= Solution
Assume a sharp interface, uniformly mixed warm upper air with fixed <reduced gravity> $g'=g\Delta\rho/\rho_0$, cool ambient air entering below, no wind, no wall heat exchange, negligible source volume, and quasi-steady <Boussinesq> flow through the small vents. Neglect the interior approach <velocity>. The hydrostatic buoyancy head between the two vents is $\rho_0g'h$.
Both openings carry the same <volume flux> $q_v$. The <discharge coefficient> law gives a pressure loss $\rho_0q_v^2/(2C_d^2A^2)$ at each opening. Adding the two losses gives $q_v=C_dA\sqrt{g'h}$, not the single-opening value with an extra factor $\sqrt2$. Cool replacement air displaces the warm layer, so $S\dot h=-q_v$. With $h(0)=H$, integrate to obtain
$$
\boxed{h(t)=\left(\sqrt H-\frac{C_dA\sqrt{g'}}{2S}t\right)^2,\quad
0\leq t\leq t_e=\frac{2S\sqrt H}{C_dA\sqrt{g'}}.}
$$
The ideal model sets $h=0$ thereafter. The formula applies only while the parenthesis is nonnegative; squaring a negative value would invent renewed filling. A finite aperture or a diffuse final interface invalidates the idealization near exhaustion.
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