V.M. Canuto: Theoretical Modelling of Convection (Review, Mon, Aug. 21)
Canuto started the symposium (239, Convection in Astrophysics) by giving a brief history of the theoretical modelling of convection. It starts in 1890 with Reynolds introducing the Reynolds stresses and continues with Boussinesq and the "down-gradient" approximation with diffusivity, which is the beginning of the Mixing Length Saga.
In 1929, Friedmann (the same as the one known from the Friedmann equations) wrote that the Navier-Stokes equations (NSE) should also yield an equation for the correleations of fluctuations, not only a dynamical equation for the mean components, which lead to the Reynolds stress model (RSM). But he was ignored - nobody continued in that direction.
Only in 1940, Chou published the first dynamical equation for the momentum Reynolds stresses. He treated only shear flows (no buoyancy) and the engineering community has since then used the RSM as a working tool. There was a nice story of how Chou was led to publish this equation, which I cannot reproduce here. The engineering community was followed by the geophysical community (early seventies) which used the RSM for global warming calculations.
Eventually, we get to stellar convection - its fortunes and misfortunes. Fortune: most of the convection zone is unstably stratified - the gradient is equal to the adiabatic one. Misfortune: the layers below the convection zone are stable but are affected by overshooting. The advantage of the convection zone is that it contains large eddies with long lifetime which carry most of the energy. One can use a mixing length invented by Prandtl (engineer, not astrophysicist!) to describe one large eddy or the Canuto & Mazzitelli model with many eddies. The disadvantage is that convection is non-local, but all models use the local approximation (which is bad because it is false). The overshooting zone is stable, contains small eddies and is therefore local, but difficult to model because of short life-times, vorticity, etc.
A very good paper was mentioned (even though it contains no equations at all), by J.D. Woods (1969, Radio Science vol. 4).
Now followed a detailed account of the Peclet number debate with lots of equations, as well as a discussion on the modifications that should be included in the equation for mixing and transport: Both shear and vorticity tensors should appear, and buoyancy and gravity waves should be included in the Reynolds stress tensor. Check out MNRAS 328, 829 (2001), which contains the complete algebraic espressions.
Further topics: "salt-fingers" (in oceans, correspond to molecular weight gradient in stars) and semi-convection and their influence on overshooting: salt-fingers cause larger overshooting, semi-convection causes smaller overshooting. Shear distroys salt-fingers (in models and experiments) showing that instabilities can act against each other.
After a few more equations, the non-local, so-called plume model by Morton, Taylor, Turner (1956) was mentioned but found to be unsuitable for astrophysics (because of assumption of small area of down plumes).
Some final words on turbulence:
It's not a source of anything! It is a very efficient distribution mechanism at zero cost (like superconductivity). Without an energy source it dies out. We need a formalism resilient enough to accommodate different processes without changing the rules of the game every time. The only formalism that can do that is the one derived directly from the NSE - the Reynolds stress model.
All this made for an exciting and at times entertaining first 40 minutes of the Symposium.