arXiv subjects

Jérémie Vidal

Publications and source records attributed to Jérémie Vidal.

At least 19 recordsLinked to original sources

arXiv2026

On gravito-inertial surface waves

In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency waves existing in the presence of a solid boundary. We consider an idealized fluid model for an incompressible fluid enclosed in a smooth compact three-dimensional domain, subject to a constant rotation vector. The fluid is also stratified in density under a constant Brunt-Väisälä frequency. The spectral problem is formulated in terms of the pressure, which satisfies a Poincaré equation within the domain, and a Kelvin equation on the boundary. The Poincaré equation is elliptic when the wave frequency is small enough, such that we can use the Dirichlet-to-Neumann operator to reduce the Kelvin equation to a pseudo-differential equation on the boundary. We find that the wave energy is concentrated on the boundary for large covectors, and can exhibit surface wave attractors for generic domains. In an ellipsoid, we show that these waves are square-integrable and reduce to spherical harmonics on the boundary.

math.AP↗

arXiv2026

Did lunar tides sustain the early Earth's dynamo?

Geological data show that, early in its history, the Earth had a large-scale magnetic field with an amplitude comparable to the one of the present geomagnetic field. However, its origin remains enigmatic and various mechanisms have been proposed to explain the Earth's field over geological time scales. Here, we critically evaluate whether tidal forcing could explain the ancient geodynamo, by combining constraints from geophysical models of the Earth-Moon system and predictions from turbulence studies. Our analysis shows that lunar tidal forcing could have been sufficiently strong before $-3.25$ Gy to trigger turbulence within the Earth's core, and potentially to sustain dynamo action during that interval. Then, we propose new scaling laws for the magnetic field amplitude $B$. We expect the latter to scale as $B \propto β^{4/3}$, where $β$ is the equatorial ellipticity of the liquid core, if the turbulence involves weak interactions of three-dimensional inertial waves. Alternatively, in the regime of strong tidal forcing, the expected scaling becomes $B \propto β$. When extrapolated to the Earth's core, it suggests that tidal forcing alone was too weak to possibly explain the ancient geomagnetic field. Therefore, our study indirectly favours another origin for the early Earth's dynamo on long time scales (e.g. exsolution of light elements atop the core, or thermal convection due to secular cooling).

physics.geo-ph↗

arXiv2025

The spectrum of the Poincar{é} operator in an ellipsoid

We study the spectrum of the Poincaré operator in triaxial ellipsoids subject to a constant rotation. As explained in the paper, this mathematical problem is interesting for many physical applications. It is known that the spectrum of this bounded self-adjoint operator is pure point with polynomial eigenvectors [Backus & Rieutord, Phys. Rev. E 95 (2017), 053116]. We give two new proofs of this result. Moreover, we describe the large-degree asymptotics of the restriction of that operator to polynomial vector fields of fixed degrees. The main tool is the microlocal analysis of the partial differential equation satisfied by the orthogonal polynomials in ellipsoids. This work also contains numerical calculations of these spectra, showing a very good agreement with the mathematical results.

math.AP↗

arXiv2025

Geophysical flows over topography, a playground for laboratory experiments

Physicists face major challenges in modelling multi-scale phenomena that are observed in geophysical flows (e.g. in the Earth's oceans and atmosphere, or liquid planetary cores). In particular, complexities arise because geophysical fluids are rotating and subject to density variations, but also because the fluid boundaries have complex geometries (e.g. the ocean floor) with wavelengths ranging from metres to thousands of kilometres. Dynamical models of planetary fluid layers are thus often constrained by observations, whose interpretation necessitates a comprehensive understanding of the underlying physics. To this end, geophysical studies often combine cutting-edge experiments across a wide range of parameters, together with theory and numerical simulations, to derive predictive scaling laws applicable for planetary settings. In this review, we discuss experimental efforts that have contributed to our understanding of geophysical flows with topography. More specifically, we focus on (i) the flow response to mechanical (orbital) forcings in the presence of a large-scale (ellipsoidal) topography, (ii) some effects of small-scale topography onto bulk flows and boundary-layer dynamics, and (iii) the interaction between convection and roughness. The geophysical context is briefly introduced for each case, and some experimental perspectives are drawn.

