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.

The seismology and physics of localized structures beneath the surface of the Sun takes on a special significance with the completion in 2006 of a solar cycle of observations by the ground-based Global Oscillation Network Group (GONG) and by the instruments on board the Solar and Heliospheric Observatory (SOHO). Of course, the spatially unresolved Birmingham Solar Oscillation Network (BiSON) has been observing for even longer. At the same time, the testing of models of stellar structure moves into high gear with the extension of deep probes from the Sun to other solar-like stars and other multi-mode pulsators, with ever-improving observations made from the ground, the success of the MOST satellite, and the recently launched CoRoT satellite. Here we report the current state of the two closely related and rapidly developing fields of helio- and asteroseimology.

The Session F round table. and audience, considered the following questions:

1. (i) what determines the differential rotation in stellar convective zones?

2. (ii) If there really is a rapid inward increase in angular velocity in giants will it have significant effects in the late stages of stellar evolution?

3. (iii) How effective is penetration above a convective core?

4. (iv) Why is the solar core nearly uniformly rotating?

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.

In continuation of our earlier studies (Singh, Roxburgh & Chan 1997a,b; hereafter SRC97a,b) we perform some further tests to study the general behaviour of penetrative convection and its scaling with rms vertical velocity by varying a number of input parameters like the aspect ratio and the positioning of the interface between the unstable-lower stable layer.

STARS is a proposed asteroseismology (and stellar activity) space mission which is currently under phase A study in the European Space Agency as a candidate for the Medium Size Mission (M3).

The integral constraint on convective overshooting (Roxburgh 1976, 1978, 1989) is re-derived for for statistically stationary two-dimensional convection between stress-free horizontal boundaries maintained at constant temperatures, the vertical boundaries being periodic. We report on numerical simulations of such two-dimensional compressible convection in a fluid where the central regions are convectively unstable and the surrounding layers are stable. The results support the conclusion by Roxburgh (1989) that the contribution from the radiative flux is well approximated by the flux carried by the mean temperature gradient, but demonstrate that viscous dissipation is important, reducing the extent of overshooting.

Long term numerical integrations of planetary orbits designed to study the stability of the Solar System over timescales comparable to its age have become very promising thanks to the availability of very powerful computers and to a substantial improvement in our methods of investigating the stability of hierarchical dynamical systems. The stability of such numerical integrations relies on the ability to control all possible sources of error. Among the errors caused by the inadequacy of the physical model are those due to the fact that Newton's theory of gravitation is used instead of general relativity. We show that the secular advance of perihelia predicted by general relativity can be simulated exactly by a 1/r2 perturbing potential with almost negligible additional cost in computer time.

Several authors (myself included!) have suggested that turbulent mixing takes place in some, if not all, stars, and in particular that such mixing can explain the low solar neutrino flux. This turbulence is thought to be caused by differential rotation produced by braking due to angular momentum loss in a stellar wind, and/or to the effect of meridional circulation currents in redistributing angular momentum. Whilst such instabilities may exist even in the presence of a stabilizing distribution of chemical composition, they do not necessarily cause mixing. To be effective in mixing, the energy available to the instability be it differential rotation or any other mechanism, has to be sufficient to lift the helium rich matter in the interior of the star to the outer regions. This requires where Erot is the kinetic energy in rotation, Eg the gravitational energy, τth the thermal time scale and τnuc the nuclear evolution time scale of the star.

The method used by Tassoul and Tassoul assumes a “turbulent viscosity” which is important in determining the dynamics, but is unimportant in heat transport. This approximation is inconsistent, the argument is as follows:

The rotation is expanded about uniform rotation in a series.

My interest in the origin of planetary nebulae goes back to 1967 when I proposed that the envelopes of giants with degenerate carbon-helium cores were dynamically unstable and would be completely ejected to form a planetary nebula. Essentially the same proposal was made almost simultaneously by Lucy (1967) and Paczyński (1967). For some years after that I worked on the dynamical ejection with W. van der Reijden, but for the last few years my interests have been in other areas, principally solar physics and gravitation theory. What then can I contribute to a discussion on Planetary Nebulae? Not a detailed review of work in this area; for that, the reader is referred to review articles by Salpeter (1971), Osterbrock (1973) and Miller (1974).

Our understanding of the internal structure of the Sun and solar type stars has been undermined by recent observations. In this paper we consider some of the puzzles and possible resolutions; the solar neutrino problem, lithium and beryllium abundance, rotation and calcium emission, variation of the solar constant, solar oscillations and the solar convective zone. The picture that emerges is one of confusion, and so it should be since we have no idea of what is going on inside the Sun and a fortiori of what is going on inside other stars.

The interaction of rotation and turbulent convection gives rise to a latitude dependent turbulent energy transport. Energy conservation demands a slow meridional circulation in the solar outer convective zone. The transport of angular momentum by this circulation is balanced in a steady state by the turbulent viscous transport across an angular velocity gradient. Models are constructed which give equatorial acceleration as observed on the sun.

Research into the effect of rotation on the internal structure of stars has a long and detailed history. The names of McLaurin, Jacobi, Kelvin and Jeans are associated with detailed work on the structure and stability of rotating liquid masses assuming uniform rotation. Much of this work is summarized in Jeans’ (1929) Astronomy and Cosmogony and more recent summaries are given in Lyttleton (1953) and Lebovitz (1967).