 Diffusion-controlled solute gravity current

When vertical diffusion equation controls depth, $H\sim \sqrt{Dt}$, while conserved excess solute mass gives $\Delta \rho R^{2}H\sim \Delta {\rho }_0{V}_0$. Ambient viscous resistance then leads to a $t^{1/2}$ spreading law. The enriched-fluid volume grows, so the fixed-volume $t^{1/5}$ law does not apply.