  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 setup however,
BPE is not uniquely defined and the method of Winters et al. (1995, J. Fluid
Mech., 289) 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.