physics.flu-dyn↗

arXiv2024

Inertia-gravity waves in geophysical vortices

Pancake-like vortices are often generated by turbulence in geophysical flows. Here, we study the inertia-gravity oscillations that can exist within such geophysical vortices, due to the combined action of rotation and gravity. We consider a fluid enclosed within a triaxial ellipsoid, which is stratified in density with a constant Brunt-Väisälä frequency (using the Boussinesq approximation) and uniformly rotating along a (possibly) tilted axis with respect to gravity. The wave problem is then governed by a mixed hyperbolic-elliptic equation for the velocity. As in the rotating non-stratified case considered by Vantieghem (2014, Proc. R. Soc. A, 470, 20140093, doi:10.1098/rspa.2014.0093), we find that the spectrum is pure point in ellipsoids (i.e. only consists of eigenvalues) with smooth polynomial eigenvectors. Then, we characterise the spectrum using numerical computations (obtained with a bespoke Galerkin method) and asymptotic spectral theory. Finally, the results are discussed in light of natural applications (e.g. for Mediterranean eddies or Jupiter's vortices).

physics.flu-dyn↗

arXiv2021

Mean zonal flows induced by weak mechanical forcings in rotating spheroids

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using boundary-layer theory, we determine the mean zonal flows generated by nonlinear interactions within the viscous Ekman layer. These flows are of interest because they survive in the relevant planetary regime of both vanishing forcings and viscous effects. We extend the theory to take into account (i) the combination of spatial and temporal perturbations, providing new mechanically driven zonal flows (e.g. driven by latitudinal librations), and (ii) the spheroidal geometry relevant for planetary bodies. Wherever possible, our analytical predictions are validated with direct numerical simulations. The theoretical solutions are in good quantitative agreement with the simulations, with expected discrepancies (zonal jets) in the presence of inertial waves generated at the critical latitudes (as for precession). Moreover, we find that the mean zonal flows can be strongly affected in spheroids. Guided by planetary applications, we also revisit the scaling laws for the geostrophic shear layers at the critical latitudes, and the influence of a solid inner core.

physics.flu-dyn↗

arXiv2021

Acoustic and inertial modes in planetary-like rotating ellipsoids

The bounded oscillations of rotating fluid-filled ellipsoids can provide physical insight into the flow dynamics of deformed planetary interiors. The inertial modes, sustained by the Coriolis force, are ubiquitous in rapidly rotating fluids and Vantieghem (2014, Proc. R. Soc. A, 470, 20140093, doi:10.1098/rspa.2014.0093) pioneered a method to compute them n incompressible fluid ellipsoids. Yet, taking density (and pressure) variations into account is required for accurate planetary applications, which has hitherto been largely overlooked in ellipsoidal models. To go beyond the incompressible theory, we present a Galerkin method in rigid coreless ellipsoids, based on a global polynomial description. We apply the method to investigate the normal modes of fully compressible, rotating and diffusionless fluids. We consider an idealized model, which fairly reproduces the density variations in the Earth's liquid core and Jupiter-like gaseous planets. We successfully benchmark the results against standard finite-element computations. Notably, we find that the quasi-geostrophic inertial modes can be significantly modified by compressibility, even in moderately compressible interiors. Finally, we discuss the use of the normal modes to build reduced dynamical models of planetary flows.

astro-ph.EP↗

arXiv2021

Kinematic dynamos in triaxial ellipsoids

Planetary magnetic fields are generated by motions of electrically conducting fluids in their interiors. The dynamo problem has thus received much attention in spherical geometries, even though planetary bodies are non-spherical. To go beyond the spherical assumption, we develop an algorithm that exploits a fully spectral description of the magnetic field in triaxial ellipsoids to solve the induction equation with local boundary conditions (i.e. pseudo-vacuum or perfectly conducting boundaries). We use the method to compute the free-decay magnetic modes and to solve the kinematic dynamo problem for prescribed flows. The new method is thoroughly compared with analytical solutions and standard finite-element computations, which are also used to model an insulating exterior. We obtain dynamo magnetic fields at low magnetic Reynolds numbers in ellipsoids, which could be used as simple benchmarks for future dynamo studies in such geometries. We finally discuss how the magnetic boundary conditions can modify the dynamo onset, showing that a perfectly conducting boundary can strongly weaken dynamo action, whereas pseudo-vacuum and insulating boundaries often give similar results.

astro-ph.EP↗

arXiv2021

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 $M_Ω\gtrsim 0.3$, 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$_8$).

