scholarly journals Gravito-inertial waves in a differentially rotating spherical shell

2016 ◽  
Vol 800 ◽  
pp. 213-247 ◽  
Author(s):  
G. M. Mirouh ◽  
C. Baruteau ◽  
M. Rieutord ◽  
J. Ballot
Keyword(s):  
Spherical Shell ◽  
Differential Rotation ◽  
Power Laws ◽  
Boussinesq Approximation ◽  
Theoretical Explanation ◽  
Heat Diffusion ◽  
Complex Spectrum ◽  
Rotating Stars ◽  
Inertial Waves ◽  
Short Period

The gravito-inertial waves propagating over a shellular baroclinic flow inside a rotating spherical shell are analysed using the Boussinesq approximation. The wave properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the instability is driven by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. The eigenvalue spectrum turns out to be very rich and complex, which lets us suppose an even richer and more complex spectrum for rotating stars or planets that own a differential rotation driven by baroclinicity.

scholarly journals Oscillations of Rapidly Rotating Stars

2000 ◽  
Vol 176 ◽  
pp. 373-373
Author(s):  
B. Dintrans ◽  
M. Rieutord
Keyword(s):  
Spherical Shell ◽  
Numerical Simulations ◽  
Boussinesq Approximation ◽  
Type Star ◽  
Second Step ◽  
Rotating Stars ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Rotating Sphere ◽  
Rapidly Rotating Stars

AbstractWe present numerical simulations of gravito-inertial waves propagating in radiative zones of rapidly rotating stars. A first model, using the Boussinesq approximation, allows us to study the oscillations of a quasi-incompressible stratified fluid embedded in a rapidly rotating sphere or spherical shell. In a second step, we investigate the case of a γ Doradus-type star using the anelastic approximation. Some fascinating features of rapidly rotating fluids, such as wave attractors, appear in both cases.

scholarly journals Asymptotic theory of gravity modes in rotating stars

2018 ◽  
Vol 615 ◽  
pp. A106 ◽  
Author(s):  
V. Prat ◽  
S. Mathis ◽  
K. Augustson ◽  
F. Lignières ◽  
J. Ballot ◽  
...  
Keyword(s):  
Dispersion Relation ◽  
Differential Rotation ◽  
Chemical Elements ◽  
Hamiltonian Formalism ◽  
Stellar Model ◽  
Rotating Stars ◽  
Inertial Waves ◽  
Local Dispersion ◽  
Propagation Of Waves ◽  
The Impact

Context. Differential rotation has a strong influence on stellar internal dynamics and evolution, notably by triggering hydrodynamical instabilities, by interacting with the magnetic field, and more generally by inducing transport of angular momentum and chemical elements. Moreover, it modifies the way waves propagate in stellar interiors and thus the frequency spectrum of these waves, the regions they probe, and the transport they generate. Aims. We investigate the impact of a general differential rotation (both in radius and latitude) on the propagation of axisymmetric gravito-inertial waves. Methods. We use a small-wavelength approximation to obtain a local dispersion relation for these waves. We then describe the propagation of waves thanks to a ray model that follows a Hamiltonian formalism. Finally, we numerically probe the properties of these gravito-inertial rays for different regimes of radial and latitudinal differential rotation. Results. We derive a local dispersion relation that includes the effect of a general differential rotation. Subsequently, considering a polytropic stellar model, we observe that differential rotation allows for a large variety of resonant cavities that can be probed by gravito-inertial waves. We identify that for some regimes of frequency and differential rotation, the properties of gravito-inertial rays are similar to those found in the uniformly rotating case. Furthermore, we also find new regimes specific to differential rotation, where the dynamics of rays is chaotic. Conclusions. As a consequence, we expect modes to follow the same trend. Some parts of oscillation spectra corresponding to regimes similar to those of the uniformly rotating case would exhibit regular patterns, while parts corresponding to the new regimes would be mostly constituted of chaotic modes with a spectrum rather characterised by a generic statistical distribution.

