# Inertial modes of a freely rotating ellipsoidal planet and their relation to
nutations

Jeremy Rekier Royal Observatory of Belgium
3 avenue circulaire
Brussels, 1180, Belgium  Santiago A. Triana Royal Observatory of Belgium
3 avenue circulaire
Brussels, 1180, Belgium  Antony Trinh Lunar and Planetary Lab
University of Arizona
Tucson, AZ 85721-0092  Véronique Dehant Royal Observatory of Belgium
3 avenue circulaire
Brussels, 1180, Belgium

(Received April 1 2020; Revised May 13 2020; Accepted May 14 2020)

###### Abstract

We compute the inertial modes of a freely rotating two-layer planetary model
with an ellipsoidal inviscid fluid core and a perfectly rigid mantle. We
present a method to derive analytical formulae for the frequencies of the
_Free Core Nutation_ (FCN) and _Chandler Wobble_ (CW) which are valid to all
orders of the dynamical flattening of the core and mantle, and we show how the
FCN and CW are the direct generalisation of the purely fluid _Spin-Over mode_
(SO) and of the _Eulerian Wobble_ (EW) to the case where the mantle can
oscillate freely around a state of steady rotation. Through a numerical
computation for an axisymmetric (oblate spheroidal) planet, we demonstrate
that all other inertial modes of the steadily rotating fluid core are also
free modes of the freely rotating two-layer planet.

††journal: PSJ

##  1 Introduction

The rotation of a planet depends on its internal structure via its moments of
inertia. In particular, the misalignment between the rotation axis of a planet
and its moment of inertia axis, and the misalignment between the rotation axis
of a planet and of its liquid core gives rise to two free modes of rotation
known as the _Chandler Wobble_ (CW) and the _Free Core Nutation_ (FCN),
respectively. In the planetary reference frame, the CW, if excited, is a long
period prograde rotation of the instantaneous axis of rotation around the
principal axis of largest moment of inertia of the planet, while the FCN, if
exited, is a retrograde motion of nearly diurnal frequency around the same
axis.111This motion is sometimes referred to as the _Nearly Diurnal Free
Wobble_ (NDFW) in the planetary frame of reference. When the gravitational
pull on the planet from the sun, the moon and other planets is taken into
account, these two modes can resonantly amplify the planet’s forced nutation.
Concerning the Earth, nutations are observed by using Very Long Baseline
Interferometry (VLBI) and modeled precisely, accounting for deformation and
coupling mechanisms at the core-mantle boundary such as pressure and
electromagnetic couplings. A solid deformable inner core inside the liquid
core is also accounted for (see e.g. Dehant and Mathews (2015)). Concerning
the planet Mars that possesses precession and nutations as well, the joint
efforts of the current RISE (Folkner et al., 2018) and future LaRa (Dehant et
al., 2020) lander missions aim to detect such amplification in the nutation
signal of Mars measured directly from its surface. Such a detection would
allow an estimate of Mars’s FCN frequency, from which Mars’ polar moment of
inertia can be deduced; this can then be used to build radial profiles of the
density structure of the planet, and in particular constrain its core size
(Dehant et al., 2020). Usually, such a computation is based on the formalism
of Sasao et al. (1980) in which the rotational dynamics of the fluid core is
described in terms of its angular momentum which is assumed to be proportional
to a mean flow resembling a solid-body rotation (see Sec. 2.3). This has a
couple of important advantages like that of being easily adaptable to
accommodate the existence of a non-rigid (elastic) mantle. However, as we show
in the present work, it is, by design, valid only to first order in the
planet’s dynamical flattening parameters. This casts doubts on the results of
previous studies attempting to push the formalism of Sasao et al. (1980)
beyond first order to evaluate the effects of trixiality (Van Hoolst and
Dehant, 2002; Chen and Shen, 2010; Dehant and Mathews, 2015; Guo and Shen,
2019; Shen et al., 2019).

In addition to the FCN and CW, the liquid core of a rotating star or planet is
known to support the existence of oscillations known as _inertial modes_
caused by the restoring effect of the Coriolis force. Experimental studies in
laboratory have shown that these inertial modes can be excited by the action
of tidal deformation at the fluid boundary, see _e.g._ Malkus (1968); Aldridge
and Toomre (1969); Kelley et al. (2007) and Morize et al. (2010). However, the
role of the inertial modes on the rotation of planets and other astrophysical
objects remains unclear. The main reason for this being that most studies, so
far, have focused on the response of the fluid core in situations where the
rotation of the planet is prescribed, _i.e._ known _a priori_ , therefore
disregarding the action of the fluid core on the rotation via the forces
acting at the _Core Mantle Boundary_ (CMB), see Le Bars et al. (2015) for a
review. In a previous work, we included the action of the fluid core on the
free rotation of a two-layer planet with a viscous liquid core (Triana et al.,
2019). We showed that the FCN and CW naturally emerge among the spectrum of
inertial modes and that complex interactions between the eigenmodes take place
when the moment of inertia of the mantle varies. Due to the difficulty in
dealing with both the presence of a thin viscous boundary layer and the
ellipticity of the CMB, we could not explore the regime of very low viscosity
relevant to planetary and astrophysical objects.

In the present work, we complement the analysis of Triana et al., focusing
specifically on the effects of the pressure torque between the fluid core and
the solid mantle. We do so by computing the free modes of rotation of a two-
layer ellipsoidal planet model with a perfectly rigid mantle and an inviscid
fluid core. When one sets the viscosity of the fluid to zero, the viscous
boundary layer at the CMB and the various _shear layers_ that it generates
inside the core disappear and the flow pattern inside the core becomes smooth.
This enables us to derive the free modes of rotation at all values of the core
and mantle’s ellipticities.

Previous studies have attempted to mathematically model the rotational modes
of a terrestrial planet based on the displacement field approach whereby the
motion of the core and mantle are computed using the set of gravito-elastic
equations which follow from the local conservation of momentum (Smith, 1974,
1977; Rogister, 2001; Rochester et al., 2014; Seyed-Mahmoud et al., 2017).
However, these are limited to considering planetary models that are only
slightly aspherical. There is also some doubt regarding the ability of these
work to capture adequately the effects of the presence of a solid inner core
which is known to generate singularities within the displacement field
(Rieutord et al., 2000).

This work is structured as follows. In Sec. 2, we present the two-layer planet
model and its equations of motion. We then consider the special case of a core
flow with a uniform vorticity and derive analytical formulae for the FCN and
CW frequencies. In Sec. 3, we compute the inertial modes of a two-layer planet
with an axisymmetric CMB numerically using the set of oblate spheroidal
coordinates and compare the value of the FCN frequency to the analytical
expressions of Sasao et al. (1980) and our own formalism. Results are
discussed in Sec. 4.

##  2 Model

Figure 1: The two-layer planet model with an ellipsoidal core and surface. The
cartesian basis
$\\{\boldsymbol{\hat{x}},\boldsymbol{\hat{y}},\boldsymbol{\hat{z}}\\}$ is
aligned with the principal axes of inertia of the core and the rigid mantle.
$\boldsymbol{\Omega}_{m}$ is the rotation vector of the mantle relative to the
inertial frame of reference and is related to the polar figure axis,
$\boldsymbol{\hat{z}}$, by Eq. (2.9), where $|\boldsymbol{m}|\ll 1$.

Fig. 1 represents the planetary model used throughout this work. The usual way
to compute the rotation of this two-layer model is via the system of equations
expressing the conservation of angular momentum for each layer. In the
reference frame attached to the mantle rotating at angular velocity
$\boldsymbol{\Omega}_{\text{m}}$ relative to the inertial frame, and in the
absence of external forces, these read:

$\displaystyle\partial_{t}\boldsymbol{L}_{\text{m}}+\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{L}_{\text{m}}$
$\displaystyle=\boldsymbol{\Gamma}~{},$ (2.1)
$\displaystyle\partial_{t}\boldsymbol{L}_{\text{c}}+\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{L}_{\text{c}}$
$\displaystyle=-\boldsymbol{\Gamma}~{},$ (2.2)

where $\boldsymbol{L}_{\text{c}}$ and $\boldsymbol{L}_{\text{m}}$ denote the
angular momentum of the core and mantle respectively and $\boldsymbol{\Gamma}$
represents the total torque that the fluid core exerts on the mantle. All
these quantity are measured relative to the inertial frame. In the absence of
other external torques, this must be balanced by the torque from the core on
the mantle. The addition of Eqs. (2.1) and (2.2) then yields:

$\partial_{t}\boldsymbol{L}+\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{L}=\boldsymbol{0}~{},$
(2.3)

where
$\boldsymbol{L}\equiv\boldsymbol{L}_{\text{c}}+\boldsymbol{L}_{\text{m}}$ is
the total angular momentum of the planet. Traditionally, one chooses to solve
Eqs. (2.2) and (2.3). The difficulty then lies in the determination of
$\boldsymbol{\Gamma}$ which expresses the coupling between the core and the
mantle. For the case of a perfect fluid with zero viscosity and in the absence
of magnetic field the only contribution comes from the pressure forces at the
CMB which give rise to the pressure torque:

$\boldsymbol{\Gamma}=\oint_{\text{CMB}}P\boldsymbol{r}\times\boldsymbol{\hat{n}}$
(2.4)

where $P$ is the pressure inside the fluid and $\boldsymbol{r}$ is the
position vector. The integration runs over the whole surface of the CMB and
$\boldsymbol{\hat{n}}$ denotes the radially-outward normal vector to this
surface. In principle, one should add the increment of gravity resulting from
the change in the repartition of mass to Eq. (2.4). In our simple model,
however, this contribution is zero as the distribution of masses is kept
constant due to the perfect rigidity of the mantle and the incompressibility
of the fluid core. Sasao et al. (1980) have shown how to avoid the computation
of the pressure torque explicitly by replacing Eq. (2.2) with an equation
involving the mean rotation of the fluid. This approach, however, is only
valid in the limit of small flattening of the CMB (see Sec. 2.3). In the
present work, we use a different approach based on the equation giving the
evolution of the flow inside the fluid core. In the special case where the
flow vorticity is spatially uniform, the problem reduces to determining the
evolution of the components of the vorticity in time. We can do so
analytically by working from the equation of vorticity (the curl of the
momentum equation) in a way that is formally similar to the formalism of Sasao
et al. but which has the advantage to be valid for all values of the
flattening.

