In this review, I discuss physical mechanisms leading to momentum and chemical transport in stars. Various instabilities leading to turbulence are discussed. I then present a self-consistent description of rotational mixing under the action of turbulence and meridional circulation in 1D models. Limitations of the model are discussed, both in terms of an extra mechanism for momentum transport in the Sun and solar-type stars (magnetic field and/or gravity waves) and in terms of our understanding of turbulent properties.

Thermal diffusivities within stars are higher by several orders of magnitude than in the earth atmosphere. This particularity of the stellar fluid has subtle effects on the dynamics of stably stratified radiative zones as it contributes to favour some flows and supress others. An asymptotic analysis of hydrodynamic equations for large values of the thermal diffusivity is presented and this results in a simplification of the dynamical effects of the thermal diffusivity. This asymptotic behavior proved to be very useful to understand shear instabilities of Kelvin-Helmholtz type.

The first entirely self-consistent 2D model of rotational mixing in a stellar radiative zone is presented. This nonlinear problem is solved numerically assuming axisymmetry of the system. The dynamical behaviour of a rotating star is found to be controlled by one parameter only, the ratio of the Eddington-Sweet timescale to the viscous timescale. In the quasi-steady state, the limit of slow rotation recovers Eddington-Sweet theory, whereas in the limit of rapid rotation, the system settles into a centrifugal equilibrium. The evolution of the dynamical structure of the star undergoing spin-down is then studied, and the relevance of these findings to observations of rotational mixing is discussed.

We have done a 2D simulation of meridional circulation in the presence of helium diffusion below stellar convective zones. For slowly rotating stars the microscopic diffusion of helium leads to a μ-gradient which has an important effect on the rotation induced mixing. We present here the results of this 2D simulation for solar type stars.

Magnetic fields can be created in stably stratified (non-convective) layers in a differentially rotating star. A magnetic instability in the toroidal field (wound up by differential rotation) replaces the role of convection in closing the field amplification loop. A dynamo model is developed from these ingredients, and applied to the problem of angular momentum transport in stellar interiors. It produces a predominantly horizontal field. The process is found to be more effective in transporting angular momentum than the known hydrodynamic mechanisms, with the possible exception of transport by internal gravity waves.

I describe three key moments in the rotational evolution of three single stars, chosen to illustrate the richness and complexity of the interaction between rotation and magnetic fields. I then piece together two general scenarii for the rotational evolution of early-type and late-type stars, in which the “fossil” magnetic field inherited by a star from its formative years basically sets its subsequent rotational fate. Since no one has yet offered a convincing scenario according to which a star could somehow get rid of its primordial fossil field, the inescapable conclusion is that magnetic fields cannot be neglected in modelling rotation, and that “rotational evolution” should really be thought of as “magnetorotational evolution”.

We have conducted preliminary numerical simulations of a core convection dynamo operating within an A-type star of two solar masses. Convection within the core clearly can admit magnetic dynamo action. Magnetic field strengths in our three-dimensional simulations grow by many orders of magnitude, from an initial seed field to kilo-Gauss levels. We discuss the differential rotation and magnetic field sustained in our simulations.

Results for multidimensional stellar model simulations of both 2D and 3D hydrodynamic models and 2D stellar evolution sequences are presented. Simulations of the highly superadiabatic region of the solar convective region provide a good example of the current status and limitations of explicit 3D finite difference methods in stellar problems. Such simulations cannot be used for stellar cores, where the motion is expected to be well subsonic. The results of some 2D fully implcit hydrodynamic simulations of convective cores and shells are given for models with and without rotation, and their effects examined through fully 2D stellar evolution sequences. One effect of moderate to rapid rotation in convective cores is to alter the convective flow pattern so that convective eddies tend to line up parallel to the rotation axis. Rotation also appears to modestly reduce the amount of convective core overshooting, at least for intermediate mass models.

The advent of massively parallel supercomputing has begun to permit explicit 3–D simulations of turbulent convection occurring within the cores of early-type main sequence stars. Such studies should complement the stellar structure and evolution efforts that have so far largely employed 1–D nonlocal mixing length descriptions for the transport, mixing and overshooting achieved by core convection. We have turned to A-type stars as representative of many of the dynamical challenges raised by core convection within rotating stars. The differential rotation and meridional circulations achieved deep within the star by the convection, the likelihood of sustained magnetic dynamo action there, and the bringing of fresh fuel into the core by overshooting motions, thereby influencing main sequence lifetimes, all constitute interesting dynamical questions that require detailed modelling of global-scale convection. Using our anelastic spherical harmonic (ASH) code tested on the solar differential rotation problem, we have conducted a series of 3–D spherical domain simulations that deal with a simplified description of the central regions of rotating A-type stars, i.e a convectively unstable core is surrounded by a stable radiative envelope. A sequence of 3–D simulations are used to assess the properties of the convection (its global patterns, differential rotation, meridional circulations, extent and latitudinal variation of the overshooting) as transitions are made between laminar and turbulent states by changing the effective diffusivities, rotation rates, and subadiabaticity of the radiative exterior. We report on the properties deduced from these models for both the extent of penetration and the profile of rotation sustained by the convection.

In this review, we describe the physical processes driving the dynamical evolution of binary stars, namely the circularization of the orbit and the synchronization of their spin and orbital rotation. We also discuss the possible role of the elliptic instability which turns out to be an unavoidable ingredient of the evolution of binary stars.

