Jérémie Vidal

David Cébron

Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity

The acoustic modes of a rotating fluid-filled cavity can be used to determine the effective rotation rate of a fluid (since the resonant frequencies are modified by the flows). To be accurate, this method requires a prior knowledge of the acoustic modes in rotating fluids. Contrary to the Coriolis force, centrifugal gravity has received much less attention in the experimental context. Motivated by on-going experiments in rotating ellipsoids, we study how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond the standard acoustic equation, which neglects solid-body rotation and gravity, by deriving an exact wave equation for the acoustic velocity. We then solve the wave problem using a polynomial spectral method in ellipsoids, which is compared with finite-element solutions of the primitive fluid-dynamic equations. We show that the centrifugal acceleration has measurable effects on the acoustic frequencies when M Ω 0.3, where M Ω is the rotational Mach number defined as the ratio of the sonic and rotational time scales. Such a regime can be reached with experiments rotating at a few tens of Hz, by replacing air with a highly compressible gas (e.g. SF 6 or C 4 F 8 ).

I. INTRODUCTION

The acoustic resonant frequencies of fluid-filled cavities have several important applications in experimental physics. The acoustic modes can, for instance, be employed to measure the thermodynamical properties of gases [START_REF] Mehl | Precision acoustic measurements with a spherical resonator: Ar and C 2 H 4[END_REF][START_REF] Moldover | Gas-filled spherical resonators: Theory and experiment[END_REF] , to determine the Boltzmann constant in metrology [START_REF] Moldover | Measurement of the universal gas constant r using a spherical acoustic resonator[END_REF][START_REF] Guianvarc'h | Acoustic field in a quasi-spherical resonator: unified perturbation model[END_REF][START_REF] Pitre | Determination of the Boltzmann constant using a quasi-spherical acoustic resonator[END_REF] , or to passively determine the temperature in sulfur plasma lamps [START_REF] Koulakis | Acoustic resonances in gas-filled spherical bulb with parabolic temperature profile[END_REF] . The acoustic modes are also known to be sensitive to the advection by background flows, which can slightly change the acoustic resonant frequencies (as evidenced by groundbreaking applications in astrophysics [START_REF] Aerts | Asteroseismology[END_REF] ). The acoustic modes have thus been recently used to remotely image the effective flow rotation rate (i.e. the differential rotation between the solid-body rotation and the flow) in rotating fluid-filled spheres 8 and spheroids [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . This imaging method promises future applications in experimental fluid mechanics, as it could be used for fluids that cannot be easily imaged with conventional velocimetry techniques (e.g. gases, which cannot be easily seeded by optical particles [START_REF] Melling | Tracer particles and seeding for particle image velocimetry[END_REF] ).

A theoretical knowledge of the diffusionless acoustic modes has proven important for the aforementioned applications, despite certain advances in computational acoustics (e.g. relying on finite-element computations [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] ). It is, for instance, still costly to numerically resolve the viscous (and thermal) boundary layers in threedimensional models, which are thus often approximated a) jeremie.vidal@univ-grenoble-alpes.fr using asymptotic theory [START_REF] Moldover | Gas-filled spherical resonators: Theory and experiment[END_REF][START_REF] Berggren | Acoustic boundary layers as boundary conditions[END_REF] . Motivated by rapidly rotating fluid-filled experiments dedicated to planetary applications (e.g. the on-going ZoRo setup [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] in spheroidal geometry), we aim to theoretically study the diffusionless acoustic modes in rapidly rotating ellipsoidal cavities subject to pressure or temperature variations. Acoustic modes with angular frequency ω are usually modeled using the equation

-ω 2 P ω = C 2 s ∇ 2 P ω , (1)
where P ω is the acoustic pressure and C s the adiabatic speed of sound. Helmholtz Eq. ( 1) results from simple manipulations of the non-rotating primitive fluid equations upon a quiescent homogeneous medium, and analytical solutions can be obtained in spheroids [START_REF] Chang | Natural resonant frequency of a prolate acoustical resonator[END_REF][START_REF] Chang | Natural resonant frequencies of an oblate acoustical resonator[END_REF] and triaxial ellipsoids [START_REF] Willatzen | Eigenmodes of triaxial ellipsoidal acoustical cavities with mixed boundary conditions[END_REF] . The standard acoustic equation can also be simply extended when the background medium exhibits weak variations of pressure but strong density variations (i.e. when |∇P 0 | |C 2 s ∇ρ 0 |, where [P 0 , ρ 0 ] are the background pressure and density). This gives the acoustic equation for waves upon isobaric states [START_REF] Bergmann | The wave equation in a medium with a variable index of refraction[END_REF] -ω

2 P ω = ρ 0 C 2 s ∇ • ρ -1 0 ∇P ω , (2)
which is used in underwater acoustics or in the presence of strong density gradient of thermal origin [START_REF] Koulakis | Acoustic resonances in gas-filled spherical bulb with parabolic temperature profile[END_REF][START_REF] Karlsen | Acoustic force density acting on inhomogeneous fluids in acoustic fields[END_REF][START_REF] Koulakis | Convective instability in a stratified ideal gas containing an acoustic field[END_REF] . Equations (1) and (2) are, however, not valid when the fluid has substantial pressure variations in an ambient gravity field, or in the presence of global rotation. A scalar acoustic equation for rotating stratified fluids has been obtained under the beta-plane approximation [START_REF] Desanto | Derivation of the acoustic wave equation in the presence of gravitational and rotational effects[END_REF] , but it does not rigorously account for global rotation in threedimensional inhomogeneous media (e.g. as encountered in the global fluid envelopes of rapidly rotating planets or stars). Rotational effects are thus usually modeled using asymptotic theory [START_REF] Backus | The rotational splitting of the free oscillations of the Earth[END_REF] as small perturbations with respect to the acoustic modes upon non-rotating inhomogeneous media [START_REF] Bergmann | The wave equation in a medium with a variable index of refraction[END_REF][START_REF] Lignières | Asymptotic analysis of highfrequency acoustic modes in rapidly rotating stars[END_REF] , but the asymptotic approach unfortunately becomes inaccurate for rapidly rotating fluids (e.g. for rotating stars [START_REF] Reese | Acoustic oscillations of rapidly rotating polytropic stars-II. Effects of the Coriolis and centrifugal accelerations[END_REF] , and even for uniform-density fluids [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] ). A non-perturbative description of global rotation in (strongly) inhomogeneous media is thus desirable for rapidly rotating experiments in ellipsoidal geometries (but also in cylinders [START_REF] Morton | Waves in a gas in solidbody rotation[END_REF][START_REF] Miles | Waves in a rapidly rotating gas[END_REF][START_REF] Dodgson | Some special wave solutions in an adiabatic gas in solid-body rotation[END_REF] ).