In the remainder of this section, we present the equations governing the
dynamics of the fluid core coupled to planetary rotation. We then present
their analytical resolution in the special case of a flow with uniform
vorticity and provide a comparison to the formalism of Sasao et al..

###  2.1 Equations for a general flow

We focus on the small departures of the flow from a solid-body rotation and
write:

$\boldsymbol{v}=\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{r}+\boldsymbol{u}~{},$
(2.5)

where $\boldsymbol{v}$ and $\boldsymbol{u}$ denote the velocity field measured
in the inertial frame and the mantle frame, respectively. Assuming that the
fluid is incompressible and has zero viscosity, the equation of motion in the
mantle frame and to first order in $\boldsymbol{u}$ reads (Triana et al.,
2019):

$\partial_{t}\boldsymbol{u}+2\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{u}+\partial_{t}\boldsymbol{\Omega}_{\text{m}}\times\boldsymbol{r}=-\boldsymbol{\nabla}p~{},$
(2.6)

where $\rho$ denotes the fluid density and we have identified the _reduced
pressure_ , $p$, as:

$p\equiv\frac{P}{\rho}-\frac{1}{2}|\boldsymbol{\Omega}_{m}\times\boldsymbol{r}|^{2}~{},$
(2.7)

where the second term represents the centrifugal potential which, in the
absence of external forces, is the only contribution to the total potential.
The flow must satisfy the _no-penetration condition_ at the CMB:

$\left.\boldsymbol{u}\cdot\boldsymbol{\hat{n}}\right|_{\text{CMB}}=0~{}.$
(2.8)

In this work, we focus on the small oscillations of the planet about a steady
rotation whereby $\boldsymbol{\Omega}_{\text{m}}$ can be decomposed as:

$\boldsymbol{\Omega}_{\text{m}}=\Omega_{0}(\boldsymbol{\hat{z}}+\boldsymbol{m})~{},$
(2.9)

where the cartesian basis vector, $\boldsymbol{\hat{z}}$, is chosen as the
mean axis of rotation (see Fig. 1) and we have $|\boldsymbol{m}|\ll 1$. In
what follows, we set $\Omega_{0}=1$ for simplicity which amounts to measure
frequencies in cycles per day. As we are looking for solutions that are
oscillatory, we use the following Fourier decomposition:

$\displaystyle\boldsymbol{u}$ $\displaystyle=\boldsymbol{u}~{}e^{i\omega
t}+\boldsymbol{u}^{*}e^{-i\omega t}~{},$ (2.10) $\displaystyle p$
$\displaystyle=p~{}e^{i\omega t}+p^{*}e^{-i\omega t}~{},$ (2.11)
$\displaystyle\boldsymbol{m}$ $\displaystyle=\boldsymbol{m}~{}e^{i\omega
t}+\boldsymbol{m}^{*}e^{-i\omega t}~{},$ (2.12)

where a $*$ denotes the complex conjugation and where we have used the same
symbol to denote a quantity and its Fourier component at frequency $+\omega$
for simplicity. Inserting Eq. (2.9) and Eqs. (2.10) to (2.12) into Eq. (A1),
keeping terms up to first order in $\boldsymbol{m}$ and $\boldsymbol{u}$
yields:

$\displaystyle
i\omega\boldsymbol{u}+2\boldsymbol{\hat{z}}\times\boldsymbol{u}+i\omega(\boldsymbol{m}\times\boldsymbol{r})$
$\displaystyle=-\boldsymbol{\nabla}p~{},$ (2.13)
$\displaystyle-i\omega\boldsymbol{u}^{*}+2\boldsymbol{\hat{z}}\times\boldsymbol{u}^{*}-i\omega(\boldsymbol{m}^{*}\times\boldsymbol{r})$
$\displaystyle=-\boldsymbol{\nabla}p^{*}~{}.$ (2.14)

From Eq. (2.13), we can isolate $\boldsymbol{u}$ and express it in terms of in
terms of $p$ and $\boldsymbol{m}$ only. Upon using the incompressibility
condition ($\boldsymbol{\nabla}\cdot\boldsymbol{u}=0$), and the condition of
no-penetration Eq. (2.8), we then arrive respectively to (see Appendix A):

$-\omega^{2}\boldsymbol{\nabla}^{2}{p}+4(\boldsymbol{\hat{z}}\cdot\boldsymbol{\nabla})^{2}p=-4(\boldsymbol{\hat{z}}\cdot\boldsymbol{m})~{},$
(2.15)

which is valid inside the whole core and

$-\omega^{2}\boldsymbol{\hat{n}}\cdot(\boldsymbol{\nabla}p+i\omega(\boldsymbol{m}\times\boldsymbol{r}))+2i\omega(\boldsymbol{\hat{z}}\times\boldsymbol{\hat{n}})\cdot(\boldsymbol{\nabla}p+i\omega(\boldsymbol{m}\times\boldsymbol{r}))+4(\boldsymbol{\hat{z}}\cdot\boldsymbol{\hat{n}})\boldsymbol{\hat{z}}\cdot(\boldsymbol{\nabla}p+i\omega(\boldsymbol{m}\times\boldsymbol{r}))|_{\text{CMB}}=0~{}.$
(2.16)

which must be satisfied at the CMB. When the mantle is in steady rotation, one
has $\boldsymbol{m}=\boldsymbol{0}$. Eq. (2.15) then reduces to the well-known
_Poincaré equation_ , the solutions of which are global oscillations of the
flow called _inertial modes_ and their frequencies satisfy $-2<\omega<2$ (see
_e.g._ Sec. 8.3.3 of Rieutord (2015)). In his classic paper, Bryan (1889) gave
the general analytical implicit expression of the inertial modes frequencies
for an axisymmetric ellipsoid (those are reproduced in Appendix A). His work
was later extended by that of Hough (1895) to the geometry of the triaxial
ellipsoid. In both cases, the pressure field associated to each mode can be
represented as a polynomial function in the cartesian coordinates (Kudlick,
1966; Zhang et al., 2001; Vantieghem, 2014), see also Rekier et al. (2018).
The inertial mode of lowest degree is called the _Spin-Over mode_ (SO) and its
flow, when it is observed from the mantle frame, resembles a solid-body
rotation around an axis perpendicular to the rotation vector of the mantle. It
is the only inertial mode that carries a net amount of total angular momentum
when integrated over the whole core and is therefore especially important for
rotation as we show later.

In order to obtain the evolution of $\boldsymbol{m}$, we must use Eq. (2.1)
and express the angular momentum of the mantle in terms of its tensor of
inertia, $\boldsymbol{I}_{\text{m}}$, and its angular velocity:

$\boldsymbol{L}_{\text{m}}=\boldsymbol{I}_{\text{m}}\cdot\boldsymbol{\Omega}_{m}~{}.$
(2.17)

To obtain the right-hand side of Eq. (2.1), we must write the pressure torque
Eq. (2.4) in terms of the reduced pressure, $p,$ and the increment of angular
velocity $\boldsymbol{m}$. From the definition Eq. (2.7), one has, to first
order in $\boldsymbol{m}$:

$P=\rho\left(p+\frac{1}{2}|\boldsymbol{\Omega}_{m}\times\boldsymbol{r}|^{2}\right)\approx\rho\;p+\frac{\rho}{2}|\boldsymbol{\hat{z}}\times\boldsymbol{r}|^{2}+\rho(\boldsymbol{\hat{z}}\times\boldsymbol{r})\cdot(\boldsymbol{m}\times\boldsymbol{r})~{}.$
(2.18)

In Sec. 3, we solve Eqs. (2.15), (2.16) and Eq. (2.1) numerically for an
axisymmetric oblate ellipsoidal fluid core using a set of oblate spheroidal
coordinates, in a manner similar to what we presented in Rekier et al. (2018).
In the remainder of the present section, we focus on the special case where
the flow inside the core has a uniform vorticity for which the free modes of
the two-layer planet can be computed analytically.

###  2.2 Flow with a uniform vorticity

In his classic paper, Poincaré (1885) looked for solutions of the fluid flow
in an ellipsoid characterised by a spatially uniform (yet possibly time-
dependent) vorticity. Denoting this vorticity by
$\boldsymbol{\nabla}\times\boldsymbol{u}=2\boldsymbol{w}$, the velocity field,
$\boldsymbol{u}$, writes as:

$\boldsymbol{u}=(\boldsymbol{w}\times\boldsymbol{r})+\boldsymbol{\nabla}\psi~{},$
(2.19)

where $\psi$ is explicitly expressed in terms of $\boldsymbol{w}$ and the
ellipsoid dimensions ($a$, $b$, $c$). This scalar function is there to adjust
the flow to the no-penetration boundary condition Eq. (2.8) and is zero for a
spherical core. The explicit expressions of $\boldsymbol{u}$ and $\psi$ are
given in Appendix B. We obtain the equation giving the time evolution of the
vorticity by taking the curl of Eq. (A1):

$\partial_{t}\boldsymbol{w}-\boldsymbol{\Omega}_{\text{m}}\cdot\boldsymbol{\nabla}\boldsymbol{u}+\partial_{t}\boldsymbol{\Omega}_{\text{m}}=\boldsymbol{0}~{}.$
(2.20)

This has the advantage to make the gradient of pressure vanish and the whole
influence of the mantle on the fluid core is accounted for entirely via the
variations of $\boldsymbol{\Omega}_{\text{m}}$. Under the assumption of
uniform vorticity, the computation of the fluid flow inside the core reduces
to the resolution of Eq. (2.20) for the three cartesian components of
$\boldsymbol{w}$. In order to obtain the coupled motion of the core and
mantle, we must add another equation giving the evolution of
$\boldsymbol{\Omega}_{\text{m}}$. We choose to use Eq. (2.3) as it does not
require any explicit computation of the torque at the CMB. It does, however,
require to evaluate the angular momentum of the whole planet relative to the
inertial frame, $\boldsymbol{L}$. Using Eq. (2.5), this writes:

$\boldsymbol{L}=(\boldsymbol{I}_{\text{c}}+\boldsymbol{I}_{\text{m}})\cdot\boldsymbol{\Omega}_{\text{m}}+\int_{\text{core}}\rho(\boldsymbol{r}\times\boldsymbol{u})~{},$
(2.21)

where the first term is proportional to the total tensor of inertia of the
whole planet and the integral runs over the whole volume of the core. As the
mantle is taken to be perfectly rigid, we can always chose the coordinates of
the mantle frame to be aligned with the principale axes of inertia of the core
and mantle at all time and parametrise the tensor of inertia as:

