= Non-Boussinesq pure-plume density transition
{title2=$\rho=\rho_0/[1+(z_b/z)^{5/3}]$}
For an ideal-gas heated <pure plume> with entrainment proportional to $w\sqrt{\rho/\rho_0}$, the excess heat flux implies conserved kinematic buoyancy flux $B$. Put $q=Q/\rho_0$, $m=M/\rho_0$. Their balances are $q'=2\alpha\sqrt m$ and $m'=Bq/m$, identical to the pi-suppressed Boussinesq pure-plume flux equations. With $d=6\alpha/5$ and $s=[25B/(48\alpha^2)]^{1/3}$, $q=d^2s z^{5/3}$, $m=d^2s^2z^{4/3}$ and $z_b^{5/3}=B/(gd^2s)$. The radius is $b=dz\sqrt{1+(z_b/z)^{5/3}}$ and reduced gravity is $g/[1+(z/z_b)^{5/3}]$. At $z=z_b$ density is half ambient; far above this height the <Boussinesq approximation> becomes accurate.
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