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 energy (IE) to mechanical energy (ME) in this approximation. To gain some insight
into these issues, we derive the evolution equations of the different forms of energy for
Boussinesq stratified flows from the Lagrangian point of view. 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. Moreover, this new analysis can be applied to many other fields such as ocean
energetics and dispersion model in atmosphere.
In this study, the vertical dispersion of fluid particles in stably stratified homogeneous turbulence with mean shear is investigated. For the clarification of the energetics, Lagrangian analysis is used since it can show the specific energy conversion among fluid particles whereas Eulerian analysis represents only the net energy transfer between internal and potential energy because it integrates whole through the volume This clarifies how the potential energy is related to internal
energy by using Lagrangian analysis. Moreover, this provides a more clear and consistent interpretation of the behavior
of the mean square vertical displacement, $\sigma_{zp}^2(t)$, which corresponds to the total potential energy (TPE) of marked fluid particles. Our analysis considers TPE in terms of the available potential energy (APE), associated with the nonequilibrium displacement, and the equilibrium potential energy (EPE), associated with the change in particle equilibrium height. The corresponding evolution equations describe the key sequence of processes. As fluid particles move away from their (original) equilibrium height, TPE increases as vertical kinetic energy is converted (reversibly) to APE. Subsequently, as fluid particles change their density through molecular diffusion, APE is dissipated (irreversibly) to IE and at the same time, IE is converted into EPE where the energy accumulates (in a Lagrangian sense). Small-scale mixing thus acts to preserve displacements and
reduce the reconversion of potential energy to kinetic energy but is not, by itself, the predominant transport mechanism.
At long time, the EPE will dominate the TPE; $\sigma_{zp}^2(t)$ is then a measure of the total APE dissipated by the flow.
The significance of this with respect to the total energy dissipated is given by the cumulative mixing efficiency, $\Omega_c$, which depends on the strength of stratification. In the case of decaying turbulence, $\sigma_{zp}^2(t)$ evolves to a constant value, proportional to $\Omega_c$. In the case of stationary turbulence, the (constant) rate of growth of $\sigma_{zp}^2(t)$ is proportional to $\Omega_c$. The analysis is demonstrated using direct numerical simulations of homogeneous shear flows with decaying, stationary, and growing turbulence. Results for the latter case show $\sigma_{zp}^2(t)$ to continually increase, and to have a reduced dependence on and reduced values of $\Omega_c$ as the strength of stratification decreases. In general, simulation results are in agreement with the analysis and confirm that, in homogeneous stratified flows with mean shear, an effective time scale for vertical dispersion at long time is that of the turbulence decay time.