2. Turbulent plumes in a convective core
A convective core is assumed to reach a statistically mean stationary state in which the motions can be divided into two distinct kind of flows: the ascending motions, represented by an unspecified number of turbulent plumes, and the interstitial background, called interplume medium. The case of descending plumes will not be considered, this choice will be argued in Sect. (4).
We assume the convective transport to be entirely carried out by the plumes. These plumes originate from sources located near the stellar center where energy generation is the largest. Local inhomogeneities are expected to produce large enough density fluctuations to initiate these plumes.
The core will be described as a fully ionized perfect gas in hydrostatic equilibrium, the radiative pressure will not be taken into account. We further simplify the treatment by assuming spherical symmetry, so rotation and magnetic fields are neglected. As mentioned previously, convection in such deep layers is very efficient and leads to a nearly isentropic stratification in the unstable domain. This result will not be put in question, the temperature gradient will be given the adiabatic value throughout the whole core.
To keep as close as possible to RZ, the same notations will be used. However, it is more convenient here to let the vertical coordinate increase in the upward direction.
2.1. Turbulent plumes features
Turbulent plumes are boundary-free shear flows that are driven by buoyancy forces. Among this family of flows, we have to distinguish between suddenly released buoyant elements, called thermals, and flows that are steadily supplied by buoyancy during their motion. Plumes are of the latter kind. However, the motion is not governed by buoyancy forces only. Shear instabilities give rise to turbulence and the flow also evolves under the influence of this turbulence. At the edge of plumes, mesoscale turbulence captures matter from the ambient medium (Turner 1986) so plumes broaden during their motion due to this turbulent entrainment. In a first approximation, the structure of a turbulent plume can be said to result from a balance between buoyancy forces and turbulent entrainment.
When steadily supplied by hot matter at its source, a plume is a long-lived structure and presents the simplifying advantage of reaching a mean stationary state. In terrestrial conditions, a self-similar régime appears in most of the flow (Turner 1969, 1986). This régime is characterized by well established features. If we consider an axisymmetric flow, the mean axial velocity inside the plume is fairly well described by a Gaussian function (List 1982). Then, in spherical coordinates, if the plume axis is set on a radius, the horizontal distribution of the mean radial velocity is given by
is a measure of the horizontal extent of the plume and will be called the effective radius; the distance from the axis of the plume is equal to . Because of the symmetry, there is no dependence on the -component.
Furthermore, the density contrast is closely related to the axial velocity and its horizontal variation inside the plume may be obtained from the same bell-shaped curve (List 1982):
the subscript 0 refers to a physical quantity outside the plume, such functions depend on r only.
The turbulent entrainment is usually taken into account in a form introduced by G.I. Taylor (Morton et al. 1956). It is written as a boundary condition at the edge of a plume; if we set this boundary to be at , the turbulent entrainment is written
According to Taylor's hypothesis, the entrainment rate is proportional to a characteristic velocity that is the mean axial velocity of the plume. The proportionality coefficient is a parameter whose value is experimentally determined. Afterwards, we will use the commonly assumed constant value (Turner 1986). The above relation states in a simple way that there is a horizontal flow of matter into the plume, whatever the plume is going up or down.
2.2. Equations of motion
Considering a single plume in a steady state, we now derive the equations governing its mean dynamics. The plumes are assumed to be far from each other so they do not have direct interactions. As in RZ, the basic equations are the stationary equations of continuity, conservation of momentum and conservation of energy where viscous terms have been neglected:
is the density, the mean velocity, the tensorial product of the mean velocity, the gravity acceleration, P the pressure, h the specific enthalpy, the energy production rate per unit mass and the radiative flux. All these quantities describe physical conditions inside the plume.
Before proceeding further, we make the following simplifications. The flow is assumed to have no component in the azimuthal direction () and to be in pressure balance with the surrounding medium, pressure fluctuations are thus neglected. Perturbations of the gravitational field, the energy production rate and the radiative flux related to density and temperature fluctuations are not taken into account either. Hence,
Dissipative processes are assumed to be small enough so they are neglected: radiative transfer from the plume to the ambient medium is not considered and the motion is adiabatic. These simplifications allow us to rewrite Eq. (2), the radial component of Eq. (3) and Eq. (4) as follows:
The enthalpy fluctuations have been related to density fluctuations through the perfect gas law. Afterwards, the contribution of the horizontal velocity to the kinetic energy will be omitted.
The next step is to integrate each equation on the horizontal section of the plume. We shall assume that the horizontal extent remains small compared to the distance traveled so that the geometry is locally flat and we can take . Plumes are narrow flows and this hypothesis is amply justified (Tennekes & Lumley 1972). Another difficulty arises when we have to define the outer limit of the turbulent plume. The colatitude should vary from 0 to that is not precisely defined. However, the edge of the flow may be understood as the place where the radial velocity becomes zero and the density contrast vanishes. Then, with the fast decrease of the Gaussian function, the upper limit of the integral may be formally taken equal to infinity.
