 Solution

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.