Point-source spherical thermal similarity ($z\propto B_0^{1/4}{t}^{1/2}$)

In the Boussinesq approximation, a spherical buoyant thermal with conserved total buoyancy $B_0$ obeys $\dot{b}=\alpha u$ and $d(Vu)/dt=B_0$, with $V=4\pi b^3/3$. A zero-volume, zero-impulse point source gives $b=\alpha z$, $K{z}^3\dot{z}=B_{0}t$, where $K=4\pi {\alpha }^3/3$. Hence $z^4=2B_0{t}^2/K$ and $u=\sqrt{B_0/(2K)}/z$. The zero-time singularity represents an ideal source limit, not a finite initially regular sphere.