physics.flu-dyn↗

arXiv2020

Acoustic spectra of a gas-filled rotating spheroid

The acoustic spectrum of a gas-filled resonating cavity can be used to indirectly probe its internal velocity field. This unconventional velocimetry method is particularly interesting for opaque fluid or rapidly rotating flows, which cannot be imaged with standard methods. This requires to (i) identify a large enough number of acoustic modes, (ii) accurately measure their frequencies, and (iii) compare with theoretical synthetic spectra. Relying on a dedicated experiment, an air-filled rotating spheroid of moderate ellipticity, our study addresses these three challenges. To do so, we use a comprehensive theoretical framework, together with finite-element calculations, and consider symmetry arguments. We show that the effects of the Coriolis force can be successfully retrieved through our acoustic measurements, providing the first experimental measurements of the rotational splitting (or Ledoux) coefficients for a large collection of modes. Our results pave the way for the modal acoustic velocimetry to be a robust, versatile, and non-intrusive method for mapping large-scale flows. Pages: 1-14

physics.flu-dyn↗

arXiv2020

The Effects of Robin Boundary Condition on Thermal Convection in a Rotating Spherical Shell

Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully developed transport properties and flow structure. We use the Boussinesq approximation, and impose a Robin boundary condition at the top. This enforces the temperature anomaly and its radial derivative to be linearly coupled with a proportionality factor $β$. Using the height H of the fluid layer, we introduce the non-dimensional Biot number Bi* = $β$H. Varying Bi* allows us to transition from fixed temperature for Bi* = +$\infty$, to fixed thermal flux for Bi* = 0. The bottom boundary of the shell is kept isothermal. We find that the onset of convection is only affected by Bi* in the non-rotating case. Far from onset, considering an effective Rayleigh number and a generalized Nusselt number, we show that the Nusselt and P{é}clet numbers follow standard universal scaling laws, independent of Bi* in all cases considered. However, the large-scale flow structure keeps the signature of the boundary condition with more vigorous large scales for smaller Bi* , even though the global heat transfer and kinetic energy are the same. Finally, for all practical purposes, the Robin condition can be safely replaced by a fixed flux when Bi* < 0.03 and by a fixed temperature for Bi* > 30.

physics.class-ph↗

arXiv2020

Efficiency of tidal dissipation in slowly rotating fully convective stars or planets

Turbulent convection is thought to act as an effective viscosity in damping equilibrium tidal flows, driving spin and orbital evolution in close convective binary systems. Compared to mixing-length predictions, this viscosity ought to be reduced when the tidal frequency $|ω_t|$ exceeds the turnover frequency $ω_{cν}$ of the dominant convective eddies, but the efficiency of this reduction has been disputed. We reexamine this long-standing controversy using direct numerical simulations of an idealized global model. We simulate thermal convection in a full sphere, and externally forced by the equilibrium tidal flow, to measure the effective viscosity $ν_E$ acting on the tidal flow when $|ω_t|/ω_{cν} \gtrsim 1$. We demonstrate that the frequency reduction of $ν_E$ is correlated with the frequency spectrum of the (unperturbed) convection. For intermediate frequencies below those in the turbulent cascade ($|ω_t|/ω_{cν} \sim 1-5$), the frequency spectrum displays an anomalous $1/ω^α$ power law that is responsible for the frequency-reduction $ν_E \propto 1/|ω_t|^α$, where $α< 1$ depends on the model parameters. We then get $|ν_E| \propto 1/|ω_t|^δ$ with $δ> 1$ for higher frequencies, and $δ=2$ is obtained for a Kolmogorov turbulent cascade. A generic $|ν_E| \propto 1/|ω_t|^{2}$ suppression is next found for higher frequencies within the dissipation range of the convection (but with negative values). Our results indicate that a better knowledge of the frequency spectrum of convection is necessary to accurately predict the efficiency of tidal dissipation in stars and planets resulting from this mechanism.

