Astron. Astrophys. 336, 553-564 (1998)

1. Introduction

Cepheid studies have acknowledged, right from the beginning, that convection should occur in Cepheid envelopes, and that they should alter the properties of both the pulsations and the underlying equilibrium models. Since the Prandtl number characteristic of stellar material is very small, , and the Rayleigh numbers associated with the partial ionization regions (PIR) are enormous, Ra , stellar convection is far into the hard turbulent regime (Krishnamurti & Howard 1983). The cooler Cepheid stars are believed to have the more extensive convection zones, leading to pulsational stability beyond the red edge of the standard instability strip (IS). Because of the estimated inefficiency of convection and because of the difficulties of treating turbulent convection, a lot of work has disregarded it altogether, the hope having been that, except for providing a red edge, convection would play a subdued role. And purely radiative models have indeed provided a reasonable overall agreement with observations.

More recently it has become increasingly clear that an impasse has been reached with radiative models, and that it is not possible to improve the agreement with observations (e.g. Buchler 1998). In particular, the amplitudes of pulsation are systematically too large. No consistent set of pseudo-viscous parameters can correctly limit the amplitudes of both fundamental and first overtone pulsations without adversely affecting the stability of the corresponding limit cycles, and the agreement with observation. The new EROS and MACHO data of the low metallicity Magellanic Cloud Cepheids (Buchler et al. 1996) reveals further disagreement between stellar evolution and radiative stellar pulsation models. Clearly some additional physical dissipation is missing in the radiative codes (see also Kovács 1990), and turbulence and convection are the primary suspects.

Simulation of even the simplest turbulent convection (TC) problem is a formidable numerical challenge; to adequately resolve the many time- and length-scales of fully turbulent 3D convection at stellar Rayleigh and Prandtl numbers is totally out of question, and astrophysicists have been looking for 1D recipes to give an approximate, but acceptable treatment of this phenomenon. Thus the 1925 mixing length picture of Prandtl was readily adapted to stellar envelopes by Böhm-Vitense, and subsequently reformulated in many variations, all of which are equivalent for time-independent problems (for an overview, see Baker 1987). But time-independent MLT was never able to resolve the red edge problem, mostly because it is time-dependent dissipation introduced by eddy viscosity that provides a clear red edge, as we will show in Sect. 4.6. Spiegel 1963 first extended MLT to the time-dependent and nonlocal cases. The earliest attempts to include a time-dependent mixing length model in pulsation (Gough 1977, Unno 1967) were not successful because they were too local in space. Subsequently, Gough's theory was extended to be less local, and eventually was used in a full linearization of Solar modes (Balmforth 1992), analogous to what we have done here. Unno's theory was also developed (Castor 1968) as a diffusion model, and simplified to a single equation for the turbulent energy by Stellingwerf (1982). More recent applications of this and similar turbulent diffusion recipes have been made by Gonczi & Osaki 1980, Bono & Stellingwerf 1994, Gehmeyr & Winkler 1992, hereafter GW) and Kuhfuß(1986). Further closely related models also appear in the literature, such as those of Canuto (1991) but have yet to be applied.

All 1D model TC equations contain several dimensionless, order unity parameters that are directly related to the physical quantities of the model. In principle, the validity of the 1D model equation can be checked against experiments or against detailed 3D simulations of turbulent convection, and values of the unknown parameters can be extracted. However, neither such experiments nor such simulations are presently available. We are obliged to calibrate these parameters indirectly with the help of observational astronomical constraints, as suggested more than a decade ago by Stellingwerf (1982). In the best outcome, observational constraints could select one model in favor of the others. On the other hand we might discover that single equation models for turbulent convection are insufficient, or that plumes (e.g. Rieutord & Zahn 1995) play an essential role in the convective transport.

This programme has recently been initiated by incorporating these recipes into hydrodynamic pulsation codes. Full nonlinear hydrodynamic calculations have already been used to determine the vibrational stability properties of convective models (e.g. Bono & Stellingwerf 1994, Bono & Marconi 1997), but this approach is inefficient, and unsuitable when a large number of models is required. The approach we follow in this paper is to compute the linear nonadiabatic properties directly by linearizing the hydrodynamic equations about the equilibrium model and by then computing the linear eigenvalues, a procedure which is relatively fast. This allows us to perform an extensive survey of the sensitivity to the TC parameters.

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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