= Ventilated filling-box relaxation
{title2=$\tau_h,\tau_b$}
For total plume flux $P(h)$, fixed ceiling exhaust $F$, zero ambient <buoyancy>, and uniform upper-layer <reduced gravity> $b$, the ideal two-layer balances are
$$
A\dot h=F-P(h),\qquad \frac{d}{dt}\bigl[A(H-h)b\bigr]=B-Fb.
$$
At $P(h_*)=F$ and $b_*=B/F$, <linearization> yields the decay times $\tau_h=A/P'(h_*)$ and $\tau_b=A(H-h_*)/F$. For $P(h)\propto h^{5/3}$, $\tau_h=3Ah_*/(5F)$. Upper-layer <buoyancy> relaxation usually dominates when $h_*\ll H$. Exact equilibrium is approached asymptotically; a fixed small relative tolerance takes order $\max(\tau_h,\tau_b)\log(1/\varepsilon)$ after the initial transient.
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