Interaction between eddies and mean flows has been one of the central themes of fluid dynamics. Theories date back to Osborne Reynolds and G.I. Taylor, but a major breakthrough occurred from the 1960s to 80s, in which a connection between the group propagation of waves and their radiation stress on the mean flow was firmly established. In the context of dynamic meteorology (my field), the counter-intuitive, anti-frictional nature of Rossby wave radiation stress came to light, and concepts like the generalized Eliassen-Palm flux and the Transformed Eulerian Mean have since become mainstay diagnostics of the general circulation.
Despite these previous efforts, there remain challenges in both (1) a gap between the theory and data, due largely to the small-amplitude assumption made for the waves in the former and (2) lack of theory for Reynolds stress. Although neither of these challenges will likely go away anytime soon, I will describe some progresses we made recently in the theory of eddy-mean flow interaction in geostrophic turbulence, where both finite-amplitude Rossby waves and turbulent eddies interact with the zonal-mean flow.