$\underbrace{\begin{pmatrix}\text{A}&0&0\\\ 0&\text{B}&0\\\
0&0&\text{C}\end{pmatrix}}_{\boldsymbol{I}}=\underbrace{\begin{pmatrix}\text{A}_{f}&0&0\\\
0&\text{B}_{f}&0\\\
0&0&\text{C}_{f}\end{pmatrix}}_{\boldsymbol{I}_{\text{c}}}+\underbrace{\begin{pmatrix}\text{A}_{m}&0&0\\\
0&\text{B}_{m}&0\\\
0&0&\text{C}_{m}\end{pmatrix}}_{\boldsymbol{I}_{\text{m}}}~{}.$ (2.22)

In Appendix B, we give the explicit expressions of the components of the core
angular momentum. In what follows, we parametrise the moments of inertia in
terms of the following _dynamical flattening_ parameters (see p.154 of Dehant
and Mathews (2015)):

$\displaystyle\alpha_{f}$
$\displaystyle\equiv\frac{\text{C}_{f}-\frac{\text{A}_{f}+\text{B}_{f}}{2}}{\frac{\text{A}_{f}+\text{B}_{f}}{2}}~{},$
$\displaystyle\beta_{f}$
$\displaystyle\equiv\frac{\text{B}_{f}-\text{A}_{f}}{\text{B}_{f}+\text{A}_{f}}~{},$
(2.23)

for the fluid core and

$\displaystyle\alpha$
$\displaystyle\equiv\frac{\text{C}-\frac{\text{A}+\text{B}}{2}}{\frac{\text{A}+\text{B}}{2}}~{},$
$\displaystyle\beta$
$\displaystyle\equiv\frac{\text{B}-\text{A}}{\text{B}+\text{A}}~{},$ (2.24)

for the whole planet.

As we are interested in the small oscillations of the flow around the axis of
rotation, we write, in analogy with Eq (2.12) and consistent with the notation
of Sasao et al. (1980):

$\boldsymbol{w}=(\boldsymbol{m}_{f}~{}e^{i\omega
t}+\boldsymbol{m}_{f}^{*}e^{-i\omega t})~{},$ (2.25)

and we assume $|\boldsymbol{m}_{f}|\ll 1$. The system of Eqs. (2.3) and (2.20)
in the first order in $\boldsymbol{m}$ and $\boldsymbol{m}_{f}$ can be put in
the following matrix form from which the components of these vectors in the
$z$-direction decouple completely:

$\scriptsize\underbrace{\left(\begin{array}[]{cccc}\omega&-\frac{i(\alpha-\beta)}{(1-\beta)}&\frac{\text{A}_{f}\left(1-\alpha_{f}\right)\left(1+\alpha_{f}-2\beta_{f}\right)}{\text{A}\left(1-\beta_{f}\right){}^{2}}~{}\omega&\frac{i\text{A}_{f}\left(1-\alpha_{f}\right)\left(1+\alpha_{f}+2\beta_{f}\right)}{\text{A}\left(1-\beta_{f}^{2}\right)}\\\
\frac{i(\alpha+\beta)}{(1+\beta)}&\omega&-\frac{i\text{A}_{f}(1-\beta)\left(1-\alpha_{f}\right)\left(1+\alpha_{f}-2\beta_{f}\right)}{\text{A}(1+\beta)\left(1-\beta_{f}\right){}^{2}}&\frac{\text{A}_{f}(1-\beta)\left(1-\alpha_{f}\right)\left(1+\alpha_{f}+2\beta_{f}\right)}{\text{A}(1+\beta)\left(1-\beta_{f}^{2}\right)}~{}\omega\\\
\omega&0&\omega&\frac{i\left(1+\alpha_{f}+2\beta_{f}\right)}{(1+\beta_{f})}\\\
0&\omega&-\frac{i\left(1+\alpha_{f}-2\beta_{f}\right)}{(1-\beta_{f})}&\omega\\\
\end{array}\right)}_{M}\left(\begin{array}[]{c}m^{x}\\\ m^{y}\\\ m_{f}^{x}\\\
m_{f}^{y}\end{array}\right)=0~{}$ (2.26)

The problem of finding the free modes frequencies $\omega$, then reduces to
solving the polynomial equation which results from imposing:

$\text{det}[M]=0~{}.$ (2.27)

This polynomial equation is in fact _biquadratic_ in the frequency $\omega$
and so its roots come in pairs that have the same magnitude and opposite
signs. The set of individual modes associated to each root can be classified
into prograde and retrograde motions (relative to the mean planetary
rotation). In the end, there are only two independent physical motions as a
prograde (respectively retrograde) mode of frequency $\omega<0$ is equivalent
to a retrograde (respectively prograde) mode of $\omega>0$. Here we follow the
convention of Dehant and Mathews (2015) and others and represent the
retrograde modes using negative frequencies. The explicit expressions of the
two independent frequencies are too lengthy to reproduce here. Instead, we
provide the expressions of their Taylor expansions in the dynamical flattening
parameters:

$\displaystyle\omega=-1-$
$\displaystyle\frac{\text{A}\alpha_{f}}{\text{A}_{m}}+\frac{\text{A}\left(2(\alpha+\beta)\left(\text{A}-\text{A}_{m}\right)\alpha_{f}-2\left(\text{A}-\text{A}_{m}\right)\alpha_{f}\beta_{f}-\left(\text{A}-4\text{A}_{m}\right)\beta_{f}^{2}\right)}{2\text{A}_{m}^{2}}+\dots~{},$
(2.28) $\displaystyle\omega$
$\displaystyle=\frac{\text{A}\sqrt{|\alpha^{2}-\beta^{2}|}}{\text{A}_{m}}-\frac{\text{A}\sqrt{|\alpha^{2}-\beta^{2}|}\left(\text{A}-\text{A}_{m}\right)\left(\beta+\alpha_{f}-\beta_{f}\right)}{\text{A}_{m}^{2}}+\dots~{}.$
(2.29)

We identify these frequencies to those of the FCN and CW, respectively. The
former presents a nearly diurnal retrograde nutation as measure in the
planetary frame of reference. The latter is a prograde nutation of small
frequency (and thus long period) in that same frame.

In anticipation of the next section, we now turn to the special case where the
planet and its liquid core are axisymmetric around the mean rotation axis,
$\boldsymbol{\hat{z}}$. In such case, $\beta=\beta_{f}=0$ and Eq. (2.26) can
be further simplified in terms of the following variables:

$\displaystyle m^{+}$ $\displaystyle=\frac{1}{\sqrt{2}}(-m^{x}+im^{y})~{},$
$\displaystyle m^{-}$ $\displaystyle=\frac{1}{\sqrt{2}}(m^{x}+im^{y})~{},$
(2.30)

and analogous expressions for the components of $\boldsymbol{m}_{f}$. These
new variables are the components of $\boldsymbol{m}$ in the cartesian
canonical basis (see _e.g._ p. 193 of Trinh (2019)). The equations for $m^{+}$
and $m_{f}^{+}$ on the one-hand and for $m^{-}$ and $m_{f}^{-}$ on the other
hand decouple completely so that we end up with:

$\left(\begin{array}[]{cc}\omega-\alpha&\frac{\text{A}_{f}\left(1-\alpha_{f}^{2}\right)}{\text{A}}\left(1+\omega\right)\\\
\omega&\omega+(1+\alpha_{f})\\\
\end{array}\right)\left(\begin{array}[]{c}m^{-}\\\
m_{f}^{-}\end{array}\right)=0~{}.$ (2.31)

Setting the determinant of the above matrix to zero gives the following two
independent frequencies:

$\omega=-\frac{\text{A}_{f}\left(1-\alpha_{f}^{2}\right)-\text{A}\left(1-\alpha+\alpha_{f}\right)\pm\sqrt{4\text{A}\alpha\left(1+\alpha_{f}\right)\left(\text{A}-\text{A}_{f}\left(1-\alpha_{f}^{2}\right)\right)+\left(\text{A}\left(1-\alpha+\alpha_{f}\right)-\text{A}_{f}\left(1-\alpha_{f}^{2}\right)\right){}^{2}}}{2\left(\text{A}-\text{A}_{f}\left(1-\alpha_{f}^{2}\right)\right)}~{}.$
(2.32)

By taking the series expansion of Eq. (2.32) in $\alpha$ and $\alpha_{f}$,
assuming that these two parameters are small quantities of the same order, one
recovers more familiar expressions for the FCN and the CW in the axisymmetric
case:

$\displaystyle\omega=-1$
$\displaystyle-\frac{\text{A}\alpha_{f}}{\text{A}_{m}}+\frac{\text{A}\alpha\left(\text{A}-\text{A}_{m}\right)\alpha_{f}}{\text{A}_{m}^{2}}-\frac{\text{A}\left(\text{A}-\text{A}_{m}\right)\alpha_{f}\left(-\text{A}_{m}\alpha_{f}^{2}+\text{A}\alpha\left(\alpha+\alpha_{f}\right)\right)}{\text{A}_{m}^{3}}+\dots~{},$
(2.33) $\displaystyle\omega=$
$\displaystyle\frac{\text{A}\alpha}{\text{A}_{m}}-\frac{\text{A}\alpha\left(\text{A}-\text{A}_{m}\right)\alpha_{f}}{\text{A}_{m}^{2}}+\frac{\text{A}\left(\text{A}-\text{A}_{m}\right)\alpha\left(-\text{A}_{m}\alpha_{f}^{2}+\text{A}\alpha_{f}\left(\alpha+\alpha_{f}\right)\right)}{\text{A}_{m}^{3}}+\dots~{}.$
(2.34)

From Eq. (2.33), one sees that the FCN has an exactly diurnal frequency in the
case where the fluid core is spherical ($\alpha_{f}$). This is true to all
orders in $\alpha$ and $\alpha_{f}$ as can be shown by working directly from
Eq. (2.32). From Eq. (2.34), we see that the frequency of the CW remains
proportional to the dynamical flattening of the whole planet, $\alpha$, in
that same limit. In Sec. 3, we demonstrate the validity of Eq. (2.32) by
comparing it to the result of a direct numerical integration of Eq. (2.15). We
also provide a comparison to the formula obtained using the formalism of Sasao
et al. (1980) to which we now turn.

