<jats:title>Abstract</jats:title> <jats:p>A generalization of the transformed Eulerian and temporal residual means is presented. The new formulation uses rotational fluxes of buoyancy, and the full hierarchy of statistical density moments, to reduce the cross-isopycnal eddy flux to the physically relevant component associated with the averaged water mass properties. The resulting eddy-induced diapycnal diffusivity vanishes for adiabatic, statistically steady flow, and is related to either the growth or decay of mesoscale density variance and/or the covariance between small-scale forcing (mixing) and density fluctuations, such as that associated with the irreversible removal of density variance by dissipation. The relationship between the new formulation and previous approaches is described and is illustrated using results from an eddying channel model. The formalism is quite general and applies to all kinds of averaging and to any tracer (not just density).</jats:p>