In this contribution, I will examine the interaction between stellar rotation and pulsation. I begin with a brief review of the non-rotating case, emphasizing the character of pulsations as azimuthally-propagating waves. I then go on to discuss how these waves are modified under the influence of the centrifugal and Coriolis forces. Through simple arguments, I outline the conditions under which each force can become significant in determining the wave dynamics. Particular attention is paid to the Coriolis force, since it is responsible for the formation of a waveguide, which confines the pulsation to a narrow band centered on the stellar equator. Using the example of a prograde sectoral pulsation mode, I explain the basic physical principles underlying this trapping.

The Coriolis force is also responsible for the existence of Rossby waves, which are not found in non-rotating stars. I demonstrate how these waves may be understood in terms of a conservation law for angular momentum, and review their most important characteristics. I then examine how rotation modifies the frequencies of pulsation, and explain how observations of such modifications can provide information regarding a star's rotation rate. To conclude, I focus on the converse of the pulsation-rotation interaction: how the transport of angular momentum by pulsation might be important in determining the evolution of a star's rotation profile.

Oscillation modes of rapidly rotating stars have not yet been calculated with precision, rotational effects being generally approximated by perturbation methods. We developed a mathematical formalism and a numerical method which fully account for the deformation of the star by the centrifugal force. The method has been first tested in the case of Maclaurin spheroids and then applied to uniformly rotating polytropic stars.

Assuming that pre-main sequence stars rotating at the deuterium-burning birthline as solid bodies with rates within the range observed in T-Tauri stars is an adequate initial condition to explain the rotational properties of stars in clusters of all ages, provided that appropriate evolutionary schemes are used. It is, in fact, on the evolution that most recent advances have been made, and a new paradigm has arisen. Specifically, stellar periods (as a function of mass) begin their evolution on one of two branches, and the fast rotators migrate to the slow branch through a short-lived gap. These branches represent different physical properties of the internal stellar dynamics, and are related to the type of dynamo mechanism which operates within the stars. This paradigm appears to have the potential to address several fundamental issues in stellar astronomy.

A relation between activity and rotation in young ~ 1Myr PMS in often not observed, suggesting that the mechanism responsible for the X-ray emission may differ from the α - ω dynamo. We re-investigate the matter utilizing recent X-ray and rotational data on the Orion Nebula Cluster (ONC).

Measurements of stars in the Orion OB association show that there is a continuous power law relationship between specific angular momentum (J/M) and mass (M) for stars on convective tracks having masses in the range ~0.5 to ~3 M⊙; this power law extends smoothly into the domain of more massive stars on the ZAMS. If we assume that stars are “locked” to circumstellar accretion disks via their magnetic fields until they are deposited on the stellar birthline, we can account for the observed slope and zero point of the power law fit to the upper envelope of the observed J/M vs M distribution.

Pre-main sequence stars with M<2 M⊙ on radiative tracks do not follow the power law relationship. A sharp decrease in J/M with decreasing mass has been recognized for more than 30 years for older field stars, but remarkably is seen already among our Orion sample of stars that are only a few million years old. We show that this break in the power law is a consequence of loss of angular momentum on convective tracks, combined with core-envelope decoupling at the time of the transition from the convective to radiative tracks.

We present some first results of stellar models of low mass, solar metallicity giants that include rotation and the mixing associated with it. We base our choice of initial conditions and input physics on observational constraints obtained for main sequence (MS) and horizontal branch (HB) stars, and obtain what can be regarded as a “best case scenario” for rotational mixing. These observations suggest local conservation of angular momentum in the cores and constant specific angular momentum in convective regions on the red giant branch (RGB). We show that under these circumstances and with moderate rotation rates at the turnoff it is possible to obtain enough mixing in order to reproduce the anomalous abundance ratios of the CNO species seen in giants. It is also shown that solid body rotation in the convective envelope fails to account for the observations. It is found that rotational mixing is inefficient on the lower RGB, so there is no need to invoke any inhibiting role of μ–gradients.

Low mass stars (< 2-2.5 M⊙) exhibit, at all the stages of their evolution, signatures of processes that require challenging modeling beyond the standard stellar theory. In this paper we focus on their peculiarities while they climb the red giant branch (RGB). We first compare the classical predictions for abundance variations due to the first dredge-up with observational data in various environments. We show how clear spectroscopic diagnostics probe the nucleosynthesis and the internal mixing mechanisms that drive RGB stars. Coherent data reveal in particular the existence of a non-standard mixing process that changes their surface abundances at the so-called RGB bump. By reviewing the models presented so far to explain the various abundance anomalies, we show that the occurrence of this extra-mixing process is certainly related to rotation. Finally we discuss the so-called Li-flash which is expected to occur at the very beginning of the extra-mixing episode.

Three critical tests are applied to the models that hypothesize the transport of angular momentum and material via turbulence. The models fail on all three counts.

We present nearly 70 new observations for M67 members at the main-sequence (MS), turn-off (TO) and post-turn-off (PTO) in addition to those already published in Melo, Pasquini & De Medeiros (2001). These additional v sin i measurements confirm that most of stars are experiencing a strong braking as they pass the turn-off. In contrast to our previous results, some spread in the v sin i is observed among the TO and PTO stars.

Abundance anomalies in the Sun, main sequence F, A and B stars, turnoff Pop II stars, the horizontal branch and white dwarfs are caused at least partly by particle transport processes. Detailed evolutionary models have been calculated for most of those objects taking into account the gravitational settling, thermal diffusion and radiative accelerations of 28 isotopes (24 atomic species). These will be used together with the observed abundances to put constraints on the mixing that rotationnally induced turbulence may lead to. The link between abundance anomalies and rotation on the HB is explained by the variation with log g of the ratio of the Eddington-Sweet and atomic diffusion velocities.