Gravito-inertial waves in a rotating stratified sphere or spherical shell

1999 ◽  
Vol 398 ◽  
pp. 271-297 ◽  
Author(s):  
B. DINTRANS ◽  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Periodic Orbit ◽  
Spherical Shell ◽  
Periodic Orbits ◽  
Point Spectrum ◽  
Boussinesq Approximation ◽  
Diffusion Limit ◽  
Order Partial Differential Equation ◽  
Inertial Waves ◽  
Zero Diffusion Limit ◽  
Hyperbolic Domain

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.

scholarly journals Inertial waves in a differentially rotating spherical shell

2013 ◽  
Vol 719 ◽  
pp. 47-81 ◽  
Author(s):  
C. Baruteau ◽  
M. Rieutord
Keyword(s):  
Spherical Shell ◽  
Solid Body ◽  
Shear Layers ◽  
Inviscid Limit ◽  
Order Partial Differential Equation ◽  
Body Rotation ◽  
Inertial Waves ◽  
Visible Absorption ◽  
Solid Body Rotation ◽  
Short Period

AbstractWe investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

scholarly journals Learning about Stellar Dynamos from Long–term Photometry of Starspots

1991 ◽  
Vol 130 ◽  
pp. 353-369 ◽  
Author(s):  
Douglas S. Hall
Keyword(s):  
Differential Rotation ◽  
Rotation Period ◽  
Turnover Time ◽  
Rotating Stars ◽  
Photometric Variability ◽  
Solar Type ◽  
Magnetic Cycles ◽  
The Many ◽  
Rapidly Rotating Stars

AbstractSpottedness, as evidenced by photometric variability in 277 late-type binary and single stars, is found to occur when the Rossby number is less than about 2/3. This holds true when the convective turnover time versus B–V relation of Gilliland is used for dwarfs and also for subgiants and giants if their turnover times are twice and four times longer, respectively, than for dwarfs. Differential rotation is found correlated with rotation period (rapidly rotating stars approaching solid-body rotation) and also with lobe-filling factor (the differential rotation coefficient k is 2.5 times larger for F = 0 than F = 1). Also reviewed are latitude extent of spottedness, latitude drift during a solar-type cycle, sector structure and preferential longitudes, starspot lifetimes, and the many observational manifestations of magnetic cycles.

scholarly journals Boussinesq convection in a gaseous spherical shell

2019 ◽  
Vol 82 ◽  
pp. 373-382
Author(s):  
L. Korre ◽  
N. Brummell ◽  
P. Garaud
Keyword(s):  
Temperature Gradient ◽  
Boundary Conditions ◽  
Spherical Shell ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boussinesq Approximation ◽  
Fixed Temperature ◽  
Planetary Interiors ◽  
Adiabatic Temperature Gradient ◽  
Relevant Quantity

In this paper, we investigate the dynamics of convection in a spherical shell under the Boussinesq approximation but considering the compressibility which arises from a non zero adiabatic temperature gradient, a relevant quantity for gaseous objects such as stellar or planetary interiors. We find that depth-dependent superiadiabaticity, combined with the use of mixed boundary conditions (fixed flux/fixed temperature), gives rise to unexpected dynamics that were not previously reported.

Inertial waves in a rotating spherical shell

1997 ◽  
Vol 341 ◽  
pp. 77-99 ◽  
Author(s):  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Spherical Shell ◽  
Cross Section ◽  
Shear Layers ◽  
Inertial Waves ◽  
Ekman Number ◽  
Distribution Of Eigenvalues ◽  
Simple Rules ◽  
Meridional Cross Section ◽  
Zero Viscosity ◽  
The Web

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

scholarly journals Differential Rotation and Magnetic Activity of the Lower Main Sequence Stars

1980 ◽  
Vol 51 ◽  
pp. 296-297
Author(s):  
G. Belvedere ◽  
L. Paterno ◽  
M. Stix
Keyword(s):  
Spherical Shell ◽  
Differential Rotation ◽  
Main Sequence ◽  
Magnetic Activity ◽  
The Sun ◽  
Coupling Constant ◽  
Main Sequence Stars ◽  
Dynamo Theory ◽  
Magnetic Cycles ◽  
Surface Convection

AbstractWe extend to the lower main sequence stars the analysis of convection interacting with rotation in a compressible spherical shell, already applied to the solar case (Belvedere and Paterno, 1977; Belvedere et al. 1979a). We assume that the coupling constant ε between convection and rotation, does not depend on the spectral type. Therefore we take ε determined from the observed differential rotation of the Sun, and compute differential rotation and magnetic cycles for stars ranging from F5 to MO, namely for those stars which are supposed to possess surface convection zones (Belvedere et al. 1979b, c, d). The results show that the strength of differential rotation decreases from a maximum at F5 down to a minimum at G5 and then increases towards later spectral types. The computations of the magnetic cycles based on the αω-dynamo theory show that dynamo instability decreases from F5 to G5, and then increases towards the later spectral types reaching a maximum at MO. The period of the magnetic cycles increases from a few years at F5 to about 100 years at MO. Also the extension of the surface magnetic activity increases substantially towards the later spectral types. The results are discussed in the framework of Wilson’s (1978) observations.

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