We use a simplified model to study wave reflection and transmission at interface of convective region and stably stratified region (e.g. radiative zone in star or stratification layer in gaseous planet). Inertial wave in convective region and gravito-inertial wave in stably stratified region are considered. We begin with polar area and then extend to any latitude. Six cases are discussed (see Table 1), and in Case 2 both waves co-exist in both regions. Four configurations are further discussed for Case 2. The angles and energy ratios of wave reflection and transmission are calculated. It is found that wave propagation and transmission depend on the ratio of buoyancy frequency to rotational frequency. In a rapidly rotating star or planet wave propagates across interface and most of energy of incident wave is transmitted to the other region, but in a slowly rotating star or planet wave transmission is inhibited.
Stably stratified layers are present in stellar interiors (radiative zones) as well as planetary interiors - recent observations and theoretical studies of the Earth's magnetic field seem to indicate the presence of a thin, stably stratified layer at the top of the liquid outer core. We present direct numerical simulations of this region, which is modelled as an axisymmetric spherical Couette flow for a stably stratified fluid embedded in a dipolar magnetic field. For strong magnetic fields, a super-rotating shear layer, rotating nearly 30% faster than the imposed rotation rate difference betw…
In this paper, the properties of stably stratified turbulence (SST) and turbulent thermal convection are contrasted. A key difference between these flows is the sign of the kinetic energy feed by buoyancy, $\mathcal{F}_B$. For SST, $\mathcal{F}_B < 0$ due to its stable nature; consequently, the kinetic energy flux $Π_u(k)$ decreases with wavenumber $k$ that leads to a steep kinetic energy spectrum, $E_u(k) \sim k^{-11/5}$. Turbulent convection is unstable, hence $\mathcal{F}_B > 0$ that leads to an increase of $Π_u(k)$ with $k$; this increase however is marginal due to relatively weak buoyancy…
We present numerical investigations into three principal properties of the small-scale dynamo in stably stratified turbulence: the onset criterion, the growth rate, and the nature of the magnetic field anisotropy in the kinematic regime. The results suggest that all three dynamo properties are controlled by the scale separation between the Ozmidov scale and the viscous or resistive scale. In addition to the critical magnetic Reynolds number, this allows for the definition of critical buoyancy and magnetic buoyancy Reynolds numbers for stratified small-scale dynamo onset in the high and low mag…
We study fully compressible convection in the context of plane-parallel, polytropically stratified atmospheres. We perform a suite of 2D and 3D simulations in which we vary the initial superadiabaticity ($ε$) and the Rayleigh number (Ra) while fixing the initial density stratification, aspect ratio, and Prandtl number. The evolved heat transport, quantified by the Nusselt number (Nu), follows scaling relationships similar to those found in the well-studied, incompressible Rayleigh-Bénard problem. This scaling holds up in both 2D and 3D and is not appreciably affected by the magnitude of $ε$.
We extend the notion of a shell model to stratified systems, and propose one that represents stratified, nonmagnetic, nonrotating convection at low Mach number. Motivated by profiles of background stratification that support convection in stars such as the Sun, we study numerical solutions corresponding to a highly unstable layer above a mildly unstable layer. We find that at low Prandtl number, convective amplitudes decrease with depth in the lower layer. This suggests that the suppression of convection in the deeper layers of the Sun's convection zone (the convective conundrum) can be addres…
The topic of turbulent transport in a stellar radiative zone is vast and poorly understood. Many physical processes can potentially drive turbulence in stellar radiative zone but the limited observational constraints and the uncertainties in modelling turbulent transport make it difficult to identify the most relevant one. Here, we focus on the effect of stable stratification on the radial turbulent transport of chemicals and more particularly on the case where the turbulence is driven by a radial shear. Results of numerical simulations designed to test phenomenological models of turbulent tra…
We have advanced the energy and flux budget (EFB) turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGW) and the shear-free stably stratified turbulence. This theory is based on the budget equation for the total (kinetic plus potential) energy of IGW, the budget equations for the kinetic and potential energies of fluid turbulence, and turbulent fluxes of potential temperature for waves and fluid flow. The waves emitted at a certain level, propagate upward, and the losses of wave energy cause the production of turbulence energy. We demonstrate …
We extend the scaling relations of stably stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $PrRb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of t…
Long-lived magnetic fields are known to exist in upper main-sequence stars, white dwarfs, and neutron stars. In order to explore possible equilibrium configurations of the magnetic field inside these stars, we have performed 3D-magnetohydrodynamic simulations of the evolution of initially random magnetic fields in stably stratified and barotropic stars with an ideal-gas equation of state using the {\sc Pencil Code}, a high-order finite-difference code for compressible hydrodynamic flows in the presence of magnetic fields. In barotropic (isentropic) stars, we confirm previous results in the sen…
