Third-order correlations and skewness in convection

I. A new approach for three-equation non-local models

¹ Faculty Comp. Science and Appl. Math., Univ. of Applied Sciences Technikum Wien, Höchstädtplatz 6, A-1200 Wien, Austria
² Wolfgang-Pauli-Institute c/o Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
³ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

^★ Corresponding author: friedrich.kupka@technikum-wien, This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 15 December 2025
Accepted: 21 February 2026

Abstract

Context. 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. However, the physical completeness of such models depends on how third-order correlations, which characterise the non-local transport processes, are expressed in terms of second-order correlations and the stellar mean structure.

Aims. Physical arguments and 3D hydrodynamical simulations are used to develop and test new closure relations for the skewness of the vertical velocity and temperature fields as well as for third-order cross-correlations with the intention to improve the predictive capabilities of non-local models of convection used in stellar astrophysics and in other disciplines such as meteorology.

Methods. We developed the structural form of a set of closure relations through a series of physical arguments. Their accuracy was evaluated through self-consistency tests based on 3D hydrodynamical simulations for the Sun and a DA-type white dwarf.

Results. The new closure relations derived for the skewness of vertical velocity and temperature fields offer improvements of up to an order of magnitude compared to previous models. This advancement enables the release of the full potential of closure relations for the vertical velocity and temperature cross-correlations previously proposed in meteorology. It also allows the construction of new, more reliable models for the third-order moments of vertical velocity and temperature in non-local models of turbulent convection.

Conclusions. New models for the skewness and third-order cross-correlations in vertical velocity and temperature enable the construction of non-local models of turbulent convection that have, among other advantages, the capability to remove several major short-comings of down-gradient approximation based three equation non-local convection models. The application of the new closure relations in stellar evolution modelling is expected to clarify whether this approach is a major step forward in the modelling of stellar convection when multidimensional hydrodynamical simulations are not affordable.