R. van der Borght

A numerical study of non-linear convection in a compressible medium

Abstract In this paper we consider the problem of non-linear convection in a compressible layer with polytropic structure. After deriving the appropriate forms of the basic conservation equations a single mode expansion is used to obtain a simpler model, within the framework of the anelastic approximation. These equations are integrated, using a band-matrix algorithm, for two characteristic density stratifications. The results are given in the form of graphs and are discussed in detail. An outline of the numerical algorithm is given in an appendix together with a discussion of its effectiveness.

A Convective Model of Solar Granulation

Some years ago the full non-linear equations, within the one mode approximation, describing finite amplitude convection in a compressive medium with polytropic structure were derived (Van der Borght 1977). It is the purpose of the present paper to report on the numerical integrations of these equations and to show how these can be used to construct a convective model of granulation. In this way one may gain some insight into the structure of the outer layers of the Sun.

Granulation and Supergranulation as a Diagnostic Test of Solar Structure

One of the most difficult tasks facing theoretical astrophysics today is to find a satisfactory model for the convective transport of energy in regions where the temperature gradient is superadiabatic. A number of such models have been proposed, such as the mixing-length theory and its extensions (Vitense 1953, Bohm-Vitense 1958, Spiegel 1963, Travis and Matsushima 1973 a, b, Parson 1969), to take into account the combined effects of convection and turbulence but it is generally agreed that, to-date, no satisfactory theory has been put forward.

A Model of the Solar Convection Zone

In a previous paper it was shown how one could improve upon the Bohm-Vitense model of the solar convection zone by the inclusion of four different length scales and by the determination of these length scales with the use of the quasi-Vitense model as developed by Unno. In this way the vertical wave number k z , associated with a characteristic eddy, can be determined by the integration of a second order differential equation. The integrations have to be started at a suitable depth and all model calculations depend critically on the assumed structure of the top layer.

The Effect of Rotation on Thermal Convection

An investigation into the influence of rotation on thermal convection has some applicability in the study of the solar convection zone. Of particular interest is the effect of rotation on the total heat transport and the cell size for maximum heat transport at high Rayleigh number, which is estimated to be as high as 10 20 for the Sun.

A Multimode Investigation of Granular and Supergranular Motions I: Boussinesq Model

Two types of large-scale convective motions, granulation and supergranulation, are observed in the outer layers of the sun and their observed characteristics can be summarized as follows:

The effect of viscous dissipation on non-linear convection at high Rayleigh number

When studying deep convection in a compressible medium the effects of viscous dissipation can become important and must be taken into account in any realistic model. But even in shallow convection, for which the Boussinesq approximation is valid, the viscous dissipation effects will become important at high Rayleigh numbers. These effects are estimated with the help of asymptotic methods and the results are compared with those obtained by numerical integration.

Convective instability of hydrogen-burning shells in

It appears that the relative number of β –Cephei variables among the early B stars is comparable to the fraction of the evolutionary lifetime spent in the secondary contraction and early shell-burning phases (Lesh and Aizenman 1973a) and it may therefore be of some interest to investigate possible instability mechanisms associated with hydrogen shell-burning in stars within the mass range of 10 to 15 M ⊙ (Lesh and Aizenman 1973b).

beta

As shown recently by Y. Osaki super-massive stars with mass M 5 M⊙ can, in the absence of rotation, reach the hydrogen-burning main sequence before the onset of general relativistic instability. Such objects are then pulsationally unstable. A considerable simplification is introduced if one considers only very massive stars, for which the relative amplitude of the fundamental mode of oscillation is practically constant. This sets a lower limit of 10 4 M⊙ to the mass that can be considered. The upper limit is also reduced to 2 × 10 5 M⊙ if one neglects the relativistic correction. One necessary step in the study of non-linear oscillations of massive stars is to derive a differential equation for the adiabatic pulsations. The relativistic correction could be taken into account in the following way.