Going beyond the scalar acoustic equation is quite unconventional in acoustic modeling, but the usefulness of employing a vector wave equation has already been recognized in several contexts [START_REF] Lynden-Bell | On the stability of differentially rotating bodies[END_REF][START_REF] Ross | A note on the Helmholtz equations for acoustic waves in inhomogeneous media[END_REF][START_REF] Komatitsch | Spectral-element simulations of global seismic wave propagation-II. Three-dimensional models, oceans, rotation and self-gravitation[END_REF] . Therefore, we derive in this work an exact vector wave equation that includes a rigorous treatment of rotation and buoyancy for fluids in isentropic and isothermal equilibrium. Such equilibrium states are indeed relevant for experiments [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF][START_REF] Menaut | Experimental study of convection in the compressible regime[END_REF] . The mathematical formulation extends our previous investigations of the acoustic modes in uniform-density fluids [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] and in a planetary context [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] , to include centrifugal gravity and isothermal reference states. We then solve the acoustic problem in ellipsoidal geometries, using a spectral description of the velocity field. The paper is organized as follows. We describe the acoustic problem in §II, then presents results in §III. We discuss our results in light of experimental applications in §IV, and we end the paper in §V.

II. FORMULATION OF THE PROBLEM

A. Primitive fluid-dynamic equations

We consider a compressible Newtonian fluid of density ρ, pressure P and temperature T , enclosed within a rigid arbitrary ellipsoidal cavity of semi-axes [a, b, c] and volume V = 4πabc/3. The cavity is subject to the constant (local) Earth's gravity g E = -g E 1 z (where 1 z is the unit vertical vector), and spins at the steady angular frequency Ω = Ω 1 z with respect to an inertial frame. To account for global rotation, we work in the frame rotating at Ω, where the ellipsoidal boundary ∂V is steady and given by (x/a) 2 + (y/b) 2 + (z/c) 2 = 1 in the Cartesian coordinates (x, y, z). Global rotation thus generates in the rotating frame the additional centrifugal acceleration g c = -Ω × (Ω × r), where r = (x, y, z) is the position vector. We model sound below as small-amplitude timedependent perturbations for the velocity u 1 , the density ρ 1 , pressure P 1 and entropy S 1 , upon a spatially inhomogeneous reference state [ρ 0 , P 0 , S 0 ]. The reference state is assumed to be in hydrostatic equilibrium

∇P 0 = ρ 0 g, g = g E + g c , (3a,b)
where g is the effective gravity. The other relation among the ambient variables is the equation of state (EoS) 31

∇P 0 = C 2 s ∇ρ 0 + ∂P ∂S ρ ∇S 0 , (4)
where S 0 is the background entropy, and C s = (∂P/∂ρ) S is the adiabatic speed of sound that can be inhomogeneous in space.

In the diffusionless theory (i.e. without attenuation due to viscosity and thermal diffusion), the smallamplitude perturbations are given in the rotating frame by the linearized fluid-dynamic equations 31

ρ 0 (∂ t u 1 + 2Ω × u 1 ) = -∇P 1 + ρ 1 g, (5a)
∂ t P 1 + u • ∇P 0 = -ρ 0 C 2 s ∇ • u 1 , (5b) ∂ t ρ 1 + ∇ • (ρ 0 u 1 ) = 0, ( 5c
)
where we have included the Coriolis term 2ρ 0 Ω × u 1 and the buoyancy force ρ 1 g in the momentum equation. Note that Eq. ( 5b) is obtained from the isentropic equation of continuity ∂ t S 1 + u 1 • ∇S 0 = 0 for diffusionless perturbations. Additionally, the perturbations must also satisfy the EoS

∂ t P 1 = C 2 s ∂ t ρ 1 + u 1 • C 2 s ∇ρ 0 -∇P 0 , (6)
which is obtained by combining Eqs. ( 4) and ( 5). The equations are also supplemented with appropriate boundary conditions (BC) on ∂V . The boundary is assumed to be rigid (considering an infinite acoustic impedance at the wall). This assumption is realistic for most experimental conditions (e.g. with a gas-filled metallic cavity [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] ). The fluid velocity thus obeys the no-penetration BC u 1 • n| ∂V = 0, where n = (x/a 2 , y/b 2 , z/c 2 ) is the (non-unit) normal vector on the boundary. For a nonrotating fluid without buoyancy, this BC is equivalent to the sound hard BC ∇P 1 • n| ∂V = 0 for the acoustic pressure. However, the no-penetration BC gives a more complicated BC for the pressure in the presence of rotation and buoyancy (which is implicitly obtained from the continuity of the normal component of the momentum equation [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF] ). The BC for the density can finally obtained from the normal component of EoS (6) if required.

B. Wave equation for the acoustic modes

The diffusionless acoustic modes are sought as harmonic perturbations [u 1 , P 1 , ρ 1 ](r, t) = [u ω , P ω , ρ ω ](r) exp(λt) with λ = iω, where ω is the angular frequency and [u ω , P ω , ρ ω ](r) represent the spatial structure of the modes. The identification of a master wave equation often allows a better understanding of the modes' properties, which are usually difficult to uncover from the primitive fluid-dynamic equations. As shown in Appendix A, it is possible to extend scalar acoustic Eqs. ( 1) and (2) to account for non-rotating inhomogenous reference states stratified in density under an external gravity field. The Coriolis force has however significant effects on the lowestfrequency acoustic modes [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF][START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] , which are important for experimental applications [START_REF] Triana | Helioseismology in a bottle: modal acoustic velocimetry[END_REF][START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] , and so should not be neglected (see below). Unfortunately, no simple scalar acoustic equation can be obtained in the presence of global rotation. To have a full account of rotational and buoyancy effects, we can fortunately combine the fluid-dynamic equations to obtain a rigorous wave equation for the velocity of the acoustic modes upon arbitrary hydrostatic reference states.

To do so, we substitute Eqs. (5b) and (5c) into the time derivative of momentum Eq. (5a). This gives

λ 2 ζ ω = -2λΩ × ζ ω -(∇ • ζ ω ) g + ∇ C 2 s ∇ • ζ ω + ∇ 1 ρ 0 ζ ω • [∇P 0 -C 2 s ∇ρ 0 ] (7)
where the fluid momentum ζ ω = ρ 0 u ω is the dynamical unknown. The wave equation is here solely supplemented with the no-penetration BC ζ ω • n| ∂V = 0, which is sufficient to solve the problem. Wave Eq. ( 7) is an exact generalization of the scalar acoustic equations, which rigorously takes global rotation and buoyancy into account for arbitrary inhomogeneous reference states in hydrostatic equilibrium (i.e. in the presence of possibly large pressure and density gradients). Note that alternative vector equations have been obtained in astrophysics [START_REF] Lynden-Bell | On the stability of differentially rotating bodies[END_REF] or seismology [START_REF] Komatitsch | Spectral-element simulations of global seismic wave propagation-II. Three-dimensional models, oceans, rotation and self-gravitation[END_REF] , in which the fluid displacement vector is instead taken as unknown.

