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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 givesBut $dV/dt=4\pi a^{2}da/dt$ and $dz/dt=W$. Therefore, wherever the thermal is rising,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.