If one strives for a reliable description of “Turbulent Mixing in Stars” one must account for a large variety of physical processes. These include non-locality, that is needed in unstably stratified regimes, overshooting, which occurs in a stably stratified regime, double-diffusion processes (semi-convection and salt-fingers), transport of angular momentum, the Li7 problem, compressibility, and magnetic fields. While phenomenological models are manifestly inadequate, LES are too computer intensive to tackle this large variety of processes. Since the requirement of completeness of the description of these processes must also result in models that are usable in stellar structure-evolution codes, we conclude that only the RSM (Reynolds Stress Model) can do so and a description of the state of the art in that field is presented in Part 2.

The Reynolds Stress Model (RSM) yields the dynamic equations for the second-order moments (e.g., heat fluxes) needed in the equations for the mean variables (e.g., mean temperature). The RSM equations are in general time dependent and non-local. We first discuss the “buoyancy only” case and the tests of the non-local model against a variety of data. We also “plumenize” the model in order to exhibit the up-down flows that characterize convection so as to show that a non-local RSM is fully equipped to account for the “plume aspect” of buoyant flows. Next, we extend the RSM to account for stable and/or unstable stratification and shear, a formalism that is needed to describe the overshooting region contributed by differentail rotation. We conclude by discussing the equation for the dissipation of turbulent kinetic energy which plays a key role in any RSM.

Turbulent convection carries most of the solar luminosity from the stable radiative interior to the visible surface over a substantial fraction of the solar radius. The turbulent motions are highly nonlinear displaying activity on several spatial and temporal scales. The effects of these motions are to redistribute angular momentum, mix light elements, excite gravity waves and power magnetic activity. In this talk, I shall review some of the recent efforts, mostly based on numerical simulations, to model turbulent convection spanning multiple scale heights. I will discuss what we have learnt about the nature of convective transport, and what the relevant timescales are on which convection operates.

In the quest for a better theoretical understanding of the dynamical processes operating in stellar envelopes, increasingly more realistic hydrodynamical models of stellar convection have been developed over the last 25 years. Based on 3D state-of-the-art simulations performed recently with the radiation hydrodynamics code CO5BOLD, we review the present status of hydrodynamical modeling of stellar surface convection and demonstrate the basic differences between 3D hydrodynamical models and 1D ‘classical’ hydrostatic atmospheres where convection is treated by the mixing-length theory. We briefly discuss some of the many possible applications across the Hertzsprung-Russell diagram, including spectroscopic abundance determinations using 3D stellar atmospheres, particularly with regard to the much debated question of the solar oxygen abundance.

This paper summarizes the results of a recent investigation on the Hanle effect in atomic and molecular lines, which indicates that there is a vast amount of “hidden” magnetic energy and (unsigned) magnetic flux in the internetwork regions of the quiet solar photosphere. This hidden magnetic energy, localized in the (intergranular) downflowing plasma of the solar photosphere, is carried mainly by tangled fields at sub-resolution scales with strengths between the equipartition field values and ∼1 kG, and is more than sufficient to compensate the radiative energy losses of the solar outer atmosphere.

The chromosphere in internetwork regions of the quiet Sun was regarded as a static and homogeneous layer for a long time. Thanks to advances in observations and numerical modelling, the wave nature of these atmospheric regions received increasing attention during the last decade. Recent three-dimensional radiation magnetohydrodynamic simulations with CO5BOLD feature the chromosphere of internetwork regions as a dynamic and intermittent phenomenon. It is a direct product of interacting waves that form a mesh-like pattern of hot shock fronts and cool post-shock regions. The waves are excited self-consistently at the top of the convection zone. In the middle chromosphere above an average height of 1000 km, plasma beta gets larger than one and magnetic fields become more important. The model chromosphere exhibits a magnetic field that is much more homogeneous than in the layers below and evolves much faster. That includes fast propagating (MHD) waves. Further improvements of the simulations like time-dependent hydrogen ionisation are currently in progress. This class of models is capable of explaining apparently contradicting diagnostics such as carbon monoxide and UV emission at the same time.

Classical theories of turbulence do not describe accurately inertial range scaling laws in turbulent convection and notably fail to model the shape of the turbulent spectrum of solar photospheric convection. To understand these discrepancies, a detailed study of scale-by-scale budgets in turbulent Rayleigh-Bénard convection is presented, with particular emphasis placed on anisotropy and inhomogeneity. A generalized Kolmogorov equation applying to convection is derived and its various terms are computed using numerical simulations of turbulent Boussinesq convection. The analysis of the isotropic part of the equation shows that the third-order velocity structure function is significantly affected by buoyancy forcing and large-scale inhomogeneities. Anisotropic contributions to this equation are also shown to be comparable to their isotropic counterpart at moderate to large scales. Implications of these results for convection in the solar photosphere, mesogranulation and supergranulation are discussed.

A summary of the round table discussion following Session A on Modelling Convection and Radiative Transfer is provided based on the video recording made during the conference.

