Armand Leclerc

Context . Thanks to helioseismology, the rotation profile of the Sun has been measured with great precision down to 20% of its total radius. This rotation profile is used as a calibration to infer the rotation of other stars as well as a test of angular momentum transport theory in stellar interiors. However, the deepest 20% of the layers remain out of reach of current observations, preventing astronomers from discriminating between currently competing angular momentum transport mechanisms. Aims . The main obstacle is that no global oscillation modes sensitive to rotation ( ℓ > 0) reaching the solar core have been detected yet, as non-radial p modes cannot reach it and g modes are evanescent at the surface and still elude detection. In this work, we propose and examine a new method to constrain the rotation of the core of the Sun, which does not require direct observation of solar g modes. Methods . Our method was based on a recent prediction that g modes in the radiative interior couple with f modes in the outer parts of the star. These mixed f – g are at the same time sensitive to the rotation of the core and able to reach the surface. These modes can be used together to build average inversion kernels and perform an inversion of the rotation of the solar core. Results . We find that the oscillation spectrum of the Sun should present six mixed f – g modes that can be used to measure the rotation rate of the Sun at r = 0.07 and 0.2 R _⊙ . We estimate that the uncertainty on the measurements should be small enough to distinguish between competing scenarios of angular momentum transport in the Sun.

Mixed modes are observed in many low-mass evolved stars. They provide information about core rotation rates of these stars, which are lower than predicted by stellar evolution models. The mixed modes themselves have been invoked as an angular momentum (AM) transport mechanism, but estimating their transport efficiency requires knowledge of their amplitudes. We constrain, for the first time, the mixed-mode amplitudes in 2D hydrodynamical simulations of a 1.3 M _⊙ red giant using the code MUSIC . We perform two simulations with outer radial truncations at fractional radii r _o / r _⋆ = 0.90 and 0.98. We compare the modes in the simulation with those found using both GYRE and a Dedalus eigenvalue solver. Excellent frequency agreement is found for all p -dominated modes, with minor discrepancies for g -dominated modes, especially in the frequency range [60, 240] μ Hz. We find excellent eigenfunction agreement for all modes except those in this frequency range. According to empirical predictions, the largest kinetic energies are located around ν max = 312.8 μ Hz , but in both simulations, the modes with frequencies of ν < 50 μ Hz have the largest kinetic energies. In the simulation with r / r _⋆ = 0.98, the simulated modes have extrapolated surface velocities comparable to the empirical predictions, with the highest surface velocities in a bell-shaped curve peaking around ν = 700 μ Hz. The extrapolated surface velocities of the low-frequency modes are small and thus hard to observe, but their large kinetic energies deeper in the interior could significantly impact AM transport, which has not yet been investigated.

Aims. We investigate signatures of pressure extrema on global oscillations in discs. Methods. We used the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients. We highlight the influence of a previously unrecognised epicyclic–acoustic frequency, and we derive an analytical criterion for the existence of a branch of modes transiting between the inertial and the pressure bands, known as topological modes. Results. We find that pressure extrema consist of wave guides in which such topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at any frequency, allowing it to resonate with any temporal forcing. Conversely, the fundamental mode associated with a pressure dip propagates at any vertical phase velocity. These specific features make them attractive candidates for future discoseismology.

