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Canuto, V.M., 1992: Turbulent convection with overshooting: Reynolds stress approach. Astrophys. J., 392, 218-232, doi:10.1086/171420.
Turbulent convection is a phenomenon relevant to both stellar structure and accretion disks. In the latter, a basic parameter such as the turbulent velocity nut is still treated phenomenologically; in the case of stellar structure, most of the work still relies on the mixing length theory (MLT) which assumes homogeneity and thus lacks diffusion terms (divergence of third-order moments like ave(w2*θ), ave(w*θ2)m ave(q2*w)). To include them, one needs a new formalism. We review and discuss the Reynolds stress approach (proven successful in other fields) which provides a set of coupled differential equations that yield all the turbulent quantities of interest. Although the system can only be solved numerically, some features can be listed:
1. The convective flux Fc = cp*ρ*ave(w*θ) is not given simply by (κt is the turbulent velocity)
Fc = FcMLT ≃ κt(∇-∇ad>/sub>).
2. Inclusion of the diffusion terms related to ave(w2*θ) and ave(w*θ2) contributes a countergradient term Γc, which may carry heat from cold to hot regions,
Fc ≃ κt(∇-∇ad>/sub>+Γc).
3. Inclusion of the diffusion term related to 0.5*ave(q2*w) (turbulent kinetic energy flux) contributes an additional term (first discussed in atmospheric turbulence by Tennekes)
Fc ≃ κt(∇-∇ad+Γc)+Fcdiff,
which is responsible for overshooting.
In addition to the convective flux, we also derive a model expression for nut as a function of both shear and buoyancy: it is needed in the numerical simulation of stellar convection and in accretion disks to replace the phenomenological expressions used thus far.
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