 Boussinesq point-source plume

A steady top-hat plume model supplied with kinematic buoyancy flux $f_0>0$ but zero source volume flux and momentum flux hasThe 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.