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

Parametrization of irreversible diapycnal diffusivity in salt-fingering turbulence using DNS

“…The third term in (2.16) is negative in a salt-fingering-favourable environment, since thermal and saline density components compensate each other to make . The magnitude of this negative term can be larger than the first two positive terms in double-diffusive fields (as discussed in Middleton & Taylor (2020)), allowing to be released even in a closed system. This property contradicts the original physical meaning of the background potential energy that was defined in a single-component system, which can only increase in a closed system due to irreversible mixing processes.…”

“…As we have discussed above, the background potential energy in a single-component system is associated with the stable rearrangement of the fluid parcels based on their densities. Middleton & Taylor (2020) directly extends this definition of background potential energy to the double-diffusive system, as where represents the vertical position in the sorted profile of the fluid parcel at and is determined by the temperature and salinity through the linear equation of state (2.1 g ). In a closed system the evolution of can be derived following the same strategy as in Winters et al.…”

“…(1995) to be where and are the components of density that are determined by the temperature field and salinity field separately. Equation (2.16) is essentially equivalent to equation (2.19) of Middleton & Taylor (2020) in a closed system. In order to make a clearer comparison, we rewrite the evolution equation for the background potential energies (2.12 c ) using and as …”

“…systems In this subsection we will discuss an alternative possibility for extending the idea of background potential energy from a single-component system to a doubly diffusive system. This approach has been discussed in detail in the recent work of Middleton & Taylor (2020) and we will present and discuss their main results using our notation in this subsection. These results will then be compared with our results presented in the previous subsection in (2.12)-(2.14), on the basis of which we will argue that the definitions of background potential energies proposed in the current paper are distinctly preferable for application to the study of irreversible mixing in doubly diffusive systems.…”

“…These distinct background potential energies will be shown to be related to irreversible heat and salt fluxes, respectively. It is worth noting that the background potential energies defined in the current work differ from the definitions employed in the recent work of Middleton & Taylor (2020). In this work the original formulae for background potential energy in Winters et al (1995) are retained in the double-diffusion case, which therefore continues to rely on sorting the density field itself.…”