Quantifying tidal interactions in close-in two-body systems is of prime interest since they have a crucial impact on the architecture and on the rotational history of the bodies. Various studies have shown that the dissipation of tides in either body is very sensitive to its structure and to its dynamics, like differential rotation which exists in the outer convective enveloppe of solar-like stars and giant gaseous planets. In particular, tidal waves may strongly interact with zonal flows at the so-called corotation resonances, where the wave's Doppler-shifted frequency cancels out. We aim to provide a deep physical understanding of the dynamics of tidal inertial waves at corotation resonances, in the presence of differential rotation profiles typical of low-mass stars and giant planets. By developping an inclined shearing box, we investigate the propagation and the transmission of free inertial waves at corotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model. We find that inertial waves can be either fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux to the fluid when damped at critical levels, or they can extract action flux to the fluid when amplified at critical levels. Both situations could lead to significant angular momentum exchange between the tidally interacting bodies.
Motivated by understanding the dynamics of stellar and planetary interiors, we have performed a set of direct numerical simulations of Boussinesq convection in a rotating full sphere. The domain is internally heated with fixed temperature and stress-free boundary conditions, but fixed heat flux and no-slip boundary conditions are also briefly considered. We particularly focus on the large-scale coherent structures and the mean zonal flows that can develop in the system. At Prandtl number of unity, as the thermal forcing (measured by the Rayleigh number) is increased above the value for the ons…
In close two-body astrophysical systems, such as binary stars or Hot Jupiter systems, tidal interactions often drive dynamical evolution on secular timescales. Many host stars and presumably giant gaseous planets feature a convective envelope. Tidal flows generated therein by the tidal potential of the companion can be dissipated through viscous friction, leading to the redistribution and exchange of angular momentum within the convective shell and with the companion, respectively. In the tightest systems, nonlinear effects are likely to have a significant impact on the tidal dissipation and t…
Tidally-excited inertial waves in stellar convective regions are a key mechanism for tidal dissipation in stars and therefore the evolution of close-in binary or planetary systems. As a first step, we explore here the impact of latitudinal differential rotation on the properties of free inertial modes and identify the different families of modes. We show that they differ from the case of solid-body rotation. Using an analytical approach as well as numerical calculations, we conclude that critical layers (where the Doppler-shifted frequency vanishes) could play a very important role for tidal d…
Tidal interactions in close star-planet or binary star systems may excite inertial waves (their restoring force is the Coriolis force) in the convective region of the stars. The dissipation of these waves plays a prominent role in the long-term orbital and rotational evolution of the bodies involved. If the primary star rotates as a solid body, inertial waves have a Doppler-shifted frequency restricted to the range $[-2Ω, 2Ω]$ ($Ω$ being the angular velocity of the star), and they can propagate in the entire convective region. However, turbulent convection can sustain differential rotation wit…
In close exoplanetary systems, tidal interactions drive orbital and spin evolution of planets and stars over long timescales. Tidally-forced inertial waves (restored by the Coriolis acceleration) in the convective envelopes of low-mass stars and giant gaseous planets contribute greatly to the tidal dissipation when they are excited and subsequently damped (e.g. through viscous friction), especially early in the life of a system. These waves are known to be subject to nonlinear effects, including triggering differential rotation in the form of zonal flows. In this study, we use a realistic tida…
Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelope, with an equatorial acceleration (as in the Sun) or deceleration, which can modify the waves' propagation properties. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assu…
Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. Their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. If the star is assumed to be rotating as a solid-body, the waves' Doppler-shifted frequency is restricted to $[-2 Ω, 2 Ω]$ ($Ω$ being the angular velocity of the star) and they can propagate in the entire convective region. However, turbulent convection can sustain differential rotation with an equatorial acceleration (as in the Sun) or deceleration that may modify waves propagation.…
Parametrisation of the wave-zonal flow interactions taking into account the full Coriolis acceleration. The necessity of going beyond the traditional approximation in the sub-inertial regime and in weakly stratified regions
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM（巴黎萨克雷大学，巴黎西岱大学，法国原子能和替代能源委员会，法国国家科学研究中心，天体物理学与分子星际介质实验室）
Star-planet tidal interactions play a significant role in the dynamical evolution of close-in planetary systems. We investigate the propagation and dissipation of tidal inertial waves in a stellar/planetary convective region. We take into account a latitudinal differential rotation for the background flow, similar to what is observed in the envelope of low-mass stars like the Sun. Previous works have shown that differential rotation significantly alters the propagation and dissipation properties of inertial waves. In particular, when the Doppler-shifted tidal frequency vanishes in the fluid, a…
