3. Application to a star
Above a mass of approximately , stars develop a convective core during the main sequence evolution. The plume model was applied to a star in an arbitrary stage of evolution during the main sequence phase. The purpose is to illustrate the dynamics of turbulent plumes in convective cores, this choice is based on no particular reasons. Relevant physical quantities that determine conditions in the stellar core are summarized in Table (1).
Table 1. Temperature, density, energy generation rate, mean opacity and mass abundances of hydrogen and helium of the stellar model at the core center and at the Schwarzschild boundary.
Two kind of calculations have been conducted during which the number of turbulent plumes has been varied, this number being a free parameter of the model; cases with and without downward counter flow were both examined (these will be called case B and case A respectively). The plumes have been followed from their sources located near the core center () up to the Schwarzschild boundary ().
There is an upper limit on the number of plumes that may be present at the same time in the convective region, imposed by the fact that plumes cannot take up more space than there is available at a given level. If we define the filling factor to be the ratio of the surface occupied by the plumes to the surface of the sphere at the given radius, this factor has to be less than unity. When flows are narrow enough, so the curvature can be locally neglected, this parameter may be defined as .
3.1. General behavior of turbulent plumes
Turbulent plumes move across the entire core and broaden during their motion due to turbulent entrainment but they remain fairly narrow flows (Fig. (1)). However, the self-similar régime encountered in terrestrial conditions or in RZ is lost: this is due to the peculiar physics of the core.
From Fig. (2), it appears that the velocity remains small compared to the sound speed (the sound speed is of the order of ), the Mach number is no where larger than 0.0005. Then, the mean structure of the convective core is not modified by the plumes and the motion is nearly incompressible. A consequence of this highly subsonic flow is the unusually small values of the density contrast and of the related temperature fluctuations (Fig. (3)): a few Kelvins are sufficient to achieve the convective transport. This result amply justifies the simplifications made so far about neglecting the pressure perturbations and the influence of density and temperature fluctuations. This further accounts for the near adiabaticity of the motion.
The variation of the velocity is controlled by the distribution of energy inside the core. As shown in Fig. (4), the radiative transport is a significant part of the whole energy transport process, its role becoming more and more important as we approach the outer edge of the core. In the first part of their motion, plumes are accelerated by the energy sources. But, at some level, the available luminosity () is no longer able to push their increasing mass up: the convective velocity reaches a maximum value and the plumes begin to decelerate (Fig. (2)).
This may be considered as an individual effect but there is also what we may call a collective effect. The convective speed also depends on the number of plumes and decreases when this number increases: the more plumes there are, the slower they move. This is easily understood by the fact that a lesser velocity is needed to carry up the same luminosity when convective elements are more numerous. In every case, plumes arrive at the Schwarzschild boundary with non-zero velocity and the convective motions overshoot into the neighbouring radiative zone.
The density contrast lessens during the motion as the plumes grow in size and as the need of convective transport lessens when going upwards, radiation becoming more and more efficient.
While the kinetic energy flux is negligible in the bulk of the unstable domain, near the core surface, it becomes of the same order, in absolute value, as the convective flux. Its highest value is reached in the vicinity of the surface while the convective luminosity is maximum somewhere deeper. Before entering the radiative domain, the convective flux changes sign to compensate the kinetic flux so as to satisfy energy conservation. At the same time, the density contrast becomes negative, the upflows are further slowed down as a result.
3.2. Plumes in a static medium (case A)
When the downward current is not taken into account, the plumes grow in size identically, whatever their number. Changing the number of plumes only modifies the radial speed and the density contrast. These quantities adjust themselves so as to reproduce the total luminosity at every depth.
Things happen as if plumes capture matter independently of their speed, contrary to what may be expected from Taylor's entrainment hypothesis (Eq. (1)). This behavior may be understood if we consider that the broadening process depends, after all, only on the conditions encountered in the interplume medium. In the present case, this medium is the same for all the plumes even when their number is changed, in particular, the downflow speed is zero. When a return flow is introduced, the downflow velocity varies with the number of plumes and the effective radius changes as well as we will see. A consequence is that the top velocities are all reached at the same height () when there is no inverse current.
The distribution of energy among the different forms of transport is not modified when the number of plumes is varied (Fig. (4)). It ensues that, in the present case, the velocity and the density contrast of a plume follow simple power laws of N:
3.3. Plumes with a return flow (case B)
In this case, every dynamical quantities depend on the number of plumes. The more plumes there are, the lesser the convective velocities, as before, and the broader the plumes. The density contrast, although smaller, are not much different from the ones obtained previously.
For a given number of plumes, the effective radius is larger than in case A and the convective speed is smaller as a result. The differences in effective radius and convective speed are small for low values of N, but they become more pronounced as this number is increased. The reason is that, now, the turbulent entrainment rate depends on the number of plumes: the downward speed scales as (Eq. (17)) and the entrainment rate is proportional to . The downward velocity is negative in our sign convention. It is equal, in absolute value, to the upward velocity when the plumes occupy half of the available area. As the number of plumes is increased, the occupation factor grows rapidly and the downflows become much faster than the upflows. In absolute value, the increase rate of the downward velocity with the number of plumes is larger than the decrease rate of the upward velocity. This accounts for a significant enhancement of the turbulent entrainment despite slower plumes and explains the marked differences in effective radius obtained for large values of N.
Due to the high level of turbulent entrainment involved in the present case, the upper limit on the number of plumes is reduced to 118, this number was equal to 491 in the previous case.
The highest convective velocities are reached at different levels for different values of the number of plumes because mass distribution in the upflows changes with this number. The largest velocities are reached deeper for larger N. Another reason that accounts for the smaller velocity is the entrainment of negative momentum from the outside (see Sect. (2.5)). However, the energy distribution is not modified significantly.
When the number of plumes is larger than approximately one hundred, the results are less consistent with the assumptions of the model. For these values, plumes are much broader and the narrow flow condition is no longer satisfied accurately enough. Furthermore, the counter flow becomes so important that not taking account of its influence on the convective transport would hardly be justified. Calling the kinetic luminosity of the inverse current, we have
When , the two fluxes are equal in absolute value, becoming more important for higher f.
Finally, given the assumptions that have been made, we argue that this model of convection seems realistic and that core convection may indeed be achieved by means of less than one hundred ascending plumes. Relevant quantities are summarized in Table (2).
Table 2. Velocities and effective radius of turbulent plumes at the Schwarzschild boundary.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998