= Solution
An <instantaneous mixing efficiency> defined as a fraction of irreversible mechanical-energy loss is
$$
\boxed{\eta(t)=\frac{M(t)}{M(t)+\epsilon(t)}.}
$$
The <cumulative mixing efficiency> over $[0,t]$ is
$$
\boxed{\eta_c(t)=\frac{\int_0^t M(s)\,ds}
{\int_0^t[M(s)+\epsilon(s)]\,ds}.}
$$
It is weighted by the loss rate, rather than being the arithmetic time average of $\eta$. The flux coefficient $\Gamma=M/\epsilon$ is also sometimes called mixing efficiency; it obeys $\eta=\Gamma/(1+\Gamma)$, and the corresponding cumulative coefficient is $\Gamma_c=\int M/\int\epsilon$. These conventions should be stated rather than interchanged. Only when $\dot A$ is negligible does $M\simeq B$, permitting the frequently used buoyancy-flux form $B/(B+\epsilon)$. Input-power efficiency $M/F$ is another quantity and generally differs during changes in stored energy.
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