A few characteristics of Eq. ( 7) are worth commenting on. The momentum can be expressed using Helmholtz decomposition [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] ζ ω = ∇Φ ω + ∇ × A ω , where Φ ω is the mass flux potential and A ω is the vector potential. Acoustic studies generally seek the momentum in the form [START_REF] Koulakis | Acoustic resonances in gas-filled spherical bulb with parabolic temperature profile[END_REF][START_REF] Karlsen | Acoustic force density acting on inhomogeneous fluids in acoustic fields[END_REF][START_REF] Pierce | Wave equation for sound in fluids with unsteady inhomogeneous flow[END_REF][START_REF] Collas | Acoustic radiation force on a particle in a temperature gradient[END_REF] ζ ω = ∇Φ ω , which is valid if the curl of all the terms in the right-hand side of Eq. ( 7) vanishes. This assumption is appropriate for non-rotating fluids with negligible pressure variations, such that we can set ∇P 0 = g = 0 in wave Eq. ( 7) that then simply reduces to acoustic Eq. ( 2) for the mass flux potential (or the hydrodynamic pressure given by [START_REF] Collas | Acoustic radiation force on a particle in a temperature gradient[END_REF] P ω = -λΦ ω ). Here, this decomposition is however not appropriate because ∇ × [2λΩ × ζ ω + (∇ • ζ ω ) g] = 0. Indeed, the rotational component ∇ × A ω must be retained in the Helmholtz decomposition of ζ ω to obtain accurate solutions of the acoustic modes in rotating systems (even for uniformdensity fluids [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] ). Therefore, we cannot reduce vector Eq. ( 7) to a simple scalar equation in the presence of global rotation.

C. Reference states

We describe the inhomogeneous reference states that we consider in the following, and which are relevant for our experimental applications. It is worth noting that the centrifugal forces do not modify here the boundary shape since the cavity is assumed to be perfectly rigid (contrary to stellar applications where free-surface flows are considered [START_REF] Aerts | Asteroseismology[END_REF] ). Hence, we can solve the hydrostatic equilibrium assuming the problem geometry. To do so, we rewrite EoS (4) in the form

C 2 s ∇ρ 0 = Γ ∇P 0 , (8)
where Γ is a thermodynamic parameter (possibly inhomogeneous in space). EoS (6) for the perturbations then reduces to 8) for the ambient quantities, which clearly indicates that the linearized perturbations do not obey the standard relationship 31 ∂ t P 1 = C 2 s ∂ t ρ 1 unless the reference state is characterized by Γ = 1. This generic EoS for the reference state, initially introduced in the planetary context [START_REF] Pekeris | Dynamics of the liquid core of the Earth[END_REF] , allows us to model two important experimental configurations.

∂ t P 1 = C 2 s ∂ t + (Γ -1) u 1 • ∇P 0 using EoS (
We can first model isentropic references states by setting Γ = 1, which develop when the fluid is well-mixed by flow motions (e.g. by convective motions [START_REF] Menaut | Experimental study of convection in the compressible regime[END_REF][START_REF] Tilgner | Convection in an ideal gas at high Rayleigh numbers[END_REF] ). Equilibrium (8) can also model isothermal (∇T 0 = 0) hydrostatic states for arbitrary fluids. Indeed, isothermal states are such that

∇S 0 = ∂S ∂P T ∇P 0 , (9)
and Eq. ( 4) can then be written in the form of Eq. ( 8) with

Γ = 1 - ∂P ∂S ρ ∂S ∂P T = 1 - α P α S . (10)
We have used in the right-hand side of Eq. ( 10) the Maxwell relations (∂P/∂S) ρ = ρ 2 (∂T /∂ρ) S and (∂S/∂P ) T = (1/ρ 2 ) (∂ρ/∂T ) P to introduce

α S = - 1 ρ ∂ρ ∂T S , α P = - 1 ρ ∂ρ ∂T P , (11a,b)
where α S and α P are respectively the isentropic [START_REF] Ray | Isentropic coefficient of expansion[END_REF][START_REF] Kouremenos | Isentropic exponents of real gases and application for the air at temperatures from 150 K to 450 K[END_REF] and isobaric coefficients of thermal expansion. The thermodynamic parameter Γ has not been directly tabulated in the literature, but it admits a simple expression. As shown in Appendix B, we have indeed Γ = γ for all fluids in isothermal equilibrium. To summarize, we have thus Γ = 1 for isentropic equilibrium, γ for isothermal equilibrium, (12) in EoS (8) for all (diffusionless) fluids.

We can now simplify wave Eq. ( 7) for isentropic and isothermal reference states, using Eq. ( 3) and EoS (8). We obtain after reduction

λ 2 ζ ω + 2λΩ × ζ ω = ∇ C 2 s ∇ • ζ ω + (1 -Γ) g • ζ ω -(∇ • ζ ω ) g. (13)
Finally, it is worth noting that Γ directly controls the density stratification of the fluid, which can affect the acoustic waves [START_REF] Lignières | Asymptotic analysis of highfrequency acoustic modes in rapidly rotating stars[END_REF] . The strength of stratification is usually measured by the squared Brunt-Väisälä frequency 28

N 2 0 = 1 ρ 0 ∇ρ 0 - ρ 0 C 2 s g • g = Γ -1 C 2 s g 2 (14)
(with g 2 = g • g), which quantifies the departure of the reference density field from an isentropic density profile. Neutral interiors N 2 0 = 0 are isentropic with Γ = 1, whereas fluids in isothermal equilibrium with Γ = 1 are stably stratified in density if N 2 0 > 0. Since Γ = γ in isothermal conditions (see in Appendix B), it shows that isothermal fluids are usually only slightly stably stratified in density (because γ ≤ 2 for most fluids in standard experimental conditions).

D. Numerical modeling

To solve wave Eq. ( 13), we have to provide mathematical expressions for the isothermal and isentropic reference states. For mathematical simplicity, we assume below that γ is constant in the isentropic and isothermal cases. For the isothermal case, we also assume that C s = C 0 is spatially homogeneous (as encountered in most experimental conditions, e.g. for an ideal gas at constant temperature T 0 ). The hydrostatic profiles for the isothermal equilibrium are then given by

[ρ 0 , P 0 ] = [ρ c , P c ] exp s 2 2H 2 Ω Γ - z + c H G Γ , (15)
where ρ c and P c = ρ c C 2 0 /Γ are respectively the density and pressure at z = -c on the rotation axis, and with the cylindrical radius s = (x 2 + y 2 ) 1/2 . We have also introduced the gravity scale height H G = C 2 0 /g E , and the rotation scale height H Ω = C 0 /Ω. The value of ρ c is obtained by the conservation of mass

V ρ 0 dV = (4/3)πabcρ m , (16)
where ρ m is the mean density of the fluid in the absence of rotation (and gravity). The ratio ρ c /ρ m , computed from Eq. ( 16), is illustrated in Fig. 1 as a function of the strength of global rotation when g E = 0. The density on the axis of rotation can be strongly weakened compared to the density at rest when the fluid is rapidly spinning. Contrary to the isothermal case, an isentropic equilibrium is characterized by an ambient inhomogeneous temperature T 0 defined by 29

∇T 0 = α P T 0 C P g. (17)
We assume below that the fluid obeys the ideal gas law P 0 = R ρ 0 T 0 in the isentropic case, where R = R/M is the specific gas constant (with the molar gas constant R and the molar mass M of the gas). We have thus α P = 1/T 0 and C P = R γ/(γ -1), where γ is the heat capacity ratio. The solution of Eq. ( 17) is then

T 0 = T c + γ -1 R γ Ω 2 2 s 2 -g E (z + c) , (18)
where T c is the temperature at s = 0 and z = -c. The adiabatic speed of sound, defined by C s = (γR T 0 ) 1/2 for an ideal gas, is thus given by the inhomogeneous profile

C 2 s = C 2 0 1 + γ -1 C 2 0 Ω 2 2 s 2 -g E (z + c) (19)
for isentropic states (with the constant C 2 0 = γR T c ). Considering the dry air-filled ZoRo experiment 9 rotating at Ω/(2π) = 50 Hz, we obtain a temperature difference of 2 K between the center and s = 0.2 m.

