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Write $G=g^{\prime }$ for the positive reduced gravity of dense fluid. With $z$ upwards, $G_z>0$ places denser fluid above lighter fluid and permits convective turbulence. A displacement over the radius scale $d$ samples a buoyancy difference of order $d{G}_z$; balancing kinetic energy per mass with buoyant work over $d$ givesThe buoyancy-gradient mixing-length closure then gives an eddy diffusivity $K\sim ud=d^2\sqrt{G_z}$. We use the paper's unit-prefactor convention for this dimensional closure and its effective cross-sectional area $d^2$; order-one mixing constants and the cylinder's geometric factor $\pi$ are suppressed in its displayed evolution equation.

The downward reduced-gravity transport magnitude and its convergence are different quantities:The printed expression labelled a flux is actually the second quantity, a flux convergence per unit height. The dimensions distinguish them: $\mathcal{F}$ has units $L^4{T}^{-3}$, whereas $F_{\mathrm{conv}}$ has units $L^3{T}^{-3}$. A constant positive gradient has nonzero transport but zero convergence, providing a direct counterexample to identifying the derivative with the transport itself.

The upward total buoyancy flux is $QG-\mathcal{F}$. Conservation therefore yieldsThis flux convergence in turbulent buoyancy mixing obtains the displayed evolution equation with the correct transport interpretation.

For the arrested cubic buoyancy profile, an undisturbed region below $-H$ has $G=0$ and no turbulent buoyancy flux. The joining condition is $G(-H)=0$ andIn a steady state, integrating the conservation equation from that front gives $QG=d^4(G_z{)}^{3/2}$. For a nonzero mixed region,and henceThe associated gradient is $Q^2(z+H{)}^2/(9d^8)$, so it matches the required zero-gradient state continuously. We take the mixed interval to start immediately above $-H$; adding a further zero interval would merely relocate the arrest front.

The source buoyancy flux is $B_s=Q_s{g}_s^{\prime }$, with $g_s^{\prime }=g\Delta \rho /\rho$ in the Boussinesq approximation. The stated top-source convention means that this entire flux enters the downward turbulent transport there: $\mathcal{F}(0)=B_s$. Since steady transport also satisfies $\mathcal{F}(0)=QG(0)$,This is the extent in the paper's dimensional-closure normalization. It presupposes positive $Q$; with no upflow there is no finite steady arrest depth. For a mixture of the two original fluids, $G(0)\le g_s^{\prime }$ is necessary, so an extrapolation to $Q_s/Q>1$ cannot represent a physical concentration profile under this top-flux boundary model. A dilute-source regime avoids appreciable source-volume corrections.

Assuming the same scalar-mixing coefficient for a passive tracer, its eddy diffusivity in the mixed region isIt vanishes at the arrest front and increases linearly towards the source. A different turbulent scalar diffusivity ratio would multiply this result by its corresponding constant.

The geometric normalization can be restored without changing the reasoning. With actual area $S=\pi d^2$ and $K=C_m{d}^2\sqrt{G_z}$, put $R=S{C}_m{d}^2$. Then $G=Q^2(z+H{)}^3/(27R^2)$, $H=3R^{2/3}{B}_s^{1/3}/Q$, and $K=Q(z+H)/(3S)$. These formulas show which numerical prefactors are convention-dependent.