Neutral height of a bubbly thermal in a stratified fluid ($(1-z/H_p)[(a_0+\alpha z{)}^4-a_0^4]=4\alpha g{\varphi }_0{a}_0^3/N^2$)

For a constant ambient buoyancy frequency, entrained liquid lags the decreasing ambient density by ${\rho }_0{N}^2[(a_0+\alpha z{)}^4-a_0^4]/[4\alpha g(a_0+\alpha z{)}^3]$. 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 $z_n=[(a_0^4+4\alpha g{\varphi }_0{a}_0^3/N^2{)}^{1/4}-a_0]/\alpha$. Actual momentum can cause overshoot, while bubble slip becomes important as the liquid stops rising.