###  2.3 The inertial torque approximation

Rather than working directly from the vorticity Eq. (2.20), most studies
prefer to compute the motion of the fluid core from the conservation of
angular momentum. In theory, this amounts to solve Eq. (2.2) which requires
the computation of the pressure torque contribution Eq. (2.4) to
$\boldsymbol{\Gamma}$. In practice, however, Sasao et al. (1980) have shown
how Eq. (2.2) can be replaced by the following:

$\partial_{t}\boldsymbol{L}_{\text{c}}-\boldsymbol{w}_{\text{T}}\times\boldsymbol{L}_{\text{c}}\approx\mathcal{O}(\epsilon^{2})~{},$
(2.35)

where $\boldsymbol{w}_{\text{T}}$ denotes the angular velocity of the
_Tisserand frame_ of the fluid core, which is _defined_ via

$\boldsymbol{L}_{\text{c}}=\boldsymbol{I}_{\text{c}}\cdot(\boldsymbol{\Omega}_{\text{m}}+\boldsymbol{w}_{\text{T}})~{}.$
(2.36)

The right-hand side of Eq. (2.35) only involves terms that are second order or
more in the flattening parameters $\alpha_{f}$ and $\beta_{f}$ here assumed to
be proportional to a single small parameter $\epsilon$ (see Dehant and Mathews
(2015), p. 271 and Trinh (2019), p. 79 for details). The usage of Eq. (2.35)
over Eq. (2.2) is usually referred to as the _inertial torque approximation_.
From the cartesian components of the core angular moment, one can verify that,
_for all considerations of angular momentum_ , $\boldsymbol{w}_{\text{T}}$ is
equivalent to $\boldsymbol{w}$ to first order in the flattening so that the
latter can be used in place of the former in Eq. (2.35) (Appendix B). In this
approximation, Eq. (2.26) gets replaced by:

$\left(\begin{array}[]{cccc}\omega&-\frac{i(\alpha-\beta)}{(1-\beta)}&\frac{\text{A}_{f}}{\text{A}}~{}\omega&\frac{i\text{A}_{f}\left(1+\beta_{f}\right)}{\text{A}\left(1-\beta_{f}\right)}\\\
\frac{i(\alpha+\beta)}{(1+\beta)}&\omega&-\frac{i\text{A}_{f}(1-\beta)}{\text{A}(1+\beta)}&\frac{\text{A}_{f}(1-\beta)\left(1+\beta_{f}\right)}{\text{A}(1+\beta)\left(1-\beta_{f}\right)}~{}\omega\\\
\omega&0&\omega&\frac{i\left(1+\alpha_{f}\right)}{(1-\beta_{f})}\\\
0&\omega&-\frac{i\left(1+\alpha_{f}\right)}{\left(1+\beta_{f}\right)}&\omega\\\
\end{array}\right)\left(\begin{array}[]{c}m^{x}\\\ m^{y}\\\ m_{f}^{x}\\\
m_{f}^{y}\end{array}\right)=0~{}.$ (2.37)

The frequencies, $\omega$, derived from Eq. (2.37) are equivalent to those
derived from Eq. (2.26) only to first order in the flattening parameters,
$\alpha$, $\beta$, $\alpha_{f}$ and $\beta_{f}$.

###  2.4 The Spin-Over mode

In Sec. 2.2, we have seen how the FCN and CW wobble are natural oscillations
of the two-layer system when the ‘wobbly’ motion of the mantle is coupled to a
core flow of uniform vorticity. From Eq. (2.26), we can illustrate the
relation between the FCN and the SO by considering the special case where
$\boldsymbol{m}=\boldsymbol{0}$. Physically, this corresponds to the situation
where the mantle is forced to remain in a state of steady rotation at all
time. Mathematically, this amounts to ignore the first two rows and columns of
$M$ so that Eq. (2.26) reduces to:

$\left(\begin{array}[]{ccc}\omega&\frac{i\left(1+\alpha_{f}+2\beta_{f}\right)}{(1+\beta_{f})}\\\
-\frac{i\left(1+\alpha_{f}-2\beta_{f}\right)}{(1-\beta_{f})}&\omega\\\
\end{array}\right)\left(\begin{array}[]{c}m_{f}^{x}\\\
m_{f}^{y}\end{array}\right)=0~{},$ (2.38)

this gives a single independent frequency (using the same convention of sign
as in Sec. 2.2):

$\omega=-\sqrt{\frac{(1+\alpha_{f})^{2}-4\beta_{f}^{2}}{(1-\beta_{f})(1+\beta_{f})}}~{}.$
(2.39)

Upon using Eqs. (C4) and (C5) expressing $\alpha_{f}$ and $\beta_{f}$ in terms
of the dimensions of the ellipsoidal core ($a$, $b$, $c$), one arrives to the
more familiar expression (see _e.g._ Eq. (3.21) of Vantieghem (2014) or Eq.
(A.23) of Rekier et al. (2018)):

$\omega=-\frac{2ab}{\sqrt{(a^{2}+c^{2})(b^{2}+c^{2})}}~{}.$ (2.40)

Note that carrying the same exercise with Eq. (2.37) gives a different, and
incorrect, expression for the SO frequency which matches Eq. (2.39) only up to
first order in the flattening parameters.

We see that, strictly speaking, the SO is a mode of the two-layer system only
in the limit where the rotation of the mantle is steady. When this is not the
case, such as in the context of planetary nutations, the SO leaves its place
to the FCN. The flow inside the core is similar for both modes, having a
uniform vorticity. The main difference is that the FCN is a combined motion of
the core _and_ mantle. When the core is axisymmetric ($\beta_{f}=0$), Eq.
(2.39) reduces to:

$\omega=-(1+\alpha_{f})~{}.$ (2.41)

In Sec. 3, we use Eq. (2.41) to evaluate the discrepancy between the
frequencies of the SO and the FCN numerically as a function of the core
flattening and we compare it to the analytical expression Eq. (2.32).

##  3 Results

In this section, we present the results from the direct numerical integration
of Eqs. (2.15), (2.16) and Eq. (2.1) for an axisymmetric oblate ellipsoidal
fluid core. This computation is based on spectral decomposition introduced in
Rekier et al. (2018) where the reduced pressure field is developed onto the
basis of spheroidal harmonics of the form:

$p(\xi,\vartheta,\varphi)=\sum_{\ell=0}^{L}\sum_{m=-\ell}^{\ell}p_{\ell,m}(\xi)\text{Y}_{\ell}^{m}(\vartheta,\varphi)~{},$
(3.1)

where $\\{\xi,\vartheta,\varphi\\}$ are the set of oblate spheroidal
coordinates and $L$ the degree of truncation. Rekier et al. have shown how to
use these coordinates to compute the inertial modes of a steadily rotating
axisymmetric ellipsoid numerically by using the expansion Eq. (3.1) and a
decomposition of the $p_{\ell,m}$ over the basis of Chebyshev polynomials,
$\text{T}_{k}(x)$:

$p_{\ell,m}(\xi)=\sum_{k=0}^{N}p_{k,\ell,m}\text{T}_{k}(x)~{},$ (3.2)

where $N$ is the degree of truncation. $x=\xi/\xi_{\text{CMB}}$ denotes the
normalised $\xi$ coordinate and $\xi_{\text{CMB}}$ is the value of $\xi$ at
the CMB. With an appropriate choice of normalisation, $\xi_{\text{CMB}}$ is a
constant that depends only on the polar flattening. It can be shown that the
component of the pressure torque Eq. (2.4) in the $z$-direction vanishes in
these coordinates, due to axial symmetry in the azimutal $\varphi$ coordinate.
The other components in the cartesian canonical basis reduce to:

$\displaystyle\Gamma^{+}$
$\displaystyle=~{}~{}\frac{4\pi}{5}\frac{ia(a^{2}-c^{2})}{\sqrt{3}}P_{2,+1}\left(\xi_{\text{CMB}}\right)~{},$
(3.3) $\displaystyle\Gamma^{-}$
$\displaystyle=-\frac{4\pi}{5}\frac{ia(a^{2}-c^{2})}{\sqrt{3}}P_{2,-1}\left(\xi_{\text{CMB}}\right)~{},$
(3.4)

where $a$ and $c$ are the semi-major and semi-minor axes of the core
respectively and the $P_{2,\pm 1}\left(\xi_{\text{CMB}}\right)$ are the
spheroidal harmonics components of the physical pressure field with $\ell=2$
and $m=\pm 1$ at the CMB. This should not be confused with the components of
the reduced pressure, $p\left(\xi_{\text{CMB}}\right)$, to which it is related
via Eq. (2.18) which gives:

$P_{2,\pm 1}\left(\xi_{\text{CMB}}\right)=\rho\left(p_{2,\pm
1}\left(\xi_{\text{CMB}}\right)-\frac{ac}{\sqrt{3}}~{}m^{\pm}\right)~{}.$
(3.5)

From Eqs. (3.3) and (3.4), we see that the geometry of the CMB restricts the
family of flows coupled to the planetary rotation to those with a non-
vanishing $\ell=2$ and $m=\pm 1$ pressure component at the CMB. In particular,
this limits the analysis to the set of modes that are antisymmetric with
respect to the equatorial plane.

Eqs. (2.1), (2.15) and (2.16) form a closed system of equations in $p$ and
$\boldsymbol{m}$ that we solve numerically as a (polynomial) eigenvalue
problem in the frequency $\omega$. For definiteness, we arbitrarily assume
that the two layers of the planet have the same density and that the semi-
major axes of the fluid core and of the whole planet are in the $\frac{1}{2}$
ratio. We also set the exterior flattening to zero, $\alpha=0$, _i.e._ the
outside surface of the planet is spherical. This leaves $\alpha_{f}$ as the
only free parameter. In Appendix C, we provide the detailed expressions of the
different moments of inertia involved in this computation.

Fig. 2 represents the partial spectrum of eigenvalues of modes with the
azimuthal wave number $m=1$ as a function of $\alpha_{f}$.

Figure 2: Partial spectrum of eigenfrequencies of the two-layer planet as a
function of the polar flattening of the fluid core. The blue circles are the
frequencies of the inertial modes computed numerically for a planet in steady
rotation ($\boldsymbol{m}=\boldsymbol{0}$). The red dots are the same thing
for a planet with a non-steadily rotating (wobbly) mantle
($\boldsymbol{m}\neq\boldsymbol{0}$). The eigenvalues corresponding to the
frequencies of inertial modes are the same in both cases except for the SO
which gets replaced by the FCN. The blue dashed lines represent the analytical
frequencies of the inertial modes of a steadily rotating planet (see Appendix
A). These are labelled with the $\ell$-number of their highest degree
component. The red curve represents the frequency of the FCN computed from Eq.
(3.6). We have subtracted the value of the SO frequency Eq. (2.41) from all
eigenvalues to accentuate the discrepancy with the FCN. All the frequencies in
the figure correspond to modes with the azimuthal wave number $m=1$.

