Non-Boussinesq top-hat plume equations

For a turbulent plume in a homogeneous ambient of density ${\rho }_0$, define actual sectional mass flux, momentum flux and density-weighted buoyancy flux by $Q=\pi b^2\rho w$, $M=\pi b^2\rho w^2$ and $F=\pi b^{2}wg({\rho }_0-\rho )$. With non-Boussinesq Batchelor entrainment $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, their balances areHere $Q^2/M$ is sectional mass, $QF/M$ is sectional density-deficit buoyancy, and the entrained ambient initially has zero vertical momentum. The last equation follows by subtracting mass conservation from ${\rho }_0$ times volume conservation. Scase, Caulfield, Dalziel and Hunt's time-dependent plume model derives the equivalent system with the common factor $\pi$ suppressed.

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 $q^{\prime }=2\alpha \sqrt{m}$ and $m^{\prime }=Bq/m$, identical to the pi-suppressed Boussinesq pure-plume flux equations. With $d=6\alpha /5$ and $s=[25B/(48{\alpha }^2){]}^{1/3}$, $q=d^{2}s{z}^{5/3}$, $m=d^2{s}^2{z}^{4/3}$ and $z_b^{5/3}=B/(g{d}^{2}s)$. 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.