 Solution

With inflow and overflow both zero and fixed reservoir volume $V$, the transient balances are $V\dot{T}=-A{F}_T$ and $V\dot{S}=-A{F}_S$. ThereforeThe last equality uses the initial steady state. Define $k_c=Ab(\alpha T_0{)}^{1/3}/(V{R}_0^2)=CQ/V$, where the latter $Q$ is the pre-shutdown flow. The post-shutdown double-diffusive reservoir cooling equation isSeparate variables. An explicit primitive of $(1-q\theta {)}^2/(1-\theta {)}^{10/3}$ isDifferentiate it to verify the three powers. HenceFor $q<1$, $\theta$ increases monotonically to one only at infinite time, with $1-\theta \sim [3(1-q{)}^2/(7k_{c}t){]}^{3/7}$ up to the choice of time origin. The reservoir approaches $T_{\infty }$, but its salinity tends to $S_i-q{S}_0$, leaving the positive salt contrast $(1-q)S_0$. Thus $R_{\rho }\to \infty$ and the double-diffusive flux shuts down asymptotically. The assumption of a stationary interface and continued applicability of this flux parameterization is part of the reservoir model, not a claim about every actual hydrothermal pool.