The solar convection zone rotates differentially, with its equatorial region rotating more rapidly than the polar regions. This form of differential rotation, also observed in many other low-mass stars, is understood to arise when Coriolis effects are stronger than those associated with buoyant driving of the convection. When buoyancy dominates, a so-called antisolar state of differential rotation results, characterized by rapidly-rotating poles and a slow equator. The transition between these two states has been shown to occur when the intensity of these two forces is roughly equal or, equiva…
Oscillations have been detected in a variety of stars, including intermediate- and high-mass main sequence stars. While many of these stars are rapidly and differentially rotating, the effects of rotation on oscillation modes are poorly known. In this communication we present a first study on axisymmetric gravito-inertial modes in the radiative zone of a differentially rotating star. These modes probe the deep layers of the star around its convective core. We consider a simplified model where the radiative zone of a star is a linearly stratified rotating fluid within a spherical shell, with di…
Under the assumption of thermal wind balance and effective entropy mixing in constant rotation surfaces, the isorotational contours of the solar convective zone may be reproduced with great fidelity. Even at this early stage of development, this helioseismology fit may be used to put a lower bound on the midlatitude {\em radial} solar entropy gradient, which in good accord with standard mixing length theory. In this paper, we generalize this solar calculation to fully convective stars (and potentially planets), retaining the assumptions of thermal wind balance and effective entropy mixing in i…
We describe a 3D finite-difference spherical anelastic MHD (FSAM) code for modeling the subsonic dynamic processes in the solar convective envelope. A comparison of this code with the widely used global spectral anlastic MHD code, ASH (Anelastic Spherical Harmonics), shows that FSAM produces convective flows with statistical properties and mean flows similar to the ASH results. Using FSAM, we first simulate the rotating solar convection in a partial spherical shell domain and obtain a statistically steady, giant-cell convective flow with a solar-like differential rotation. We then insert buoya…
Magnetic fields of low-mass stars and planets are thought to originate from self-excited dynamo action in their convective interiors. Observations reveal a variety of field topologies ranging from large-scale, axial dipole to more structured magnetic fields. In this article, we investigate more than 70 three-dimensional, self-consistent dynamo models obtained by direct numerical simulations. The control parameters, the aspect ratio and the mechanical boundary conditions have been varied to build up this sample of models. Both, strongly dipolar and multipolar models have been obtained. We show …
The relation of period spacing ($ΔP$) versus period ($P$) of dipole prograde g modes is known to be useful to measure rotation rates in the g-mode cavity of rapidly rotating $γ$ Dor and slowly pulsating B (SPB) stars. In a rapidly rotating star, an inertial mode in the convective core can resonantly couple with g modes propagative in the surrounding radiative region. The resonant coupling causes a dip in the $P$-$ΔP$ relation, distinct from the modulations due to the chemical composition gradient. Such a resonance dip in $ΔP$ of prograde dipole g modes appears around a frequency corresponding …
We present numerical simulations, using two complementary setups, of rotating Boussinesq thermal convection in a three-dimensional Cartesian geometry with misaligned gravity and rotation vectors. This model represents a small region at a non-polar latitude in the convection zone of a star or planet. We investigate the effects of rotation on the bulk properties of convection at different latitudes, focusing on determining the relation between the heat flux and temperature gradient. We show that our results may be interpreted using rotating mixing length theory (RMLT). The simplest version of RM…
The precise mechanism that forms jets and large-scale vortices on the giant planets is unknown. An inverse cascade has been suggested. Alternatively, energy may be directly injected by small-scale convection. Our aim is to clarify whether an inverse cascade feeds zonal jets and large-scale eddies in a system of rapidly rotating, deep, geostrophic spherical-shell convection. We analyze the nonlinear scale-to-scale transfer of kinetic energy in such simulations as a function of the azimuthal wave number, m. We find that the main driving of the jets is associated with upscale transfer directly fr…
We study rotating thermal convection in spherical shells. We base our analysis on a set of about 450 direct numerical simulations of the (magneto)hydrodynamic equations under the Boussinesq approximation. The Ekman number ranges from $10^{-3}$ to $10^{-5}$. The supercriticality of the convection reaches about 1000 in some models. Four sets of simulations are considered: non-magnetic simulations and dynamo simulations with either free-slip or no-slip flow boundary conditions. The non-magnetic setup with free-slip boundaries generates the strongest zonal flows. Both non-magnetic simulations with…
Anti-solar differential rotation profiles have been found for decades in numerical simulations of convective envelopes of solar-type stars. These profiles are characterized by a slow equator and fast poles (i.e., reversed with respect to the Sun) and have been found in simulations for high Rossby numbers (slow rotators). Rotation profiles like this have been reported observationally in evolved stars, but have never been unambiguously observed for cool solar-type stars on the main sequence. In this context, detecting this regime in main-sequence solar-type stars would improve our understanding …
We study differential rotation in late-stage shell convection in a 3D hydrodynamic simulation of a rapidly rotating $16M_\odot$ helium star with a particular focus on the convective oxygen shell. We find that the oxygen shell develops a quasi-stationary pattern of differential rotation that is neither described by uniform angular velocity as assumed in current stellar evolution models of supernova progenitors, nor by uniform specific angular momentum. Instead, the oxygen shell develops a positive angular velocity gradient with faster rotation at the equator than at the pole by tens of percent.…
