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

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.

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.