Convective Shutdown in a Porous Medium

Convective flow in a porous medium, driven by a buoyancy source along one boundary, is found in many geophysical and industrial processes, and has recently been investigated in the context of CO$_2$ sequestration. If the domain is closed then the convective flux soon starts to decrease due to the slow evolution of the average interior density. We reveal a close link between such a ``one-sided'' shutdown system and the ``two-sided'' statistically steady Rayleigh--B\'enard cell. We present high-resolution numerical simulations of convective shutdown at high Rayleigh number $Ra$ in a two-dimensional porous medium. A simple analytic box model of the shutdown system is constructed, with time-dependent Rayleigh and Nusselt numbers, which is based on measurements of the convective flux from a Rayleigh--B\'enard cell (Hewitt {\it et al.} Phys. Rev. Lett. 2012) and gives excellent quantitative agreement with numerical results. These ideas are generalised to model fluids with a power-law equation of state. The dynamical structure of high-$Ra$ shutdown flow is dominated by vertical columnar flow in the interior, and the evolving horizontal wavenumber $k[Ra(t)]$ of the columns gives extremely good agreement with similar measurements of $k(Ra)$ from the columnar flow in a Rayleigh--B\'enard cell.