This meeting, which deals with turbulence in stars, opens with a review on thermal convection. There is no better way to state from the start that among all instabilities that are likely to arise in stars, it is thermal convection which is the most firmly established as a cause for the turbulence that we observe on their surface. Our confidence in this comes mainly from the theoretical prediction that convective instability sets in whenever the density stratification becomes superadiabatic, as is expected in late type stars whose outer layers are very opaque, due to the ionization of the two most abundant elements, hydrogen and helium. And, in these stars at least, thermal convection occurs close enough to the photosphere to influence, be it indirectly, the profile of spectral lines.

The density in the outer layers of stars varies by several orders of magnitude and it is desirable to include the full effects of compressibility in any study of instabilities arising in stellar convection zones. In an unstable compressible fluid-layer that is thermally conducting both the oscillatory and non-oscillatory motions can simultaneously arise. The conditions under which the oscillatory acoustic modes can be overstabilized in a polytropic atmosphere are examined. It is argued that the linearized perturbation theory breaks down when applied to an inviscid complete polytrope which has vanishing density and temperature at the top for both optically-thin and optically-thick approximations. However, the linearized theory is demonstrated to be self-consistent when viscosity and thermal conductivity are included in the study of complete polytropes.

Two regions of convective instability are present in the photosphere of a typical B-star (Teff = 30,000 K Log g = 4.0). One is the usual thermal instability caused by the helium ionization. The other is driven by the continuum radiation pressure in a thermally stable layer. Mixing length and anelastic modal representations of these unstable regions show that the rapid radiative cooling of temperature fluctuations limits the velocities to an amplitude of a few meters per second, much too small to account for the observed line broadening and asymmetries.

We derive expressions for the turbulent viscosity and turbulent conductivity applicable to convection zones of rotating stars. We assume that the dimensions of the convective cells are known and derive a simple distribution function for the turbulent convective velocities under the influence of rotation. From this distribution function (which includes, in particular, the stabilizing effect of rotation on convection) we calculate in the mixing-length approximation: i) the turbulent Reynolds stresstensor and ii) the expression for the heat flux in terms of the superadiabatic gradient. The contributions of the turbulent convective motions to the mean momentum and energy equation are treated consistently, and assumptions about the turbulent viscosity and heat transport are replaced by assumptions about the turbulent flow itself. The free parameters in our formalism are the relative cell sizes and their dependence on depth and latitude.

Spectrograms of Mg I-absorptions lines have been registrated at different positions of the damping wings. The corresponding heights within the atmosphere have been approximated by the Eddington-Barbier approximation. We calculated the coherence between the intensity fluctuations in the continuum and those of the higher layers. We found a rather flat gradient up to a height of 100 km and above this a steep decrease of coherence. From this result we can conclude that above a height of 100 km any fluctuations of intensity are not due to ordinary convective processes. However, up to a height of 400 km we found a coherence between the velocity of granulation and its intensity fluctuations.

Numerical simulations of the convective granular motions in the solar photosphere are presented. Realistic background physics allows a detailed comparison with observed characteristics of the solar granulation. The numerical methods are based on a bivariate Fourier representation in the horizontal plane, combined with a cubic spline representation in the vertical direction. Using a numerical grid with 16×16×16 grid points, which cover a unit cell of dimension ⋍ 3600×3600×1500 km, granular motions have been followed over several turnover times. The simulated motion shows the characteristics of granular motion. The evolution of large granules into bright rings (“Exploding granules”) is a consequence of the accumulating excess pressure at the granule center, necessary to support the horizontal velocities required by the continuity equation. The increasing pressure evenutally inhibits further upward motion at the granule center, which cools radiatively and shows up as a dark center in the expanding granule.

The topic I was originally assigned for this colloquium was “Generation of Non Thermal, Non Oscillatory Motions”. Being basically a fluid dynamicist, at first I thought this meant I was supposed to talk about the origin of motions which are not thermally driven, i.e., I should not talk about convection. But then I realized all that was meant was that I was to talk about bulk fluid motions, rather than the molecular “thermal” motion of stellar gas that defines its temperature. Obviously the original question was posed by a stellar spectroscopist! Having surmounted that small semantic hurdle, I began to think about all the ways circulatory motions might be generated in a star. All manner of fluid dynamical instabilities come to mind--not only convective instability, but also barotropic or inertial, baroclinic, Kelvin-Helmholz, Rayleiqh-Taylor, Goldreich-Shubert, Solberg-Hoiland, etc. The list is large, overlapping, I am sure confusing to an observer (and to many a theoretician). Then there are Eddington-Sweet currents, and several additional motions arising from the presence of magnetic fields--fields which give rise to magnetic buoyancy of flux tubes, and large collection of magnetohydrodynamic instabilities.

