Stability of columnar convection in a porous medium

Convection in a porous medium at high Rayleigh number Ra exhibits a striking quasisteady columnar structure with a well-defined and Ra-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber k and amplitude Aˆ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter A=AˆRa/k. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of A. The growth rate of the most unstable mode scales with A4/9 for A≫1, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high Ra, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers k greater than k∼Ra5/14 as Ra→∞. This stability criterion is consistent with hitherto unexplained numerical measurements of k in a Rayleigh–Darcy cell