Passive scalar dispersion along porous stratum with natural convection

We investigate horizontal dispersion of a passive scalar in a porous stratum with Rayleigh-Darcy convection initiated by a geothermal gradient. While increasing Rayleigh number ($Ra$) keeps enhancing convection, the horizontal dispersion coefficient ($\widehat{D}$) only scales with Ra in a narrow range, and saturates at $Ra\gtrsim 2500$. We rationalize this two-stage behavior: at low Ra, passive scalar migrates through bulk circumflex; at high Ra, boundary micro-plume layer becomes the dominant scalar pathway. Theory gives saturate $\widehat{D}$ value proportional to the Lewis number $Le$ and molecular diffusivity $D_0$.