Energy pathways and mixing transitions in stratified shear instabilities

Stratified shear instabilities and associated turbulence can undergo an evolution cycle where the flow initially develops in two dimensions and then transitions to three dimensions before decaying and restratifying. While past studies have explored the overall energy evolution through this process, the role of pressure-mediated mechanisms, such as the redistribution of energy among different flow components, remain less understood. In this study, we expand the classical energy framework by separately tracking the energy in each directional component of the two-dimensional (2-D) and three-dimensional (3-D) flows. This is done by analysing energy budgets based on direct numerical simulations of Kelvin–Helmholtz and Holmboe instabilities. We identify four key physical pathways through which energy can be transferred from the mean flow into 2-D motions and ultimately into 3-D turbulent mixing. These pathways involve both the well-studied shear production across different flow dimensions and the less-studied pressure–strain redistribution within a given dimension. Specifically, this pressure redistribution is instrumental in generating vertical kinetic energy for both types of instabilities, a prerequisite for the subsequent generation of available potential energy (APE) and irreversible mixing of APE into background potential energy (BPE). Although more energy is transferred from the background flow into 3-D motions, we find that the 2-D APE generation pathway contributes most significantly to the irreversible mixing of APE into BPE. Our analysis reveals new insights into the coupling between flow dimensionality, pressure dynamics and energy exchange in stratified environments.