Topologie des ondes pour l'astérosismologie: L'objectif général de cette thèse a été de diriger la lumière de la topologie des ondes vers une sélection de problèmes ondulatoires en milieux astrophysiques fluides — une approche encore inexplorée jusqu'à présent. Deux grandes questions ont guidé ce travail. La première consistait à déterminer si certaines ondes, dans certains milieux astrophysiques, sont soumises à des contraintes topologiques sous-jacentes. Ces ondes seraient alors dites topologiquement non triviales, au sens défini et discuté au Chapitre II. Les travaux présentés ici portent principalement sur les ondes se propageant dans les intérieurs planétaires et stellaires, dont la grande diversité de structures, de dynamiques et de processus thermodynamiques en fait un terrain fertile pour l'émergence de propriétés topologiques. La seconde question fondamentale était de comprendre ce que l'on peut apprendre sur les ondes — et sur les objets dans lesquels elles se propagent — à partir de leurs propriétés topologiques. En d'autres termes : que peut-on prédire du fait qu'une onde est topologiquement non triviale ? Les outils de la topologie permettent d'imposer des contraintes sur la localisation spatiale des modes, leur robustesse face aux perturbations, et parfois sur leur sens de propagation, notamment dans le cas de modes unidirectionnels, dits modes topologiques. Lorsqu'ils sont identifiés, ces modes possèdent des propriétés uniques qui peuvent être mises à profit pour mieux comprendre certaines phénoménologies propres aux objets astrophysiques qui les hébergent. Par exemple, les modes topologiques identifiés par Delplace, Marston et Venaille en 2017 sont connus pour se propager uniquement vers l'ouest dans l'océan équatorial. Cette impossibilité de propagation vers l'est est liée au phénomène El Niño — cette vague de chaleur partant de l'Asie vers les Amériques. La topologie des ondes fournit un cadre pertinent pour analyser si des ondes peuvent soutenir un tel comportement unidirectionnel.Les ondes en milieux astrophysiques sont étudiées depuis longtemps, car elles constituent à la fois un outil d'observation indirecte des corps célestes (comme les étoiles et les planètes), et un acteur essentiel de leur dynamique. Les instabilités linéaires, le transport d'énergie et de moment angulaire influencent la formation et l'évolution des étoiles, des disques d'accrétion et protoplanétaires, des galaxies, des planètes et de leurs atmosphères. Chaque communauté travaillant sur ces objets étudie également les ondes qui s'y propagent. La topologie des ondes, en tant que méthode d'analyse nouvelle et complémentaire, reste encore peu exploitée dans ce contexte. Développer cette approche ouvre ainsi la voie à une meilleure compréhension de nombreux phénomènes astrophysiques. L'organisation des chapitres de ce manuscrit ne suit pas la chronologie des travaux effectués au cours de cette thèse, mais a été pensée pour proposer une progression dans la complexité de l'analyse topologique. Le Chapitre III introduit un problème d'ondes inertielles avec un seul monopôle, où un flux spectral clair est observé. Le Chapitre IV traite de la dynamique de trois monopôles susceptibles de fusionner, avec la possibilité d'une disparition du flux spectral. Le Chapitre V explore les ondes dans les intérieurs stellaires, mettant en évidence le rôle de la géométrie sphérique et la présence de deux monopôles de charges opposées. Le Chapitre VI se penche sur ces ondes à travers une simulation hydrodynamique de l'intérieur solaire, où les modes topologiques apparaissent relativement délocalisés — un comportement peu commun dans la littérature en physique topologique. Le Chapitre VII approfondit également l'étude de l'intérieur solaire, en soulignant cette fois l'importance de la phase de Berry associée aux modes acoustiques. Enfin, les Chapitres VIII et IX abordent les instabilités linéaires sous l'angle de la topologie non hermitienne et du rôle des symétries discrètes.

This study investigates the signatures of pressure extrema on global oscillations in discs. To this end, we use the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients. We highlight the influence of a previously unrecognized epicyclic-acoustic frequency and derive an analytical criterion for the existence of a branch of modes transiting between the inertial and the pressure bands. We find that pressure extrema consist of wave guides in which such topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at all frequencies, allowing it to resonate with any temporal forcing, while the mode associated with a pressure gap propagates at a fixed frequency, propagates with arbitrary vertical phase velocity. These specific features make them attractive candidates for future discoseismology.

Helioseismology has revolutionized our understanding of the Sun by analyzing its global oscillation modes. However, the solar core remains elusive, limiting a full understanding of its evolution. In this work, we study a previously unnoticed global oscillation mode of the Sun using a fully compressible hydrodynamical simulation of the solar interior and assess that it is a mixed f/g-mode with a period of about 1 hr. This is the first global stellar hydrodynamics simulation to successfully couple compressible and gravity modes. To understand this coupling, we invoke a recent theory on the nature of f-modes seen through the prism of wave topology, characterizing their ability to propagate deep into stellar interiors. We demonstrate that the mixed f/g-mode is highly sensitive to the core’s rotation rate, providing a new promising pathway to explore the Sun’s core.

An analytic expression for the frequencies of standing waves in stars, applicable to any radial order n, is derived from ray-tracing equations by the mean of Wigner–Weyl calculus. A correction to previous formulas currently employed in asteroseismology is identified as the Berry phase, which accounts for the vectorial nature of wave propagation in stars. Accounting for this quantity significantly improves upon previous laws for low n modes of the Sun, and we show that the Berry phase is indeed present in the available observational data of solar modes. This phase is due to inhomogeneities of the medium.

An analytic expression for the frequencies of standing waves in stars, applicable to any radial order n, is derived from ray-tracing equations by the mean of Wigner-Weyl calculus. A correction to previous formulas currently employed in asteroseismology is identified as the Berry phase, which accounts for the vectorial nature of wave propagation in stars. Accounting for this quantity significantly improves upon previous laws for low n modes of the Sun, and we show that the Berry phase is indeed present in the available observational data of solar modes. This phase is due to inhomogeneities of the medium.