astro-ph.SR↗

arXiv2020

Turbulent viscosity acting on the equilibrium tidal flow in convective stars

Convection is thought to act as a turbulent viscosity in damping tidal flows and in driving spin and orbital evolution in close convective binary systems. This turbulent viscosity should be reduced, compared to mixing-length predictions, when the forcing (tidal) frequency $|ω_t|$ exceeds the turnover frequency $ω_{cv}$ of the dominant convective eddies. However, two contradictory scaling laws have been proposed and this issue remains highly disputed. To revisit this controversy, we conduct the first direct numerical simulations (DNS) of convection interacting with the equilibrium tidal flow in an idealized global model of a low-mass star. We present direct computations of the turbulent effective viscosity, $ν_E$, acting on the equilibrium tidal flow. We unexpectedly report the coexistence of the two disputed scaling laws, which reconciles previous theoretical (and numerical) findings. We recover the universal quadratic scaling $ν_E \propto (|ω_t|/ω_{cv})^{-2}$ in the high-frequency regime $|ω_t|/ω_{cv} \gg 1$. Our results also support the linear scaling $ν_E \propto (|ω_t|/ω_{cv})^{-1}$ in an intermediate regime with $1 \leq |ω_t|/ω_{cv} = \mathcal{O}(10)$. Both regimes may be relevant to explain the observed properties of close binaries, including spin synchronization of solar-type stars and the circularization of low-mass stars. The robustness of these two regimes of tidal dissipation, and the transition between them, should be explored further in more realistic models. A better understanding of the interaction between convection and tidal flows is indeed essential to correctly interpret observations of close binary stars and short-period planetary orbits.

astro-ph.SR↗

arXiv2019

Fossil field decay due to nonlinear tides in massive binaries

Surface magnetic fields have been detected in 5 to 10% of isolated massive stars, hosting outer radiative envelopes. They are often thought to have a fossil origin, resulting from the stellar formation phase. Yet, magnetic massive stars are scarcer in (close) short-period binaries, as reported by the BinaMIcS (Binarity and Magnetic Interaction in various classes of Stars) collaboration. Different physical conditions in the molecular clouds giving birth to isolated stars and binaries are commonly invoked. In addition, we propose that the observed lower magnetic incidence in close binaries may be due to nonlinear tides. Indeed, close binaries are probably prone to tidal instability, a fluid instability growing upon the equilibrium tidal flow via nonlinear effects. Yet, stratified effects have hitherto been largely overlooked. We theoretically and numerically investigate tidal instability in rapidly rotating, stably stratified fluids permeated by magnetic fields. We use the short-wavelength stability method to propose a comprehensive (local) theory of tidal instability at the linear onset, discussing damping effects. Then, we propose a mixing-length theory for the mixing generated by tidal instability in the nonlinear regime. We successfully assess our theoretical predictions against proof-of-concept, direct numerical simulations. Finally, we compare our predictions with the observations of short-period, double-lined spectroscopic binary systems. Using new analytical results, cross-validated by a direct integration of the stability equations, we show that tidal instability can be generated by nonlinear couplings of inertia-gravity waves with the equilibrium tidal flow in short-period massive binaries, even against the Joule diffusion. In the nonlinear regime, a fossil magnetic field can be dissipated by the turbulent magnetic diffusion induced by the saturated tidal flows. We predict that the turbulent Joule diffusion of fossil fields would occur in a few million years for several short-period massive binaries. Therefore, turbulent tidal flows could explain the observed dearth of some short-period magnetic binaries.

astro-ph.SR↗

arXiv2019

Rotating double-diffusive convection in stably stratified planetary cores

In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection. We investigate rotating double-diffusive convection (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compositional source terms. We focus on the finger regime, in which the thermal gradient is stabilising whereas the compositional one is destabilising. First, we perform a global linear stability analysis in spheres. The critical Rayleigh numbers drastically drop for stably stratified fluids, yielding large-scale convective motions where local analyses predict stability. We evidence the inviscid nature of this large-scale double-diffusive instability, enabling the determination of the marginal stability curve at realistic planetary regimes. In particular, we show that in stably stratified spheres, the Rayleigh numbers $Ra$ at the onset evolve like $Ra \sim Ek^{-1}$, where $Ek$ is the Ekman number. This differs from rotating convection in unstably stratified spheres, for which $Ra \sim Ek^{-4/3}$. The domain of existence of inviscid convection thus increases as $Ek^{-1/3}$. Second, we perform nonlinear simulations. We find a transition between two regimes of RDDC, controlled by the strength of the stratification. Furthermore, far from the RDDC onset, we find a dominating equatorially anti-symmetric, large-scale zonal flow slightly above the associated linear onset. Unexpectedly, a purely linear mechanism can explain this phenomenon, even far from the instability onset, yielding a symmetry breaking of the nonlinear flow at saturation. For even stronger stable stratification, the flow becomes mainly equatorially-symmetric and intense zonal jets develop. Finally, we apply our results to the early Earth core. Double diffusion can reduce the critical Rayleigh number by four decades for realistic core conditions. We suggest that the early Earth core was prone to turbulent RDDC, with large-scale zonal flows.

