= Solution
The <Boussinesq approximation> retains small density differences only in the gravitational buoyancy force and uses a constant reference density in inertia and volume conservation. The <entrainment coefficient> $\alpha$ specifies the inward velocity of ambient fluid relative to the thermal boundary, $u_e=\alpha W$. Turbulent interface motions engulf ambient water and mix it into the coherent rising volume.
For a spherical <buoyant thermal>, $V=4\pi a^3/3$. To leading order in the dilute gas fraction, expansion of the gas does not dominate the total-volume balance, so <fluid entrainment> gives
$$
\frac{dV}{dt}=4\pi a^2\alpha W.
$$
But $dV/dt=4\pi a^2 da/dt$ and $dz/dt=W$. Therefore, wherever the thermal is rising,
$$
\boxed{\frac{da}{dz}=\alpha,\qquad a=a_0+\alpha z,\qquad 2a=2a_0+2\alpha z.}
$$
This growth law for a <dilute bubbly thermal> is independent of stratification under the constant-$\alpha$ model. Stratification changes the speed and eventual stopping height; it does not enter this volume-per-distance relation. In real flows the entrainment coefficient itself can vary with stratification and flow structure.
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