arXiv subjects

Michel Rieutord

Publications and source records attributed to Michel Rieutord.

At least 19 recordsLinked to original sources

arXiv2026

Fundamental properties of two rapidly rotating stars: Rasalhague and Alkaid

Abridged : The fundamental parameters of rapidly rotating stars are key quantities to understand the impact of rotation on stellar evolution. A few nearby early-type stars offer the possibility of precise measurements of these parameters, which will help us constrain newly available two-dimensional models. We propose a method to retrieve the fundamental parameters of a fast rotating star (mass, rotation rate, and age), and also the inclination of its rotation axis on the line of sight, using five spectrophotometric observables along with a set of steady 2D-models. Using the temperature derived by the infrared flux method, the Vsini and the apparent luminosity, along with a grid of 2D steady state models, we select models that are compatible with all observational constraints, and derive the most probable mass, rotation rate, core hydrogen mass fraction relative to that of the envelope, and inclination of the rotation axis on the line of sight of the targetted star. We apply this method to two stars: Rasalhague (alpha Oph) and Alkaid (eta UMa). We confirm and improve the fundamental parameters of Rasalhague and provide a new determination of its rotation axis inclination on the line-of-sight, which we find to be $\sim69$ degrees. Concerning Alkaid, we infer a mass of $5.071\pm0.023 M_\odot$, a rotation rate corresponding to an equatorial rotation period of 14.6 hours. We also find an inclination of the rotation axis of $\sim42$ degrees. We show that Alkaid is a very young star, presumably between 2 and 8 Myrs off the Zero-Age Main Sequence. As a side result, using high resolution spectra and the Least Square Deconvolution method, we determined a precise value of the Vsini of Rasalhague, namely $224.3\pm2.6$ km/s. Similarly, we find $V\sin i=154.3\pm9.1$ km/s for Alkaid.

astro-ph.SR↗

arXiv2025

An Educational Guide for 2D Stellar Structure Calculations of Rapidly Rotating Stars using the ESTER code

The Evolution STEllaire en Rotation (ESTER) code is the first 2D stellar structure code to be made open-source and freely available to the astronomy and astrophysics community. An important and novel advancement of this code is that it can reproduce the distorted shape and observable signatures (e.g., gravity darkening) of rapidly rotating stars. ESTER also calculates the steady-state large-scale flows within the star, namely their differential rotation and associated meridional circulation. In this report, we explore and document the physics implemented within version 1.1.0rc2 of the ESTER code, in a way that complements published descriptions. We illustrate this physics by plotting how stellar structure parameters vary through stellar interiors at a range of latitudes and at different angular velocities. We investigate how the thin convective envelopes of intermediate mass stars vary with latitude when rapidly rotating, becoming deeper and thicker near the equator. Simple comparisons of ESTER model predictions (e.g., central temperature and density, luminosity) with the output from the Modules for Experiments in Stellar Astrophysics (MESA) code [Paxton et al., 2010] shows generally good agreement. Additional comparisons provide important benchmarking and verification for ESTER as a comparatively young code. Finally, we provide a guide for installing and running the code on our local university cluster, aimed at helping students to begin work.

astro-ph.SR↗

arXiv2025

Angular momentum relaxation in models of rotating early-type stars

The rotational evolution of stars is still an open question of stellar physics because of the numerous phenomena that can contribute to the distribution of angular momentum. This paper aims at determining the time scale over which a rotating early-type star relaxes to a steady baroclinic state or, equivalently, in which case its nuclear evolution is slow enough to let the evolution of the star be modelled by a series of quasi-steady states. We investigate the damping time scale of baroclinic and viscous eigenmodes that are potentially excited by the continuous forcing of nuclear evolution. We first investigate this problem with a spherical Boussinesq model. Since much of the dynamics is concentrated in the radiative envelope of the star, we then improve the realism of the modelling by using a polytropic model of the envelope that takes into account a realistic density profile. The polytropic model of the envelope underlines the key role of the region at the core-envelope interface. The results of evolutionary models recently obtained with two-dimensional axisymmetric ESTER models turn out to be a consequence of the slow damping of viscous modes. Using a vanishing Prandtl number appears to be too strong an approximation to explain the dynamics of the models. Baroclinic modes previously thought as good candidates of this relaxation process turn out to be too quickly damped. The dynamical response of rotating stars to the slow forcing of its nuclear evolution appears as a complex combination of non-oscillating eigenmodes. Simple Boussinesq approach are not realistic enough to explain this reality. The present work underlines the key role of layers near the core-envelope interface in an early-type star and also the key role of any angular momentum transport mechanism, here played by viscosity, for early-type stars to reach critical rotation, presumably associated with the Be phenomenon.