Convective dynamo simulations are performed in local Cartesian geometry. We report the first successful simulation of a large-scale oscillatory dynamo in rigidly rotating convection without stably stratified layers. A key requirement for exciting the large-scale dynamo is a sufficiently long integration time comparable to the ohmic diffusion time. By comparing two models with and without stably stratified layers, their effect on the large-scale dynamo is also studied. The spatiotemporal evolution of the large-scale magnetic field is similar in both models. However, it is intriguing that the ma…
In this paper we advance physical background of the energy- and flux-budget turbulence closure based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time-scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth rotation. In accordance to modern experimental evidence, the closure implies maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the tw…
We present a numerical study of turbulence and dynamo action in stratified shearing boxes with zero magnetic flux. We assume that the fluid obeys the perfect gas law and has finite (constant) thermal diffusivity. We choose radiative boundary conditions at the vertical boundaries in which the heat flux is propor- tional to the fourth power of the temperature. We compare the results with the corresponding cases in which fixed temperature boundary conditions are applied. The most notable result is that the formation of a fully convective state in which the density is nearly constant as a function…
We report the first successful simulation of spontaneous formation of surface magnetic structures from a large-scale dynamo by strongly-stratified thermal convection in Cartesian geometry. The large-scale dynamo observed in our strongly-stratified model has physical properties similar to those in earlier weakly-stratified convective dynamo simulations, indicating that the $α^2$-type mechanism is responsible for it. Additionally to the large-scale dynamo, we find that large-scale structures of the vertical magnetic field are spontaneously formed in the convection zone surface only for the case …
In this paper, we study the inertial and gravity wave transmissions near the radiative-convective boundaries in the {\it f}-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or In this paper, we study inertial and gravity wave transmissions near radiative-convective boundaries on the {\it f}-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or the other way around. It has been found that waves prefer to survive at low latitudes when…
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…
We study low frequency waves that propagate in a region of layered semi-convection. Layered semi-convection is predicted to be present in stellar and planetary interiors and can significantly modify the rate of thermal and compositional mixing. We derive a series of analytical dispersion relations for plane-parallel layered semi-convection in the Boussinesq approximation using a matrix transfer formalism. We find that like a continuously stratified medium, a semi-convective staircase -- in which small convective regions are separated by sharp density jumps -- supports internal gravity waves (g…
The dynamics of stably stratified stellar radiative zones is of considerable interest due to the availability of increasingly detailed observations of Solar and stellar interiors. This article reports the first non-axisymmetric and time-dependent simulations of flows of anelastic fluids driven by baroclinic torques in stably stratified rotating spherical shells -- a system serving as an elemental model of a stellar radiative zone. With increasing baroclinicity a sequence of bifurcations from simpler to more complex flows is found in which some of the available symmetries of the problem are bro…
Typical flows in stellar interiors are much slower than the speed of sound. To follow the slow evolution of subsonic motions, various sound-proof equations are in wide use, particularly in stellar astrophysical fluid dynamics. These low-Mach number equations include the anelastic equations. Generally, these equations are valid in nearly adiabatically stratified regions like stellar convection zones, but may not be valid in the sub-adiabatic, stably stratified stellar radiative interiors. Understanding the coupling between the convection zone and the radiative interior is a problem of crucial i…
Recent observations of Jupiter and Saturn suggest that heavy elements may be diluted in the gaseous envelope, providing a compositional gradient that could stabilise ordinary convection and produce a stably-stratified layer near the core of these planets. This region could consist of semi-convective layers with a staircase-like density profile, which have multiple convective zones separated by thin stably-stratified interfaces, as a result of double-diffusive convection. These layers could have important effects on wave propagation and tidal dissipation that have not been fully explored. We an…
A monomodal model for stellar and planetary convection is derived for the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale as a function of rotation rate as well as with thermal and viscous diffusivities. The convection model is used as a boundary condition for a linearization of the equations of motion in the transition region between convectively unstable and stably-stratified regions, yielding the depth to which convection penetrates into the stable region and establishing a relationship between that depth and the local convective Rossby number, di…
The present study aims at studying the flow and field produced by a stellar radiative zone which is initially made to rotate differentially in the presence of a large-scale poloidal magnetic field threading the whole domain. We focus both on the axisymmetric configurations produced by the initial winding-up of the magnetic field lines and on the possible instabilities of those configurations. We aim in particular at assessing the role of the stratification at stabilising the system. We perform 2D and 3D global Boussinesq numerical simulations started from an initial radial or cylindrical diffe…
