Dilute bubbly thermal ($a=a_0+\alpha z,\varphi ={\varphi }_0{a}_0^3/[a^3(1-z/H_p)]$)

A dilute bubbly thermal is a rising, entraining fluid volume whose gas bubbles provide part of its buoyancy. With negligible bubble slip and gas loss, Boyle's law gives gas volume proportional to $(1-z/H_p{)}^{-1}$, where $H_p=p_0/({\rho }_{0}g)$. Spherical fluid entrainment with normal entrainment velocity $\alpha W$ gives $da/dz=\alpha$. Bubble concentration follows by dividing the gas volume by the growing thermal volume. A diameter-based Froude number closure is $W=F\sqrt{2a{g}^{\prime }}$.

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.