astro-ph.SR↗

arXiv2024

An improved asteroseismic age of the rapid rotator Altair from TESS data

Understanding the effects of rotation in stellar evolution is key to modelling early-type stars, half of which have equatorial velocities over 100 km/s. The nearby star Altair is an example of such fast-rotating stars, and furthermore, it has the privilege of being modelled by a detailed 2D concordance model that reproduces most of its observables. The aim of this paper is to include new asteroseismic frequencies to improve our knowledge of Altair, especially its age. We processed images of Altair obtained during July 2022 by the Transiting Exoplanet Survey Satellite using the halo photometry technique to obtain its light curve over this observation period. By analysing the light curve, we derived a set of 22 new frequencies in the oscillation spectrum of Altair and confirmed 12 previously known frequencies. Compared with model predictions, we could associate ten frequencies with ten axisymmetric modes. This identification is based on the modelled visibility of the modes. Moreover, nine of the modelled frequencies can be adjusted to simultaneously match their corresponding observed frequencies, once the core hydrogen mass fraction of the concordance model is set to $X_{\rm core}/X_{\rm ini}\simeq0.972$, with $X_{\rm ini}=0.739$. Using the combined results of a 1D MESA model computing the pre-main sequence and a 2D time-dependent ESTER model computing the main sequence, we find that this core hydrogen abundance sets the age of Altair to 88$\pm$10 Myrs, which is slightly younger than previous estimates.

astro-ph.SR↗

arXiv2024

Photometric determination of rotation axis inclination, rotation rate, and mass of rapidly rotating intermediate-mass stars

Intermediate-mass stars are often fast rotators, and hence are centrifugally flattened and affected by gravity darkening. To analyse this kind of stars properly, one must turn to 2D models to compute the visible radiative flux and to take the geometrical effect of the star inclination into account. Assuming a given stellar age and chemical composition, we aim to derive the mass and rotation rates of main sequence fast rotating stars, along with their inclination, from photometric quantities. We chose three observables that vary with mass, rotation, and inclination: the infrared flux method temperature T_IRFM, the Strömgren c1 index, and a second index c2 built in the same way, but sensitive to the UV side of the Balmer jump. These observables are computed from synthetic spectra produced with the PHOENIX code and rely on a 2D stellar structure from the ESTER code. These quantities are computed for a grid of models in the range 2 to 7~M_Sun, and rotation rates from 30% to 80% of the critical rate. Then, for any triplet (T_IRFM, c1, c2), we try to retrieve the mass, rotation rate, and inclination using a Levenberg-Marquardt scheme, after a selection step to find the most suitable starting models. Hare-and-hound tests showed that our algorithm can recover the mass, rotation rate, and inclination with a good accuracy. The difference between input and retrieved parameters is negligible for models lying on the grid and is less than a few percent otherwise. An application to the real case of Vega showed that the u filter is located in a spectral region where the modelled and observed spectra are discrepant, and led us to define a new filter. Using this new filter and subsequent index, the Vega parameters are also retrieved with satisfactory accuracy. This work opens the possibility to determine the fundamental parameters of rapidly rotating early-type stars from photometric space observations.

astro-ph.SR↗

arXiv2024

A two-dimensional perspective of the rotational evolution of rapidly rotating intermediate-mass stars

