 Solution

Let $V=4\pi b^3/3$ and $B=Vg({\rho }_0-\rho )/{\rho }_{\mathrm{ref}}$. The four balances describe, respectively, fluid entrainment through the spherical surface at normal speed $\alpha u$, the increase of vertical momentum due to buoyancy, conservation of total buoyancy in a uniform ambient, and motion of the thermal centre. Entrained ambient fluid initially has no vertical momentum, so the momentum equation includes the inertia needed to accelerate it. This model neglects added mass and drag; these effects would alter the numerical similarity coefficients.

With $|\rho -{\rho }_0|\ll {\rho }_0$, the Boussinesq approximation replaces inertial density by ${\rho }_0$ while retaining the density difference in buoyancy. Take ${\rho }_{\mathrm{ref}}={\rho }_0$ to this accuracy. ThenFor a point source, $b(0)=z(0)=0$ and the initial integrated impulse is zero. Therefore $b=\alpha z$ and $Vu=B_{0}t$. Integrating $b^3\dot{b}=3\alpha B_{0}t/(4\pi )$ yieldsThe mass density follows from conserved buoyancy:Thus the radius and rise height grow as $t^{1/2}$, the rise speed decays as $t^{-1/2}$, and the density deficit decays as $t^{-3/2}$. In height coordinates, put $K=4\pi {\alpha }^3/3$ and $C=\sqrt{B_0/(2K)}$; then $V=K{z}^3$ and $u=C/z$.

If a reference density different from the inertial ambient density is retained, $d(Vu)/dt=({\rho }_{\mathrm{ref}}/{\rho }_0)B_0$: replace $B_0$ in the kinematic formulas by $({\rho }_{\mathrm{ref}}/{\rho }_0)B_0$, while retaining the original $B_0$ in the density-deficit formula. This keeps reference-density normalization separate from the physical acceleration.

The point-source spherical thermal similarity is singular at $t=0$ and cannot satisfy the small-density-difference condition arbitrarily close to its formal origin. A finite-radius, zero-impulse regularization has $b=b_s+\alpha z$ and $b^4=b_s^4+3\alpha B_0{t}^2/(2\pi )$, with $u=B_{0}t/V$. This displays the required initial data rather than attributing the singularity to a physical infinite initial speed.