On the dynamics of radiative zones in rotating stars

In this lecture I try to explain the basic dynamical processes at work in a radiative zone of a rotating star. In particular, the notion of baroclinicity is thoroughly discussed. Attention is specially directed to the case of circulations and the key role of angular momentum conservation is stressed. The specific part played by viscosity is also explained. The old approach of Eddington and Sweet is reviewed and criticized in the light of the seminal papers of Busse 1981 and Zahn 1992. Other examples taken in the recent literature are also presented; finally, I summarize the important points.Comment: 21 pages 5 figure

Dynamics of the radiative envelope of rapidly rotating stars: Effects of spin-down driven by mass loss

(abridged) This paper aims at deciphering the dynamics of the envelope of a rotating star when some angular momentum loss due to mass loss is present. We especially wish to know when the spin-down flow forced by the mass loss supersedes the baroclinic flows that pervade the radiative envelope of rotating stars. We consider a Boussinesq fluid enclosed in a rigid sphere whose flows are forced both by the baroclinic torque, the spin-down of an outer layer, and an outward mass flux. The spin-down forcing is idealized in two ways: either by a rigid layer that imposes its spinning down velocity at some interface or by a turbulent layer that imposes a stress at this same interface to the interior of the star. In the case where the layer is rigid and imposes its velocity, we find that, as the mass-loss rate increases, the flow inside the star shows two transitions: the meridional circulation associated with baroclinic flows is first replaced by its spin-down counterpart, while at much stronger mass-loss rates the baroclinic differential rotation is superseded by the spin-down differential rotation. In fact, we find three wind regimes: weak (or no wind), moderate, and strong. In the first case, the flow in the radiative envelope is of baroclinic origin. In the moderate case, the circulation results from the spin-down while the differential rotation may either be of baroclinic or of spin-down origin, depending on the coupling between mass and angular momentum losses. For fast rotating stars, our model says that the moderate wind regime starts when mass loss is higher than ~1e-11 Msun/yr. In the strong wind case, the flow in the radiative envelope is mainly driven by angular momentum advection. This latter transition depends on the mass and the rotation rate of the star, being around 1e-8 Msun/yr for a 3 Msun ZAMS star rotating at 200 km/s according to our model.Comment: 13 pages, 9 figures, to appear in Astronomy and Astrophysic

Inertial waves in a differentially rotating spherical shell

We 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 behavior 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.Comment: 31 pages, 16 figures, accepted for publication in Journal of Fluid Mechanic

More concerning the anelastic and subseismic approximations for low-frequency modes in stars

Two approximations, namely the subseismic approximation and the anelastic approximation, are presently used to filter out the acoustic modes when computing low frequency modes of a star (gravity modes or inertial modes). In a precedent paper (Dintrans & Rieutord 2001), we observed that the anelastic approximation gave eigenfrequencies much closer to the exact ones than the subseismic approximation. Here, we try to clarify this behaviour and show that it is due to the different physical approach taken by each approximation: On the one hand, the subseismic approximation considers the low frequency part of the spectrum of (say) gravity modes and turns out to be valid only in the central region of a star; on the other hand, the anelastic approximation considers the Brunt-Vaisala frequency as asymptotically small and makes no assumption on the order of the modes. Both approximations fail to describe the modes in the surface layers but eigenmodes issued from the anelastic approximation are closer to those including acoustic effects than their subseismic equivalent. We conclude that, as far as stellar eigenvalue problems are concerned, the anelastic approximation is better suited for simplifying the eigenvalue problem when low-frequency modes of a star are considered, while the subseismic approximation is a useful concept when analytic solutions of high order low-frequency modes are needed in the central region of a star.Comment: 5 pages 3 fig, to appear in MNRA

The Sun's Supergranulation

Supergranulation is a fluid-dynamical phenomenon taking place in the solar photosphere, primarily detected in the form of a vigorous cellular flow pattern with a typical horizontal scale of approximately 30--35~megameters, a dynamical evolution time of 24--48~h, a strong 300--400~m/s (rms) horizontal flow component and a much weaker 20--30~m/s vertical component. Supergranulation was discovered more than sixty years ago, however, explaining its physical origin and most important observational characteristics has proven extremely challenging ever since, as a result of the intrinsic multiscale, nonlinear dynamical complexity of the problem concurring with strong observational and computational limitations. Key progress on this problem is now taking place with the advent of 21st-century supercomputing resources and the availability of global observations of the dynamics of the solar surface with high spatial and temporal resolutions. This article provides an exhaustive review of observational, numerical and theoretical research on supergranulation, and discusses the current status of our understanding of its origin and dynamics, most importantly in terms of large-scale nonlinear thermal convection, in the light of a selection of recent findings.Comment: Major update of 2010 Liv. Rev. Sol. Phys. review. Addresses many new theoretical, numerical and observational developments. All sections, including discussion, revised extensively. Also includes previously unpublished results on nonlinear dynamics of convection in large domains, and lagrangian transport at the solar surfac

Completeness of Inertial Modes of an Incompressible Non-Viscous Fluid in a Corotating Ellipsoid

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid and contained in a rigid container, these modes satisfy Poincar\'e's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. In this paper we investigate the mathematical side of this problem. We first show that the Poincar\'e problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincar\'e operator is bounded and self-adjoint and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric then the base can be identified with the set of Poincar\'e modes, first obtained by Bryan (1889), and completed with the geostrophic modes.Comment: 19 pages, 1 figure, to appear in Physical Review

Two-dimensional models of early-type fast rotating stars: new challenges in stellar physics

Two-dimensional models of rapidly rotating stars are already unavoidable for the interpretation of interferometric or asteroseismic data of this kind of stars. When combined with time evolution, they will allow the including of a more accurate physics for the computation of element transport and the determination of surface abundances. In addition, modeling the evolution of rotation will improve gyrochronology. Presently, two-dimensional ESTER models predict the structure and the large-scale flows (differential rotation and meridional circulation) of stars with mass larger than 1.7Msun at any rotation rate. Main sequence evolution can be mimicked by varying the hydrogen content of the convective core. Models have been successfully tested on half a dozen of nearby fast rotating stars observed with optical or infra-red interferometers. They are now the right tool to investigate the oscillation spectrum of early-type fast rotators.Comment: 10 pages, to appear in the proceedings of the conference "New advances in stellar physics: from microscopic to macroscopic processes", Roscoff, May 201

Tracking granules at the Sun's surface and reconstructing velocity fields. II. Error analysis

The determination of horizontal velocity fields at the solar surface is crucial to understanding the dynamics and magnetism of the convection zone of the sun. These measurements can be done by tracking granules. Tracking granules from ground-based observations, however, suffers from the Earth's atmospheric turbulence, which induces image distortion. The focus of this paper is to evaluate the influence of this noise on the maps of velocity fields. We use the coherent structure tracking algorithm developed recently and apply it to two independent series of images that contain the same solar signal. We first show that a k-\omega filtering of the times series of images is highly recommended as a pre-processing to decrease the noise, while, in contrast, using destretching should be avoided. We also demonstrate that the lifetime of granules has a strong influence on the error bars of velocities and that a threshold on the lifetime should be imposed to minimize errors. Finally, although solar flow patterns are easily recognizable and image quality is very good, it turns out that a time sampling of two images every 21 s is not frequent enough, since image distortion still pollutes velocity fields at a 30% level on the 2500 km scale, i.e. the scale on which granules start to behave like passive scalars. The coherent structure tracking algorithm is a useful tool for noise control on the measurement of surface horizontal solar velocity fields when at least two independent series are available.Comment: in press in Astronomy and Astrophysics, 9 page