physics.flu-dyn↗

arXiv2018

Magnetic fields driven by tidal mixing in radiative stars

Stellar magnetism plays an important role in stellar evolution theory. Approximatively 10% of observed main sequence (MS) and pre-main-sequence (PMS) radiative stars exhibit surface magnetic fields above the detection limit, raising the question of their origin. These stars host outer radiative envelopes, which are stably stratified. Therefore, they are assumed to be motionless in standard models of stellar structure and evolution. We focus on rapidly rotating, radiative stars which may be prone to the tidal instability, due to an orbital companion. Using direct numerical simulations in a sphere, we study the interplay between a stable stratification and the tidal instability, and assess its dynamo capability. We show that the tidal instability is triggered regardless of the strength of the stratification (Brunt-V{ä}is{ä}l{ä} frequency). Furthermore, the tidal instability can lead to both mixing and self-induced magnetic fields in stably stratified layers (provided that the Brunt-V{ä}is{ä}l{ä} frequency does not exceed the stellar spin rate in the simulations too much). The application to stars suggests that the resulting magnetic fields could be observable at the stellar surfaces. Indeed, we expect magnetic field strengths up to several Gauss.Consequently, tidally driven dynamos should be considered as a (complementary) dynamo mechanism, possibly operating in radiative MS and PMS stars hosting orbital companions. In particular, tidally driven dynamos may explain the observed magnetism of tidally deformed and rapidly rotating Vega-like stars.

astro-ph.SR↗

arXiv2017

Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits

We consider the hydrodynamic stability of homogeneous, incompressible and rotating ellipsoidal fluid masses. The latter are the simplest models of fluid celestial bodies with internal rotation and subjected to tidal forces. The classical problem is the stability of Roche--Riemann ellipsoids moving on circular Kepler orbits. However, previous stability studies have to be reassessed. Indeed, they only consider global perturbations of large wavelength or local perturbations of short wavelength. Moreover many planets and stars undergo orbital motions on eccentric Kepler orbits, implying time-dependent ellipsoidal semi-axes. This time dependence has never been taken into account in hydrodynamic stability studies. In this work we overcome these stringent assumptions. We extend the hydrodynamic stability analysis of rotating ellipsoids to the case of eccentric orbits. We have developed two open-source and versatile numerical codes to perform global and local inviscid stability analyses. They give sufficient conditions for instability. The global method, based on an exact and closed Galerkin basis, handles rigorously global ellipsoidal perturbations of unprecedented complexity. Tidally driven and libration-driven elliptical instabilities are first recovered and unified within a single framework. Then we show that new global fluid instabilities can be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits. Their existence is confirmed by a local analysis and direct numerical simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital eccentricity may have a strong destabilising effect in celestial fluid bodies, which may lead to space-filling turbulence in most of the parameters range.

physics.class-ph↗

arXiv2017

Experimental and Theoretical Study of Magnetohydrodynamic Ship Models

Magnetohydrodynamic (MHD) ships represent a clear demonstration of the Lorentz force in fluids, which explains the number of students practicals or exercises described on the web. However, the related literature is rather specific and no complete comparison between theory and typical small scale experiments is currently available. This work provides, in a self-consistent framework, a detailed presentation of the relevant theoretical equations for small MHD ships and experimental measurements for future benchmarks. Theoretical results of the literature are adapted to these simple battery/magnets powered ships moving on salt water. Comparison between theory and experiments are performed to validate each theoretical step such as the Tafel and the Kohlrausch laws, or the predicted ship speed. A successful agreement is obtained without any adjustable parameter. Finally, based on these results, an optimal design is then deduced from the theory. Therefore this work provides a solid theoretical and experimental ground for small scale MHD ships, by presenting in detail several approximations and how they affect the boat efficiency. Moreover, the theory is general enough to be adapted to other contexts, such as large scale ships or industrial flow measurement techniques.

physics.class-ph↗