= Kinematic plume fluxes
For an axisymmetric <top-hat plume model> of radius $b$, uniform upward <velocity> $w$ and <reduced gravity> $g'=g(\rho_0-\rho)/\rho_0$, the full-cross-section kinematic fluxes are
$$
q=\pi b^2w,\qquad m=\pi b^2w^2,\qquad f=\pi b^2wg'.
$$
They are respectively <volume flux>, kinematic <momentum flux> and kinematic <buoyancy flux>, with units $\mathrm{m^3\,s^{-1}}$, $\mathrm{m^4\,s^{-2}}$ and $\mathrm{m^4\,s^{-3}}$. 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 $\rho_0q$, $\rho_0m$ and $\rho_0f$. 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.
Back to article page
