Double-diffusive overflow reservoir (${\theta }_s=C(1-{\theta }_s{)}^{10/3}/(1-{\eta }_s{)}^2$)

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 measureswith 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. 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 $Q(T_i-T)=A{F}_T$ and $Q(S_i-S)=A{F}_S$. 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 $0\le R_F<1$ the interface is statically stable while preferential heat transfer can drive double-diffusive convection.

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 measureswith 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. 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.