Turbulent Convection with Overshooting: Reynolds Stress Approach. II.
Abstract
We derive a new nonlocal model for turbulent convection which incorporates recent advances from laboratory, planetary, and numerical simulation of turbulence, and we show how previous prototypic models can be recovered. The new model is given by five coupled differential equations, equations (81)(84) and (100), that yield: convective flux, temperature variance, turbulent kinetic energy in the zdirection, total turbulent kinetic energy, and (rate of) dissipation of kinetic energy. The solution of these five equations yields all the turbulent quantities as a function of the temperature gradient. The latter is then obtained by solving the flux conservation law, equation (96), which we derive anew to account for the kinetic energy flux. The main features of the new model are as follows.
1. Nonlocality. This basic feature is represented by the thirdorder moments that enter the governing equations (81)(84) and (100). In all nonlocal models thus far, these moments were treated with the diffusion approximation. Since the latter yields incorrect results in the case of the convective boundary layer, we avoid it. We derive the dynamic equations for all the thirdorder moments and solve them analytically.
2. Gravity waves, stable stratification. The fundamental feature of the overshooting (OV) region is that the flow is stably stratified, Δ  Δ_{ad} < 0. Under such circumstances, the Kolmogorov spectrum is no longer valid since eddies, working against gravity, lose a fraction of their kinetic energy which goes to generate "gravity waves." To fully account for the appearance of a "buoyancy subrange" E(k) ∼ k^{3} in lieu of the Kolmogorov spectrum ∼ k^{5/3}, we adopt a recent model for stably stratified turbulence which has been successfully tested against convective boundary layer data.
3. Dissipation ɛ.  The process of dissipation of turbulent kinetic energy has been neglected for many years, but is now viewed as crucial for a proper quantification of OV. The assumption ɛ = 0 not only violates the energy conservation law, but overestimates the extent of the OV region. When ɛ is included, it is generally computed locally with a mixing length l. If the description of l is difficult in the main convective region, it is all the more so in the OV region where the concept of a mixing length loses its physical content. We avoid the use of a mixing length in both the convective and the OV region by introducing a differential equation for the dissipation ɛ, equation (100), which, being nonlocal, accounts for the fact that turbulent kinetic energy created at one point in the flow may be dissipated somewhere else, in accordance with the nonlocal nature of turbulent convection.
4. Pressure forces, anisotropy. The stably stratified turbulence found in the OV region is experimentally known to be highly anisotropic since negative buoyancy suppresses the eddy vertical motion. Thus, pressure velocity and pressuretemperature correlations, which help restore isotropy, play a crucial role.
5. The Boussinesq Approximation is avoided.
6. The turbulent kinetic energy flux.  A new flux conservation law, equation (96), is derived which includes the turbulent kinetic energy flux recently found to be up to 50% of the total flux for Sunlike stars.
7. A new hydrostatic equilibrium equation, equation (103), is derived which, in addition to a turbulent pressure, also includes buoyancy effects.
The next step is to couple the new model to a stellar structure code.
 Publication:

The Astrophysical Journal
 Pub Date:
 October 1993
 DOI:
 10.1086/173238
 Bibcode:
 1993ApJ...416..331C
 Keywords:

 CONVECTION;
 STARS: INTERIORS;
 TURBULENCE