Stratified mixing energy budget ($\dot{K}=F-B-ϵ,\dot{A}=B-M$)

With boundary fluxes suppressed, let $B$ transfer kinetic energy to potential energy, $M$ transfer available potential energy irreversibly to background potential energy, and $W$ transfer internal energy to potential energy. Then $\dot{P}=B+W$, ${\dot{P}}_b=M+W$, $\dot{A}=B-M$, and $\dot{K}=F-B-ϵ$. The internal reservoir has $\dot{I}=ϵ-W$, conserving total energy apart from forcing $F$. The distinction between reversible buoyancy exchange and irreversible mixing is essential outside a quasi-steady negligible-$A$ regime.

Instantaneous mixing efficiency is the fraction of instantaneous irreversible mechanical-energy loss that raises background potential energy, after subtracting internal-to-potential exchange. The ratio $\Gamma =M/ϵ$ is a different quantity, related by $\eta =\Gamma /(1+\Gamma )$. Replacing $M$ by buoyancy exchange $B$ additionally assumes negligible change of available potential energy.

Cumulative mixing efficiency is the ratio of total irreversible density-mixing conversion to total irreversible mechanical-energy loss over an interval. It is a loss-rate-weighted average of instantaneous mixing efficiency, not an unweighted time average. Stored kinetic energy, changes in available potential energy and boundary work must be accounted for before replacing its denominator by integrated forcing.