## Abstract

Three alternative methods of averaging the general conservation equation of a fluid property in a turbulent flow in the Boussinesq approximation are compared: Lagrangian, residual, and isopycnal (or semi-Lagrangian) mean. All methods differentiate consistently but in different ways between effects of advection and irreversible changes of the average property. Because the three average properties differ, the mean transport velocities and the mean irreversible changes in the mean conservation equation differ in general.

The Lagrangian and the semi-Lagrangian (or isopycnal) mean frameworks are shown to be approximately equivalent only for weak irreversible changes, small amplitudes of the turbulent fluctuations, and particle excursion predominantly along the mean property gradient. In that case, the divergent Stokes velocity of the Lagrangian mean framework can be replaced in the Lagrangian mean conservation equation by a nondivergent, three-dimensional version of the quasi-Stokes velocity of T. J. McDougall and P. C. McIntosh, for which a closed analytical form for the streamfunction in terms of Eulerian mean quantities is given.

*Corresponding author address:*Carsten Eden, KlimaCampus, University of Hamburg, Bundesstrasse 53, Hamburg 20146, Germany. E-mail: carsten.eden@zmaw.de