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Turbulent mixing exerts a significant influence on many physical processes in the ocean. In a stably stratified Boussinesq fluid, this irreversible mixing describes the conversion of available potential energy (APE) to background potential energy (BPE). In some settings the APE framework is difficult to apply and approximate measures are used to estimate irreversible mixing. For example, numerical simulations of stratified turbulence often use triply periodic domains to increase computational efficiency. In this set-up, however, BPE is not uniquely defined and the method of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128) cannot be directly applied to calculate the APE. We propose a new technique to calculate APE in periodic domains with a mean stratification. By defining a control volume bounded by surfaces of constant buoyancy, we can construct an appropriate background buoyancy profile $b_\ast (z,t)$ and accurately quantify diapycnal mixing in such systems. This technique also permits the accurate calculation of a finite-amplitude local APE density in periodic domains. The evolution of APE is analysed in various turbulent stratified flow simulations. We show that the mean dissipation rate of buoyancy variance $\chi$ provides a good approximation to the mean diapycnal mixing rate, even in flows with significant variations in local stratification. When quantifying measures of mixing efficiency in transient flows, we find significant variation depending on whether laminar diffusion of a mean flow is included in the kinetic energy dissipation rate. We discuss how best to interpret these results in the context of quantifying diapycnal diffusivity in real oceanographic flows.
A conceptual framework for analysing the energetics of density-stratified Boussinesq fluid flows is discussed. The concept of gravitational available potential energy is used to formulate an energy budget in which the evolution of the background potential energy, i.e. the minimum potential energy attainable through adiabatic motions, can be explicitly examined. For closed systems, the background potential energy can change only due to diabatic processes. The rate of change of background potential energy is proportional to the molecular diffusivity. Changes in the background potential energy provide a direct measure of the potential energy changes due to irreversible diapycnal mixing. For open systems, background potential energy can also change due to boundary fluxes, which can be explicitly measured. The analysis is particularly appropriate for evaluation of diabatic mixing rates in numerical simulations of turbulent flows. The energetics of a shear driven mixing layer is used to illustrate the analysis.