= Post-shutdown double-diffusive reservoir cooling
{title2=$\eta=q\theta,\quad q=R_F/R_0$}
Let $T_i,S_i$ be the <temperature> and <salinity> of the original supplied liquid, and $T_\infty,S_\infty$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measures
$$
\theta=\frac{T_i-T}{T_i-T_\infty},\qquad
\eta=\frac{S_i-S}{S_i-S_\infty},
$$
with positive reference differences. Here $\alpha$ is the <thermal expansion coefficient>, $\beta$ the <haline contraction coefficient>, $A$ the interface <area> and $R_\rho=\beta(S-S_\infty)/[\alpha(T-T_\infty)]$ the <density ratio (double-diffusive convection)>. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed <double-diffusive convection> closure.
With a constant buoyancy-flux ratio and the preceding steady initial state, the heat and salt balance equations give $d\eta/d\theta=q$ and $\eta_s=q\theta_s$, yielding $\eta=q\theta$ after shutdown.
After inflow and overflow cease, a fixed-volume reservoir initially in the preceding steady state obeys $\eta=q\theta$ and $\dot\theta=k(1-\theta)^{10/3}/(1-q\theta)^2$, with $k=Ab[\alpha(T_i-T_\infty)]^{1/3}/(V R_0^2)$. Integrating gives $k(t-t_0)=G(\theta)-G(\theta_s)$, where $G=3(1-q)^2(1-\theta)^{-7/3}/7+3q(1-q)(1-\theta)^{-4/3}/2+3q^2(1-\theta)^{-1/3}$. For $q<1$, heat contrast tends to zero only at infinite time, while a nonzero salt contrast remains.
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