Convective differential rotation in stars and planets – II. Observational and numerical tests

ABSTRACT Differential rotation is central to a great many mysteries in stars and planets. In part I, we predicted the order of magnitude and scaling of the differential rotation in both hydrodynamic and magnetohydrodynamic convection zones. Our results apply to both slowly and rapidly rotating systems, and provide a general picture of differential rotation in stars and fluid planets. We further calculated the scalings of the meridional circulation, entropy gradient, and baroclinicity. In this companion paper, we compare these predictions with a variety of observations and numerical simulations. With a few exceptions, we find that these are consistent in both the slowly rotating and rapidly rotating limits. Our results help to localize core–envelope shear in red giant stars, suggest a rotation-dependent frequency shift in the internal gravity waves of massive stars, and potentially explain observed deviations from von Zeipel’s gravity darkening in late-type stars.

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Convective differential rotation in stars and planets – I. Theory

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa2323 ◽

2020 ◽

Vol 498 (3) ◽

pp. 3758-3781 ◽

Cited By ~ 1

Author(s):

Adam S Jermyn ◽

Shashikumar M Chitre ◽

Pierre Lesaffre ◽

Christopher A Tout

Keyword(s):

Differential Rotation ◽

Detailed Comparison ◽

Companion Paper ◽

Angular Frequency ◽

Circulation Rate ◽

Rotation Rates ◽

Rotating Systems ◽

Order Of Magnitude ◽

The Magnetic Field ◽

Hydrodynamic Systems

ABSTRACT We derive the scaling of differential rotation in both slowly and rapidly rotating convection zones using order of magnitude methods. Our calculations apply across stars and fluid planets and all rotation rates, as well as to both magnetized and purely hydrodynamic systems. We find shear |R∇Ω| of order the angular frequency Ω for slowly rotating systems with Ω ≪ |N|, where N is the Brünt–Väisälä frequency, and find that it declines as a power law in Ω for rapidly rotating systems with Ω ≫ |N|. We further calculate the meridional circulation rate and baroclinicity and examine the magnetic field strength in the rapidly rotating limit. Our results are in general agreement with simulations and observations and we perform a detailed comparison with those in a companion paper.

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Collective instability of salt fingers

Journal of Fluid Mechanics ◽

10.1017/s0022112069001066 ◽

1969 ◽

Vol 35 (2) ◽

pp. 209-218 ◽

Cited By ~ 124

Author(s):

Melvin E. Stern

Keyword(s):

Gravity Waves ◽

Laboratory Experiments ◽

Internal Gravity Waves ◽

Asymptotic Dependence ◽

Salinity Gradients ◽

Salt Fingers ◽

Order Of Magnitude ◽

Ocean Observations ◽

Molecular Salt ◽

Steady Laminar

We first consider a steady laminar model of salt fingers and show that the latter become unstable with respect to internal gravity waves when the finger Reynolds number exceeds a critical value. The criterion is then used in speculations about the statistically steady state in a fully developed similarity model where horizontally averaged temperature and salinity gradients are constant at all depths. Dimensional reasoning is used to obtain the asymptotic dependence of the turbulent flux on the molecular salt diffusivity. From this and other relationships order-of-magnitude estimates are obtained and compared with laboratory experiments and ocean observations.

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Interplay between magnetic fields and differential rotation in a stably stratified stellar radiative zone

Astronomy and Astrophysics ◽

10.1051/0004-6361/202037828 ◽

2020 ◽

Vol 641 ◽

pp. A13 ◽

Cited By ~ 1

Author(s):

L. Jouve ◽

F. Lignières ◽

M. Gaurat

Keyword(s):

