The equations of motion, thermodynamics and scalar concentration are averaged separately over infra-grid scales (comprising roughly ocean microstructure scales) and sub-grid scales (encompassing 10–100 km scales), the latter average being carried out on constant potential temperature surfaces. (Since variation of salinity has been neglected, potential temperature is synonymous with entropy.) These methods of averaging are used to lend precision to statements about oceanic turbulent diffusion: that infra-grid scales are primarily responsible for dientropic diffusion; and that sub-grid scales are responsible for along-isentropic diffusion of passive scalars. Equivalent sets of averaged equations expressed either in isentropic coordinates or quasi-Cartesian coordinates can be obtained. Diffusion tensors for potential temperature express only infra-grid effects. For other passive tracers, diffusion caused by sub-grid scales of motion is also felt, whose effects are shown by scale analysis to be oriented principally along infra-grid averaged isentropes.