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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.

Let $P$ denote total gravitational potential energy, $P_b$ the background potential energy of the adiabatically sorted density field, and $A=P-P_b$ the available potential energy. Define $B$ positive for reversible transfer from kinetic energy to $P$, $ϵ$ as viscous kinetic-energy loss, $F$ as imposed mechanical work, $W$ as molecular transfer from internal energy to potential energy, and $M$ as the irreversible conversion of $A$ to $P_b$. With boundary contributions removed, the stratified mixing energy budget isFor an unforced flow set $F=0$. In an alternative notation, if ${\Phi }_d={\dot{P}}_b$ is the full diffusive increase of background energy and ${\Phi }_i=W$, then $M={\Phi }_d-{\Phi }_i$ and $\dot{A}=B+{\Phi }_i-{\Phi }_d$. This distinguishes irreversible density mixing from reversible buoyancy exchange. The internal reservoir obeys $\dot{I}=ϵ-W$, so summing the kinetic, potential and internal budgets gives $d(K+P+I)/dt=F$ as required by conservation of energy.

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.