arXiv subjects
Publications and source records attributed to Michel Rieutord.
In this lecture I present the way stars can be modeled in two dimensions and especially the fluid flows that are driven by rotation. I discuss some of the various ways of taking into account turbulence and conclude this contribution by a short presentation of some of the first results obtained with the ESTER code on the modeling of interferometrically observed fast rotating early-type stars.
Context.Interpretation of light curves of many types of binary stars requires the inclusion of the (cor)relation between surface brightness and local effective gravity. Until recently, this correlation has always been modeled by a power law relating the flux or the effective temperature and the effective gravity, namely T_eff α g_eff^β. Aims. We look for a simple model that can describe the variations of the flux at the surface of stars belonging to a binary system. Methods. This model assumes that the energy flux is a divergence-free vector anti-parallel to the effective gravity. The effective gravity is computed from the Roche model. Results. After explaining in a simple manner the old result of Lucy (1967), which says that β=0.08 for solar type stars, we first argue that one-dimensional models should no longer be used to evaluate gravity darkening laws. We compute the correlation between log T_eff and log g_eff using a new approach that is valid for synchronous, weakly magnetized, weakly irradiated binaries. We show that this correlation is approximately linear, validating the use of a power law relation between effective temperature and effective gravity as a first approximation. We further show that the exponent β of this power law is a slowly varying function, which we tabulate, of the mass ratio of the binary star and the Roche lobe filling factor of the stars of the system. The exponent β remains mostly in the interval (0.20, 0.25) if extreme mass ratios are eliminated. Conclusions. For binary stars that are synchronous, weakly magnetized and weakly irradiated, the gravity darkening exponent is well constrained and may be removed from the free parameters of the models.
Modelling isolated rotating stars at any rotation rate is a challenge for the next generation of stellar models. These models will couple dynamical aspects of rotating stars, like angular momentum and chemicals transport, with classical chemical evolution, gravitational contraction or mass-loss. Such modelling needs to be achieved in two dimensions, combining the calculation of the structure of the star, its mean flows and the time-evolution of the whole. We present here a first step in this challenging programme. It leads to the first self-consistent two-dimensional models of rotating stars in a steady state generated by the ESTER code. In these models the structure (pressure, density and temperature) and the flow fields are computed in a self-consistent way allowing the prediction of the differential rotation and the associated meridian circulation of the stars. After a presentation of the physical properties of such models and the numerical methods at work, we give the first grid of such models describing massive and intermediate-mass stars for a selection of rotation rates up to 90% of the breakup angular velocity.
Spherical Couette flow (flow between concentric rotating spheres) is one of flows under consideration for the laboratory magnetic dynamos. Recent experiments have shown that such flows may excite Coriolis restored inertial modes. The present work aims to better understand the properties of the observed modes and the nature of their excitation. Using numerical solutions describing forced inertial modes of a uniformly rotating fluid inside a spherical shell, we first identify the observed oscillations of the Couette flow with non-axisymmetric, retrograde, equatorially anti-symmetric inertial modes, confirming first attempts using a full sphere model. Although the model has no differential rotation, identification is possible because a large fraction of the fluid in a spherical Couette flow rotates rigidly. From the observed sequence of the excited modes appearing when the inner sphere is slowed down by step, we identify a critical Rossby number associated with a given mode and below which it is excited. The matching between this critical number and the one derived from the phase velocity of the numerically computed modes shows that these modes are excited by an instability likely driven by the critical layer that develops in the shear layer staying along the tangent cylinder of the inner sphere.
Interpretation of interferometric observations of rapidly rotating stars requires a good model of their surface effective temperature. Until now, laws of the form T_eff \propto g_eff^β have been used, but they are only valid for slowly rotating stars. We propose a simple model that can describe the latitudinal variations in the flux of rotating stars at any rotation rate. This model assumes that the energy flux is a divergence-free vector that is antiparallel to the effective gravity. When mass distribution can be described by a Roche model, the latitudinal variations in the effective temperature only depend on a single parameter, namely the ratio of the equatorial velocity to the Keplerian velocity. We validate this model by comparing its predictions to those of the most realistic two-dimensional models of rotating stars issued from the ESTER code. The agreement is very good, as it is with the observations of two rapidly rotating stars, α Aql and α Leo. We suggest that as long as a gray atmosphere can be accepted, the inversion of data on flux distribution coming from interferometric observations of rotating stars uses such a model, which has just one free parameter.
