= Top-hat plume in constant unstable stratification
{title2=$b=2\alpha z/3,\quad w=Gz/4,\quad g'=G^2z/4$}
For prescribed $N^2=-G^2<0$, the steady <Boussinesq top-hat plume in a stratified ambient> has kinematic fluxes $Q=\alpha^2Gz^3/9$, $M=\alpha^2G^2z^4/36$, and $F=\alpha^2G^3z^4/36$. Balancing powers in $Q_z=2\alpha\sqrt M$, $M_z=FQ/M$, and $F_z=G^2Q$ gives exponents $(3,4,4)$ and these positive amplitudes. The ambient's static instability is prescribed without allowing it to drive ambient motion; the Boussinesq approximation additionally requires small density differences over the modeled height range.
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