Thermal overstability of non-radial eigenmodes of stars is discussed as one of possible causes for generating non-thermal motions in the stellar atmosphere. The nature of oscillatory motions in stars is first considered both in the local and the global stand points. Then, the excitation of eigen-oscillations is discussed and results of numerical studies so far made are reviewed for the vibrational stability of various stellar models against non-radial oscillations. It is found that many of non-radial p-modes of high tesseral harmonics are likely excited in various stars of the HR diagram and that they possibly manifest themselves as non-thermal velocity fields in the stellar atmosphere.

High-resolution photographic spectra of the center of the solar disk have been obtained with the Vacuum Tower Telescope at Sacramento Peak Observatory. Two weak iron lines and the neighboring continuum were recorded with 40 sec time resolutions and better than 1” spatial resolution over a period of 40 min. Intensity and velocity fluctuations were obtained in the two lines and continuum as a function of time and space, and 300 sec oscillations were filtered out. The resulting fluctuations, due solely to granulation, were assembled into an ensemble average of the center of a granule and the center of an intergranular lane, as a function of time. The intensity-fluctuation data have been analyzed through calculation of model line profiles to yield temperature fluctuations in a granule as functions of time and height. We find that the line parameters are distinctly out of phase with continuum brightness, so that, for example, maximum brightness at line center occurs approximately 100 sec prior to maximum continuum brightness. A series of one-dimensional model atmospheres representing the granule at various stages of its lifetime is presented.

Convective motions are manifest in the solar spectrum as slight spectral line asymmetries and wavelength shifts. These have been studied for 311 Fe I lines. Most lines are blueshifted because the larger contribution of blueshifted photons from bright and rising granules statistically dominates over the contribution from dark and sinking intergranular lanes. Fainter lines (formed deeper) show larger blueshifts than strong lines; high-excitation lines usually are more blueshifted (preferentially formed in the hotter granules) and short-wavelength lines are more blueshifted because of increased granulation contrast there. Detailed studies of line bisector behavior as function of line parameters permit the construction of model atmospheres incorporating convective hydrodynamics. The method does not require spatially resolved observations and can be extended to studies of stellar convection.

The differential line shifts method for studying convective type motions has been described in the review of Glebocki and Stawikowski at this colloquium. Here the results of differential line shifts for twenty late type stars are presented. Line shifts were determined either from published wavelength measurements (α CMi, β Peg, α Boo, α Ser, α UMa, α Per, α Sco, γ Cyg α Car and HD 19445) or from my own wavelength measurements on 8.1 Å/mm dispersion spectra (56 Ori, η Peg, η Psc, 61 Cyg A, ε Eri, δ Eri, o Tau, ε Cyg, α Aqr and ι Ari). Because of low accuracy of the wavelength determinations, mean shifts (VR) were calculated for narrow ranges of lower excitation potentials (LEP). These shifts were plotted against excitation potential and VR-LEP relations were obtained for all analysed stars. Because of the scatter of points in these diagrams linear approximation was assumed and the slopes (denoted A and expressed in Km/s/eV) of the VR-LEP relations were calculated. The stellar parameters and the derived slopes for the whole sample of investigated stars were analysed, and correlations with microturbulence, luminosity and gravity were found. These correlations demonstrate that the slope of VR-LEP relation describes some component of velocity field in late type stars.

The detailed description of the procedure and results will be published in Acta Astronomica.

Analyses of some 300 hours of time sequences of solar EUV line profiles obtained with 0S0-8 show large fluctuations in line widths. At a given location on the sun, line widths fluctuate temporally on time scales ranging from less than a minute to over an hour. At any given time, line widths fluctuate spatially on a variety of scales ranging from active region size to arc second size. Temporal and spatial fluctuations are of approximately the same amplitude. Thus, the sun can be characterized by an aggregate of small cells in each of which line widths are fluctuating in time and which have random phases with respect to each other.

Spatial fluctuations in line width are correlated with large scale spatial fluctuations in brightness for some lines but not for others. Temporal fluctuations in width are sometimes correlated with either Doppler shifts or intensity fluctuations, but more often such correlations are absent.

For a given line, the line width varies through an extreme range of about a factor of two. Nonthermal components of line width vary from approximately the local sound speed to a small fraction of the sound speed.

The shapes of weak fraunhofer lines, including their asymmetry, are explained by the influence of acoustic waves and granular motions. The following pattern of granula is adopted: the ascending (granular) and descending (intergranular) velocities V1 and V2 as well as the relevant temperatures T1 and T2 are maximum (minimum) at the center of granula (intergranula) and decrease (increase) sinusoidally outwards. There is also tangential outflow of the matter from the center of granula with the velocity V3 = const. and temperature T3 equal to the temperature of the photospheric model T0. The areas occupied by ascending, descending, and tangential motions are S1, S2, S3 respectively.