Recently, the first successful attempt at computing stellar models in two dimensions has been presented with models that include the centrifugal deformation and self-consistently compute the velocity field. This paper aims at studying the rotational evolution of 2D models of stars rotating at a significant fraction of their critical angular velocity. From the predictions of these models, we aim to improve our understanding of the formation of single Be stars. Using the ESTER code that solves the stellar structure of a rotating star in two dimensions with time evolution, we have computed evolution tracks of stars between 4 and 10Msun for initial rotation rates ranging between 60 and 90% the critical rotation rate. A minimum initial rotation rate at the start of the main sequence is needed to spin up the star to critical rotation within its main sequence lifetime. This threshold depends on the stellar mass, and increases with increasing mass. The models do not predict any stars above 8Msun to reach (near) critical rotation during the main sequence. Furthermore, we find the minimum threshold of initial angular velocity is lower for SMC metallicity compared to Galactic metallicity, which is in agreement with the increased fraction in the number of observed Be stars in lower metallicity environments. Self-consistent 2D stellar evolution provide more insight into the rotational evolution of intermediate-mass stars, and our predictions are consistent with observations of velocity distributions and fraction of Be stars amongst B-type stars. We find that stars with a mass above 8Msun do not increase their fraction of critical rotation during the main sequence. Since a fraction of stars above 8Msun have been observed to display the Be phenomenon, other processes or formation channels must be at play, or critical rotation is not required for the Be phenomenon above this mass.

astro-ph.SR↗

arXiv2023

Hydrodynamic modelling of dynamical tides dissipation in Jupiter's interior as revealed by Juno

The Juno spacecraft has acquired exceptionally precise data on Jupiter's gravity field, offering invaluable insights into Jupiter's tidal response, interior structure, and dynamics, establishing crucial constraints. We develop a new model for calculating Jupiter's tidal response based on its latest interior model, while also examining the significance of different dissipation processes for the evolution of its system. We study the dissipation of dynamical tides in Jupiter by thermal, viscous and molecular diffusivities acting on gravito-inertial waves in stably stratified zones and inertial waves in convection ones. We solve the linearised equations for the equilibrium tide. Next, we compute the dynamical tides using linear hydrodynamical simulations based on a spectral method. The Coriolis force is fully taken into account, but the centrifugal effect is neglected. We study the dynamical tides occurring in Jupiter using internal structure models that respect Juno's constraints. We study specifically the dominant quadrupolar tidal components and our focus is on the frequency range that corresponds to the tidal frequencies associated with Jupiter's Galilean satellites. By incorporating the different dissipation mechanisms, we calculate the total dissipation and determine the imaginary part of the tidal Love number. We find a significant frequency dependence in dissipation spectra, indicating a strong relationship between dissipation and forcing frequency. Furthermore, our analysis reveals that, in the chosen parameter regime in which kinematic viscosity, thermal and molecular diffusivities are equal, the dominant mechanism contributing to dissipation is viscosity, exceeding in magnitude both thermal and chemical dissipation. We find that the presence of stably stratified zones plays an important role in explaining the high dissipation observed in Jupiter.

astro-ph.EP↗

arXiv2023

Hydrodynamical modelling of tidal dissipation in gas giant planets at the time of space missions

Gas giant planets are differentially rotating magnetic objects that have strong and complex interactions with their environment. In our Solar system, they interact with their numerous moons while exoplanets with very short orbital periods (hot Jupiters), interact with their host star. The dissipation of waves excited by tidal forces in their interiors shapes the orbital architecture and the rotational dynamics of these systems. Recently, astrometric observations of Jupiter and Saturn systems have challenged our understanding of their formation and evolution, with stronger tidal dissipation in these planets than previously predicted, in contrast to what appears to be weaker in gas giant exoplanets. These new constraints are motivating the development of realistic models of tidal dissipation inside these planets. At the same time, the Juno and Cassini space missions have revolutionised our knowledge of the interiors of Jupiter and Saturn, whose structure is a combination of stably stratified zones and convective regions. In this work, we present results of hydrodynamical calculations modelling tidal waves and their dissipation in Jupiter, taking for the first time the latest, state-of-the-art interior model of the planet. We performed 2D numerical simulations of linear tidal gravito-inertial waves that propagate and dissipate within Jupiter interior by taking into account viscous, thermal and chemical diffusions. This new model allows us to explore the properties of the dissipation and the associated tidal torque as a function of all the key hydrodynamical and structural parameters.

astro-ph.EP↗

arXiv2023

The first two-dimensional stellar structure and evolution models of rotating stars

