In this lecture I introduce the basic concepts needed to understandfluid motions at a macroscopic level. Hence, the idea of a continuousmedium and its kinematical properties are presented. Then the physicallaws governing the motion (mass, momentum and energy equation) arederived and discussed. The concept of rheological law is introduced andillustrated. I end this lecture with a discussion of the boundaryconditions that are needed to complete the equations of motion.

In this lecture I propose a little tour of thermal convection and itsapplications in astrophysics. The first part of the lecture is devotedto a qualitative introduction to the convective instability using theSchwarzschild criterion; then, concentrating on the equations governingthe fluid motions, I introduce the Boussinesq and anelasticapproximations which are so often used in these problems. The followingpart focuses on the Rayleigh-Bénard model which is worked out indetail up to the Landau equation and the Lorenz strange attractor.Finally, I briefly sketch out some results on turbulent convection andend the lecture with the case of stellar convection.

This chapter is primarily intended as an introductory text, a pedagogical platform, on the phenomenon ofturbulence in fluids. For the sake of simplicity, the discussion is mostly limited to the case of anincompressible (constant-property) newtonian fluid in simple three-dimensional turbulent flows. Additionalcomplexities related to thermal convection, magnetic forces, nuclear reactions and so forth are ignored onpurpose.The main motivation is to exhibit the general problematic of turbulence in an as simple as possible physicalsetting. Modelling prospecting, which aims at elaborating numerically tractable mathematical models of turbulence,is also brought up.

This lecture provides some introduction to perfect fluid dynamics within theframework of general relativity. The presentation is based on theCarter-Lichnerowicz approach. It has the advantage over the more traditionalapproach of leading very straightforwardly to important conservation laws, such as the relativistic generalizations ofBernoulli's theorem or Kelvin's circulation theorem. It also permits to get easily first integrals of motion which are particularly usefulfor computing equilibrium configurations of relativistic stars in rotation or in binary systems.The presentation is relatively self-contained and does not require any a prioriknowledge of general relativity. In particular, the three types of derivatives involved in relativistic hydrodynamics are introducedin detail: this concerns the Lie, exterior and covariant derivatives.

An analytical theory is developed that describes theprocess of nonlinear saturation of the magnetorotational instability. Thetheory employs the shearing sheet approximation with arbitrary (but linear)shear flow, and is asymptotically exact in a well-defined parameter regime. In this regime the instability saturates by extracting energy from the shear,and modifying the shear responsible for it. The saturation process requires both viscous and ohmic dissipation. The theory also describes the approach from small amplitude perturbations to the final strongly nonlinear saturated state.

In this course, I will review the main properties of internal gravity waves, andexamine how they can lead to particle and angular momentum transport in stars. Iwill then present some recent results that have been obtained for angularmomentum transport which show that waves are an important ingrediant of stellar modelling.

Using a variety of laboratory analogues and some simple models, Itry to portray the fluid dynamics of very hot stars and disks. By analogy to fluidizedbeds, we may expect the formation of photon bubbles that stir things up intoa sort of turbulence. Because of rotation, the bubbles merge tomake vortices whose presence causes inhomogeneous radiative outflowsand produces spots. These phenomena will allow the Eddington limit to be exceeded and, where the radiative outflows are locally large, mass loss will be encouraged.

Turbulent combustion is the basic physical phenomenon responsiblefor efficient energy release by any internal combustion engine. Howeverit is accompanied by other undesirable phenomena such as noise, pollutantspecies emission or damaging instabilities that may even lead tothe system desctruction. It is then crucial to control this phenomenon,to understand all its mecanisms and to master it in industrial systems.For long time turbulent combustion has been explored only through theoryand experiment. But the rapid increase of computers power during thelast years has allowed an important development of numerical simulation,that has become today an essential tool for research and technicaldesign. Direct numerical simulation has then allowed to rapidlyprogress in the knowledge of turbulent flame structures, leading tonew modelisations for steady averaged simulations. Recently large eddysimulation has made a new step forward by refining the descriptionof complex and unsteady flames. The main problem that arises whenperforming numerical simulation of turbulent combustion is linkedto the description of the flame front. Being very thin, it can nothowever be reduced to a simple interface as it is the location of intensechemical transformation and of strong variations of thermodynamicalquantities. Capturing the internal structure of a zone with a thicknessof the order of 0.1 mm in a computation with a mesh step 10 timeslarger being impossible, it is necessary to model the turbulentflame. Models depend on the chemical structure of the flame, on theambiant turbulence, on the combustion regime (flamelets, distributedcombustion, etc.) and on the reactants injection mode (premixedor not). One finds then a large class of models, from the most simplealgebraic model with a one-step chemical kinetics, to the most complexmodel involving probablity density functions, cross-correlations andmultiple-step or fully complex chemical kinetics.