Having prescribed the reference state, Eq. ( 13) is a quadratic eigenvalue problem of unknowns [λ, ζ ω ], which can be solved in ellipsoids as follows. We seek ζ ω using the finite-dimensional expansion ζ ω = j Λ j e j , where Λ j are complex-valued coefficients and e j are vector elements that exactly satisfy the no-penetration BC e j • n| ∂V = 0. The elements e j are made of suitable combinations of Cartesian monomials x i y j z k of maximum degree i + j + k ≤ n, which are sought using the Helmholtz decomposition 30 e j = ∇Φ + ∇ × Ψ (where Φ is the velocity potential, and Ψ is the vector potential). The vortical part ∇ × Ψ, which is usually neglected in acoustics, must be retained in the presence of global rotation [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] or buoyancy since ∇ × (λ 2 ζ ω ) = 0 in the vector wave equation. We then substitute the polynomial expansion into wave Eq. ( 13), and employ a projection (Galerkin) method to minimize the residuals with respect to every basis function e i (using the volume-averaged inner product e i , e j = V e i • e j dV ). This gives a matrix quadratic eigenvalue problem λ 2 M + λC + K Λ = 0 for the eigenvalue λ and the unknown vector Λ = (Λ 1 , Λ 2 , . . . ) , which can be solved using standard numerical algorithms. We have implemented the aforementioned spectral method in our bespoke compressible numerical code shine [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] , which has been extended to account for centrifugal gravity and isothermal reference states when Γ = 1. We truncate the polynomial expansion at maximum degree n = 20, which is sufficient to have a good frequency (and spatial) convergence [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] for the large-scale acoustic modes presented below. The above algorithm has already been thoroughly validated against theoretical and standard numerical computations for uniform-density [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] and inhomogeneous [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] fluids, but neither in the presence of centrifugal effects nor when Γ = 1 in Eq. ( 13). To validate the algorithm in such conditions, we have also employed standard finite-element computations performed with comsol (see the numerical details in Appendix C).

III. RESULTS

We present below numerical computations of the resonant acoustic frequencies. To survey the parameter space, it is advantageous to non-dimensionalize the physical variables. We use the semi-axis a as the length scale, the typical value C 0 of the speed of sound as the velocity scale, the sonic timescale a/C 0 , the density ρ c at s = 0 and z = -c as the density scale, and P c as the pressure scale. Rotational and buoyancy effects are then controlled by the dimensionless numbers

M Ω = aΩ C 0 = a H Ω , M G = √ ag E C 0 = a H G , (20a,b)
where M Ω is the rotational Mach number (which compares the sonic time scale and the rotational time scale Ω -1 ), and M G is the gravitational Mach number (i.e. the ratio of the free-fall velocity √ ag E and the speed of sound). In dimensionless form, the Coriolis force scales as M Ω while the centrifugal acceleration varies as M 2 Ω , and the buoyancy force due to Earth's gravity evolves as M 2 G . Typical experimental values give M Ω ≤ 1 and M G 1. The dimensionless variables are written below with an asterisk * for clarity (to distinguish them from their dimensional counterparts).

Since the angular frequencies come in pairs ±ω * (by virtue of the symmetries of the problem), we only consider the solutions ω * > 0 in the following. To compute the acoustic modes, an ellipsoidal geometry must be specified. Solutions of acoustic Eq. ( 1) have angular frequencies that exhibit an azimuthal degeneracy in nonrotating spheres (i.e. the modes with different azimuthal wave numbers, but with the same meridional structure, have identical angular frequencies). However, an ellipsoidal deformation is known to lift the azimuthal degeneracy of the acoustic frequencies, which allows a simple identification of the acoustic modes in the ellipsoid using their observed frequencies [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . Moreover, ellipsoidal geometries are also relevant for planetary applications [START_REF] Bars | Flows driven by libration, precession, and tides[END_REF] . The effects of ellipsoidal deformation have already been thoroughly studied elsewhere [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF][START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] (albeit without centrifugal gravity), and so will not be further presented here. We consider below the ellipsoidal geometry of the ZoRo apparatus 9 , a spheroid with dimensionless axes a * = b * = 1 and c * = 0.95, and we have checked that similar conclusions are obtained in other geometries (not shown).

A. Effects of Earth's gravity

Earth's gravity is usually neglected in the description of the non-rotating acoustic modes [START_REF] Bergmann | The wave equation in a medium with a variable index of refraction[END_REF][START_REF] Pierce | Wave equation for sound in fluids with unsteady inhomogeneous flow[END_REF] , though it may modify the acoustic modes at rest for sufficient large values of M G (due to the presence of buoyancy terms in the wave equation). We explore in Fig. 2 how the angular frequencies of the non-rotating acoustic modes upon isothermal reference states vary with M 2 G (for dry air with Γ = 1.4 and water with Γ 1). We have also shown the comsol solutions for Γ = 1 obtained by solving Eq. (A5) with g = g E , which are in perfect agreement with the polynomial solutions. We find that the modes are almost unaffected by Earth's gravity for small values M 2 G 1 (as expected), but Earth's gravity is able to modify the acoustic frequencies when typically M G O(1) for all the modes (the precise value is mode-dependent). The acoustic frequencies are also more affected for air than for liquid water (as expected since air is more compressible). Another interesting point in the figure is the presence of non-acoustic modes when Γ = 1, which have non-discrete frequencies in the spectrum (contrary to the acoustic modes). These modes are internal gravity modes that exist because the fluid is stably stratified in density, as evidenced here by the non-zero value of the Brunt-Väisälä frequency

N * 0 = M 2 G √ Γ -1.
Since the spectrum of internal gravity modes is bounded 39 by |ω * | ≤ N * 0 , they can have frequencies comparable to the acoustic ones when M G is sufficiently large.

We have shown that Earth's gravity has measurable effects when M G O(1), that is when ag E C 2 0 . Such an extreme regime is valid for atmospheric conditions where the length scale is very large, but it cannot be obtained in experimental conditions (even if a strongly compressible gas is used such as SF 6 , see its properties in Table I). Therefore, we discard Earth's gravity in the following and set M G = 0 in all the computations.

