Stability of two-layer miscible convection

The dynamics at the edge of fusion confinement devices involves the motion of plasma across two regions—the core and the scrape-off layer—that may have very different properties. Motivated by this problem, and exploiting the analogy between the equations governing plasma interchange dynamics and those of classical Rayleigh-Bénard convection, we consider the linear stability of two-dimensional, two-layer miscible convection. We focus specifically on the influence of three particular parameters: the ratio of the viscosities in the two layers, the ratio of the thermal diffusivities, and the ratio of the depths of the two layers. The key result is that, depending on the parameters of the problem, the most unstable mode can take one of three quite distinct forms: whole layer solutions, in which the eigenfunctions of the stream function and temperature extend over both layers of fluid; localized solutions, with the velocity cells or the temperature perturbation (or both) confined to just one of the layers; and segregated solutions, in which the fluid motion and temperature perturbation are confined to different fluid layers.