Two possible initial paths of transition to turbulence in simpleshear flows are examined. The first is the – by now classical – transient (or algebraic) growth scenario which may have an important rolein the by-pass transition of those flows for which traditional eigen-analysis predicts asymptotic stability. Transient growth isoptimally excited by certain initial disturbances now known as"optimal perturbations"; they can be found through a classical variationalanalysis initiated by Farrell (1988). The second path starts with theexponential amplification, in nominally subcritical conditions, ofmodal disturbances developing over a base flow mildly distorted withrespect to its idealized counterpart. The base flow distortion of given norm that excites the largest growth of the instability wave is called the “minimal defect", and its study was initiated by Bottaro et al. (2003). Both paths provide feasible initial conditions for the transition process and it is likely that in most practical situations algebraic and exponential growth mechanisms are concurrently at play in provoking transition to turbulence in shear flows.

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then presented. The next section is devoted to one dimensional equation solvers using only one domain. Three different methods are described. Techniques using several domains are shown in the last section of this paper and their various merits discussed.

The purpose of this lecture is two fold: first, to describe a powerful numerical technic, namely the spectral method, to solve the compressible (anelastic) magnetohydrodynamic (MHD) equations in spherical geometry and then to discuss some recent numerical applications to study stellar dynamics and magnetism. We thus start by describing the semi-implicit, anelastic spherical harmonic (ASH) code. In this code, the main field variables are projected into spherical harmonics for their horizontal dimensions and into Chebyshev polynomials for their radial direction. We then present, high resolution 3–D MHD simulations of the convective region of A- and G-type stars in spherical shells. We have chosen to model A and G-type stars because they represent good proxies to study and understand stellar dynamics and magnetism given their strikingly different internal “up-side-down” structure and magnetic activity level. In particular, we discuss the nonlinear interactions between turbulent convection, rotation and magnetic fields and the possibility for such flows and fields to lead to dynamo action. We find that both core and envelope turbulent convective zones are efficient at inducing strong mostly non-axisymmetric fields near equipartition but at the expense of damping the differential rotation present in the purely hydrodynamic progenitor solutions.

The purpose of this lecture is to give an overview of the subgrid scale models used for Large Eddy Simulations (LES).The first models where essentially eddy-viscosity models,where the turbulent viscosity coefficient was found by simple dimensional considerations (the mixing length theory) ortaking into account Kolmogorov theory (theSmagorinsky model).Since then there have been many improvements, and now some of the physical ingredients of turbulence, such as the localbackscatter and the anisotropy of small scale structures, canbe successfully taken into account.Far from being a black boxtool to obtain answers toturbulent flow problems, the Large Eddy Simulation techniqueis still a work in progress with many odds and ends. We give in this lecture an overview of the historicaldevelopment of turbulence models and wepoint out the lights and shadows of them with attention to the open problems that still remain to be solved.

The dynamics of solar and stellar surfaces is very important to understand the dynamicsof their interior as well as their activity cycles. This paper describes the dynamics observed at the solar surface from small scales (granulation) to the largestscales (differential rotation and meridional circulation).

Solar magnetic fields are produced by dynamo action in or just below the solar convection zone. After an account of observations of solar magnetic activity, the fundamentals of dynamo theory are reviewed, and the ideas applied to a discussion of different models of dynamo action in the Sun.