B. Effects of centrifugal gravity

Contrary to Earth's gravity, we cannot a priori neglect the effects of centrifugal gravity on the acoustic modes in rapidly rotating configurations. The centrifugal acceleration has indeed a maximum amplitude Ω 2 a, which can be hundred times larger than Earth's gravity for rapid rotation (when Ω/(2π) ≥ 11 Hz for the ZoRo apparatus [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] with a = 0.2 m). Actually, the centrifugal and Coriolis effects have received much attention in the modeling of stellar oscillations. Stellar models usually neglect the Coriolis force in the wave equation whereas the buoyancy force (including centrifugal gravity) is retained, because the Coriolis force is expected to have a weak influence on the high-frequency acoustic modes that are used in astrophysics [START_REF] Lignières | Asymptotic analysis of highfrequency acoustic modes in rapidly rotating stars[END_REF] . We assess the validity of this approximation in Fig. 3 for dry air in isentropic equilibrium. We show the evolution of ω * as a function of M Ω for some acoustic modes obtained from vector Eq. ( 13), either without approximation (i.e. with a rigorous description of the Coriolis and centrifugal effects) or artificially neglecting the Coriolis force. We also show the comsol solutions of acoustic Eq. (A5) with g = g c , which are in excellent quantitative agreement with the solutions of the vector wave equation without the Coriolis force (as expected). The results clearly show that neglecting the Coriolis force is physically incorrect when M Ω ≥ O(10 -2 ), since the lowest-frequency modes of experimental interest [START_REF] Triana | Helioseismology in a bottle: modal acoustic velocimetry[END_REF][START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] are strongly impacted by the Coriolis force in this range [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] . Note that some acoustic modes can also hybridize, when M Ω ≥ 1, with the inertial modes [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF] . The latter are fluid modes sustained by the Coriolis force, which are characterized by the frequency spectrum |ω * | ≤ 2M Ω in compressible fluids [START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] . The inertial modes can thus interact with some low-frequency acoustic modes in that range, which would make the identification of the proper acoustic modes more difficult for experimental applications.

The relative importance of centrifugal gravity must be finally addressed, since it could have weaker effects than reported in stellar applications (since centrifugal acceleration does not deform here the rigid cavity). In our case, it could indeed be argued that centrifugal gravity would only slightly modify the acoustic resonant frequencies, since the buoyancy force ρ 1 g c in the momentum equation has the typical amplitude M 2 Ω (which is usually smaller than the amplitude of the Coriolis force M Ω ). As previously reported in astrophysics [START_REF] Reese | Acoustic oscillations of rapidly rotating polytropic stars-II. Effects of the Coriolis and centrifugal accelerations[END_REF] , we find that the centrifugal effects are more pronounced for the highfrequency acoustic modes (in particular because they have smaller-scale structures). As shown in Fig. 4, centrifugal gravity can have measurable effects onto the acoustic frequencies of the high-frequency modes as soon as M Ω 0.3. We have illustrated in the figure a few highfrequency branches with ω * 10, which approximately correspond to the highest-frequency modes that can be currently detected in rotation using the acoustic device of the ZoRo apparatus [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . The frequencies of the illustrated modes is erroneously shifted towards higher values if centrifugal gravity is disregarded, and we obtain crossings of acoustic branches that occur at wrong values of M Ω . Erroneous mode crossings can also occur for lowerfrequency modes (e.g. in Fig. 6 below). Mode crossing is a phenomenon that considerably complicates the identification of the modes in the experimental spectrum [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] , and will also raise mathematical issues for the interpretation of the observed resonant frequencies in terms of flow structures for modal acoustic velocimetry [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] . A theoretical model accounting for centrifugal gravity is thus appropriate to obtain accurate predictions for the acoustic frequencies, and also to precisely locate the possible mode crossings in the parameter space.

IV. DISCUSSION

A. Effects of attenuation

We have shown that centrifugal gravity has nonnegligible effects on the diffusionless acoustic frequencies when M Ω 0.3. The validity of the diffusionless theory may however be questioned for experimental applications, where fluid viscosity and thermal diffusion are present (see some typical values in Table I). The predicted centrifugal effects might indeed be buried in the experimental noise (e.g. if diffusive effects were stronger than centrifugal effects). Hence, it is important to explore the effects of diffusion on the acoustic modes.

To address this point, we have computed acoustic modes from the primitive fluid-dynamic equations in the presence of viscosity and thermal diffusion, for dry air at ambient temperature T 0 = 20 • C using comsol (see details in Appendix C). We have considered acoustic modes with large-scale azimuthal wave numbers |m| ≤ 6 and angular frequencies |ω * | ≤ 10, which correspond to the modes that could be identified with the acoustic apparatus of the ZoRo experiment [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . The eigenvalue of a diffusive mode becomes λ * = σ * + iω * d , where σ * < 0 is the damping rate of the mode (due to viscous effects and thermal diffusion) and ω * d is the diffusive angular frequency. We illustrate in Fig. 5 the evolution of the quality factor

Q = |ω * d |/|σ * | (21)
as a function of M Ω . Very large quality factors Q ≥ 10 3 are found when M Ω ≤ 1, showing that these modes can be measured in experimental conditions. In addition to the damping σ * , attenuation is also known to slightly reduce the acoustic resonant frequencies. Diffusion is responsible for small shifts in frequency 0 ≤ δ *

g c = 0 comsol (m = 3) comsol (m = 4) comsol (m = 5)
FIG. 6. (Color online) Comparison between diffusionless (left vertical axis, colored curves) and diffusive (right vertical axis, red empty squares) angular frequencies of some acoustic modes in a rotating spheroid with dimensionless polar axis c * = 0.95, as a function of MΩ for isothermal states with Γ = 1.4 (dry air). Olive solid curves are diffusionless solutions of vector wave Eq. ( 13). Empty symbols are comsol solutions with azimuthal wave numbers |m| ≤ 6 of the exact diffusive fluid-dynamic equations for dry air at ambient temperature T0 = 20 • C (see parameters in Table I). For graphical purposes, the diffusive frequencies ω * d have been shifted by the amount 3 × 10 -3 on the right vertical axis (to be superimposed with the diffusionless frequencies). In the right panel, dashed colored curves indicate diffusionless solutions without the centrifugal force for the perturbation (as in Fig. 4).

the weak attenuation regime, such that the actual diffusive acoustic frequency is ω * d ≈ ω *δ * ω at leading asymptotic order [START_REF] Moldover | Gas-filled spherical resonators: Theory and experiment[END_REF][START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . The comparison between the diffusionless theory and the diffusive solutions of the exact diffusive equations is illustrated in Fig. 6. To ease the comparison between our diffusionless and diffusive computations, we have not directly shown in the figure the diffusive frequencies ω * d but instead ω * d + δ * ω , where δ * ω 3 × 10 -3 is the typical shift of frequency due to attenuation. An excellent quantitative agreement is then observed between the diffusionless frequencies and diffusive frequencies, showing that attenuation actually only very weakly depends on M Ω for the modes with |ω * | ≤ 10. Moreover, as further illustrated in the right panel for a few high-frequency modes with |ω * | 7, centrifugal effects do persist in the presence of attenuation when M Ω 0.2 -0.3. Therefore, the effects of centrifugal gravity must be considered for rapidly rotating experiments.

