= Solution
The source supplies the kinematic <buoyancy flux> $B_0=F_0/\rho_0$. The unstratified <line plume> velocity scale is $B_0^{1/3}$. Comparing this with the stratification time scale $N_0^{-1}$ gives the <stratified line-plume height scale>
$$
\boxed{z_s\sim\frac{B_0^{1/3}}{N_0}
=\left(\frac{F_0}{\rho_0N_0^3}\right)^{1/3}.}
$$
If $z_s\gg H$, the initial plume is only weakly affected by the <stable density stratification> and reaches the ceiling much like an unstratified wall plume. If $z_s\sim H$, its <buoyancy flux> falls substantially during the rise; it approaches neutral buoyancy near the upper room, overshoots because of its <momentum flux>, and spreads as a horizontal <buoyant intrusion>. If $z_s\ll H$, the plume reaches neutral buoyancy low in the room and forms a low intrusion after a modest overshoot. In each sketch the plume widens by <entrainment>, while increasing $N_0$ lowers the neutral-buoyancy and overshoot heights.
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