Magnetic Field ◽

Differential Rotation ◽

Large Scale ◽

Stable Stratification ◽

Rotational Instability ◽

Intermediate Mass ◽

Poloidal Magnetic Field ◽

Giant Stars ◽

Radiative Zone ◽

Red Giant

Context. The interactions between magnetic fields and differential rotation in stellar radiative interiors could play a major role in achieving an understanding of the magnetism of intermediate-mass and massive stars and of the differential rotation profile observed in red-giant stars. Aims. The present study is aimed 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 investigate in detail the effects of the stable stratification and thermal diffusion and we aim, in particular, to assess the role of the stratification at stabilising the system. Methods. We performed 2D and 3D global Boussinesq numerical simulations started from an initial radial or cylindrical differential rotation and a large-scale poloidal magnetic field. Under the conditions of a large rotation frequency compared to the Alfvén frequency, we built a magnetic configuration strongly dominated by its toroidal component. We then perturbed this configuration to observe the development of non-axisymmetric instabilities. Results. The parameters of the simulations were chosen to respect the ordering of time scales of a typical stellar radiative zone. In this framework, the axisymmetric evolution is studied by varying the relative effects of the thermal diffusion, the Brunt-Väisälä frequency, the rotation, and the initial poloidal field strength. After a transient time and using a suitable adimensionalisation, we find that the axisymmetric state only depends on tes/tAp the ratio between the Eddington–Sweet circulation time scale and the Alfvén time scale. A scale analysis of the Boussinesq magnetohydrodynamical equations allows us to recover this result. In the cylindrical case, a magneto-rotational instability develops when the thermal diffusivity is sufficiently high to enable the favored wavenumbers to be insensitive to the effects of the stable stratification. In the radial case, the magneto-rotational instability is driven by the latitudinal shear created by the back-reaction of the Lorentz force on the flow. Increasing the level of stratification then leaves the growth rate of the instability mainly unaffected while its horizontal length scale grows. Conclusions. Non-axisymmetric instabilities are likely to exist in stellar radiative zones despite the stable stratification. They could be at the origin of the magnetic dichotomy observed in intermediate-mass and massive stars. They are also unavoidable candidates for the transport of angular momentum in red giant stars.

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Excitation of internal gravity waves by penetrative convection

EAS Publications Series ◽

10.1051/eas/1982024 ◽

2019 ◽

Vol 82 ◽

pp. 247-251

Author(s):

C. Pinçon ◽

K. Belkacem ◽

M.J. Goupil

Keyword(s):

Angular Momentum ◽

Gravity Waves ◽

Internal Gravity Waves ◽

Penetrative Convection ◽

Low Mass Stars ◽

Low Mass ◽

Red Giant ◽

Excitation Model ◽

Giant Branch Stars ◽

Do So

We investigate the ability of internal gravity waves that are generated by penetrative convection to redistribute angular momentum in the internal radiative zone of evolved low-mass stars. To do so, we use the semianalytical excitation model recently proposed by Pinçon et al. 2016. We briefly report the preliminary results of the study focusing on the subgiant and red giant branch stars.

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Modelling differential rotation of red giants: the case of the evolved sun

Monthly Notices of the Royal Astronomical Society Letters ◽

10.1093/mnrasl/slz150 ◽

2019 ◽

Vol 490 (1) ◽

pp. L71-L75

Author(s):

Leonid Kitchatinov ◽

Alexander Nepomnyashchikh

Keyword(s):

Angular Momentum ◽

Differential Rotation ◽

Convective Transport ◽

Rotational State ◽

Red Giants ◽

Solar Mass ◽

Angular Momentum Transport ◽

Stellar Convection ◽

Order Of Magnitude ◽

Red Giant

ABSTRACT Asteroseismology has revealed that cores of red giants rotate about one order of magnitude faster than their convective envelopes. This paper attempts an explanation for this rotational state in terms of the theory of angular momentum transport in stellar convection zones. A differential rotation model based on the theory is applied to a sequence of evolutionary states of a red giant of one solar mass. The model computations show a rotation of about ten times faster in the cores compared to the stellar surface. This rotational state is caused by the non-diffusive downward convective transport of angular momentum. The contrast in rotational rates between core and envelope increases with the radius (age) of the star. Seismologically detected scaling for the spin-down of the giants’ cores is also reproduced.

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Can plume-induced internal gravity waves regulate the core rotation of subgiant stars?

Astronomy and Astrophysics ◽

10.1051/0004-6361/201730998 ◽

2017 ◽

Vol 605 ◽

pp. A31 ◽

Cited By ~ 15

Author(s):

C. Pinçon ◽

K. Belkacem ◽

M. J. Goupil ◽

J. P. Marques

Keyword(s):