B. Parameter regimes for experiments

It is worth estimating if the regime M Ω 0.3 where centrifugal effects are significant is relevant for on-going laboratory experiments dedicated to planetary applications. Measuring the resonant acoustic frequencies can indeed be used to reconstruct the effective rotation profile of the fluid 8 , which requires prior accurate predictions of the resonant frequencies in the presence of solid-body rotation. Existing rotating experiments in ellipsoidal geometries are usually filled with water [START_REF] Lemasquerier | Libration-driven flows in ellipsoidal shells[END_REF] or dry air [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . The typical parameters for water-filled experiments are a 0.2 m and Ω/(2π) ≤ 4 Hz, such that M Ω 10 -2 . The effects of centrifugal gravity on the acoustic frequencies is thus entirely negligible for such experiments (as well as the Coriolis effects [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF] ). The gas-filled acoustic experiment ZoRo, with a = 0.2 m and Ω/(2π) ≤ 70 Hz, can reach larger values M Ω ≤ 0.25 when filled with dry air at ambient temperature, but this is still insufficient to be sensitive to the centrifugal effects.

However, it is currently under consideration to fill the ZoRo experiment with gases having much smaller speeds of sound than air, such as SF 6 or C 4 F 8 with respectively 41 C 0 133.9 m.s -1 and C 0 110.3 m.s -1 at the usual experimental conditions of pressure P 0 = 10 5 Pa and temperature T 0 = 20 • C. Much larger values M Ω ≤ 0.7 could thus be reached with the ZoRo experiment rotating at high rotation rates and filled with such gases. Therefore, it will be imperative to account for centrifugal gravity in forthcoming experiments that will consider gases more compressible than air in the reachable rapidly rotating regime M Ω 0.3. Using other gases would also allow us to explore different regimes for the flow dynamics. For instance, SF 6 is characterized by a much smaller value of the kinematic viscosity ν = µ/ρ m 2.5 × 10 -6 (compared to ν 1.5 × 10 -5 for dry air). Changing ν will indeed give a different force balance between viscous diffusion and rotation, as mea- sured by the Ekman number E = ν/(a 2 Ω) that plays a central role in the theory of rotating fluids [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF] . Since it is very small in planetary bodies (typically E 10 -10 ), reducing ν will significantly enhance the planetary relevance of the ZoRo experiment. SF 6 also has a smaller thermal diffusivity κ = k/(ρ m C P ) than air (see in Table I), which would allow us to change the value of Prandtl number P r = ν/κ (which strongly impacts the outcome of buoyancy-driven flows, e.g. for thermal convection [START_REF] Kaplan | Subcritical thermal convection of liquid metals in a rapidly rotating sphere[END_REF] ). Therefore, using different gases is highly desirable for planetary modeling.

Finally, it is also worth considering the effects of the ambient temperature for experimental applications. In dimensionless variables, varying the temperature only amounts to change the values of the heat capacity ratio γ ≥ 1 and of M Ω for the isothermal regime. The variations with M Ω have already been considered above, so it only remains to explore the effects of γ. The variations of ω * with γ in the rapidly rotating regime are illustrated in Fig. 7. Only small departures are found from the value at γ = 1, with relative errors always smaller than a percent in the range of interest γ ≤ 2. Hence, varying γ by changing the temperature will only have weak effects on the acoustic frequencies for usual experimental conditions with γ ≤ 2. However, the relevance of the isothermal assumption could also be questioned, since it is often difficult to keep a constant fluid temperature in experimental conditions. Slight variations of temperature (e.g. in the room, or due to acoustic transducers [START_REF] Koulakis | Convective instability in a stratified ideal gas containing an acoustic field[END_REF] ) may indeed break the assumption of isothermal equilibrium, such that Eq. ( 13) may lead to inaccurate predictions. Order-of-magnitude arguments show that the density variations due to hydrostatic effects would be smaller than the temperature-induced ones if where ∆ρ 0 is the typical variation of density across the fluid due to the temperature difference ∆T and g the strength of the effective local gravity. If condition ( 22) is fulfilled, it could be more appropriate to use wave Eq. ( 7), with possibly ∇P 0 = g = 0 to disregard the hydrostatic pressure variations due to gravity. If the ZoRo experiment is rotating at moderate rotation rates Ω/(2π) 15 Hz, temperature variations of only ∆T 1 K in air, and ∆T 5 K in SF 6 , could be sufficient to break the centrifugally-driven isothermal equilibrium. However, condition (22) would require ∆T ≥ 10 K in air and ∆T ≥ 60 K in SF 6 for |C 2 s ∇ρ 0 | to violate the (nearly) isothermal assumption for the gas-filled ZoRo experiment rotating at Ω/(2π) = 50 Hz (with a = 0.2 m and the centrifugal acceleration g = g c ∼ Ω 2 a). Therefore, in the rapidly rotating regime, slight variations of temperature in ZoRo are not expected to disturb the (centrifugally-modified) acoustic resonant frequencies obtained by assuming isothermal equilibrium.

V. CONCLUDING REMARKS

Motivated by laboratory experiments of compressible rotating flows dedicated to planetary applications [START_REF] Triana | Helioseismology in a bottle: modal acoustic velocimetry[END_REF][START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] , we have investigated in this paper how the resonant acoustic frequencies of rotating fluid-filled ellipsoids are modified by centrifugal gravity. We have shown that, in the diffusionless theory (i.e. without attenuation), the primitive fluid-dynamic equations are exactly reducible to a single vector wave equation for the velocity perturbation upon inhomogeneous reference states in isothermal (or isentropic) equilibrium. Although such an approach is unconventional in acoustic modeling, it is essential to rigorously account for the effects of global rotation and buoyancy onto the acoustic perturbations. The acoustic problem has been numerically solved using an efficient spectral polynomial method in ellipsoids [START_REF] Vidal | Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry[END_REF][START_REF] Vidal | Acoustic and inertial modes in planetary-like rotating ellipsoids[END_REF] , which has been validated against standard finite-element computations performed with a commercial software. We have shown that centrifugal gravity has non-negligible effects on the acoustic frequencies when M Ω 0.3 (at least for the high-frequency modes). Such a regime can be achieved with a rapidly rotating experiment filled with a highly compressible gas (i.e. with a speed of sound lower than that of air, such as SF 6 or C 4 F 8 ). We have finally shown that measuring centrifugal effects would be possible in real experimental conditions, despite viscous effects and thermal diffusion.

