Towards consistency in simple prescriptions for stellar convection⋆

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Abstract
We describe a simple rule-of-thumb procedure by which the normal fluid dynamics equations can be averaged, for a turbulent convective region in a star (a) to give the variation of mean quantities, such as temperature, density and composition, as a function of depth, (b) to give the variation of the rms values of fluctuating quantities, such as the temperature, density and composition excess of turbulent eddies relative to the mean, as a function of depth. The averaging procedure is not capable of being justified in detail by analysis; its virtue is that, when applied systematically to the normal equations for (i) composition, (ii) heat and (iii) velocity, it leads straightforwardly and consistently to the kinds of terms that have commonly been expected on physical grounds. We show the following three points: for a region of non-uniform composition, we get stability, overstability and instability according to the criteria of M. Schwarzschild and of Ledoux; for steady convection in a region of uniform composition, we get the usual mixing length theory if we ignore non-local terms; and for nuclear evolution we get a diffusion approximation for the mixing effect of convection. The procedure we describe leads to a time-dependent, non-local, ‘theory’ of convection which, we suggest, is less arbitrary and more consistent than others which have been proposed.

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