A general criterion for the release of background potential energy through double diffusion

Double diffusion occurs when the fluid density depends on two components that diffuse at different rates (e.g. heat and salt in the ocean). Double-diffusive fluids display forms of convection not present in single-component fluids as well as modifying the effects of canonical environmental flows (gravity currents, jets etc.). Energetically, double diffusion can lead to an up-gradient buoyancy flux which may drive motion at the expense of potential energy. Here, we follow the work of Lorenz 1955 and Winters et al. 1995, for a single-component fluid and define the background potential energy (BPE) as the energy associated with an adiabatically sorted density field and derive its budget for a double-diffusive fluid. We find that double diffusion can convert BPE into available potential energy (APE), unlike in a single-component fluid, where the transfer of APE to BPE is irreversible. We also derive an evolution equation for the sorted buoyancy in a double-diffusive fluid, extending the work of Winters & D’Asaro 1996, and Nakamura 1996. The criterion we develop for a release of BPE can be used to analyse the energetics of mixing and double diffusion in the ocean and other multiple-component fluids. We illustrate its application using two-dimensional simulations of salt fingering.