Future applications of our work concern the on-going ZoRo experiment [START_REF] Su | Acoustic spectra of a gas-filled rotating spheroid[END_REF] . Beyond the isothermal situation, our results for isentropic equilibrium will also be useful to predict the expected resonant acoustic frequencies upon fully turbulent convection states (since the adiabatic temperature gradient is established in compress-ible convection [START_REF] Menaut | Experimental study of convection in the compressible regime[END_REF][START_REF] Tilgner | Convection in an ideal gas at high Rayleigh numbers[END_REF] ). Our results will thus be used to interpret the observed slight departures of the acoustic frequencies from our isentropic predictions in terms of convection-driven temperature anomalies upon the adiabatic gradient. We believe that combining experimental and theoretical works about rotating compressible flows is a promising avenue for the next generation planetarydriven convection models. Beyond the ZoRo experiment, our wave equation could also be considered for different boundary conditions [START_REF] Lebovitz | The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases[END_REF][START_REF] Vidal | Polynomial description of acoustic modes in fluid ellipsoids[END_REF] , or for other acoustic problems with rotation (e.g. gyrometers in cylindrical geometries [START_REF] Bruneau | A rate gyro based on acoustic mode coupling[END_REF][START_REF] Ecotiere | Inertial coupling of resonant normal modes in rotating cavities: Acoustic gyrometers for high rotation rates[END_REF] , which could probe the regime M Ω ≤ 10 to measure very high rotation rates up to 10 5 degrees per second). Other applications of our theory could also concern (rotating) fluids that are not in isothermal nor isentropic equilibrium (e.g. with a conductive temperature profile due to internal heating [START_REF] Koulakis | Acoustic resonances in gas-filled spherical bulb with parabolic temperature profile[END_REF][START_REF] Koulakis | Convective instability in a stratified ideal gas containing an acoustic field[END_REF] ), for which wave Eq. ( 7) is appropriate. We thus hope that our results will be useful for other acoustic problems.

ACKNOWLEDGMENTS

We are indebted to the other members of the ZoRo team (S. Su, H.-C. Nataf, P. Cardin, M. Solazzo and Y. Do) for helpful discussions about the experiment and future prospects (rapid rotation and change of gas). We also acknowledge the two referees for their valuable comments, which helped us to significantly improved the quality of the manuscript. The source code shine is available at https://bitbucket.org/vidalje/shine/ download. The CoolProp library 41 is available at https: //www.coolprop.org/. This work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 847433, theia project).

APPENDIX A: GENERALIZED ACOUSTIC EQUATIONS

Mathematical formulation

We present here a generalization of the scalar acoustic equations to account for arbitrary inhomogeneous states (i.e. possibly non-hydrostatic), but still without taking the Coriolis force into account. It has been initially discovered in acoustics [START_REF] Bergmann | The wave equation in a medium with a variable index of refraction[END_REF] , and later rediscovered in astrophysics for hydrostatic reference states [START_REF] Lignières | Asymptotic analysis of highfrequency acoustic modes in rapidly rotating stars[END_REF] .

To derive the equation, we take the time derivative of Eqs. (5b)-(5c), and replace λu ω with Eq. (5a) in which the Coriolis force is neglected. This gives

λ 2 ρ ω + ∇ • (ρ ω g) = ∇ 2 P ω , (A1a)
λ 2 P ω -C 2 s ρ ω = C 2 s S 0 • [ρ ω g -∇P ω ] , (A1b)
where the vector S 0 , defined by 15

S 0 = 1 ρ 0 ∇ρ 0 - ∇P 0 C 2 s , (A2)
is related to the gradient of entropy ∇S 0 in EoS (4) such as ∇S 0 = (α S C 2 s ) S 0 . We then obtain from Eq. (A1) the density perturbation

ρ ω = λ 2 P ω + C 2 s S 0 • ∇P ω C 2 s (λ 2 + S 0 • g) . (A3)
Finally, we can substitute Eq. (A3) into Eq. (A1a) to obtain the wave equation for the acoustic pressure P ω .

The pressure equation has to be supplemented by a BC, which is obtained by taking the normal component of Eq. (5a) on the rigid boundary ∂V (where the velocity satisfies the no-penetration BC). This gives the Robin BC

n • ∇P ω - P ω C 2 s g ∂V = 0. (A4)
In isentropic interiors where ∇P 0 = C 2 s ∇ρ 0 and P ω = C 2 s ρ ω from EoS (6), we obtain from (A1b) the equation

λ 2 P ω = C 2 s ∇ • ∇P ω - P ω C 2 s g . ( A5
)
The latter equation thus extends Eqs. ( 1) and ( 2) to include the effects of buoyancy for isentropic reference states (which exhibit substantial pressure variations).

For non-isentropic reference states in hydrostatic equilibrium, we have ∇P 0 = ρ 0 g ⇒ ∇ρ 0 ×g = 0 (i.e. barotropic fluids) and the acoustic equation reduces in this case to 20

λ 2 (λ 2 P ω + C 2 s S 0 • ∇P ω ) = C 2 s (λ 2 + N 2 0 ) ∇ 2 P ω -C 2 s (λ 2 + N 2 0 ) ∇ • λ 2 P ω + C 2 s S 0 • ∇P ω C 2 s (λ 2 + N 2 0 ) g , (A6)
where S 0 = (N 2 0 /g 2 ) g is directly related to the square of the Brunt-Väisälä frequency N 2 0 = S 0 • g.

Range of validity of the equations

We can now discuss the relevance of the various acoustic equations for experimental modeling. When the inhomogeneous reference state is isentropic, which occurs for instance in the presence of vigorous convection [START_REF] Menaut | Experimental study of convection in the compressible regime[END_REF][START_REF] Tilgner | Convection in an ideal gas at high Rayleigh numbers[END_REF] (as in planetary or stellar interiors), then Eq. (A5) can be employed. If the reference state is not isentropic (i.e. ∇P 0 = C 2 s ∇ρ 0 ), we can compare the magnitude of the hydrostatic pressure |∇P 0 | = |ρ 0 g| with respect to the amplitude of |C 2 s ∇ρ 0 |. If the fluid is in isothermal equilibrium with Γ∇P 0 = C 2 s ∇ρ 0 (i.e. N 2 0 = 0), then generalized acoustic Eq. (A6) must be considered. The latter equation is however difficult to solve, because of the fourth-order polynomial degree in λ (albeit it only involves second-order spatial derivatives for P ω ). This is another reason showing that using vector wave Eq. ( 13) is numerically advantageous, even in the non-rotating case, as it only involves a second-order time derivative.

As discussed in the main text, the isothermal assumption could become inappropriate if condition ( 22) is fulfilled. In such a regime, the hydrostatic pressure variations due to gravity could be disregarded to recover wave Eq. ( 2) by setting ∇P 0 = g = 0 in Eq. (A1b). For nonrotating fluids subject to Earth's gravity, we would have ∆ρ 0 /ρ m 10 -4 for dry air with a = 1 m. Such small density variations can be obtained for instance with a temperature difference of ∆T 1 K in air [START_REF] Koulakis | Convective instability in a stratified ideal gas containing an acoustic field[END_REF] , and so using Eq. ( 2) appears suitable in such contexts. However, we have shown in the main text that the Coriolis force can have significant effects on the low-frequency acoustic modes, such that Eq. ( 2) is inappropriate in the presence of global rotation. In the rapidly rotating regime, Eq. ( 13) can be safely considered for nearly isothermal equilibrium. Otherwise, Eq. ( 7) should be considered for non-isothermal fluids in the presence of global rotation.

