The Lagrangian energetics of stably stratified turbulence in the Boussinesq approximation with non-linear equation of state

There has been a recent resurgence of interest in determining the consistency of the Boussinesq approximation to describe the coupling of the dynamics and thermodynamics of turbulent stratified flows. In particular, there is some debate over how energy is converted from internal to mechanical energy in this approximation. Moreover, the effect of the non-linear equation of state and the strength of stratification on the internal energy is still unclear. To gain some insight into these issues, we derive the evolution equations of the different forms of energy for Boussinesq stratified flows with variable volumetric expansion coefficient in the Lagrangian frame. This analysis allows us better physical insight into these issues and allows us to show explicitly how energy is converted between internal and mechanical energy and how significant the internal energy is under these conditions. The physical significance of these results will be discussed.