This study reviews the system governed by the thickness-weighted average (TWA) equation of motion, considering energetics. It is known that the TWA equation of motion based on the primitive equation describes the fluid motion with the residual mean velocity defined as the TWA velocity and is written in the same form as the nondissipative primitive equation, except that the eddy momentum fluxes (the interfacial form stress and Reynolds flux associated with eddy motion) are embedded in this equation. Also, incompressibility and density (buoyancy) conservation in the adiabatic condition hold in this system. In this study, considering that the TWA system satisfies a time mean energy conservation of the primitive equation system, we obtain an energy equation showing that the rate of change of eddy energies (the sum of the kinetic and potential energies of the eddies) along pathlines with the residual mean velocity is caused by the work done by the eddy momentum fluxes. This relation is analogous to the relation between internal energy and the dissipation function in a viscous fluid. This study also reconsiders the TWA system in terms of Hamiltonian dynamics. Regarding the eddy energies and the eddy momentum fluxes as analogous to the internal energy and the viscous momentum fluxes, respectively, the methodology of the variational principle for a viscous fluid can be applied to the TWA system. The Lagrangian density in this system is defined as the mean kinetic energy minus the mean potential energy and the eddy energies. Minimizing this Lagrangian density integrated over space and time under the constraints of the incompressibility equation, the buoyancy equation, and the equation of the eddy energies yields the TWA equation of motion. If we neglect the eddy energies in the Lagrangian density and the constraint of the equation of the eddy energies, the resulting equation in the variational calculus is merely the nondissipative primitive equation. This suggests that considering these is essential for describing the motion in the TWA system. Moreover, we inferred from the equation of the eddy energy that the TWA equation of motion can be expressed in a different form in which the isotropic component of the eddy momentum fluxes is included as a part of the pressure. Applying this modified equation to the issue of downstream decaying mechanism of the western boundary current extension jets, it can be interpreted that the deceleration of the jet is caused by the pressure induced by the eddies.
Aoki, Kunihiro. 2014. "A constraint on the thickness-weighted average equation of motion deduced from energetics." Journal of Marine Research 72, (5). https://elischolar.library.yale.edu/journal_of_marine_research/397