Broadening and asymmetry of spectral lines in slowly rotating late A-type stars provide evidence for high-amplitude convective motions. The properties of turbulence observed in the A-star atmospheres are not understood theoretically and contradict results of previous numerical simulations of convection. Here we describe an ongoing effort to understand the puzzling convection signatures of A stars with the help of 3-D hydrodynamic simulations. Our approach combines realistic spectrum synthesis and non-grey hydrodynamic models computed with the CO5BOLD code. We discuss these theoretical predictions and confront them with high-resolution spectra of A stars. Our models have, for the first time, succeeded in reproducing the observed profiles of weak spectral lines without introducing fudge broadening parameters.

We successfully ported the suite of codes developed by R. L. Kurucz for stellar atmosphere modelling, abundance determination and synthetic spectra calculation, to run under GNU-Linux. The ported codes include ATLAS 9 and ATLAS 12 for 1-D plane-parallel atmosphere model calculation, DFSYNTHE, which calculates the Opacity Distribution Functions (ODF) to be used with ATLAS 9, WIDTH to derive chemical abundances from measured line Equivalent Widths (EW) and SYNTHE to calculate synthetic spectra. The codes input and output files remain fully compatible with the VMS versions, while the computation speed has been greatly increased due to the high efficiency of modern PC CPUs. As an example, ATLAS 9 model calculations and the computation of large (e.g. 10 nm) synthetic spectra can be executed in a matter of minutes on any mainstream laptop computer. Arbitrary chemical compositions can be used in calculations (by using ATLAS 12 through opacity sampling or by calculating ad-hoc ODFs for ATLAS 9). The large set of scripting languages existing under Linux (shell, perl, python. . .) and the availability of low-cost multiprocessor Linux architectures (such as Beowulf) makes the port highly effective to build model farms to produce large quantities of atmosphere models or synthetic spectra (e.g. for the production of integrated light synthetic spectra). The port is hosted on a dedicated website including a download section for source codes, precompiled binaries, needed data (opacities, line lists and so on), sample launch scripts and documentation.

We use numerical simulations of granulation in the Sun and a K dwarf to study the effects of coherent structures on higher order moments. The latter need to be calculated in non-local Reynolds stress models of turbulent convection. Models that explicitly account for the asymmetry between up- and downflows as well as hot and cold drafts provide a substantial improvement over traditional ones, such as the quasi-normal approximation, which is only able to provide order of magnitude estimates for this type of flow.

We investigate the properties of non-local Reynolds stress models of turbulent convection in a spherical geometry. Regularity at the centre r=0 places constraints on the behaviour of 3rd order moments. Some of the down-gradient and algebraic closure models have inconsistent behaviour at r=0. A combination of down-gradient and algebraic closures gives a consistent prescription that can be used to model convection in stellar cores.

We introduce an extended 3D numerical simulation study of Reynolds stress models of stellar convection and probe fluxes as well as mean temperature gradient profiles.

We provide results from an extended numerical simulation study of Reynolds stress models of stellar convection and probe the modelling of third order moments and non-locality.

We provide results from an extended 3D numerical simulation study of Reynolds stress models of stellar convection and probe the modelling of compressibility, pressure fluctuations, and dissipation of turbulent kinetic energy.

We present the essential features of the ANTARES code. ANTARES has been developed to perform the simulation of astrophysical flows in one, two, or three dimensions. Using, in particular, the option of grid refinement, we present results for solar granulation achieved at very high spatial resolution.

We discuss some benefits and pitfalls when combining conceptually different types of closure approximations into complete Reynolds stress models of stellar convection.

The comparison between the observed and theoretical p-mode oscillation frequencies of many orders and degrees allows a detailed determination of the internal solar structure (Christensen-Dalsgaard et al. 1996). The most recent analysis shows that one source of error in the modern description of the Sun is located in the convective layers close to the surface, in the superadiabatic region (Gabriel 1994).

The standard mixing length model of convection is ill behaved at the centre of a star since the pressure scale height Hp=P/(ρg)→∞ as r→0, and the convective flux remains non zero at r=0. We propose a modifcation of this model of convection that has the correct behaviour in the central regions of a star and smoothly changes to the standard MLT away from the centre.

Convection occurs in the visible photospheric layers of most stars having Te less than about 10000 K, and in some hotter stars. The solar granulation pattern is a symptom of this, as is the non-zero microturbulent velocity often required in abundance analysis to make both weak and strong lines yield the same abundance.

In very sharp-lined stars, the presence of a non-thermal velocity field in the visible stellar atmosphere leads to several other effects which may be detected in spectral line profiles. These include radial velocities that vary systematically with equivalent width, distortions of the line profile as compared to a profile computed with a Voigt profile and rotational broadening (“macroturbulence”), and asymmetries with respect to the line centre (“bisector curvature”).

Detection and interpretation of these effects, with the goal of obtaining empirical information about a velocity field present in the visible layers, requires comparison with calculated synthetic spectra which incorporate model velocity fields. Thus, this review will summarize some of the observational clues concerning photospheric velocity fields, as well as modelling aimed at interpreting these data.