Rotation is a key ingredient in the theory of stellar structure and evolution. Until now, stellar evolution codes operate in a 1-D framework for which the validity domain in regards to the rotation rate is not well understood. This letter aims at presenting the first results of self-consistent stellar models in two spatial dimensions that compute the time evolution of a star and its rotation rate along the main sequence together with a comparison to observations. We make use of an extended version of the ESTER code that solves the stellar structure of a rotating star in two dimensions with time evolution, including chemical evolution, and an implementation of rotational mixing. We have computed evolution tracks for a 12Msun model, once for an initial rotation rate equal to 15% of the critical frequency, and once for 50%. We first show that our model initially rotating at 15% of the critical frequency is able to reproduce all the observations of the $β$ Cephei star HD 192575 recently studied by Burssens et al. with asteroseismology. Beyond the classical surface parameters like effective temperature or luminosity, our model also reproduces the core mass along with the rotation rate of the core and envelope at the estimated age of the star. This particular model also shows that the meridional circulation has a negligible influence on the transport of chemical elements, like nitrogen, for which the abundance may be increased at the stellar surface. Furthermore, it shows that in the late main sequence, nuclear evolution is faster than the relaxation time needed to reach a steady state of the star angular momentum distribution. We have demonstrated that we have successfully taken the new step towards 2-D evolutionary modelling of rotating stars. It opens new perspectives on the understanding of the dynamics of fast rotating stars and on the way rotation impacts stellar evolution.

astro-ph.SR↗

arXiv2023

Internal shear layers in librating spherical shells: the case of attractors

Following our previous work on periodic ray paths (He et al, 2022), we study asymptotically and numerically the structure of internal shear layers for very small Ekman numbers in a three-dimensional (3D) spherical shell and in a two-dimensional (2D) cylindrical annulus when the rays converge towards an attractor. We first show that the asymptotic solution obtained by propagating the self-similar solution generated at the critical latitude on the librating inner core describes the main features of the numerical solution. The internal shear layer structure and the scaling for its width and velocity amplitude in $E^{1/3}$ and $E^{1/12}$ respectively are recovered. The amplitude of the asymptotic solution is shown to decrease to $E^{1/6}$ when it reaches the attractor, as it is also observed numerically. However, some discrepancies are observed close to the particular attractors along which the phase of the wave beam remains constant. Another asymptotic solution close to those attractors is then constructed using the model of Ogilvie (2005). The solution obtained for the velocity has an $O(E^{1/6})$ amplitude, but a different self-similar structure than the critical-latitude solution. It also depends on the Ekman pumping at the contact points of the attractor with the boundaries. We demonstrate that it reproduces correctly the numerical solution. Surprisingly, the solution close to an attractor with phase shift (that is an attractor that touches the axis in 3D or in 2D with a symmetric forcing) is found to be much weaker.

physics.flu-dyn↗

arXiv2021

Predictions for gravity-mode periods and surface abundances in intermediate-mass dwarfs from shear mixing and radiative levitation

The treatment of chemical mixing in the radiative envelopes of intermediate-mass stars has hardly been calibrated so far. Recent asteroseismic studies demonstrated that a constant diffusion coefficient in the radiative envelope is not able to explain the periods of trapped gravity modes in the oscillation spectra of $γ$ Doradus pulsators. We present a new generation of MESA stellar models with two major improvements. First, we present a new implementation for computing radiative accelerations and Rosseland mean opacities that requires significantly less CPU time. Second, the inclusion of shear mixing based on rotation profiles computed with the 2D stellar structure code ESTER is considered. We show predictions for the mode periods of these models covering stellar masses from 1.4 to 3.0${\rm M_\odot}$ across the main sequence (MS), computed for different metallicities. The morphology of the chemical mixing profile resulting from shear mixing in combination with atomic diffusion and radiative levitation does allow for mode trapping, while the diffusion coefficient in the outer envelope is large ($>10^{6}\,{\rm cm^2\,s^{-1}}$). Furthermore, we make predictions for the evolution of surface abundances for which radiative accelerations can be computed. We find that the N/C and C/O abundance ratios correlate with stellar age. We predict that these correlations are observable with precisions $\lesssim 0.1$ dex on these ratios, given that a precise age estimate can be made.

