Third order correlations and skewness in convection. I. A new approach suitable for three-equation non-local models

Abstract

Non-local models of stellar convection can account for mixing effects in regions adjacent to convectively unstable layers and for changes to the mean temperature structure caused by free, buoyancy driven convection. The physical completeness of such models, however, depends on how third order correlations, which characterize the non-local transport processes, are expressed in terms of second order correlations and the stellar mean structure. Physical arguments and 3D hydrodynamical simulations were used to develop and test new closure relations for the skewness of the vertical velocity and temperature fields and third order cross-correlations to improve the predictive capabilities of non-local models of convection used in stellar astrophysics and in other disciplines such as meteorology. The structural form of the closure correlations was developed by a series of physical arguments and their accuracy was evaluated through self-consistency tests based on 3D hydrodynamical simulations for the Sun and a DA type white dwarf. The new closure relations derived for the skewness of vertical velocity and temperature fields provided improvements of up to an order of magnitude compared to previous models. This allows releasing the full potential of closure relations for the vertical velocity and temperature cross-correlations previously proposed in meteorology as well as the construction of new, more reliable models for the third order moments of vertical velocity and temperature in non-local models of turbulent convection. The new models for the skewness and third order cross-correlations of vertical velocity and temperature permit the construction of non-local models of turbulent convection which remove, among others, several major short-comings of three equation non-local convection models that are based on the downgradient approximation.