Transient growth in linearly stable gravity-driven flow in porous media

In this paper we study gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at the horizontal surface of a porous medium. Van Duijn et al. [P.A.C. Raats, D. Smiles, and A.W. Warrick (Eds.), Environmental Mechanics - Water, Mass and Energy Transfer in the Biosphere - The Philip Volume, Geophys. Monographs, Vol. 129, American Geophysical Union, 2002, pp. 155-169], derived stability bounds by means of linear stability analysis and an (improved) energy method. These bounds do not coincide, i.e. there exists a subcritical region or stability gap in the system parameter space which is due to the asymmetry of the linear part of the perturbation equations. We show that the linear operator can be symmetrized by means of a similarity transformation. For system parameter values in the stability gap, we show that there exist initial perturbations for which the linearly stable system exhibits transient growth. We show that transient growth is norm dependent by considering weighted norms, which are induced by a one-parameter family of similarity transformations.