Angular Momentum ◽

Differential Rotation ◽

Internal Gravity Waves ◽

Low Mass Stars ◽

Rotation Rates ◽

The Core ◽

Low Mass ◽

Red Giant ◽

Core Rotation ◽

Subgiant Stars

Context. The seismic data provided by the space-borne missions CoRoT and Kepler enabled us to probe the internal rotation of thousands of evolved low-mass stars. Subsequently, several studies showed that current stellar evolution codes are unable to reproduce the low core rotation rates observed in these stars. These results indicate that an additional angular momentum transport process is necessary to counteract the spin up due to the core contraction during the post-main sequence evolution. For several candidates, the transport induced by internal gravity waves (IGW) could play a non-negligible role. Aims. We aim to investigate the effect of IGW generated by penetrative convection on the internal rotation of low-mass stars from the subgiant branch to the beginning of the red giant branch. Methods. A semi-analytical excitation model was used to estimate the angular momentum wave flux. The characteristic timescale associated with the angular momentum transport by IGW was computed and compared to the contraction timescale throughout the radiative region of stellar models at different evolutionary stages. Results. We show that IGW can efficiently counteract the contraction-driven spin up of the core of subgiant stars if the amplitude of the radial-differential rotation (between the center of the star and the top of the radiative zone) is higher than a threshold value. This threshold depends on the evolutionary stage and is comparable to the differential rotation rates inferred for a sample of subgiant stars observed by the satellite Kepler. Such an agreement can therefore be interpreted as the consequence of a regulation mechanism driven by IGW. This result is obtained under the assumption of a smooth rotation profile in the radiative region and holds true even if a wide range of values is considered for the parameters of the generation model. In contrast, on the red giant branch, we find that IGW remain insufficient, on their own, to explain the observations because of an excessive radiative damping. Conclusions. IGW generated by penetrative convection are able to efficiently extract angular momentum from the core of stars on the subgiant branch and accordingly have to be taken into account. Moreover, agreements with the observations reinforce the idea that their effect is essential to regulate the amplitude of the radial-differential rotation in subgiant stars. On the red giant branch, another transport mechanism must likely be invoked.

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Statically unstable layers produced by overturning internal gravity waves

Journal of Fluid Mechanics ◽

10.1017/s002211209400354x ◽

1994 ◽

Vol 260 ◽

pp. 333-350 ◽

Cited By ~ 20

Author(s):

S. A. Thorpe

Keyword(s):

Gravity Waves ◽

Growth Rates ◽

Large Amplitude ◽

Subsequent Development ◽

Internal Gravity Waves ◽

Companion Paper ◽

Static Stability ◽

Secondary Instability ◽

Tube Length ◽

Uniform Density

Internal waves in a uniformly stratified fluid of sufficiently large amplitude develop tilted layers in which the fluid is statically unstable. To investigate the evolution and subsequent development of this structure, experiments are made in which a horizontal rectangular tube containing a fluid of uniform density gradient is gently rocked at a selected frequency about a horizontal axis normal to the tube length. Large-amplitude standing internal gravity waves of the first mode are generated, and these steepen and overturn, the isopycnal surfaces folding to produce a vertically thin and horizontally extensive layer in which the fluid is statically unstable. In experiments with relatively small forcing, the layer persists for some 6 buoyancy periods, with no detected evidence of secondary instability, and static stability is re-established as the periodic flow reverses. The layer however breaks down, with consequent diapycnal mixing, when greater forcing is applied.The scale and growth rates of instability in the overturning internal gravity waves are estimated using the theory developed in a companion paper by Thorpe (1994a). For the parameters of the laboratory experiments with relatively small forcing, the growth rates are small, consistent with the absence of signs of secondary instability. Larger growth rates and disturbance amplification factors of about 70 are predicted for the conditions in the experiment in which mixing was observed to occur. The experimental observations are consistent with an instability having a longitudinal structure.We conclude that the form and development of breaking in internal gravity waves will vary according to the circumstances in which waves break, but depend on the Prandtl number of the fluid and, in particular, on the Rayleigh and Reynolds numbers of regions of static instability which develop as the waves overturn.

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Biharmonic Attractors of Internal Gravity Waves

Fluid Dynamics ◽

10.1134/s0015462821030046 ◽

2021 ◽

Vol 56 (3) ◽

pp. 403-412

Author(s):

D. A. Ryazanov ◽

M. I. Providukhina ◽

I. N. Sibgatullin ◽

E. V. Ermanyuk

Keyword(s):

Gravity Waves ◽

High Frequency ◽

Large Time ◽

Internal Gravity Waves ◽

Wave Pattern ◽

Mean Value ◽

Order Of Magnitude ◽

The Mean ◽

Low Amplitude ◽

Large Time Scale

Abstract— The hydrodynamic system that admits the development of internal wave attractors under biharmonic forcing is investigated. It is shown that in the case of low amplitude of external forcing the wave pattern consists of two attractors that interact between themselves only slightly: the total energy of the system is equal to the sum of energies of the components with high accuracy. In the nonlinear case the attractors interact in the more complex way which leads to the development of a cascade of triad interactions generating a rich set of time scales. In the case of closely adjacent frequencies of the components of a biharmonic perturbation, the nonlinear “beating” regime develops, namely, the mean energy of the system of coupled attractors performs oscillations at a large time scale that corresponds to the beating period. It is found that the high-frequency energy fluctuations corresponding to the same mean energy can differ by an order of magnitude depending on whether the envelope of the mean value increases or decreases.

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