Energy pathways and mixing transitions in stratified shear instabilities

Stratified shear instabilities and associated turbulence can undergo an evolution cycle where the flow initially evolves 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-related 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- and three-dimensional flows. This is done by analyzing energy budgets based on direct numerical simulations. We identify key physical pathways through which energy is transferred from the mean flow into two-dimensional motions and ultimately into turbulent mixing. These pathways include both shear production across different flow dimensions and pressure-induced redistribution within the same dimension. Our analysis reveals new insights into the coupling between flow dimensionality, pressure dynamics, and mixing efficiency in stratified environments. This refined understanding helps improve predictions of turbulent mixing in geophysical and environmental flows.