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In stable density stratification, ambient density decreases with height. A rising buoyant thermal encounters progressively lighter surroundings and entrains denser fluid from below, so its net buoyancy decreases. It reaches neutral buoyancy, overshoots through its remaining momentum, and eventually stops rising, spreads as a buoyant intrusion or excites internal gravity waves. The finite rise is a consequence of this restoring effect, not just drag.

Total buoyancy has dimensions $[B_0]=L^4{T}^{-2}$ and buoyancy frequency has $[N]=T^{-1}$. For an ideal point release with no other dimensional source length or time, dimensional analysis therefore givesThe dimensionless coefficient depends on entrainment coefficient, shape and the momentum model. A finite source radius or impulse supplies additional dimensionless parameters; dimensional analysis cannot determine the coefficient by itself.

Within precisely the spherical model, $b=\alpha z$ and the evolving thermal buoyancy satisfies $dB/dz=-N^{2}V=-K{N}^2{z}^3$, so $B(z)=B_0-K{N}^2{z}^4/4$. Neutrality occurs at $z_n^4=4B_0/(K{N}^2)$. For $P=Vu$, the momentum equation and $\dot{z}=P/V$ giveThe first nonzero-height stopping point is $z_{\max }^4=8B_0/(K{N}^2)$, so $z_{\max }=2^{1/4}{z}_n$. This illustrates both the dimensional scale and the distinction between neutral height and maximum rise; the precise coefficients rely on the specified ideal fluid entrainment and impulse model.

For a very deep release, $z/H_p\ll 1$ over the rise, so bubble expansion is negligible. The neutral-height equation reduces toThusIf the stopping thermal is also much larger than its initial size, the leading form isThe first expression does not require this extra entrainment-dominated limit.

Near the neutral height the quasi-steady speed tends to zero. Momentum can produce some overshoot and subsequent adjustment or oscillation. The water spreads horizontally as a buoyant intrusion near its equilibrium density level. Bubble slip then becomes important relative to the slowing liquid, so bubbles can separate and continue rising toward the surface, reducing the residual liquid's buoyancy. Loss of bubbles and further mixing alter the final intrusion level; the retained-bubble model no longer applies indefinitely after stopping.

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 scaleIf $z_s\gg H$, 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.

A line plume of kinematic buoyancy flux $B_0$ rising through uniform buoyancy frequency $N$ has the dimensional height scaleIt measures the height over which ambient stratification removes an order-one fraction of the plume buoyancy before the plume reaches neutral buoyancy and spreads as a buoyant intrusion.