= Neutral height of a bubbly thermal in a stratified fluid
{title2=$(1-z/H_p)[(a_0+\alpha z)^4-a_0^4]=4\alpha g\phi_0a_0^3/N^2$}
= Bubbly thermal neutral height
{synonym}
For a constant ambient <buoyancy frequency>, entrained liquid lags the decreasing ambient density by $\rho_0N^2[(a_0+\alpha z)^4-a_0^4]/[4\alpha g(a_0+\alpha z)^3]$. Balancing this negative buoyancy with expanding-bubble buoyancy gives the displayed fifth-degree equation. The first physically admissible root is the neutral height in the quasi-steady rise-speed model. In a very deep release, bubble expansion is negligible and $z_n=[(a_0^4+4\alpha g\phi_0a_0^3/N^2)^{1/4}-a_0]/\alpha$. Actual momentum can cause overshoot, while bubble slip becomes important as the liquid stops rising.
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