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From the definitions of mass and momentum fluxes,The density relation follows by solving the conserved-buoyancy identity in the preceding part. Hence $wb\sqrt{\rho }=\sqrt{M}$ andThe second equality uses $B=g({\rho }_0-\rho )Q/({\rho }_0\rho )$ and $b^2=Q^2/(\rho M)$.

Set $q=Q/{\rho }_0$ and $m=M/{\rho }_0$. The equations become $q^{\prime }=2\alpha \sqrt{m}$, $m^{\prime }=Bq/m$, exactly the same flux system as for the pi-suppressed Boussinesq pure plume. For the heated point source take zero initial mass and momentum flux and the finite conserved heat/buoyancy supply. DefineThe resulting mass and momentum fluxes areSubstituting $Q$ into the density relation gives the non-Boussinesq pure-plume density transitionThe two expressions for $z_b$ agree because $B=4d^2{s}^3/3$. Density increases from the very dilute near-source regime toward the ambient value as entrainment increases the mass flux.