astro-ph.SR↗

arXiv2021

Seismology of Altair with MOST

Altair is the fastest rotating star at less than 10 parsecs from the Sun. Its precise modelling is a landmark for our understanding of stellar evolution with fast rotation, and all observational constraints are most welcome to better determine the fundamental parameters of this star. We wish to improve the seismic spectrum of Altair and confirm the $δ$-Scuti nature of this star. We used the photometric data collected by the Microvariability and Oscillations of STars (MOST) satellite in the form of a series of Fabry images to derive Altair light curves at four epochs, namely in 2007, 2011, 2012, and 2013. We first confirm the presence of $δ$-Scuti oscillations in the light curves of Altair. We extend the precision of some eigenfrequencies and add new ones to the spectrum of Altair, which now has 15 detected eigenmodes. The rotation period, which is expected at $\sim$7h46min from models reproducing interferometric data, seems to appear in the 2012 data set, but it still needs confirmation. Finally, Altair modal oscillations show noticeable amplitude variations on a timescale of 10 to 15 days, which may be the signature of a coupling between oscillations and thermal convection in the layer where the kappa-mechanism is operating.The Altair oscillation spectrum does not contain a large number of excited eigenmodes, which is similar to the fast rotating star HD220811. This supports the idea that fast rotation hinders the excitation of eigenmodes as already pointed out by theoretical investigations.

astro-ph.SR↗

arXiv2020

Stress-driven spin-down of a viscous fluid within a spherical shell

We investigate the linear properties of the steady and axisymmetric stress-driven spin-down flow of a viscous fluid inside a spherical shell, both within the incompressible and anelastic approximations, and in the asymptotic limit of small viscosities. From boundary layer analysis, we derive an analytical geostrophic solution for the 3D incompressible steady flow, inside and outside the cylinder $\mathcal{C}$ that is tangent to the inner shell. The Stewartson layer that lies on $\mathcal{C}$ is composed of two nested shear layers of thickness $O(E^{2/7})$ and $O(E^{1/3})$. We derive the lowest order solution for the $E^{2/7}$-layer. A simple analysis of the $E^{1/3}$-layer laying along the tangent cylinder, reveals it to be the site of an upwelling flow of amplitude $O(E^{1/3})$. Despite its narrowness, this shear layer concentrates most of the global meridional kinetic energy of the spin-down flow. Furthermore, a stable stratification does not perturb the spin-down flow provided the Prandtl number is small enough. If this is not the case, the Stewartson layer disappears and meridional circulation is confined within the thermal layers. The scalings for the amplitude of the anelastic secondary flow have been found to be the same as for the incompressible flow in all three regions, at the lowest order. However, because the velocity no longer conforms the Taylor-Proudman theorem, its shape differs outside the tangent cylinder $\mathcal{C}$, that is, where differential rotation takes place. Finally, we find the settling of the steady-state to be reached on a viscous time for the weakly, strongly and thermally unstratified incompressible flows. Large density variations relevant to astro- and geophysical systems, tend to slightly shorten the transient.

physics.flu-dyn↗

arXiv2019

Classical and general relativistic post-Keplerian effects in binary pulsars hosting fast rotating main sequence stars

We consider a binary system composed of a pulsar and a massive, fast rotating, highly distorted main sequence star as a potential scenario to dynamically put to the test certain post-Keplerian effects of both Newtonian and post-Newtonian nature. We numerically produce time series of the perturbations $Δ\left(δτ\right)$ of the Rømer-like, orbital component of the pulsar's time delay $δτ$ induced over 10 years by the pN gravitoelectric mass monopole, quadrupole, gravitomagnetic spin dipole and octupole accelerations along with the Newtonian quadrupolar one. We do not deal with the various propagation time delays due to the travelling electromagnetic waves. It turns out that, for a Be-type star with $M = 15\ \textrm{M}_\odot$, $R_\textrm{e} = 5.96\ \textrm{R}_\odot$, $ν= 0.203$, $S = 3.41\times 10^{45}\ \textrm{J}\ \textrm{s}$, $J_2 = 1.92\times 10^{-3}$ orbited by a pulsar with an orbital period $P_\textrm{b}\simeq 40-70\ \textrm{d}$, the classical oblateness-driven effects are at the $\lesssim 4-150\ \textrm{s}$ level, while the pN shifts are of the order of $\lesssim 1.5-20\ \textrm{s}\ \left(GMc^{-2}\right)$, $\lesssim 10-40\ \textrm{ms}\ \left(GMR^2_\textrm{e} J_2 c^{-2}\right)$, $\lesssim 0.5 - 6\ \textrm{ms}\ \left(GSc^{-2}\right)$, $\lesssim 5 - 20\ μ\textrm{s}\ \left(GSR^2_\textrm{e} \varepsilon^2 c^{-2}\right)$, depending on their orbital configuration. The root-mean-square (rms) timing residuals $σ_τ$ of almost all the existing non-recycled, non-millisecond pulsars orbiting massive, fast rotating main sequence stars are $\lesssim\textrm{ms}$. Thus, such kind of binaries have the potential to become interesting laboratories to measure, or, at least, constrain, some Newtonian and post-Newtonian key features of the distorted gravitational fields of the fast rotating stars hosted by them [Abridged].