The energy equation needs more care than the other two equations. If we integrate the right-hand side of Eq. (7) the same way as its left-hand side, we would just take account of the nuclear energy generated inside the plume. In our model, N plumes are said to transport the whole energy. To reproduce this assumption, the r.h.s should be replaced, after integration, by where is the total energy produced on the sphere of radius r. This way, the whole luminosity is equally shared among the N plumes and each plume is treated in the same fashion as the others. Such considerations also apply to the radiative flux. A straightforward integration would forget the contribution of the interplume medium to the radiative transfer. This problem may be solved by using the same expedient: the integrated radiative term is replaced by .
Finally, the integrated equations lead to the following set of ordinary differential equations where the unknowns are , b and V:
represents the density contrast, is the radiative luminosity and and are the convective and kinetic luminosities carried by N turbulent plumes. Equations (8) and (9) are the same as Eqs. (4) and (5) in RZ; Eq. (10) differs from Eq. (7) in RZ because, here, the sum of convective and kinetic luminosities is no more constant with depth. This last equation integrated in the radial direction yields
This relation states that the total luminosity, , is transported partly by the radiation, partly by the plumes, in kinetic and enthalpy fluxes form.
From now on, we will neglect the density contrast except in the buoyancy and enthalpy terms: is set equal to unity everywhere else.
2.3. The isentropic static core
Quantities corresponding to physical conditions outside the plume flows appear in Eqs. (8)-(10), the solution of the system requires a specification of the stratification of the static medium surrounding the plumes. This medium may be fairly well modeled by an isentropic perfect gas in hydrostatic equilibrium. This gas is in a fully ionized state due to the high temperatures pertaining to stellar cores. Such a medium is a polytrope of index (Kippenhahn & Weigert 1991). Temperature and density distributions are obtained from the well-known Lane-Emden equation:
Temperature and density distributions are respectively given by
where the subscript c denotes the central value of the given quantity; x is the non-dimensional radius defined by
For a perfect monoatomic gas, the equation of state and specific heat under constant pressure are
Here, µ is the mean molecular weight of a fully ionized medium, X and Y are the respective mass abundances of hydrogen and helium. The chemical composition will be considered homogeneous through the entire core, as a result of the mixing induced by the convective motions. Therefore, the specific heat is a constant.
Once temperature and density variations are known, gravity and radiative flux may be determined. In spherical symmetry, the gravitational acceleration is related to the mass distribution in a simple way:
As stellar interiors are optically thick, the radiative flux may be expressed in the Eddington approximation:
Both the mean opacity, , and thermonuclear reaction rates vary with density, temperature and chemical composition. Whereas approximate analytic power laws are available, we preferred to make use of subroutines written for the stellar evolution code CESAM (Morel 1993; G. Berthomieu et al. 1993), these subroutines rely on tables from Caughlan & Fowler (1988) for the nuclear reactions and on opacities from Iglesias et al. (1992). This choice has been dictated by the fact that if the opacity is underestimated, the radiative flux is overestimated in consequence and the boundary of the convective core is not accurately enough determined.
The physical constants , G and are, respectively, the perfect gas constant, the constant of gravitation and the Stefan-Boltzmann constant.
2.4. Boundary conditions
Near the central singularity, thermodynamical quantities are nearly constant and may be taken equal to their central values. It follows that
Near the source of a plume, the kinetic energy may be neglected before the enthalpy term because the velocity is small. Then, Eqs. (8)-(10) imply:
In this region, the plumes present a self-similar régime. These relations define the conditions at the source of a turbulent plume, they will be used as lower boundary conditions.
The upper boundary of the convective core is reached when , following the classical Schwarzschild criterion.
2.5. Introduction of a counter flow
The model above is incomplete. It lacks a description of the dynamics in the interplume medium. In fact, the plumes transport both energy and mass, thus, there is a mass flux in the upward direction. If we require that the whole core should satisfy mass conservation, we have to introduce motions in the downward direction. This can be easily done if we consider a uniform downflow in the interplume medium. The amplitude of this inverse flow, denoted , is given by the following continuity equation:
Although such an approach may seem crude, this relation gives the correct order of magnitude of the mean downward speed that is needed to keep the mass of the core constant. For simplicity, these motions are assumed not to modify the isentropic stratification, they are strictly adiabatic and do not take part in the energy transport.
The presence of this counter flow modifies slightly the equations governing the dynamics of plumes. The velocity of a plume relative to its environment is changed from V to . Hence, in the mass conservation equation (Eq. (8)), should be replaced by . The momentum equation is also modified by the addition of the quantity in the r.h.s of Eq. (9): now, plumes capture non-zero momentum from the outside. On the other hand, the energy equation remains unchanged as we assumed the energy transport to be unaffected by the inverse current.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998