O-bearing Molecules in Carbon-rich Proto-Planetary Objects

We use high-precision multiband photometric data of the first-overtone RR Lyrae star U Comae to investigate the predictive capability of full-amplitude, nonlinear, convective hydrodynamical models. The main outcome of this investigation is that theoretical predictions properly account for the luminosity variations along a full pulsation cycle. Moreover, we find that this approach, because of the strong dependence of this observable and of the pulsation period on stellar parameters, supplies tight constraints on stellar mass, effective temperature, and distance modulus. Pulsational estimates of these parameters appear in good agreement with empirical ones. Finally, a well-defined bump just before the luminosity maximum gave the unique opportunity to calibrate the turbulent convection model adopted for handling the coupling between pulsation and convection.     

On the effect of turbulent anisotropy on pulsation stability of stars

Within the framework of a non-local time-dependent stellar convection theory, we study in detail the effect of turbulent anisotropy on stellar pulsation stability. The results show that anisotropy has no substantial influence on pulsation stability of g modes and low-order(radial order n_r 5) p modes.The effect of turbulent anisotropy increases as the radial order increases. When turbulent anisotropy is neglected, most high-order(n r 5) p modes of all low-temperature stars become unstable. Fortunately,within a wide range of the anisotropic parameter c_3, stellar pulsation stability is not sensitive to the specific value of c_3. Therefore it is safe to say that calibration errors of the convective parameter c_3 do not cause any uncertainty in the calculation of stellar pulsation stability.     

Nonlinear coupling between pulsation and convection in late type stars

A simple idealized nonlinear model applicable to long period variable stars has been formulated that assumes the convective envelope ofM giants is composed of giant convection cells, which are comparable in size to the stellar radius. The simplicity of this model essentially constitutes a physical analog to the strong dynamic coupling that occurs if the convective envelope of the star undergoes both modes of motion. As shown implicitly in the time scales associated with these motions, the coupling produces asymmetrical fluctuations of the entire star, the mean velocity of which is comparable to the escape velocity of the star at particular values of the ratio of the pulsation and convection time scales. It is suggested that this can account for the mass loss from late type stars, and the circumstellar dust shells that are associated extensively with long period variables.For critical values of the pulsation and convection time scales, the solutions correspond to the rapid expansion of the entire convective envelope, and is the basis of a new mechanism that simulates the manner in which pulsating stars ballistically accelerate their convective shells to form planetary nebulae.     

On the interaction between differential rotation and magnetic fields in the Sun

We have performed 3-D numerical simulations of compressible convection under the influence of rotation and magnetic fields in spherical shells. They aim at understanding the subtle coupling between convection, rotation and magnetic fields in the solar convection zone. We show that as the magnetic Reynolds number is increased in the simulations, the magnetic energy saturates via nonlinear dynamo action, to a value smaller but comparable to the kinetic energy contained in the shell, leading to increasingly strong Maxwell stresses that tend to weaken the differential rotation driven by the convection. These simulations also indicate that the mean toroidal and poloidal magnetic fields are small compared to their fluctuating counterparts, most of the magnetic energy being contained in the non-axisymmetric fields. The intermittent nature of the magnetic fields generated by such a turbulent convective dynamo confirms that in the Sun the large-scale ordered dynamo responsible for the 22-year cycle of activity can hardly be located in the solar convective envelope.     

Turbulence properties in the solar convection envelope： properties of the overshooting regions

We apply the turbulent convection model (TCM) to investigate properties of tur-bulence in the solar convective envelope, especially in overshooting regions. The results show TCM gives negative turbulent heat flux uγ′T′in overshooting regions, which is sim-ilar to other nonlocal turbulent convection theories. The turbulent temperature fluctuation T′T′shows peaks in overshooting regions. Most important, we find that the downward overshooting region below the base of the solar convection zone is a thin cellular layer filled with roll-shaped convective cells. The overshooting length for the temperature gradi-ent is much shorter than that for element mixing because turbulent heat flux of downward and upward moving convective cells counteract each other in this cellular overshooting region. Comparing the models' sound speed with observations, we find that raking the convective overshooting into account helps to improve the sound speed profile of our nonlocal solar models. Comparing the p-mode oscillation frequencies with observations,we validated that increasing the diffusion parameters and decreasing the dissipation pa-rameters of TCM make the p-mode oscillation frequencies of the solar model be in betteragreement with observations.     

Effect of the radiative background flux in convection

Numerical simulations of turbulent stratified convection are used to study models with approximately the same convective flux, but different radiative fluxes. As the radiative flux is decreased, for constant convective flux: the entropy jump at the top of the convection zone becomes steeper, the temperature fluctuations increase and the velocity fluctuations decrease in magnitude, and the distance that low entropy fluid from the surface can penetrate increases. Velocity and temperature fluctuations follow mixing length scaling laws. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Note on the model oscillators of irregular stellar variability

A model oscillator for irregular stellar variability presented by Tanaka and Takeuti has only one singularity in the original form. We show it has three different types of singular points when damping terms are ignored. The nonlinear pulsation of Baker's one-zone stellar models is discussed in comparison with the properties of the Tanaka-Takeuti oscillator.     

