The time-mean flow of eddying wind-driven gyres is used as a tutorial example of the relationship between Eulerian and Lagrangian mean flow. The Eulerian mean in the far field of the baroclinically unstable jet is shown to be well represented as the rectified flow of Rossby basin modes. The Stokes drift of particles released in the wave field all but cancel out the Eulerian mean, resulting in vanishingly small Lagrangian mean flow. By reformulating the interaction between eddies and mean flow in terms of, so-called, residual mean velocities and residual eddy fluxes, it becomes clear that only the component of the eddy potential vorticity flux that crosses mean potential vorticity gradients, must be parameterized.
Marshall, John. 2011. "A pedagogical example of Eulerian and Lagrangian flows induced by Rossby wave rectification in an ocean basin." Journal of Marine Research 69, (4). https://elischolar.library.yale.edu/journal_of_marine_research/321