= Dilute bubbly thermal
{title2=$a=a_0+\alpha z,\quad\phi=\phi_0a_0^3/[a^3(1-z/H_p)]$}
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 $H_p=p_0/(\rho_0g)$. 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 $W=F\sqrt{2ag'}$.
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