Nonlinear hydrodynamical models of stellar convective zones

Large-eddy simulation (LES) of turbulent convection is discussed in various versions (mixing-length theory, modal theory and spectral theory) in respect to the application to stellar convective zones. For the model construction, the non-local mixing-length formalism is suitable. However, for the determination of basic flow patterns and of mixing-length, the quasi-linear and nonlinear modal theories are useful. The eddy diffusivities are essential in these theories, and the nonlinear treatment of convection consistent with turbulent diffusivities (of effective Reynolds number of about 20 and Prandtl number of 0.4) offers a simple method of constructing stellar models without the use of the mixing-length.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Anisotropy and Dissipation of Turbulence and Their Effects on Solar Models

1 INTRODUCTIONThe maing-length theory (MLT) is the most commonly used approach to calculate convective energy transport in stars and other astrophysical situations. Based on the original idea ofPrandtl (1952) that turbulent parcels trallsfer heat in a similar way as molecules of gas do inthermal conduction, the MLT assumes that convection cells, drived by buoyancy, move thlougha ~ng length 1 and release the heat they carry when they merge with their environment. Themost widely adopted f…     

Inhomogeneous convection and the equatorial acceleration of the sun

The interaction of rotation and turbulent convection is assumed to give rise to an inhomogeneous, but isotropic, latitude dependent turbulent energy transport, which is described by a convective conduction coefficient _c which varies with latitude. Energy balance in the convective zone is then possible only with a slow meridian circulation in the outer convective zone of the sun. The angular momentum transported by this circulation is balanced in a steady state by turbulent viscous transport down an angular velocity gradient. A detailed model is constructed allowing for the transition from convective transport to radiative transport at the boundaries of the convective zone, by using a perturbation analysis in which the latitude variation of _c is small. The solution for a thin compressible shell gives equatorial acceleration and a hotter equator than pole, assuming that the convection is preferentially stabilised at the equator. For agreement with the sun's equatorial acceleration the model predicts an equatorial temperature excess of 70 K and a surface meridional velocity of 350 cm/sec from pole to equator.     

A model oscillator of irregular stellar variability

We have studied a set of equations with nonlinear and nonadiabatic terms which describes a simple oscillator. The equations have only one fixed point located at the origin. It is found that the oscillator shows the sequence of the period-doubling for the change of a parameter and results in chaotic oscillation. We illustrated the behaviour of the oscillator for several set of parameters and showed that the equations of the oscillator can be reduced to the one-zone model of stellar pulsation with simple nonlinear terms. It is suggested that the stellar irregular variability is resulted from the chaotic motion due to the nonlinear effect.     

Non-adiabatic oscillations of red giants

Using our non-local time-dependent theory of convection, the linear non-adiabatic oscillations of 10 evolutionary model series with masses of  1–3 M_⊙  are calculated. The results show that there is a red giant instability strip in the lower temperature side of the Hertzsprung–Russell diagram which goes along the sequences of the red giant branch and the asymptotic giant branch. For red giants of lower luminosities, pulsation instability is found at high order overtones; the lower order modes from the fundamental to the second overtone are stable. Towards higher luminosity and lower effective temperature, instability moves to lower order modes, and the amplitude growth rate of oscillations also grows. At the high luminosity end of the strip, the fundamental and the first overtone become unstable, while all the modes above the fourth order become stable. The excitation mechanisms have been studied in detail. It is found that turbulent pressure plays a key role for excitation of red variables. The frozen convection approximation is unavailable for the low temperature stars with extended convective envelopes. In any case, this approximation can explain neither the red edge of the Cepheid instability strip, nor the blue edge of the pulsating red giant instability strip. An analytic expression of a pulsation constant as a function of stellar mass, luminosity and effective temperature is presented from this work.     

Testing turbulent convection theory in solar models – I. Structure of the solar convection zone

Turbulent convection models (TCMs) based on hydrodynamic moment equations are compared with the classical mixing-length theory (MLT) in solar models. The aim is to test the effects of some physical processes on the structure of the solar convection zone, such as the dissipation, diffusion and anisotropy of turbulence that have been ignored in the MLT. Free parameters introduced by the TCMs are also tested in order to find appropriate values for astrophysical applications. It is found that the TCMs usually give larger convective heat fluxes than the MLT does, and the heat transport efficiency is sensitively related to the dissipation parameters used in the TCMs. As a result of calibrating to the present solar values, our solar models usually have rather smaller values of the mixing length to local pressure scaleheight ratio than the standard solar model. The turbulent diffusion is found to have important effects on the structure of the solar convection zone. It leads to significantly lowered and expanded profiles for the Reynolds correlations, and a larger temperature gradient in the central part of the superadiabatic convection region but a smaller one near the boundaries of the convection zone. It is interesting to note that, due to a careful treatment of turbulence developing towards isotropic state, our non-local TCM results in radially dominated motion in the central part and horizontally dominated motion near the boundaries of the convection zone, just as what has been observed in many 3D numerical simulations. Our solar models with the TCMs give small but meaningful differences in the temperature and sound speed profiles compared with the standard solar model using the MLT.     