The blue circles are the inertial modes frequencies when the planet is in
steady rotation ($\boldsymbol{m}=\boldsymbol{0}$). The red dots are the same
thing for a planet with a non-steadily rotating (wobbly) mantle
($\boldsymbol{m}\neq\boldsymbol{0}$). The blue dashed lines represent the
analytical frequencies of the inertial modes of a steadily rotating planet
(see Appendix A). These are labelled with the $\ell$-number of their highest
degree component. The frequencies are the same in both cases, the only
exception being the SO which disappears from the spectrum when the planet has
a non-steady rotation and is replaced by the FCN. In order to accentuate the
discrepancy in the frequencies of these two modes, we have subtracted the SO
frequency Eq. (2.41) from all the frequencies in the plot. The red curve
represents the frequency of the FCN computed from Eq. (2.32) which, for
$\alpha=0$, simplifies to:

$\omega=-1-\frac{\text{A}\alpha_{f}}{\text{A}_{m}+(\text{A}-\text{A}_{m})\alpha_{f}^{2}}~{}.$
(3.6)

It is in very good agreement with the values computed numerically even for
large values of $\alpha_{f}$. We observe that, the non-steady rotation of the
mantle has no effect on the frequencies of the inertial modes other than the
SO. This can be confirmed by looking at the ratio of the kinetic energy
densities of the mantle and the core measured from the steadily rotating
reference frame (see Appendix D) as shown on Fig. 3.

Figure 3: Ratio of the kinetic energy densities of the mantle and the fluid
core as measured from the steadily rotating frame. The red line corresponds to
the analytical value computed for the solution of Eq. (2.31) with $\alpha=0$
and with a frequency given by Eq. (3.6).

As we can see, the ratio is strictly zero for all modes other than the FCN.
The red line corresponds to the analytical value computed from the resolution
of Eq. (2.31) with $\alpha=0$ and with a frequency equal to Eq. (3.6). It is
in good agreement with the numerical computation even for large values of the
flattening.

We can further illustrate the agreement between the numerical and analytical
computations by looking at Fig. 4 which shows the same thing as Fig. 2 but
focuses on the frequency of the FCN up to very large values of the core
flattening, $\alpha_{f}\approx 1$.

Figure 4: Frequency of the FCN as a function of the core flattening. The red
dots are the results from direct numerical integration and the red curve
corresponds to the analytical formula of Eq. (3.6). The numerical and
analytical results agree well for all values of $\alpha_{f}$. The blue dashed
line represents the value of the frequency computed using the inertial torque
approximation Eq. (3.7) which predicts the wrong value of the FCN frequency
for large values of the flattening.

The red curve represents the frequency obtained from Eq. (3.6) and the dots
shows the numerical computation. The blue dashed line represents the value of
the frequency computed using the inertial torque approximation described in
Sec. 2.3 which, for $\alpha=0$, predicts:

$\omega=-1-\frac{\text{A}}{\text{A}_{m}}\alpha_{f}~{}.$ (3.7)

We see that, in the example of the figure, the numerical computation starts to
deviate markedly from Eq. (3.7) for $\alpha_{f}\gtrsim 0.1$. The precise value
of this threshold depends on the values of the moments of inertia, A and
$\text{A}_{m}$. We also see that Eq. (3.6) agrees well with the numerical
computation for all values of $\alpha_{f}$ thus proving the validity of the
computation of Sec. 2.2 based on the vorticity equation.

##  4 Discussion

We have presented a method to compute the free rotation of a simplified two-
layer planet with an inviscid and incompressible fluid core and a perfectly
rigid ellipsoidal mantle. This method is based on the resolution of the
modified Poincaré Eq. (2.15) coupled to the Liouville Eq. (2.1) which
describes the ‘wobbling’ motion of the mantle subjected to the pressure torque
from the fluid at the CMB. We have found that the inertial modes of the fluid
core enclosed in a steadily rotating mantle are also modes of the freely
rotating two-layer planet. The only exception being the Spin-Over mode – the
simplest of the inertial modes – which disappears from the spectrum and is
replaced by the Free Core Nutation (Fig. 2). The fact that this is the only
mode that can influence the rotation of the planet (or be influenced by it)
can be traced back to the fact that it is the only inertial mode that exerts a
net pressure torque at the CMB, something that was already noted by Toomre
(1974). This can be verified by looking at Eqs. (3.3), (3.4) and (3.5) after
realising that the reduced pressure components $p_{2,1}$ and $p_{2,-1}$ are
zero at the CMB for all modes except the Spin-Over. This result demonstrated
here for a freely rotating spheroid remains valid for a triaxial ellipsoid or
when there are external tidal forces acting on the planet. In which case, the
torque from the external potential must be added to the total torque Eq. (2.4)
but it must also be included in the definition of the reduced pressure Eq.
(2.7) so that nothing is changed.

Looking at Fig. 2 showing the evolution of the spectrum of eigenvalues as a
function of the core flattening, we observe that the frequency of the FCN
crosses path with that of the other inertial modes when $\alpha_{f}$ varies.
In their study of the completeness of the set of inertial modes in a steadily-
rotating ellipsoid, Backus and Rieutord (2017) showed that the modes remain
orthogonal even when such accidental degeneracies happen. In the same paper,
the authors also demonstrated that the Poincaré Eq. (2.15) and the boundary
condition Eq. (2.16) form a _self-adjoint_ system222For an introductory
discussion on the ‘self-adjointness’ of differential operators and its
implication, see _e.g_ Sec. 10.1 and 10.2 of Arfken and Weber (2005) when
$\boldsymbol{m}=\boldsymbol{0}$, thus providing the mathematical explanation
for the realness of the eigenfrequencies. Our work extends their conclusions
to the case where the motion of the mantle is non-steady. Triana et al. (2019)
have shown that the pictures drawn out from Fig. 2 changes significantly when
the viscosity of the fluid is taken into account. The eigenvalues are then
complex numbers and their imaginary parts represent the damping of the
corresponding eigenmodes. Instead of crossing, the eigenvalues _avoid_ each
other in the complex plane in a complicated manner that is not yet fully
elucidated (see their Fig. 3 & 4). A similar example of this phenomenon
referred to as _avoided crossing_ was previously given by Rogister and Valette
(2009) who studied the influence of a thermally stratified liquid core on the
rotational modes of the Earth. In both cases, avoided crossings result from
the introduction of a non-adiabatic diffusion term in the dynamical equations
(respectively viscous or thermal) which breaks their ‘self-adjointness’. Birch
(2002) gave a similar interpretation in their study of the magneto-
hydrodynamical equations for a one dimensional stratified medium where the
avoided crossings result from the introduction of radiative cooling into the
model. It should be noted that avoided crossings can also take place in
systems that are self-adjoint. This is the case, for example, for the
eigenfrequencies of the spheroidal oscillations of stars both in the adiabatic
and non-adiabatic case. See _e.g._ Chap. 5 of Christensen-Dalsgaard (2014) or
Sec. 11.3 of Smeyers and Van Hoolst (2010) for a review. See also
Triantafyllou and Triantafyllou (1991) for a more general description of the
phenomenon and applications to problems in engineering.

We have given the exact analytical expressions of FCN and the CW frequencies
of a triaxial two-layer planet. To second order in the flattening, these are
given as Eqs. (2.28) and (2.29) which derive from the joint resolution of the
Liouville equation for the mantle Eq. (2.1) and the equation of vorticity for
the fluid core Eq. (2.20) based on the assumption that flow vorticity is
uniform throughout the core. The task then reduces to solving Eq. (2.26). We
have shown that the resulting formula for the FCN is valid to all orders in
the flattening parameters, by comparing it to the numerical solution for a
spheroidal core. This is illustrated on Fig. 4 where we have also plotted the
same result based on the formalism of Sasao et al. (1980) summarised in Sec.
2.3. In the same section, we have shown how that formalism, based on Eq.
(2.35), is only valid to first order in the flattening parameters.

Based on Eq. (2.26), valid for core flows of uniform vorticity, we have shown
how the FCN reduces to the SO in the limit where the rotation of the mantle is
steady. We can understand this by considering the fact that the pressure
torque Eq. (2.4) vanishes identically when the CMB is spherical. This limit is
equivalent to setting $a=c$ in Eqs. (3.3) and (3.4) for an axi-symmetric core.
The motion of the mantle then becomes independent to that of the fluid core
and _vice versa_ and the motion of the planet is derived from Eq. (2.26) after
setting $\boldsymbol{m}_{f}=\boldsymbol{0}$. The solution is that of fully
rigid solid-body rotation with frequency:

$\omega=\sqrt{\frac{\alpha^{2}-\beta^{2}}{1-\beta^{2}}}=\sqrt{\frac{(\text{C}-\text{A})(\text{C}-\text{B})}{\text{A}\text{B}}}~{}.$
(4.1)

We recognise the expression on the right-hand side as the frequency of the
free _Euler Wobble_ (EW). If we imagine increasing the flattening of the CMB
from zero, the core and mantle become coupled and the EW and SO disappear from
the solutions to be replaced by the CW and the FCN. Both of which are coupled
motions of the core and mantle, even though the CW is mostly a motion of the
mantle and the FCN is, as its name indicates, mostly a motion of the fluid
core.

In Sec. 2.4, we have shown that, while the SO and FCN share a similar type of
motion in the fluid core (one of uniform vorticity), it is improper to speak
of the SO of a freely rotating planet. Some publications prefer to use the
term _Tilt-Over Mode_ (TOM) (Toomre, 1974; Noir et al., 2003; Cébron et al.,
2010), but this has the disadvantage to be ambiguous with yet another mode of
the same name which results from the mismatch between the polar axis of a
planet and its instantaneous rotation axis (Dehant and Capitaine, 1996). This
mode has a purely diurnal frequency in the mantle frame, regardless of the
planet’s internal structure. We give a short mathematical description of the
TOM in Appendix D in which we derive the expression for the mantle’s rotation
vector with respect to the steadily-rotating frame.

