Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous media

The linear stability of transient diffusive boundary layers in porous media has been studied extensively for its applications to carbon dioxide sequestration. The onset of nonlinear convection, however, remains understudied because the transient base state invalidates the traditional stability methods that are used for autonomous systems. We demonstrate that the onset time of nonlinear convection, t = t(on), can be determined from an expansion that is two orders of magnitude faster than a direct numerical simulation. Using the expansion, we explore the sensitivity of t(on) to the initial perturbation magnitude and wavelength, as well as the initial time at which a perturbation is initiated. We find that there is an optimal initial time and wavelength that minimize t(on), and we obtain analytical relationships for these parameters in terms of aquifer properties and initial perturbation magnitude. This importance of the initial perturbation time and magnitude is often overlooked in previous studies. To investigate perturbation evolution at late-times, t > t(on), we perform direct numerical simulations that reveal two unique features of transient diffusive boundary layers. First, when a boundary layer is perturbed with a single horizontal Fourier mode, nonlinear mechanisms generate a zero-wavenumber response whose magnitude eventually surpasses that of the fundamental mode. Second, when a boundary layer is simultaneously perturbed with many Fourier modes, the late-time perturbation magnitude is concentrated in the zero-wavenumber mode, and there is no clearly dominant, non-zero, wavenumber. These unique results are further interpreted by comparison with direct numerical simulations of Rayleigh-Benard convection.