This letter expands the stability criterion for radially stratified, vertically {unstratified} accretion disks incorporating thermal relaxation. We find a linear amplification of epicyclic oscillations in these disks that depends on the effective cooling time, i.e. an overstability. The growth rates of the overstability vanish for both extreme cases, e.g. infinite cooling time and instantaneous cooling, i.e. the adiabatic and fully isothermal cases. However, for thermal relaxation times $τ$ on the order of the orbital frequency, $τΩ\sim 1$, modes grow at a rate proportional to the square of th…
Planetary rotation organizes fluid motions into coherent, long-lived swirls, known as large-scale vortices (LSVs), which play an important role in the dynamics and long-term evolution of geophysical and astrophysical fluids. Here, using direct numerical simulations, we show that LSVs in rapidly rotating mixed convective and stably stratified fluids, which approximates the two-layer, turbulent-stratified dynamics of many geophysical and astrophysical fluids, have a generic shape and that their size can be predicted. We show that LSVs emerge in the convection zone from upscale energy transfers a…
Nonlinear interaction between dynamo-generated magnetic fields, mean flows and internal gravity waves in stellar stably-stratified layers: From 3D to 1D
Laboratoire de Physique de l’Ecole normale superieure ENS Universite PSL CNRS Sorbonne Universite Universite de Paris（巴黎高等师范学院物理实验室）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) Northwestern University（西北大学天体物理学跨学科探索与研究中心）；Laboratoire d’Instrumentation et de Recherche en Astrophysique (LIRA) Observatoire de Paris Université PSL Sorbonne Université Université Paris Cité CY Cergy Paris Université CNRS（巴黎天文台天体物理仪器与研究实验室）；Université Paris-Saclay CNRS Institut d’astrophysique spatiale（巴黎萨克雷大学空间天体物理研究所）；Departament d’Astronomia i Astrofisica Universitat de Valencia（瓦伦西亚大学天文学与天体物理学系）；Institut Universitaire de France (IUF)（法国高等研究院）
In a rigidly-rotating magnetohydrodynamic (MHD) system with convective turbulence, a large-scale dynamo, categorized as the $α^2$-type, can be excited when the spin rate is large enough. In this paper, the rotational dependence of the $α^2$-type dynamo and the cause of it are explored by mean-field (MF) dynamo models coupled with direct numerical simulations (DNSs) of MHD convections. Bearing the application to the solar/stellar dynamo in mind, we adopt a strongly-stratified polytrope as a model of the convective atmosphere. Our DNS models show that the $α^2$-type dynamo is excited when ${\rm …
In this paper, we have studied the convectively coupled equatorially trapped waves in rotating stars, with and without magnetic field. The equatorial trapped HD and MHD Poincaré, Rossby, mixed Rossby-Poincaré, and Kelvin waves were identified. The effects of stratification and non-traditional Coriolis force terms have been investigated. When the flow is strongly stratified, the wave frequencies of the convectively coupled model are almost the same as those of shallow water model. However, when the flow is weakly stratified, the wave frequencies are constrained by the buoyancy frequency. The no…
We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. Al…
We perform linear analysis of axisymmetric vertical normal modes in stratified compressible self-gravitating polytropic discs in the shearing box approximation. We study specific dynamics for subadiabatic, adiabatic and superadiabatic vertical stratifications. In the absence of self-gravity, four well-known principal modes can be identified in a stratified disc: acoustic p-, surface gravity f-, buoyancy g- and inertial r-modes. After characterizing modes in the non-self-gravitating case, we include self-gravity and investigate how it modifies the properties of these modes. We find that self-gr…
We explore the effects of an outer stably stratified coronal envelope on rotating turbulent convection, differential rotation, and large-scale dynamo action in spherical wedge models of the Sun. We solve the compressible magnetohydrodynamic equations in a two-layer model with unstable stratification below the surface, representing the convection zone, and a stably stratified coronal envelope above. The interface represents a free surface. We compare our model to models that have no coronal envelope. The presence of a coronal envelope is found to modify the Reynolds stress and the $Λ$ effect re…
We present results from numerical simulations of the interaction of internal gravity waves (IGW) with magnetic fields in the radiative interior of the Sun. In this second paper, the waves are forced self-consistently by an overlying convection zone and a toroidal magnetic field is imposed in the stably stratified layer just underneath convection zone. Consistent with the results of previous analytic and simple numerical calculations, we find a strong wave-field interaction, in which waves are reflected in the field region. The wave-field interaction and wave reflection depend on the field stre…
We review the current state of knowledge of magnetic fields inside stars, concentrating on recent developments concerning magnetic fields in stably stratified (zones of) stars, leaving out convective dynamo theories and observations of convective envelopes. We include the observational properties of A, B and O-type main-sequence stars, which have radiative envelopes, and the fossil field model which is normally invoked to explain the strong fields sometimes seen in these stars. Observations seem to show that Ap-type stable fields are excluded in stars with convective envelopes. Most stars cont…
A simplified model for stellar and planetary convection is derived for the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale with Rossby number as well as with thermal and viscous diffusivities. Integrating the convection model into a linearization of the dynamics in the transition region between convectively unstable and stably-stratified region yields a Rossby number, diffusivity, and pressure scale height dependent convective penetration depth into the stable region. This may have important consequences for mixing along the evolution of rotating sta…
