= Double-diffusive overflow reservoir
{title2=$\theta_s=C(1-\theta_s)^{10/3}/(1-\eta_s)^2$}
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.
The steady subscript denotes balance between supply, overflow and interfacial transfer. A model of this kind separates the stabilizing bulk salinity contrast from the faster transfer of heat across the interface.
A steady well-mixed hot salty reservoir receives volume flux $Q$ and loses the same flux over a weir. Heat and salt balances are $Q(T_i-T)=AF_T$ and $Q(S_i-S)=AF_S$. If $\beta F_S/(\alpha F_T)=R_F$ and $\alpha F_T=b(\alpha\Delta T)^{4/3}/R_\rho^2$, define $R_0=\beta(S_i-S_\infty)/[\alpha(T_i-T_\infty)]$. Then $\eta_s=(R_F/R_0)\theta_s$ and $C=Ab[\alpha(T_i-T_\infty)]^{1/3}/(Q R_0^2)$. For $R_0>1$ and $0\leq R_F<1$ the interface is statically stable while preferential heat transfer can drive <double-diffusive convection>.
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