Context: Supergranulation is a pattern of the velocity field at the surface of the Sun, which has been known about for more than fifty years, however, no satisfactory explanation of its origin has been proposed. Aims: New observational constraints are therefore needed to guide theoretical approaches which hesitate between scenarios that either invoke a large-scale instability of the surface turbulent convection or a direct forcing by buoyancy. Method: Using the 14-Mpixel CALAS camera at the Pic-du-Midi observatory, we obtained a 7.5h-long sequence of high resolution images with unprecedented field size. Tracking granules, we have determined the velocity field at the Sun's surface in great detail from a scale of 2.5Mm up to 250Mm. Results: The kinetic energy density spectrum shows that supergranulation peaks at 36Mm and spans on scales ranging between 20Mm and 75Mm. The decrease of supergranular flows in the small scales is close to a $k^{-2}$-power law, steeper than the equipartition Kolmogorov one. The probability distribution function of the divergence field shows the signature of intermittency of the supergranulation and thus its turbulent nature.
An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.
Context: The observations of rapidly rotating stars are increasingly detailed and precise thanks to interferometry and asteroseismology; two-dimensional models taking into account the hydrodynamics of these stars are very much needed. Aims: A model for studying the dynamics of baroclinic stellar envelope is presented. Methods: This models treats the stellar fluid at the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field along with the rotation of the system generate the baroclinic flow. Results: We manage to give an analytical solution to the asymptotic problem at small Ekman and Prandtl numbers. We show that, provided the Brunt-Vaisala frequency profile is smooth enough, differential rotation of a stably stratified envelope takes the form a fast rotating pole and a slow equator while it is the opposite in a convective envelope. We also show that at low Prandtl numbers and without $μ$-barriers, the jump in viscosity at the core-envelope boundary generates a shear layer staying along the tangential cylinder of the core. Its role in mixing processes is discussed. Conclusions: Such a model provides an interesting tool for investigating the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or, even more, of a dynamo effect.
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.
We present results concerning the linear (radial and non-radial) oscillations of non-rotating superfluid neutron stars in Newtonian physics. We use a simple two-fluid model to describe the superfluid neutron star, where one fluid consists of the superfluid neutrons,while the second fluid contains all the comoving constituents (protons, electrons). The two fluids are assumed to be "free" in the sense of absence of vortex-mediated forces like mutual friction or pinning, but they can be coupled by the equation of state, in particular by entrainment.We calculate numerically the eigen-frequencies and -modes of adiabatic oscillations, neglecting beta-reactions that would lead to dissipation.We find a doubling of all acoustic-type modes (f-modes, p-modes), and confirm the absence of g-modes in these superfluid models. We show analytically and numerically that only in the case of non-stratified background models (i.e. with no composition gradient) can these doublets of acoustic modes be separated into two distinct families, which are characterised by either co- or counter-moving fluids respectively, and which are sometimes referred to as "ordinary" and "superfluid" modes. In the general, stratified case, however, this separation is not possible, and these acoustic modes can not be classified as being either purely "ordinary" or "superfluid". We show how the properties of the two-fluid modes change as functions of the coupling by entrainment. We find avoided mode-crossings for the stratified models, while the crossings are not avoided in the non-stratified, separable case. The oscillations of normal-fluid neutron stars are recovered as a special case simply by locking the two fluids together.In this effective one-fluid case we find the usual singlet f- and p-modes, and we also find the expected g-modes.
We 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 Gamma-Doradus star using the anelastic approximation. Some fascinating features of rapidly rotating fluids, such as wave attractors, appear in both cases.
After showing that the anelastic approximation is better than the subseismic one to filter out acoustic waves when studying low-frequency stellar oscillations, we compute gravito-inertial modes of a typical Gamma-Doradus star using this approximation. We show that eigenmodes can be regular or singular, according to the possible focusing towards attractors of the underlying characteristics. Consequences on the oscillations spectrum are then discussed.
A comparative study of the properties of the anelastic and subseismic approximations is presented. The anelastic approximation is commonly used in astrophysics in compressible convection studies whereas the subseismic approximation comes from geophysics where it is used to describe long-period seismic oscillations propagating in the Earth's outer fluid core. Both approximations aim at filtering out the acoutic waves while retaining the density variations of the equilibrium configuration. However, they disagree on the form of the equation of mass conservation. We show here that the anelastic approximation is in fact the only consistent approximation as far as stellar low-frequency oscillations are concerned. We also show that this approximation implies Cowling's approximation which neglects perturbations of the gravity field. Examples are given to assert the efficiency of the anelastic approximation.
Recently, eigenmodes of rotating fluids, namely inertial modes, have received much attention in relation to their destabilization when coupled to gravitational radiation within neutron stars. However, these modes have been known for a long time by fluid dynamicists. We give a short account of their history and review our present understanding of their properties. Considering the case of a spherical container, we then give the exact solution of the boundary (Ekman) layer flow associated with inertial r-modes and show that previous estimations all underestimated the dissipation by these layers. We also show that the presence of an inner core has little influence on this dissipation. As a conclusion, we compute the window of instability in the Temperature/rotation plane for a crusted neutron star when it is modeled by an incompressible fluid.