In the context of geologic sequestration of carbon dioxide in saline aquifers, much interest has been focused on the process of density-driven convection resulting from dissolution of CO₂in brine in the underlying medium. Recent investigations have studied the time and length scales characteristic of the onset of convection based on the framework of linear stability theory. It is well known that the non-autonomous nature of the resulting matrix does not allow a normal mode analysis and previous researchers have either used a quasi-static approximation or solved the initial-value problem with arbitrary initial conditions. In this manuscript, we describe and use the recently developed non-modal stability theory to compute maximum amplifications possible, optimized over all possible initial perturbations. Non-modal stability theory also provides us with the structure of the most-amplified (or optimal) perturbations. We also present the details of three-dimensional spectral calculations of the governing equations. The results of the amplifications predicted by non-modal theory compare well to those obtained from the spectral calculations.