Convective Overshooting in Extreme Horizontal-branch Stars Using MESA with the k-omega Model*

To explore overshoot mixing beyond the convective core in core helium-burning stars, we use the k−ω model, which is incorporated into the Modules of Experiments in Stellar Astrophysics to investigate overshoot mixing in the evolution of subdwarf B (sdB) stars. Our results show that the development of the convective core can be divided into three stages. The mass of the convective core increases monotonically when the radiative temperature gradient, ∇rad, monotonically decreases outwardly, and overshoot mixing presents an exponential decay similar to Herwig. The splitting of the convective core occurs repeatedly when the minimum value of ∇rad near the convective boundary is smaller than the adiabatic temperature gradient, ∇ad. The mass at the outer boundary of the convective shell M sc can exceed 0.2 M ⊙ after the central helium abundance drops to about Y c ≈ 0.45. It is close to the convective core masses derived by asteroseismology for younger models (0.22 to ∼0.28 M ⊙). In the final stage, “core breathing pulses” occurred two or three times. Helium was injected into the convective core by overshoot mixing and increased the lifetime of sdB stars. The mass of the mixed region M mixed can rise to 0.303 M ⊙ by the end. The oxygen content in the central core of our g-mode sdB models is about 80% by mass. The high amounts of oxygen deduced from asteroseismology may be evidence supporting the existence of core breathing pulses.

Self-Consistent Thermodynamic Parameters of Diopside at High Temperatures and High Pressures: Implications for the Adiabatic Geotherm of an Eclogitic Upper Mantle

Using an iterative numerical approach, we have obtained the self-consistent thermal expansion, heat capacity, and Grüneisen parameters of diopside (MgCaSi2O6) over wide pressure and temperature ranges based on experimental data from the literature. Our results agree well with the published experimental and theoretical data. The determined thermodynamic parameters exhibit nonlinear dependences with increasing pressure. Compared with other minerals in the upper mantle, we found that the adiabatic temperature gradient obtained using the thermodynamic data of diopside is larger than that of garnet while lower than that of olivine, when ignoring the Fe incorporation. Combining our results with thermodynamic parameters of garnet obtained in previous studies, we have estimated the adiabatic temperature gradient and geotherm of an eclogitic upper mantle in a depth range of 200–450 km. The results show that the estimated adiabatic temperature gradient of the eclogite model is ~16% and ~3% lower than that of the pyrolite model at a depth of 200 km and 410 km, respectively. However, the high mantle potential temperature of the eclogite model leads to a similar temperature as the pyrolite model in a depth range of 200–410 km.

Boussinesq convection in a gaseous spherical shell

In this paper, we investigate the dynamics of convection in a spherical shell under the Boussinesq approximation but considering the compressibility which arises from a non zero adiabatic temperature gradient, a relevant quantity for gaseous objects such as stellar or planetary interiors. We find that depth-dependent superiadiabaticity, combined with the use of mixed boundary conditions (fixed flux/fixed temperature), gives rise to unexpected dynamics that were not previously reported.

On the boundaries of a convective zone and the extent of overshooting

In this work, we will show that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is convectively unstable region when the flux is positive, and it is convective overshooting zone when the flux becomes negative. In our nonlocal convection theory, convection is already sub-adiabatic (∇ < ∇ad) far before reaching the unstable boundary; while in the overshooting zone below the convective zone, convection is sub-adiabatic and super-radiative (∇rad < ∇ < ∇ad). The transition between the adiabatic temperature gradient and the radiative one is continuous and smooth instead of a sudden switch. In the unstable zone, the temperature gradient is approaching radiative rather than going to adiabatic. The distance of convective overshooting is different for different physical quantities. The overshooting distance in the context of stellar evolution, measured by the extent of mixing of stellar matter, should be more extended than that of other physical quantities. It is estimated as large as 0.25–1.7 Hp depending on the evolutionary timescale.

Convection and its Stability in the Equatorial Regions of the Convection Zone

The problem of the existence, evolution, and stability of spatial structures in convection is of considerable importance to astrophysics as well as to geophysical hydrodynamics. The Boussinesq approximation will be used because the considered motions in stars are sufficiently slow. The system of hydrodynamic equations describing convection in a rotating inhomogeneous medium has the form: Here Dt is the total time derivative, U the velocity, P, T, and C the deviations of the pressure, temperature, and helium abundance (by mass) from the basic equilibrium values, ρm, νm, χm, and Dm the values averaged over the considered layer of the density, viscosity, thermal and helium diffusivities, βT and βc the averaged coefficients of the thermal and helium expansions, g and Ω the gravitational acceleration and angular velocity, ∇Tb, and ∇Cb the values of the basic equilibrium temperature and helium gradients, and ñTad the adiabatic temperature gradient.

Nonlinear thermal convection in an elasticoviscous layer heated from below

The linear and nonlinear stabilities of a horizontal layer of an elasticoviscous fluid, whose stress-rate-of-strain relations are due to Oldroyd (1958), are studied. In the linear theory it is already shown that steady convection (the situation generally referred to as the exchange of stability) is preferred for all relevant values of the Prandtl number (which is the ratio of the kinematic viscosity to the thermal diffusivity). The study of nonlinear effects for slightly supercritical Rayleigh number (which measures the temperature contrast across the layer) shows that plane disturbances for the case where the exchange of stability is valid and plane or centred disturbances for the case of overstability are governed by equations similar to those derived by Hocking, Stewartson & Stuart (1972) for plane Poiseuille flow. The influence of elasticity is to give rise to a burst only when the principle of exchange of stability is valid and provided certain conditions relating to the elastic parameters of the fluid are satisfied. The effect of the adiabatic temperature gradient is also discussed. It is shown that it stabilizes the layer in the linear theory. However, in the nonlinear theory it can destabilize the layer if the ratio the mean temperature of the layer to the temperature difference across the layer is large enough. For most practical purposes it does not influence the conditions necessary for a burst to occur.