= Solution
For a very deep release, $z/H_p\ll1$ over the rise, so bubble expansion is negligible. The neutral-height equation reduces to
$$
(a_0+\alpha z_n)^4-a_0^4=\frac{4\alpha g\phi_0a_0^3}{N^2}.
$$
Thus
$$
\boxed{z_n=\frac{\left(a_0^4+4\alpha g\phi_0a_0^3/N^2\right)^{1/4}-a_0}{\alpha}.}
$$
If the stopping thermal is also much larger than its initial size, the leading form is
$$
\boxed{z_n\sim\left(\frac{4g\phi_0a_0^3}{N^2\alpha^3}\right)^{1/4}.}
$$
The 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.
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