenspan, H. 1968, The Theory of Rotating Fuilds, ed. Cambridge: Cambridge University Press
  * Guenther & Gilman (1985) Guenther, D. B. & Gilman, P. A. 1985, ApJ, 295, 195
  * Iglesias & Rogers (1996) Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943
  * Kurtz et al. (2014) Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS, 444, 102
  * Lee & Saio (1987) Lee, U. & Saio, H. 1987, MNRAS, 224, 513
  * Lee et al. (1992) Lee, U., Strohmayer, T. E., & van Horn, H. M. 1992, ApJ, 397, 674
  * Li et al. (2019) Li, G., Van Reeth, T., Bedding, T. R., Murphy, S. J., & Antoci, V. 2019, MNRAS, 487, 782
  * Lockitch & Friedman (1999) Lockitch, K. H. & Friedman, J. L. 1999, ApJ, 521, 764
  * Marques et al. (2013) Marques, J. P., Goupil, M. J., Lebreton, Y., et al. 2013, A&A, 549, A74
  * Mathis et al. (2018) Mathis, S., Prat, V., Amard, L., et al. 2018, A&A, 620, A22
  * Miglio et al. (2008) Miglio, A., Montalbán, J., Noels, A., & Eggenberger, P. 2008, MNRAS, 386, 1487
  * Mosser et al. (2012) Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A, 540, A143
  * Murphy et al. (2016) Murphy, S. J., Fossati, L., Bedding, T. R., et al. 2016, MNRAS, 459, 1201
  * Ogilvie & Lin (2004) Ogilvie, G. I. & Lin, D. N. C. 2004, ApJ, 610, 477
  * Ouazzani et al. (2012) Ouazzani, R.-M., Dupret, M.-A., & Reese, D. R. 2012, A&A, 547, A75
  * Ouazzani et al. (2019) Ouazzani, R. M., Marques, J. P., Goupil, M. J., et al. 2019, A&A, 626, A121
  * Ouazzani et al. (2015) Ouazzani, R.-M., Roxburgh, I. W., & Dupret, M.-A. 2015, A&A, 579, A116
  * Ouazzani et al. (2017) Ouazzani, R.-M., Salmon, S. J. A. J., Antoci, V., et al. 2017, MNRAS, 465, 2294
  * Papaloizou & Pringle (1981) Papaloizou, J. & Pringle, J. E. 1981, MNRAS, 195, 743
  * Papaloizou & Savonije (1997) Papaloizou, J. C. B. & Savonije, G. J. 1997, MNRAS, 291, 651
  * Pinçon et al. (2017) Pinçon, C., Belkacem, K., Goupil, M. J., & Marques, J. P. 2017, A&A, 605, A31
  * Prat et al. (2016) Prat, V., Lignières, F., & Ballot, J. 2016, A&A, 587, A110
  * Rieutord (1991) Rieutord, M. 1991, Geophysical and Astrophysical Fluid Dynamics, 59, 185
  * Rieutord & Valdettaro (2018) Rieutord, M. & Valdettaro, L. 2018, Journal of Fluid Mechanics, 844, 597
  * Rogers & Nayfonov (2002) Rogers, F. J. & Nayfonov, A. 2002, ApJ, 576, 1064
  * Saio et al. (2018) Saio, H., Bedding, T. R., Kurtz, D. W., et al. 2018, MNRAS, 477, 2183
  * Saio et al. (2015) Saio, H., Kurtz, D. W., Takata, M., et al. 2015, MNRAS, 447, 3264
  * Salmon et al. (2014) Salmon, S. J. A. J., Montalbán, J., Reese, D. R., Dupret, M.-A., & Eggenberger, P. 2014, A&A, 569, A18
  * Scuflaire et al. (2008a) Scuflaire, R., Montalbán, J., Théado, S., et al. 2008a, Ap&SS, 316, 149
  * Scuflaire et al. (2008b) Scuflaire, R., Théado, S., Montalbán, J., et al. 2008b, Ap&SS, 316, 83
  * Townsend (2003) Townsend, R. H. D. 2003, MNRAS, 340, 1020
  * Unno et al. (1989) Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 1989, Nonradial oscillations of stars
  * Van Reeth et al. (2018) Van Reeth, T., Mombarg, J. S. G., Mathis, S., et al. 2018, A&A, 618, A24
  * Van Reeth et al. (2016) Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, aap, 593, A120
  * Wu (2005) Wu, Y. 2005, ApJ, 635, 674
  * Zahn (1992) Zahn, J. 1992, A&A, 265, 115

##  Appendix A Ellipsoidal coordinates

The Poincaré equation becomes separable using the ellipsoidal coordinates
$(x_{1},x_{2},\phi)$ where $\phi$ is the azimuthal angle of the spherical
coordinates and $x_{1}$ and $x_{2}$ relations with the cartesian coordinates
read :

$\displaystyle x$ $\displaystyle=$
$\displaystyle\left(\frac{(1-x_{1}^{2})(1-x_{2}^{2})}{1-\mu^{2}}\right)^{1/2}\cos(\phi)$
(5) $\displaystyle y$ $\displaystyle=$
$\displaystyle\left(\frac{(1-x_{1}^{2})(1-x_{2}^{2})}{1-\mu^{2}}\right)^{1/2}\sin(\phi)$
(6) $\displaystyle z$ $\displaystyle=$ $\displaystyle\frac{x_{1}x_{2}}{\mu}$
(7)

where $x_{1}\in[\mu,1]$ and $x_{2}\in[-\mu,\mu]$, and distances have been
normalized with the sphere radius. In these coordinates, the surface of the
sphere is given by
$\\{x_{1}\in[\mu,1],x_{2}=\mu\\}\cup\\{x_{1}=\mu,x_{2}\in[-\mu,\mu]\\}$.

##  Appendix B Inertial modes of the full sphere

Figure 8:  Inertial modes in a uniform density sphere, as computed with the
method given in Sect. 2. Each of these modes have a small spatial matching
with the dipolar gravito-inertial mode of the same spin parameter. The dotted
line specifies the critical latitude $\theta_{c}=\arccos(\omega/(2\Omega))$.
The top panels show the zonal modes with low coupling coefficients presented
in Tab.1, whereas the middle and the bottom panels show the tree modes with
the highest coupling coefficients for the prograde (m=-1), and the retrograde
(m=1) modes.