In this lecture I try to explain the basic dynamical processes atwork in a radiative zone of a rotating star. In particular, the notionof baroclinicity is thoroughtly discussed. Attention is speciallydirected to the case of circulations and the key role of angular momentumconservation is stressed. The specific part played by viscosity isalso explained. The old approach of Eddington and Sweet is reviewed andcriticized 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.

The structure of radiative shock waves propagating through partially ionized hydrogen gas in stellar atmospheres is discussed. Basic equations including the radiation transfer and the method of their self-consistent solution are described. The most striking result is that the ratio of the radiative flux to the total energy flux of the shock wave very rapidly enlarges with increasing upstream velocity, so that for Mach number larger than 7, the major part of the shock energy is irreversibly lost due to dissipation processes. The understanding of the “missing temperature”,called microturbulence by the astrophysicists, which appears when we wantto modelling the width of stellar line profiles, is discussed.It is shown that the turbulenceamplification in the atmosphere of a radially pulsating star is not onlydue to the global compression of the atmosphere during thepulsation. Strong shock waves propagating from the deep atmosphereto the very low density layers also play a role in the turbulencevariation, especially when they become very strong i.e., hypersonic.For shocks, the predicted turbulence amplification predicted by classicalmodels is overestimated with respect to stellar observationswhen the compression rate becomes larger than 2 which corresponds to alimit Mach number near 2. Thus, when radiative effects take place, thepresent turbulence amplification theory breaks down. A new approach isrequired.

The observed brightness fluctuations and large-scale structures on thesurface of the Red Supergiant Betelgeuse make the star a prime targedfor future interferometric measurements. At the same time, they openthe possibility to resolve these structures in numerical radiationhydrodynamics simulations of the entire star. After some general remarksabout the possibility and difficulties of 3D simulations of stars ingeneral results of such calcuations of the outer convective envelope andthe atmosphere of a Red Supergiant and a star on the Asmyptotic GiantBranch are presented. These show that numerous short-lived small-scalesurface granules coexist with a few long-lived large-scale envelopeconvection cells. Pressure fluctuations deform the star and influencethe surface convection. The convective “granulation" pattern differsfrom the solar one. Convection and pulsations produce large-scalehigh-contrast brightness fluctuations that might explain the observedluminosity variations and surface “spots”. Shock waves and supersonicatmospheric velocities manifest themselves in broad line profiles.

Direct observations of gravitational waves will open in the nearfuture new windows on the Universe. Among the expected sources,instabilities of rotating compact astrophysical objects are waited tobe detected with some impatience as this will sign the birth of“gravitational waves asteroseismology”, a crucial way to improve ourknowledge of matter equation of state in conditions that cannot bereproduced in a lab. However, the theoretical work needed to reallyget informations from to-be-detected signals is still quite large, numericalsimulations having become a necessary key ingredient. This articletries to provide a short overview of the main physical topics involvedin this field (general relativity, gravitational waves, instabilitiesof rotating fluids, etc.), concluding with a brief descriptionof the work that was done in Paris-Meudon Observatory by SilvanoBonazzola and collaborators.

Although discovered almost forty years ago, pulsars still rank amongst the most fascinating astrophysical objects in the universe. They are believed to be strongly magnetised rotating neutron stars, whose existence had already been predicted soon after the discovery of the neutron in the beginning of the 30 s. However, relatively little progress has been made in understanding the fundamental physical mechanisms at work. Due to the lack of a satisfactory equation of state at very high densities, the internal structure of the neutron star remains enigmatic. Moreover, a global self-consistent picture of the close surrounding of the star, responsible for the emission of electromagnetic waves as well as for the launch of the pulsar wind and the energy loss due to interaction with the ambient medium, has not yet been proposed. Here, we give a brief review on the theory of pulsar's electrosphere (and magnetosphere) as well as on some recent developments. After a historical introduction, we recall some basic properties of these compact objects and the main observational characteristics, describing our present knowledge about pulsars. We then present our understanding of the pulsar magnetosphere, explaining in some detail a few theoretical ideas such as the polar cap and outer gap models. Some promising recent developments will be discussed: the electrospheric structure and the striped wind model.