APPENDIX B: Γ FOR ISOTHERMAL EQUILIBRIUM

We can obtain an explicit expression of the parameter Γ in EoS (8) for fluids in isothermal equilibrium as shown below. We rewrite Eq. (5b) as a function of the temperature perturbation T 1 upon the isothermal temperature T 0 in the form

ρ 0 C P ∂ t T 1 = α P T 0 [∂ t P 1 + u 1 • ∇P 0 ] , (B1)
together with the linearized EoS for the perturbations

ρ 1 = γ C 2 s P 1 -ρ 0 α P T 1 , α P = C P (γ -1) C 2 s T 0 , (B2a,b)
where C P is the specific heat at constant pressure. Then, substituting Eq. (B1) in the time derivative of Eq. (B2a) yields an EoS in the form (6) with Γ = 1 + (α P C s ) 2 T 0 /C P = γ ≥ 1 (B3) for isothermal equilibrium. Expression (B3) is thus valid for all (diffusionless) fluids in isothermal equilibrium. Note that a more straightforward derivation can be done for an ideal gas. We have indeed α S = 1/[(1γ)T 0 ] and α P = 1/T 0 for the ideal gas law [START_REF] Ray | Isentropic coefficient of expansion[END_REF][START_REF] Kouremenos | Isentropic exponents of real gases and application for the air at temperatures from 150 K to 450 K[END_REF] , which gives Γ = γ according to definition (10).

6 FIG. 1 .

 61 FIG. 1. (Color online) Ratio ρc/ρm given by conservation of mass (16), as a function of MΩ, for a rotating spheroid a * = b * = 1 and c * = 0.95 in isothermal equilibrium with MG = 0.

FIG. 2 .

 2 FIG. 2. (Color online) Dimensionless angular frequency ω * of the lowest-frequency acoustic modes, as a function of M 2 G for isothermal reference states in a non-rotating spheroid with dimensionless polar axis c * = 0.95. Red empty squares are the comsol solutions of Eq. (A5) with g = gE and Γ = 1, for azimuthal wave numbers |m| ≤ 8. Brown area represents internal gravity modes, which exist when Γ = 1 in the range |ω * | ≤ N * 0 with N * 0 = M 2 G √ Γ -1.

FIG. 3 .

 3 FIG. 3. (Color online) Dimensionless angular frequency ω * of acoustic modes in a rotating spheroid with dimensionless polar axis c * = 0.95, as a function of MΩ for dry air (γ = 1.4) in isentropic equilibrium. Olive solid curves represent numerical solutions of exact Eq. (13) without approximation, whereas dashed curves indicate solutions without the Coriolis force. In the presence of the Coriolis force, some acoustic modes can hybridize with inertial modes that exist when 30 |ω * | ≤ 2MΩ (brown area). Red empty squares are comsol solutions of scalar Eq. (A5) with g = gc and azimuthal wave numbers |m| ≤ 6.

FIG. 4 .

 4 FIG. 4. (Color online) Effects of centrifugal gravity on the dimensionless angular frequency ω * of a few high-frequency acoustic branches in a rotating spheroid with dimensionless polar axis c * = 0.95, as a function of MΩ for dry air (γ = 1.4) in isothermal equilibrium. Olive solid curves represent solutions of exact Eq. (13) without approximation, whereas dashed curves indicate solutions without the centrifugal force for the perturbation (i.e. setting gc = 0 in the equation).

9 ×

 9 TABLE I. Physical properties of fluids usually used in laboratory experiments, at pressure 10 5 Pa and ambient temperature T0 = 20 • C. Heat capacity ratio γ, specific gas constant R , mean density at rest ρm, typical adiabatic speed of sound C0, dynamic shear viscosity µ, thermal conductivity k, specific heat at constant pressure CP , and dimensionless Prandtl number P r = ν/κ = Cpµ/k where ν = µ/ρm is the kinematic viscosity and κ = k/(ρmCP ) is the thermal diffusivity. Properties of SF6 are taken from the library CoolProp 41 . 10 -4 1.8 × 10 -5 1.5 × 10 -5 k (W.K -1 .m -1 ) 0.6 2.58 × 10 -2 1.26 × 10 -2 CP (J.kg -1 .K -1 ) 4.18 × 10 3 1005.42 660.99 αP (K -1 ) 2.07 × 10 -4 3.42 × 10 -3 3.55 × 10 -3

FIG. 5 .

 5 FIG. 5. (Color online) Quality factor Q, as a function of MΩ, for the diffusive acoustics modes with azimuthal wave numbers |m| ≤ 6 in a rotating spheroid with dimensionless polar axis c * = 0.95. comsol solutions are obtained from the diffusive fluid-dynamic equations for dry air in isothermal equilibrium at ambient temperature T0 = 20 • C (see in TableI). Color bar shows the diffusive angular frequency ω * d .

ωO( 1 )

 1 of the diffusionless frequencies ω * in

FIG. 7 .

 7 FIG. 7. (Color online) Relative error δω * = |ω * -ω * (γ = 1)|/|ω * (γ = 1)|, as a function of γ -1, for diffusionless acoustic modes upon isothermal states in a rotating spheroid with dimensionless polar axis c * = 0.95 and MΩ = 0.5. Color bar indicates the angular frequency ω * (γ = 1) for the value γ = 1.

  17 |ρ 0 g| |C 2 s ∇ρ 0 |, that is when ∆ρ 0 /ρ 0 ≥ ag/C 2 s , ∆ρ 0 /ρ 0 α P ∆T, (22a,b)

J. Acoust. Soc. Am. / 31 August 2021 Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity

J. Acoust. Soc. Am. / 31 August 2021 Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity

APPENDIX C: FINITE-ELEMENT MODELING

To validate the spectral method presented in the main text, we have employed standard finite-element computations (performed with the commercial software comsol) in two cases. First, we have computed the diffusionless solutions of Eqs. (A4) and (A5), using the built-in acoustic module. Second, we have computed the diffusive acoustic modes for an isothermal background temperature T 0 and a homogeneous speed of sound C 0 , by solving the primitive fluid-dynamic equations in the presence of viscosity and thermal diffusion.

To do so, we include in the right-hand side of Eq. (5a) the viscous force

, where µ and µ B are respectively the dynamic shear and bulk viscosities (which are both assumed to be homogeneous in space). We also replace Eq. (5b) by the linearized heat equation including thermal diffusion, which gives for isothermal equilibrium

together with linearized EoS (B2) for the perturbations, where C P is assumed to be homogeneous, k is the (homogeneous) thermal conductivity, and α P is the (homogeneous) coefficient of thermal expansion at constant pressure defined by Eq. (11b) and given in the general case by Eq. (B2b). The diffusive fluid-dynamic equations are supplemented with the no-slip BC u 1 | ∂V = 0 for the velocity perturbation on the boundary, and the isothermal BC T 1 | ∂V = 0 for the temperature.

The equations are discretized in comsol by representing the ellipsoidal geometry with an unstructured mesh made of tetrahedral finite elements. To compute the diffusionless solutions of acoustic problem (A5), we have used quintic Lagrange elements for the pressure. For the diffusive modes, we have used cubic Lagrange elements for the pressure, and quartic Lagrange elements for the velocity and the temperature. For the computations of air-filled cavities, we have adopted the built-in comsol values for the physical parameters at the ambient temperature T 0 = 20 • C (see also values in Table I), and we have considered 48 µ B = 0.6µ.