Kinematic plume fluxes

For an axisymmetric top-hat plume model of radius $b$, uniform upward velocity $w$ and reduced gravity $g^{\prime }=g({\rho }_0-\rho )/{\rho }_0$, the full-cross-section kinematic fluxes areThey are respectively volume flux, kinematic momentum flux and kinematic buoyancy flux, with units ${\mathrm{m}}^3{\mathrm{s}}^{-1}$, ${\mathrm{m}}^4{\mathrm{s}}^{-2}$ and ${\mathrm{m}}^4{\mathrm{s}}^{-3}$. Some plume equations suppress the common factor $\pi$; their flux variables are then these quantities divided by $\pi$.

In the Boussinesq approximation, using constant inertial mass density ${\rho }_0$ makes the corresponding density-weighted fluxes ${\rho }_{0}q$, ${\rho }_{0}m$ and ${\rho }_{0}f$. Outside that approximation, the actual mass flux is $\pi b^2\rho w$. Dividing it by ambient mass density ${\rho }_0$ gives $(\rho /{\rho }_0)q$, which is generally different from the physical volume flux. Both scaled-mass and volume conventions are useful, but their definitions must accompany a non-Boussinesq calculation.