 Arrested cubic buoyancy profile

An upward volume flux $Q$ and a downward turbulent buoyancy transport $d^4(G_z{)}^{3/2}$ balance in a steady reactor. Matching $G=G_z=0$ below $-H$ gives $QG=d^4(G_z{)}^{3/2}$. Integrating yields the cubic profile above the front. If downward source buoyancy flux is $B_s$, then $H=3d^{8/3}{B}_s^{1/3}/Q$. The corresponding eddy diffusivity is $K=Q(z+H)/(3d^2)$. All these prefactors use the same area and mixing-length normalization; an actual cross-sectional area $S$ replaces the denominator in $K$ by $3S$.