We present the first global 3D simulations of thermal convection in the oblate envelopes of rapidly-rotating solar-type stars. This has been achieved by exploiting the capabilities of the new Compressible High-ORder Unstructured Spectral difference (CHORUS) code. We consider rotation rates up to 85\% of the critical (breakup) rotation rate, which yields an equatorial radius that is up to 17\% larger than the polar radius. This substantial oblateness enhances the disparity between polar and equatorial modes of convection. We find that the convection redistributes the heat flux emitted from the …
Stars on the lower main sequence (F-type through M-type) have substantial convective envelopes beneath their stellar photospheres. Convection in these regions can couple with rotation to build global-scale structures that may be observable by interferometers that can resolve stellar disks. Here I discuss predictions emerging from 3D MHD simulations for solar-type stars with the anelastic spherical harmonic (ASH) code and how these predictions may be observationally tested. The zonal flow of differential rotation is likely the most easily observable signature of dynamics occurring deep within t…
We present an extensive study of Boussinesq thermal convection including a temperature-dependent internal heating source, based on numerical three-dimensional simulations. The temperature dependence mimics triple-$α$ nuclear reactions and the fluid geometry is a rotating spherical shell. These are key ingredients for the study of convective accreting neutron star oceans. A dimensionless parameter ${\rm Ra}_n$, measuring the relevance of nuclear heating, is defined. We explore how flow characteristics change with increasing ${\rm Ra}_n$ and give an astrophysical motivation. The onset of convect…
We seek to characterize how the change of global rotation rate influences the overall dynamics and large scale flows arising in the convective envelopes of stars covering stellar spectral types from early G to late K. We do so through numerical simulations with the ASH code, where we consider stellar convective envelopes coupled to a radiative interior with various global properties. As solar-like stars spin down over the course of their main sequence evolution, such change must have a direct impact on their dynamics and rotation state. We indeed find that three main states of rotation may exi…
The Sun rotates differentially with a fast equator and slow pole. Convection in the solar interior is thought to maintain the differential rotation. However, although many numerical simulations have been conducted to reproduce the solar differential rotation, previous high-resolution calculations with solar parameters fall into the anti-solar (fast pole) differential rotation regime. Consequently, we still do not know the true reason why the Sun has a fast-rotating equator. While the construction of the fast equator requires a strong rotational influence on the convection, the previous calcula…
To explore the physics of large-scale flows in solar-like stars, we perform 3D anelastic simulations of rotating convection for global models with stratification resembling the solar interior. The numerical method is based on an implicit large-eddy simulation approach designed to capture effects from non-resolved small scales. We obtain two regimes of differential rotation, with equatorial zonal flows accelerated either in the direction of rotation (solar-like) or in the opposite direction (anti-solar). While the models with the solar-like differential rotation tend to produce multiple cells o…
The solar convection zone exhibits a strong level of differential rotation, whereby the rotation period of the polar regions is about 25-30% longer than the equatorial regions. The Coriolis force associated with these zonal flows perpetually "pumps" the convection zone fluid, and maintains a quasi-steady circulation, poleward near the surface. What is the influence of this meridional circulation on the underlying radiative zone, and in particular, does it provide a significant source of mixing between the two regions? In Paper I, we began to study this question by assuming a fixed meridional f…
We analyze the simulation result shown in Hotta & Kusano, 2021 in which the solar-like differential rotation is reproduced. The Sun is rotating differentially with the fast equator and the slow pole. It is widely thought that the thermal convection maintains the differential rotation, but recent high-resolution simulations tend to fail to reproduce the fast equator. This fact is an aspect of one of the biggest problems in solar physics called the convective conundrum. Hotta & Kusano, 2021 succeed in reproducing the solar-like differential rotation without using any manipulation with unpreceden…
Rotation is one of the key physical mechanisms that deeply impact the evolution of stars. Helio- and asteroseismology reveal a strong extraction of angular momentum from stellar radiation zones over the whole Hertzsprung-Russell diagram. Turbulent transport in differentially rotating stably stratified stellar radiation zones should be carefully modeled and its strength evaluated. Stratification and rotation imply that this turbulent transport is anisotropic. Only phenomenological prescriptions have been proposed for the transport in the horizontal direction, which however constitutes a corners…
(abridged) Context: Late-type stars such as the Sun rotate differentially due to the interaction of turbulent convection and rotation. Aims: The aim of the study is to investigate the effects of the thermal Prandtl number on the transition from anti-solar (slow equator, fast poles) to solar-like (fast equator, slow poles) differential rotation. Methods: Three-dimensional hydrodynamic and magnetohydrodynamic simulations in semi-global spherical wedge geometry are used to model convection zones of solar-like stars. Results: The overall convective velocity amplitude increases as the Prandtl numbe…