At present, the main interest of the results presented here is purely
conceptual as our model cannot, at the moment, account for the elasticity of
the mantle which induces corrections to the FCN and CW frequencies that are
first order in the core ellipticity and therefore dominates over the second
order corrections computed here. As it stands now, our model can nevertheless
provide a useful baseline for comparison with studies performed in triaxial
geometries.

Other possible extensions to the present work include taking into
consideration the effects of density stratification inside the core. Toomre
(1974) argued that a radial density profile could lead to couple the rotation
to inertial modes other than the SO. For the Earth, the inclusion of a
magnetic field can also potentially alter the spectrum of inertial modes. This
is especially important for the Earth, as the ohmic power dissipation is
estimated to be one of the main sources of damping of the FCN (Koot et al.,
2010). Finally, the presence of a solid inner core should be taken into
account in order to model the _Free Inner Core Nutation_ of the Earth. This
proves to be a significant numerical challenge as this no longer permits the
usage a single set of oblate spheroidal coordinates covering the whole volume
of the core. Concerning the planet Mars, it is considered (but not yet
demonstrated) that there is no inner core and that the main dissipation
mechanism at the core-mantle boundary will be induced by the viscosity, also
thought to be a very important factor for the Earth [Triana et al. 2020, in
prep]. Furthermore, even in the simplified spherical shell geometry, the
elliptical nature of the Poincaré equation is known to cause singularities in
the fluid velocity (and pressure) field (Rieutord et al., 2000). These
singularities can be regularised by reintroducing viscosity into the picture,
with the well-known numerical limitations that it entails.

## acknowledgement

The authors would like to thank T. Van Hoolst for the useful discussions in
the writing of this work and the two reviewers, M. Efroimsky and M. Dumberry
whose comments helped to improve its presentation significantly. The research
leading to the results presented here has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Advanced Grant agreement No. 670874).

## References

  * Aldridge and Toomre [1969] K. D. Aldridge and A. Toomre.  Axisymmetric inertial oscillations of a fluid in a rotating spherical container.  _Journal of Fluid Mechanics_ , 37(2):307–323, 1969.  10.1017/S0022112069000565.
  * Arfken and Weber [2005] G. Arfken and H. Weber.  _Mathematical Methods for Physicists_.  Elsevier, 2005.  ISBN 9780120885848.
  * Backus and Rieutord [2017] G. Backus and M. Rieutord.  Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid.  _Physical Review E_ , 95(5):1–16, 2017.  ISSN 24700053.  10.1103/PhysRevE.95.053116.
  * Birch [2002] A. C. Birch.  _Wave propagation in the sun and the interpretation of helioseismic data_.  PhD thesis, Stanford University, 2002.
  * Bryan [1889] G. H. Bryan.  The waves on a rotating liquid spheroid of finite ellipticity.  _Philosophical Transactions of the Royal Society of London. A_ , 180:187–219, 1889.
  * Cébron et al. [2010] D. Cébron, M. Le Bars, and P. Meunier.  Tilt-over mode in a precessing triaxial ellipsoid.  _Physics of Fluids_ , 22(11):1–12, 2010.  10.1063/1.3504356.
  * Chen and Shen [2010] W. Chen and W. Shen.  New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth.  _Journal of Geophysical Research_ , 115(B12):B12419, dec 2010.  ISSN 0148-0227.  10.1029/2009JB007094.
  * Christensen-Dalsgaard [2014] J. Christensen-Dalsgaard.  Lecture notes on stellar oscillations.  May 2014.  URL https://users-phys.au.dk/~jcd/oscilnotes/.
  * Dehant and Capitaine [1996] V. Dehant and N. Capitaine.  On the precession constant: Values and constraints on the dynamical ellipticity; Link with oppolzer terms and tilt-over-mode.  _Celestial Mechanics and Dynamical Astronomy_ , 65(4):439–458, 1996.  ISSN 09232958.  10.1007/bf00049506.
  * Dehant and Mathews [2015] V. Dehant and P. M. Mathews.  _Rigid Earth precession and nutation_ , pages 121–209.  Cambridge University Press, 2015.  10.1017/CBO9781316136133.006.
  * Dehant et al. [2020] V. Dehant, S. Le Maistre, R. M. Baland, N. Bergeot, Ö. Karatekin, M. J. Péters, A. Rivoldini, L. Ruiz Lozano, O. Temel, T. Van Hoolst, M. Yseboodt, M. Mitrovic, A. S. Kosov, V. Valenta, L. Thomassen, S. Karki, K. Al Khalifeh, C. Craeye, L. I. Gurvits, J. C. Marty, S. W. Asmar, and W. M. Folkner.  The radioscience LaRa instrument onboard ExoMars 2020 to investigate the rotation and interior of mars.  _Planetary and Space Science_ , 180(October 2019), 2020\.  ISSN 00320633.  10.1016/j.pss.2019.104776.
  * Folkner et al. [2018] W. M. Folkner, V. Dehant, S. Le Maistre, M. Yseboodt, A. Rivoldini, T. Van Hoolst, S. W. Asmar, and M. P. Golombek.  The Rotation and Interior Structure Experiment on the InSight Mission to Mars.  _Space Science Reviews_ , 214(5):1–16, 2018.  ISSN 15729672.  10.1007/s11214-018-0530-5.
  * Guo and Shen [2019] Z. Guo and W. Shen.  Formulation of a Triaxial Three-Layered Earth Rotation: I. Theory and Rotational Normal Mode Solutions.  pages 1–35, 2019.  URL http://arxiv.org/abs/1901.10066.
  * Hough [1895] S. S. Hough.  The oscillations of a rotating ellipsoidal shell containing fluid.  _Philosophical Transactions of the Royal Society of London. A_ , 186:469–506, 1895.
  * Kelley et al. [2007] D. H. Kelley, S. A. Triana, D. S. Zimmerman, A. Tilgner, and D. P. Lathrop.  Inertial waves driven by differential rotation in a planetary geometry.  _Geophysical and Astrophysical Fluid Dynamics_ , 101(5-6):469–487, 2007.
  * Koot et al. [2010] L. Koot, M. Dumberry, A. Rivoldini, O. De Viron, and V. Dehant.  Constraints on the coupling at the core-mantle and inner core boundaries inferred from nutation observations.  _Geophysical Journal International_ , 182(3):1279–1294, 2010.  ISSN 0956540X.  10.1111/j.1365-246X.2010.04711.x.  URL http://gji.oxfordjournals.org/cgi/doi/10.1111/j.1365-246X.2010.04711.x.
  * Kudlick [1966] M. D. Kudlick.  _On transient motions in a contained, rotating fluid._ PhD thesis, Massachusetts Institute of Technology, 1966.
  * Lai et al. [2010] W. M. Lai, D. Rubin, and E. Krempl.  _Chapter 2 - Tensors_ , pages 3 – 68.  Butterworth-Heinemann, Boston, fourth edition edition, 2010.  ISBN 978-0-7506-8560-3.  https://doi.org/10.1016/B978-0-7506-8560-3.00002-5.
  * Le Bars et al. [2015] M. Le Bars, D. Cébron, and P. Le Gal.  Flows Driven by Libration, Precession, and Tides.  _Annual Review of Fluid Mechanics_ , 47(1):163–193, 2015.  ISSN 0066-4189.  10.1146/annurev-fluid-010814-014556.
  * Malkus [1968] W. Malkus.  Precession of the earth as the cause of geomagnetism: Experiments lend support to the proposal that precessional torques drive the earth’s dynamo.  _Science_ , 160(3825):259–264, 1968.
  * Morize et al. [2010] C. Morize, M. Le Bars, P. Le Gal, and A. Tilgner.  Experimental determination of zonal winds driven by tides.  _Physical review letters_ , 104(21):214501, 2010\.
  * Noir et al. [2003] J. Noir, P. Cardin, D. Jault, and J. P. Masson.  Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid.  _Geophysical Journal International_ , 154(2):407–416, 2003.  ISSN 0956540X.  10.1046/j.1365-246X.2003.01934.x.
  * Poincaré [1885] H. Poincaré.  Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation.  _Acta mathematica_ , 7(1):259–380, 1885.
  * Rekier et al. [2018] J. Rekier, A. Trinh, S. A. Triana, and V. Dehant.  Inertial modes in Near-spherical geometries.  _Geophysical Journal International_ , pages 777–793, 2018.
  * Rieutord [2015] M. Rieutord.  _Rotating Fluids_ , pages 291–321.  Springer International Publishing, Cham, 2015.  ISBN 978-3-319-09351-2.  10.1007/978-3-319-09351-2-8.  URL https://doi.org/10.1007/978-3-319-09351-2-8.
  * Rieutord et al. [2000] M. Rieutord, B. Georgeot, and L. Valdettaro.  Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum.  _Journal of Fluid Mechanics_ , 435:42, 2000.  ISSN 00221120.  10.1017/S0022112097005491.  URL http://dx.doi.org/10.1017/S0022112001003718{%}5Cnhttp://arxiv.org/abs/physics/0007007.
  * Rochester et al. [2014] M. G. Rochester, D. J. Crossley, and Y. L. Zhang.  A new description of Earth’s wobble modes using Clairaut coordinates: 1. Theory.  _Geophysical Journal International_ , 198(3):1848–1877, 2014.  ISSN 1365246X.  10.1093/gji/ggu226.
  * Rogister [2001] Y. Rogister.  On the diurnal and nearly diurnal free modes of the Earth.  _Geophysical Journal International_ , 144(2):459–470, 2001.  ISSN 0956540X.  10.1046/j.1365-246X.2001.00359.x.
  * Rogister and Valette [2009] Y. Rogister and B. Valette.  Influence of liquid core dynamics on rotational modes.  _Geophysical Journal International_ , 176(2):368–388, 2009.  ISSN 0956540X.  10.1111/j.1365-246X.2008.03996.x.
  * Sasao et al. [1980] T. Sasao, S. Okubo, and M. Saito.  A simple theory on dynamical effects of stratified fluid core upon nutational motion of the Earth.  1980\.
  * Seyed-Mahmoud et al. [2017] B. Seyed-Mahmoud, M. G. Rochester, and C. M. Rogers.  Truncation effects in computing free wobble/nutation modes explored using a simple Earth model.  _Geophysical Journal International_ , 209(3):1455–1461, 2017.  ISSN 0956-540X.  10.1093/gji/ggx101.
  * Shen et al. [2019] W. Shen, Z. Yang, Z. Guo, and W. Zhang.  Numerical solutions of rotational normal modes of a triaxial two-layered anelastic Earth.  _Geodesy and Geodynamics_ , 10(2):118–129, 2019\.  ISSN 16749847.  10.1016/j.geog.2019.03.001.  URL https://doi.org/10.1016/j.geog.2019.03.001.
  * Smeyers and Van Hoolst [2010] P. Smeyers and T. Van Hoolst.  _Classification of the Spheroidal Normal Modes (continued)_ , pages 205–227.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.  ISBN 978-3-642-13030-4.
  * Smith [1974] M. L. Smith.  The Scalar Equations of Infinitesimal Elastic‐Gravitational Motion for a Rotating, Slightly Elliptical Earth.  _Geophysical Journal of the Royal Astronomical Society_ , 37(3):491–526, 1974.  ISSN 1365246X.  10.1111/j.1365-246X.1974.tb04099.x.
  * Smith [1977] M. L. Smith.  Wobble and nutation of the Earth.  _Geophysical Journal of the Royal Astronomical Society_ , 50(1):103–140, 1977.  ISSN 1365246X.  10.1111/j.1365-246X.1977.tb01326.x.
  * Toomre [1974] A. Toomre.  On the ‘Nearly Diurnal Wobble’ of the Earth.  _Geophysical Journal of the Royal Astronomical Society_ , 38(2):335–348, 1974.  ISSN 1365246X.  10.1111/j.1365-246X.1974.tb04126.x.
  * Triana et al. [2019] S. A. Triana, J. Rekier, A. Trinh, and V. Dehant.  The coupling between inertial and rotational eigenmodes in planets with liquid cores.  _Geophysical Journal International_ , 218(2):1071–1086, 2019.  10.1093/gji/ggz212.
  * Triantafyllou and Triantafyllou [1991] M. S. Triantafyllou and G. S. Triantafyllou.  Frequency coalescence and mode localization phenomena: A geometric theory.  _Journal of Sound and Vibration_ , 150(3):485–500, 1991.  ISSN 10958568.  10.1016/0022-460X(91)90899-U.
  * Trinh [2019] A. Trinh.  _The Interior Structure of Enceladus : A combined view from Gravity, Topography and Libration_.  PhD thesis, UCLouvain, Presse Universitaire de Louvain, 2019.
  * Van Hoolst and Dehant [2002] T. Van Hoolst and V. Dehant.  Influence of triaxiality and second-order terms in flattenings on the rotation of terrestrial planets.  _Physics of the Earth and Planetary Interiors_ , 134(1-2):17–33, nov 2002.  ISSN 00319201.  10.1016/S0031-9201(02)00068-7.  URL https://linkinghub.elsevier.com/retrieve/pii/S0031920102000687.
  * Vantieghem [2014] S. Vantieghem.  Inertial modes in a rotating triaxial ellipsoid.  In _Proc. R. Soc. A_ , volume 470, page 20140093. The Royal Society, 2014.
  * Zhang et al. [2001] K. Zhang, P. Earnshaw, X. Liao, and F. H. Busse.  On inertial waves in a rotating fluid sphere.  _Journal of Fluid Mechanics_ , 437:103–119, 2001.  ISSN 0022-1120.  10.1017/S0022112001004049.  URL http://www.journals.cambridge.org/abstract{_}S0022112001004049.

