A Dissection of Energetics of the Geostrophic Flow: Reconciliation of Rossby Wave Energy Flux and Group Velocity

It is shown in this paper that there is no ambiguity in the final form of the governing equations of a quasi-geostrophic (QG) model after partitioning the total flow into the geostrophic, balanced ageostrophic, and unbalanced ageostrophic components. The uniqueness of the QG model formulation ensures that the energetics of a QG model is the same as that derived from the QG potential vorticity equation. Particularly, the well-known but somewhat mysterious “missing term” in the energetics of Rossby waves, identified in the literature as the difference between the pressure work and the energy flux transported at the group velocity, can be easily recovered. The “missing term” is the pressure work on the convergence of the balanced ageostrophic flow, representing a “hidden” conversion between kinetic and potential energy of the geostrophic flow that excites the unbalanced flow. This energy conversion equals the convergence of a one-directional energy flux that always transports energy westward at the zonal phase speed of Rossby waves. The pressure work on the divergence of the unbalanced flow does the actual conversion between kinetic and potential energy of the geostrophic flow and the pressure work on the unbalanced flow causes energy propagation in other directions. Therefore, it is the pressure work on the unbalanced flow that causes Rossby waves to be dispersive, leading to the downstream development. The sum of the energy transported at the zonal phase speed of Rossby waves and the pressure work on the unbalanced flow exactly equals the energy transported at the group velocity of Rossby waves.