Layering in double-diffusive convection

The formation of layered profiles (or 'staircases') in temperature, salinity and density, resulting from double-diffusive convection (DDC), is a widely observed oceanographic phenomenon. In the past two decades, new developments, both in numerical simulations and theory, have significantly advanced our under-standing of this fascinating process. Nonetheless, obtaining a complete description of layering-precisely when and why it occurs-still presents a sizeable theoretical challenge to fluid dynamicists. Here, we review the current state of the field, summarizing briefly the observations and simulations, before concentrating on simplified theoretical modelling approaches. Layering is described as an essentially one-dimensional 'anti-diffusive' process acting on a quasi-homogeneous background state (which may result from a classical double-diffusive instability). A key ingredient of the models discussed is the dependence of the mixing length on an emergent, or dynamical, scale. An instability of the basic turbulent state leads initially to regular sinusoidal perturbations of the buoyancy gradient, which, over time, can sharpen, leading to the formation of a staircase comprised of well-mixed layers separated by sharp interfaces. The long-time evolution is marked by a gradual coarsening of the staircase through layer mergers. The reduced models perform well in replicating the qualitative behaviour exhibited by direct numerical simulations. This article is part of the theme issue 'Anti-diffusive transport and layering: from sub-cellular to astrophysical scales'.