= 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 $[B_0]=L^4T^{-2}$ and <buoyancy frequency> has $[N]=T^{-1}$. For an ideal point release with no other dimensional source length or time, <dimensional analysis> therefore gives
$$
\boxed{z_{\max}=C_z\left(\frac{B_0}{N^2}\right)^{1/4}}.
$$
The 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 $dB/dz=-N^2V=-KN^2z^3$, so $B(z)=B_0-KN^2z^4/4$. Neutrality occurs at $z_n^4=4B_0/(KN^2)$. For $P=Vu$, the momentum equation and $\dot z=P/V$ give
$$
\frac{dP^2}{dz}=2VB,\qquad P^2=\frac{KB_0z^4}{2}-\frac{K^2N^2z^8}{16}.
$$
The first nonzero-height stopping point is $z_{\max}^4=8B_0/(KN^2)$, 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.
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