APE dissipation is a form of Joule heating. It is irreversible, not reversible

Abstract: Available Potential Energy (APE) dissipation plays a central role in the description of mixing in turbulent stratified fluids. The dominant paradigm is that it converts APE into background gravitational potential energy ${\rm GPE}_r$, and that the APE thus converted can be infinitely recycled back into APE by external buoyancy fluxes such as high-latitude cooling in the oceans. In this paper, we argue that such a paradigm is unphysical, because its corollary is that APE dissipation is neither truly dissipative nor irreversible, while also violating energy conservation in more subtle ways. In this paper, we prove from first principles that in reality, APE dissipation is a form of Joule heating, which --- like viscous dissipation --- can only increase ${\rm GPE}_r$ via locally expanding the fluid parcels, a tiny effect. ${\rm GPE}_r$ thus primarily increases at the expense of the exergy of the stratification, a subcomponent of the background internal energy, regardless of whether the flow is laminar or turbulent. The results greatly clarify the energetics of mechanically- and buoyancy-driven circulations. As a side benefit, our results yield a new physical principle justifying why turbulent mixing tends to homogenise the fluid's materially conserved variables rather than relax the fluid towards thermodynamic equilibrium.