Quantifying mixing and available potential energy in vertically periodic simulations of stratified flows

TL;DR

This paper introduces a new method to accurately calculate available potential energy in periodic stratified flow simulations, enabling better quantification of mixing processes in oceanographic models.

Contribution

A novel technique for computing available potential energy in periodic domains with mean stratification, applicable to turbulent stratified flow simulations.

Findings

Mean buoyancy dissipation rate approximates diapycnal mixing rate.

Significant variation in mixing efficiency depending on flow conditions.

Method improves quantification of mixing in oceanographic models.

Abstract

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.