##  Appendix A Derivation of the Poincaré equation and solutions for the fluid
spheroid

From the momentum Eq. (2.13):

$\sigma\boldsymbol{u}+2\hat{\boldsymbol{z}}\times\boldsymbol{u}+\underbrace{(\boldsymbol{\nabla}p+\sigma\boldsymbol{m}\times\boldsymbol{r})}_{\textbf{w}}=\boldsymbol{0}~{},$
(A1)

where we have set $\sigma\equiv i\omega$ for simplicity. Taking the cross
product of Eq. (A1) with $\hat{\boldsymbol{z}}$ and injecting the result back
into Eq. (A1), we find:

$\displaystyle\boldsymbol{u}-\frac{4}{\sigma^{2}}\left(\hat{\boldsymbol{z}}\times(\hat{\boldsymbol{z}}\times\boldsymbol{u})\right)-2~{}\frac{\hat{\boldsymbol{z}}\times\textbf{w}}{\sigma^{2}}+\frac{\textbf{w}}{\sigma}$
$\displaystyle=\boldsymbol{0}~{},$ (A2)
$\displaystyle\leftrightarrow~{}\boldsymbol{u}-\frac{4}{\sigma^{2}}\left(\hat{\boldsymbol{z}}(\hat{\boldsymbol{z}}\cdot\boldsymbol{u})-\boldsymbol{u}\right)-2~{}\frac{\hat{\boldsymbol{z}}\times\textbf{w}}{\sigma^{2}}+\frac{\textbf{w}}{\sigma}$
$\displaystyle=\boldsymbol{0}~{}.$ (A3)

From Eq. (A1), one also has:

$\hat{\boldsymbol{z}}\cdot\boldsymbol{u}=-\frac{(\hat{\boldsymbol{z}}\cdot\textbf{w})}{\sigma}~{},$
(A4)

which can then be used to rewrite the second term of Eq. (A3). This allows one
to isolate $\boldsymbol{u}$:

$\boldsymbol{u}=\frac{\sigma}{\sigma^{2}+4}\left(-\textbf{w}+\frac{2}{\sigma}(\hat{\boldsymbol{z}}\times\textbf{w})-\frac{4}{\sigma^{2}}\hat{\boldsymbol{z}}(\hat{\boldsymbol{z}}\cdot\textbf{w})\right)~{}.$
(A5)

Eqs. (2.15) and (2.16) follow immediately from the condition of
incompressibility and the no-penetration condition respectively.

In Sec. 3, we compare the inertial modes of the freely rotating two-layer
planet with an axi-symmetric fluid core to those of a fluid core in steady
rotation around the polar axis, $\boldsymbol{\hat{z}}$. Those are the results
of Eqs (2.15) and (2.16) with $\boldsymbol{m}=\boldsymbol{0}$. In such case,
Bryan [1889] showed that the inertial modes can be written as a sum of product
of two _associated Legendre polynomials_ (see Eq. (A.13) of [Rekier et al.,
2018]). The frequencies of the inertial modes, $\omega$, then derive from the
solutions of :

${\text{P}_{\ell}^{m}}^{\prime}(x)-\frac{2mx}{\omega(1-x^{2})}{\text{P}_{\ell}^{m}}(x)=0~{},$
(A6)

where
$\text{P}_{\ell}^{m}(x)=\frac{(-1)^{m}}{2^{\ell}\ell!}(1-x^{2})^{\frac{m}{2}}\frac{d^{\ell+m}}{dx^{\ell+m}}(x^{2}-1)^{\ell}~{},$
denotes the associated Legendre polynomial of degree $\ell$ and order $m$ and
where we wrote :

$x=\omega\sqrt{\frac{1-e^{2}}{4-e^{2}\omega^{2}}}~{},$ (A7)

with $e\equiv\sqrt{1-\frac{c^{2}}{a^{2}}}$ denoting the (geometrical)
eccentricity of the fluid core.

##  Appendix B The Poincaré flow and its associated angular momentum

Eq. (2.19), gives the general expression of a flow with a uniform vorticity.
The scalar function, $\psi$, must be chosen so as to accommodate the no-
penetration boundary condition at the CMB, Eq. (2.8). Following Poincaré
[1885], we do so by operating the following coordinates transform :

$\\{x,y,z\\}\rightarrow\\{ax,by,cz\\}~{},$ (B1)

where $a$, $b$ and $c$ denote the dimensions of the core (Fig. 1). If one
defines the rescaled position vector in terms of its cartesian coordinates as
$\boldsymbol{r}^{\prime}=(\frac{x}{a},\frac{y}{b},\frac{z}{c})^{\text{T}}$,
the implicit equation of the ellipsoid surface reduces to
$|\boldsymbol{r}^{\prime}|^{2}=1$. The most general flow of uniform vorticity
satisfying Eq. (2.8) is then :

$\boldsymbol{u}=\boldsymbol{w}^{\prime}\times\boldsymbol{r}^{\prime}~{},$ (B2)

where $\boldsymbol{w}^{\prime}$ is a constant vector in space related to the
flow vorticity, $\boldsymbol{w}$, through the condition that
$(\boldsymbol{\nabla}\times\boldsymbol{u})=2\boldsymbol{w}$. In components,
this yields

$\displaystyle u^{x}$ $\displaystyle=w^{\prime y}\frac{a}{c}z-w^{\prime
z}\frac{a}{b}y=2a^{2}(\frac{w^{y}z}{a^{2}+c^{2}}-\frac{w^{z}y}{a^{2}+b^{2}})~{},$
(B3) $\displaystyle u^{y}$ $\displaystyle=w^{\prime z}\frac{b}{a}x-w^{\prime
x}\frac{b}{c}z=2b^{2}(\frac{w^{z}x}{a^{2}+b^{2}}-\frac{w^{x}z}{b^{2}+c^{2}})~{},$
(B4) $\displaystyle u^{z}$ $\displaystyle=w^{\prime x}\frac{c}{b}y-w^{\prime
y}\frac{c}{a}x=2c^{2}(\frac{w^{x}y}{b^{2}+c^{2}}-\frac{w^{y}x}{a^{2}+c^{2}})~{},$
(B5)

from which we obtain the expression of $\psi$ by comparison with Eq. (2.19) :

$\psi=\omega^{x}~{}\frac{c^{2}-b^{2}}{c^{2}+b^{2}}~{}yz+\omega^{y}~{}\frac{a^{2}-c^{2}}{a^{2}+c^{2}}~{}zx+\omega^{z}~{}\frac{b^{2}-a^{2}}{b^{2}+a^{2}}~{}xy~{}.$
(B6)

Using Eqs. (B3) to (B6), we can finally compute the angular momentum of the
core from its definition as measured from the inertial frame :

$\displaystyle\boldsymbol{L}_{\text{core}}$
$\displaystyle=\boldsymbol{I}_{\text{c}}\cdot\boldsymbol{\Omega}_{\text{m}}+\int_{\text{core}}\rho(\boldsymbol{r}\times\boldsymbol{u})$
(B7)
$\displaystyle=\boldsymbol{I}_{\text{c}}\cdot(\boldsymbol{\Omega}_{\text{m}}+\boldsymbol{w})+\int_{\text{core}}\rho(\boldsymbol{r}\times\boldsymbol{\nabla}\psi)~{},$
(B8)

where $\boldsymbol{I}_{\text{c}}$ denotes the tensor of inertia of the fluid
core and the integrals run over the whole volume of the core. In the cartesian
components, this yields :