The inhomogeneities caused by granulation are perturbed by the acoustic waves with the velocity amplitude V0 = 0.4 km/s and with the period T = 300 sec. The formula for the absorption line coefficient was derived, and the center-to-limb profiles of 11 weak fraunhofer lines were calculated. The comparison of calculated and observed line profiles have resulted in determining “best fit” values of V1, V2, V3, S1, S2, S3 and ΔT = T1 - T2. Tangential motions occupy the larger part of the solar surfaces S3 = 0.7, V3 ≈ 2.7 km/s, V ≈ 1.0 km/s, ΔT ranges between 270° - 400°.

In this review the problem of measuring velocity field in stellar atmospheres from the observer’s point of view is presented. Our purpose here is to discuss observational methods in which detailed analysis of line profiles is not necessary, i.e. methods based on measurements of total absorption of lines (curve of growth and narrowband photometry), methods based on measurements of widths of lines (Goldberg-Unno method and curve of line width correlation) and measurements of differential line shifts. Therefore, our review will be limited to the discussion of basic assumptions of each method, to the analysis of their advantages and disadvantages, to a specification of a quantity which can be derived from a given method and finally to a brief presentation of the results. Usually, the low resolution methods provide an information about a particular component of the stellar velocity field. But the problem is more complicated as we do not know if particular components can be isolated from the stellar velocity field, thus, we do not know the real physical meaning of the measurable parameters. In this paper we shall adopt classical concept of micro- and macro-turbulence and convective type velocity. This simplified picture was criticized since 15 years on every colloquium devoted to hydrodynamic phenomena in the stellar atmospheres, but until now the theoreticians have not succeded in developing a theory which would satisfactorily interpret the observations.

High resolution implies that we obtain some information on spectral line shapes. In late-type stars, we need to measure velocities of a few km/sec to accomplish this. Increasing the spectral resolution and the signal to noise ratio allows us to progress step by step toward deeper physical understanding. The steps we take often lead to good debate and “stimulate” our lives. I am sometimes amused at the urgency we feel to press on to the next step. We rarely seem to pause and enjoy the completion of previous steps. Perhaps this is because we always see shortcomings in completed work. Quite typically one will “discover” the importance of some physical phenomenon (It makes little difference how many others already know about it.), and in his eyes everything done previously becomes wrong because this phenomenon was not included. We used to hear how Milne-Eddington or Schuster-Schwarzschild model atmospheres were inadequate -we had to use instead properly computed depth dependence. We used to hear how LTE models were no good - we had to use more detailed physics. Now we talk about line analyses being inadequate because it has not included velocity fields. The curious thing is that we believe that including our pet phenomenon gives the correct models. We ignore all those other phenomena as yet unseen! (Is this a mechanism for maintaining sanity?) I think it really amounts to a statement of what we are able to measure, compute, or understand.

On the sun we can identify many of the motions derived from stellar spectral analysis. A summary is given of the observed solar velocity phenomena. Many of these (e.g. meridional flow, giant cells, solar differential rotation, supergranulation) are of great interest in astrophysics especially for interior structure and chromospheric and coronal structuring but contribute virtually nothing to the velocities derived from a solar irradiance spectrum analysis. Others (granulation, very small scale motions and to a lesser extent, oscillations) do contribute substantially to the integrated sun velocity analysis. Some of the properties of these motion fields are described.

The Fe II λ3969.4 line is one of the weak lines in the wings of Ca II H and K that appear in emission near the solar limb, and in the flux spectra of cool giants. In spatially resolved solar spectrograms the line shows very pronounced small-scale spatial intensity variation, which is strongly correlated to the line structure of the local H-wing background, and not at all to the chromospheric structure seen in the H & K cores. A 15-level atomic model computation for iron shows that this behaviour is due to pumping by photons in the wings of the strong Fe II resonance lines near 2600 Å, in the deep photosphere. The λ3969.4 line is therefore deeply controlled, with large sensitivity to photospheric inhomogeneities, while its background is formed much higher. This makes the line a useful diagnostic of stellar photospheric line structure, in contrast to the adjacent H core for which emission indicates chromospheric line structure.

It has for long been realized that the interpretation of stellar spectral lines normally requires broadening velocities well in excess of purely thermal motions. The origin and structure of such velocity fields, which have usually been summarized as turbulence, still appear to be a subject of controversy.

The basic premise of this paper is that the atmospheres of supergiants of spectral type 0, B, and A are not homogeneous static and stable. Observations show effects in the spectrum which seem to indicate velocity fields of many different scales, which probably are variable with depth and are certainly variable with time. Some effects can be understood in terms of global, steady, symmetric velocities, namely rotation and expansion. Those effects which still cannot be accounted for are generally attributed to turbulent velocities.