= Boussinesq point-source plume
{c}
A steady <top-hat plume model> supplied with kinematic <buoyancy flux> $f_0>0$ but zero source <volume flux> and <momentum flux> has
$$
b=az,\qquad w=Kz^{-1/3},\qquad g'=\frac43K^2z^{-5/3},\qquad
a=\frac{6\alpha}{5},\quad K^3=\frac{25f_0}{48\alpha^2}.
$$
The <Batchelor entrainment hypothesis> and momentum balance yield these constants. Its <Froude number> satisfies $w^2/(g'b)=5/(8\alpha)$. The <Boussinesq approximation> requires $g'/g\ll1$ and therefore fails at the ideal point source itself.
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