= Solution
Use the steady <Boussinesq top-hat plume in a stratified ambient>, with negligible ambient vertical momentum and no externally added heat or particles. The <Batchelor entrainment hypothesis> supplies inward <velocity> $\alpha w$ across circumference $2\pi b$. <Volume conservation> gives $\pi Q_z=2\pi b\alpha w$, and $bw=\sqrt M$, hence
$$
\boxed{Q_z=2\alpha\sqrt M.}
$$
The <momentum flux> increases by the integrated <buoyancy> force. Writing $g'=g\Delta\rho/\rho_o$, the force per reference density and per $\pi$ is $g'b^2$. Since $F=g'Q$ and $b^2=Q^2/M$, this gives
$$
\boxed{M_z=g'b^2=\frac{FQ}{M}.}
$$
For the <buoyancy flux>, entrained material has the local environmental <mass density>. Let $\rho_p$ be the plume's averaged density. Its scalar <mass conservation> is $(\rho_pQ)_z=\rho_a Q_z$. Since $F=(g/\rho_o)(\rho_a-\rho_p)Q$, differentiating cancels both <entrainment> terms and gives
$$
\boxed{F_z=\frac g{\rho_o}\left[(\rho_aQ)_z-(\rho_pQ)_z\right]=Q\frac g{\rho_o}\rho_a'.}
$$
In <stable density stratification>, $\rho_a'<0$ and a rising plume loses <buoyancy flux>. These are kinematic fluxes with the common factor $\pi\rho_o$ removed, so no additional factor of $\pi$ belongs in the three differential equations.
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