Boussinesq point-source plume

A steady top-hat plume model supplied with pi-suppressed kinematic buoyancy flux $f_0=b^{2}w{g}^{\prime }>0$ but zero source volume flux and momentum flux hasThe full-cross-section buoyancy flux is $\pi f_0$. The Batchelor entrainment hypothesis and momentum balance yield these constants. Its Froude number satisfies $w^2/(g^{\prime }b)=5/(8\alpha )$. The Boussinesq approximation requires $g^{\prime }/g\ll 1$ and therefore fails at the ideal point source itself.