Emergence and coalescence of zonal jets: A quasilinear Rossby wave-mean flow interaction model

Zonal jets are a fascinating natural example of how a rapidly rotating turbulent flow self-organizes at large scale in the presence of a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>β</a:mi> </a:math> effect. Understanding the long-term, nonlinear equilibration of zonal jets and the feedback with the underlying turbulence and waves is still a challenge. Following a similar approach as in the Holton-Lindzen-Plumb model for mean flow reversals in stratified fluids, this study describes a novel, quasilinear semianalytical model to discuss the emergence and coalescence of zonal winds from the radiation of Rossby waves. This model emphasizes the role of Rossby waves in exchanging momentum with the zonal flow and the feedback of the zonal flow on the waves. It employs a Wentzel-Kramers-Brillouin expansion of the wave field to obtain an explicit expression for the Reynolds stress, leading to a closed mean flow equation. Two key feedback effects control the properties of the wave-driven zonal flow: the Doppler shift, leading to the emergence of critical latitudes, and the modification of the background <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>β</b:mi> </b:math> effect by the zonal flow curvature. Motivated by previous experimental observations, we integrate this quasilinear model in time with an increasing number of latitudes of wave radiation. We observe a transition between locally driven jets that remain individual and globally driven jets that coalesce and equilibrate at a new scale. In the weak wave damping limit, coalescence occurs when critical latitudes of neighboring jets overlap. This is the first purely zonal closure which self-consistently leads to an equilibrium flow with zonal jets separated by a Rhines scale. These results are further supported by quantitative comparison with experiments and nonlinear direct numerical simulations.