gr-qc↗

arXiv2018

The evolved fast rotator Sargas. Stellar parameters and evolutionary status from VLTI/PIONIER and VLT/UVES

Gravity darkening (GD) and flattening are important consequences of stellar rotation. The precise characterization of these effects across the HRD is crucial to a deeper understanding of stellar structure and evolution. We seek to characterize such important effects on Sargas, an evolved, fast-rotating, intermediate-mass star, located in a region of the HRD where they have never been directly measured as far as we know. We use our numerical model CHARRON to analyze interferometric (VLTI/PIONIER) and spectroscopic (VLT/UVES) observations through a MCMC model-fitting procedure. The visibilities and closure phases from the PIONIER data are particularly sensitive to rotational flattening and GD. Adopting the Roche approximation, we investigate two GD models: (1) the beta-model (classical von Zeipel's law), and (2) the omega-model. Using this approach we measure several physical parameters of Sargas, namely, equatorial radius, mass, equatorial rotation velocity, mean Teff, inclination and position angle of the rotation axis, and beta. In particular, we show that the measured beta leads to a surface flux distribution equivalent to the one given by the omega-model. Thanks to our results, we also show that Sargas is most probably located in a rare and interesting region of the H-R diagram: within the Hertzsprung gap and over the hot edge of the instability strip. These results show once more the power of optical/IR long-baseline interferometry, combined with high-resolution spectroscopy, to directly measure fast-rotation effects and stellar parameters, in particular GD. As was the case for a few fast rotators previously studied by interferometry, the omega-model provides a physically more profound description of Sargas' GD, without the need of a beta exponent.

astro-ph.SR↗

arXiv2018

The Sun's Supergranulation

Supergranulation is a fluid-dynamical phenomenon taking place in the solar photosphere, primarily detected in the form of a vigorous cellular flow pattern with a typical horizontal scale of approximately 30--35~megameters, a dynamical evolution time of 24--48~h, a strong 300--400~m/s (rms) horizontal flow component and a much weaker 20--30~m/s vertical component. Supergranulation was discovered more than sixty years ago, however, explaining its physical origin and most important observational characteristics has proven extremely challenging ever since, as a result of the intrinsic multiscale, nonlinear dynamical complexity of the problem concurring with strong observational and computational limitations. Key progress on this problem is now taking place with the advent of 21st-century supercomputing resources and the availability of global observations of the dynamics of the solar surface with high spatial and temporal resolutions. This article provides an exhaustive review of observational, numerical and theoretical research on supergranulation, and discusses the current status of our understanding of its origin and dynamics, most importantly in terms of large-scale nonlinear thermal convection, in the light of a selection of recent findings.

astro-ph.SR↗

arXiv2018

Mode identification in rapidly rotating stars from BRITE data

Apart from recent progress in Gamma Dor stars, identifying modes in rapidly rotating stars is a formidable challenge due to the lack of simple, easily identifiable frequency patterns. As a result, it is necessary to look to observational methods for identifying modes. Two popular techniques are spectroscopic mode identification based on line profile variations (LPVs) and photometric mode identification based on amplitude ratios and phase differences between multiple photometric bands. In this respect, the BRITE constellation is particularly interesting as it provides space-based multi-colour photometry. The present contribution describes the latest developments in obtaining theoretical predictions for amplitude ratios and phase differences for pulsation modes in rapidly rotating stars. These developments are based on full 2D non-adiabatic pulsation calculations, using models from the ESTER code, the only code to treat in a self-consistent way the thermal equilibrium of rapidly rotating stars. These predictions are then specifically applied to the BRITE photometric bands to explore the prospects of identifying modes based on BRITE observations.

astro-ph.SR↗

arXiv2018

Axisymmetric inertial modes in a spherical shell at low Ekman numbers

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin(π/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

physics.flu-dyn↗