The initial transient period of gravitationally unstable diffusive boundary layers developing in porous media

Abstract

Gravitationally unstable, transient, diffusive boundary layers play an important role in carbon dioxide sequestration. Though the linear stability of these layers has been studied extensively, there is wide disagreement in the results, and it is not clear which methodology best reflects the physics of the instability. We demonstrate that this disagreement stems from an inherent sensitivity of the problem to how perturbation growth is measured. During an initial transient period, the concentration and velocity fields exhibit different growth rates and these rates depend on the norm used to measure perturbation amplitude. This sensitivity decreases at late times as perturbations converge to dominant quasi-steady eigenmodes. Therefore, we characterize the linear regime by measuring the duration of the initial transient period, and we interpret the convergence process by examining the growth rates and non-orthogonality of the quasi-steady eigenmodes. To judge the relevance of various methodologies and perturbation structures to physical systems, we demonstrate that every perturbation has a maximum allowable initial amplitude above which the sum of the base-state and perturbation produces unphysical negative concentrations. We then perform direct numerical simulations to demonstrate that optimal perturbations considered in previous studies cannot support finite initial amplitudes. Consequently, convection in physical systems is more likely triggered by "sub-optimal" perturbations that support finite initial amplitudes. © 2013 AIP Publishing LLC.