Negative APE dissipation as the fundamental criterion for double diffusive instabilities

Understanding the energetics and mechanisms of double diffusive instabilities in stratified fluids remains challenging. These instabilities can arise from a mechanically unforced stably stratified state with no available potential energy (APE), thus requiring the background potential energy (BPE) to be dynamically active rather than passive. Middleton and Taylor proposed a criterion linking BPE extraction to APE with the diapycnal component of the diffusive buoyancy flux, but it only predicts diffusive convection instability, not salt finger instability. We argue that the extraction of BPE into APE is better determined by the sign of the net APE dissipation rate. Our theory predicts that in doubly-diffusive fluids, the APE dissipation rate can be enhanced, suppressed, or even have the opposite sign compared to a simple fluid, depending on the diffusivity ratio, density ratio, and a spiciness parameter measuring density-compensated thermohaline variations. In the laminar or weakly-turbulent regime, negative APE dissipation predicts the occurrence of both salt finger and diffusive convection instability. However, only diffusive convection instability persists in the turbulent regime. To sustain salt finger instability in turbulent conditions, density-compensated thermohaline (spiciness) variations are essential. We derive new criteria for the range of density ratios where double diffusive instabilities are active, resembling those in laminar regimes where the Prandtl number is replaced by the inverse of the mixing efficiency. Interestingly, the APE dissipation rate can be made to appear as one of the terms contributing to the entropy budget, which establishes that it should be regarded as irreversible only when positive