Some properties of the kinetic energy flux and dissipation in turbulent stellar convection zones

We investigate simulated turbulent flow within thermally driven stellar convection zones. Different driving sources are studied, including cooling at the top of the convectively unstable region, as occurs in surface convection zones; and heating at the base by nuclear burning. The transport of enthalpy and kinetic energy, and the distribution of turbulent kinetic energy dissipation are studied. We emphasize the importance of global constraints on shaping the quasi-steady flow characteristics, and present an analysis of turbulent convection which is posed as a boundary value problem that can be easily incorporated into standard stellar evolution codes for deep, efficient convection. Direct comparison is made between the theoretical analysis and the simulated flow and very good agreement is found. Some common assumptions traditionally used to treat quasi-steady turbulent flow in stellar models are briefly discussed. The importance and proper treatment of convective boundaries are indicated.     

Convection in a rotating deep compressible spherical shell: Application to the Sun

In this paper we study the interaction of rotation with convection in a deep compressible spherical shell as the Sun's convection zone. We examine how the energy transport and the large scale motions can be affected by rotation. In particular we study how a large scale meridional circulation can give rise to variations of angular velocity with latitude and depth.It is assumed that the energy transport is only due to convection and that the mixing-length theory gives an adequate representation of it. Furthermore we assume that rotation acts as a perturbation of the turbulent convective flux through its transport coefficient.The equations involved in the model are integrated numerically in the limit of large viscosity and slow rotation. After having expanded all physical quantities to the first order in terms of Legendre polynomials, the fitting with the observed solar differential rotation gives the expansion parameter, which represents the coupling constant between rotation and convection.The results show a three-cell circulation extending from the poles to the equator. The first one is located in the lower half of the convection zone with the fluid rising at the equator and sinking at the poles. In the second one the direction of the motion is reversed while the third one, located in a thin upper layer, shows the same characteristics of the first one. The meridional velocities at the surface are directed towards the poles and are about 20 cm s^-1. In the other cells the meridional velocities are typically of a few cm s^-1 while the radial velocities are of the order of a few tenths of cm s^-1.The heat flux relative variation at the surface is about 10^-4 (3 × 10^-3 at the bottom) with a polar excess. The temperature variation at the surface is of the same order, with an equatorial excess however. The convection seems to be stabilized stronger at the equator. The angular velocity increases inwards and varies about 6% between the surface and the bottom of the convection zone.An attempt is made for explaining the picture which emerges. In particular the negligible flux and temperature variations at the surface are explained in terms of equalization by the particular structure of the latitudinal flow. This configuration of large scale circulation is attributed to the high stratification of the convection zone with depth.     

Magnetic field transfer by two-dimensional convection and solar ‘semi-dynamo’

A comparison of the equations for the magnetic field transfer and for the heat transfer by two-dimensional turbulent convection of a conducting compressible medium shows the magnetic field to be transported as a scalar admixture provided it is parallel to the convective rolls. At high magnetic Reynolds numbers the field strength in a convective zone varies proportionally to the density of the medium.A study of the distribution and amplification of the poloidal field in the two-dimensional convection zone of the Sun lying under the supergranulation, together with the processes of field pumping and amplification in other zones, reveals the importance of considering generation mechanisms of thesemi-dynamo type where the amplifying field is excited independently by weak e.m.f.'s of non-electric origin with no feedback which would otherwise produce MHD self-excitation of the field.An illustrative calculation of the solar poloidal field maintained by a weak Coriolis e.m.f. acting in a thin external layer of the convective envelope yields for the general near-polar field, if one somehow takes into account (1) field pumping by three-dimensional supergranulation, (2) field transfer and amplification by two-dimensional convection, and (3) ohmic diffusion of the field into a stable core, a value of the order of 10^–1 gauss.     

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The red edge of the instability strip of RR Lyrae stars

The thermodynamical compling between radial pulsation and convection is studied in accordance with Unno's theory of time-dependent convection. Linear, non-adiabatic calculations are made for 11 series of RR Lyrae models to illuminate the dependence of the red edge of their instability strip on mass, luminosity, helium abundance and the convective parameter C₁.     

Granulation and supergranulation as convective modes in the solar envelope

The stability of linear convective modes in the solar convection zone is investigated by incorporating the mechanical and thermal effects of turbulence through the eddy transport coefficients. The inclusion of turbulent thermal conductivity and viscosity, calculated in the framework of the mixing length approximation, is demonstrated to have a profound influence on the convective growth rates. The solar envelope model of Spruit (1977) is used to show that that most rapidly growing fundamental mode and the first harmonic are in reasonable accord with the observed features of granulation and supergranulation, respectively.On leave of absence from Govt. Digvijai College, Rajnandgaon 491441, India.