$\displaystyle L_{\text{core}}^{x}$
$\displaystyle=\text{A}_{f}~{}\Omega_{\text{m}}^{x}+\frac{(\text{A}_{f}+\text{B}_{f}-\text{C}_{f})(\text{C}_{f}+\text{A}_{f}-\text{B}_{f})}{\text{A}_{f}}w^{x}~{},$
(B9) $\displaystyle L_{\text{core}}^{y}$
$\displaystyle=\text{B}_{f}~{}\Omega_{\text{m}}^{y}+\frac{(\text{B}_{f}+\text{C}_{f}-\text{A}_{f})(\text{A}_{f}+\text{B}_{f}-\text{C}_{f})}{\text{B}_{f}}w^{y}~{},$
(B10) $\displaystyle L_{\text{core}}^{z}$
$\displaystyle=\text{C}_{f}~{}\Omega_{\text{m}}^{z}+\frac{(\text{C}_{f}+\text{A}_{f}-\text{B}_{f})(\text{B}_{f}+\text{C}_{f}-\text{A}_{f})}{\text{C}_{f}}w^{z}~{}.$
(B11)

Using the definitions of the dynamical flattening parameters Eqs. (2.23),
these read :

$\displaystyle L_{\text{core}}^{x}$
$\displaystyle=\text{A}_{f}~{}\Omega_{\text{m}}^{x}+\text{A}_{f}\frac{\left(1-\alpha_{f}\right)\left(1+\alpha_{f}-2\beta_{f}\right)}{\left(1-\beta_{f}\right){}^{2}}w^{x}\approx\text{A}_{f}~{}(\Omega_{\text{m}}^{x}+w^{x})+\mathcal{O}(\epsilon^{2})~{},$
(B12) $\displaystyle L_{\text{core}}^{y}$
$\displaystyle=\text{B}_{f}~{}\Omega_{\text{m}}^{y}+\text{B}_{f}\frac{\left(1-\alpha_{f}\right)\left(1+\alpha_{f}+2\beta_{f}\right)}{(1+\beta_{f})^{2}}w^{y}\approx\text{B}_{f}~{}(\Omega_{\text{m}}^{y}+w^{y})+\mathcal{O}(\epsilon^{2})~{},$
(B13) $\displaystyle L_{\text{core}}^{z}$
$\displaystyle=\text{C}_{f}~{}\Omega_{\text{m}}^{z}+\text{C}_{f}\frac{\left(1+\alpha_{f}-2\beta_{f}\right)\left(1+\alpha_{f}+2\beta_{f}\right)}{\left(1+\alpha_{f}\right)^{2}}w^{z}\approx\text{C}_{f}~{}(\Omega_{\text{m}}^{z}+w^{z})+\mathcal{O}(\epsilon^{2})~{}.$
(B14)

where, in the rightmost equalities, we have assumed that $\alpha_{f}$ and
$\beta_{f}$ are small quantities proportional to a single parameter
$\epsilon$.

##  Appendix C Tensor of inertia

The principle moments of inertia of the fluid ellipsoid of equation
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ and with a
homogeneous density $\rho_{f}$ are :

$\displaystyle\text{A}_{f}$
$\displaystyle=\frac{4\pi}{3}\rho_{f}~{}abc~{}\frac{(b^{2}+c^{2})}{5}~{},$
(C1) $\displaystyle\text{B}_{f}$
$\displaystyle=\frac{4\pi}{3}\rho_{f}~{}abc~{}\frac{(a^{2}+c^{2})}{5}~{},$
(C2) $\displaystyle\text{C}_{f}$
$\displaystyle=\frac{4\pi}{3}\rho_{f}~{}abc~{}\frac{(a^{2}+b^{2})}{5}~{}.$
(C3)

After introduction into the definitions of the dynamical flattening parameters
Eqs. (2.23), we find :

$\displaystyle\alpha_{f}$
$\displaystyle=\frac{a^{2}+b^{2}-2c^{2}}{a^{2}+b^{2}+2c^{2}}~{},$ (C4)
$\displaystyle\beta_{f}$
$\displaystyle=\frac{a^{2}-b^{2}}{a^{2}+b^{2}+2c^{2}}~{}.$ (C5)

Substituting Eqs. (C4) and (C5) into Eqs. (C1) to (C3), yields :

$\displaystyle\text{A}_{f}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{f}\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1-\beta_{f})~{},$
(C6) $\displaystyle\text{B}_{f}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{f}\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1+\beta_{f})~{},$
(C7) $\displaystyle\text{C}_{f}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{f}\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1+\alpha_{f})~{}.$
(C8)

We can obtain the expressions of the moments of inertia of the mantle by
working directly from the above expressions. If we define the ratio between
the semi-major axes of the core and of the whole planet as $\eta$, the
principle moments of inertia of the mantle read :

$\displaystyle\text{A}_{m}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{m}\left(\frac{\sqrt{(1-\alpha)(\alpha-2\beta+1)}}{(\alpha+2\beta+1)^{2}}\frac{(1-\beta)}{\eta^{5}}-\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1-\beta_{f})\right)~{},$
(C9) $\displaystyle\text{B}_{m}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{m}\left(\frac{\sqrt{(1-\alpha)(\alpha-2\beta+1)}}{(\alpha+2\beta+1)^{2}}\frac{(1+\beta)}{\eta^{5}}-\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1+\beta_{f})\right)~{},$
(C10) $\displaystyle\text{C}_{m}$
$\displaystyle=\frac{8\pi}{15}a^{5}\rho_{m}\left(\frac{\sqrt{(1-\alpha)(\alpha-2\beta+1)}}{(\alpha+2\beta+1)^{2}}\frac{(1+\alpha)}{\eta^{5}}-\frac{\sqrt{(1-\alpha_{f})(\alpha_{f}-2\beta_{f}+1)}}{(\alpha_{f}+2\beta_{f}+1)^{2}}(1+\alpha_{f})\right)~{}.$
(C11)

##  Appendix D The steadily rotating frame and the Tilt-Over mode

On Fig. 3, we have represented the ratio between the kinetic energy density of
the mantle and that of the fluid core as measured in the _steadily rotating
frame_ , which corresponds to the frame rotating with a constant velocity
around the mean axis of rotation of the planet. In their Appendix C, Triana et
al. [2019] gave some details on how the values of the quantities computed in
that frame relate to their value in the mantle frame. We now provide a more
detailed explanation.

We can think of the $z$-axis in the inertial frame, which we denote by
$\boldsymbol{\hat{z}}_{\text{IF}}$ as the mean rotation axis of the planet
perpendicular to the equator (at J2000). The angular velocity of the mantle
relative to the steadily rotating frame therefore reads (assuming
$\Omega_{0}=1$):

$\boldsymbol{\Omega}_{\text{m}|\text{SRF}}=(\boldsymbol{\hat{z}}+\boldsymbol{m})-\boldsymbol{\hat{z}}_{\text{IF}}~{},$
(D1)

where $\boldsymbol{\hat{z}}$ denotes the instantaneous $z-$axis of the mantle
frame. In order to write this in coordinates, we need to express
$\boldsymbol{\hat{z}}_{\text{IF}}$ in the mantle’s coordinates. We can do this
by using the three Euler angles that relate the mantle to the steadily
rotating frame. We use the following convention:

$\underbrace{\left(\begin{array}[]{c}\cdot\\\ \cdot\\\ \cdot\\\
\end{array}\right)}_{\text{mantle coordinates}}=\underbrace{R_{z}(\gamma)\cdot
R_{y}(\beta)\cdot R_{x}(\alpha)\cdot
R_{z}(t)}_{T}\underbrace{\left(\begin{array}[]{c}\cdot\\\ \cdot\\\ \cdot\\\
\end{array}\right)}_{\text{inertial frame coordinates}}~{},$ (D2)

where the $R_{i}$ denotes the ordinary (three dimensional) rotation matrices
around the axis $i$. The arguments of these matrices define the Euler angles
$\alpha$, $\beta$ and $\gamma$ which are all harmonic functions of $t$ with
frequency $\omega$, and should not be confused with the flattening parameters
of Eqs. (2.24). From the transformation matrix, $T$, we can construct the
antisymmetric rotation tensor:

$\boldsymbol{R}\equiv\frac{dT}{dt}\cdot T^{\text{T}}~{}.$ (D3)

The dual vector of $\boldsymbol{R}$ then defines the angular velocity of the
rotation (see _e.g._ , p. 46 of Lai et al. [2010]) which must be equal to the
angular velocity of the mantle. In components and assuming that $\alpha$,
$\beta$ and $\gamma$ are small, this gives:

$\begin{cases}m^{x}=i\omega\alpha-\beta\\\ m^{y}=i\omega\beta-\alpha\\\
m^{z}=i\omega\gamma~{}.\end{cases}$ (D4)

In order to obtain the components of $\boldsymbol{\hat{z}}_{\text{IF}}$ in the
mantle frame, we use the transformation matrix:

$T\cdot\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right)=\left(\begin{array}[]{c}-\beta\\\ \alpha\\\ 1\\\
\end{array}\right)~{}.$ (D5)

Finally, after solving Eq. (D4) for $\alpha$ and $\beta$ we obtain the
expression of the angular velocity of the mantle relative to the steadily
rotating frame in the mantle’s coordinates (in terms of $m^{+}$ and $m^{-}$):

$\boldsymbol{\Omega}_{\text{m}|\text{SRF}}=\left(\begin{array}[]{c}\frac{m^{+}\omega}{\sqrt{2}(1-\omega)}+\frac{m^{-}\omega}{\sqrt{2}(1+\omega)}\\\
\frac{im^{+}\omega}{\sqrt{2}(1-\omega)}-\frac{im^{-}\omega}{\sqrt{2}(1+\omega)}\\\
m^{z}\\\ \end{array}\right)+\emph{c.c.}$ (D6)

Eq. (D6) presents a clear resonance when $\omega=\pm 1$. These values can be
interpreted as new eigenfrequencies which are added to the spectrum when we
solve the first two equations of Eq. (D4) for $\alpha$ and $\beta$
simultaneously with Eqs. (2.26). They correspond to the motion known as the
_Tilt-Over mode_ which has a purely diurnal frequency independent to the
interior structure of the planet.