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Long-Range Dynamics of a Shallow Water Triad: Renormalization, Modulation, and Cyclogenesis

Abstract Long-time-range atmospheric variability has many sources, and nonlinear mode interactions could play an important role. This paper sets out to explore such internal mechanisms of the long-range variability for the rotating shallow water (RSW) system, at its lowest spectral resolution (triad), but for long (decadal) timescales, about 105 Coriolis units. A small Rossby–Froude number e sets multiple timescales for the system, ranging from the fast gravity–oscillation scale t to the slow eddy-turnover scale τ = et and yet slower (climatological) scales. A two-step averaging/renormalization procedure is developed to study underlying mechanisms of long-range variability. It explains the basic phases (relaxation, intensification) and the role of wave–vortex interactions, and reveals a novel mechanism of long-range variability, due to modulation of vortex motions [potential vorticity (PV) modes] by gravity waves. The long cycles [on timescales O(104)] are characterized by relative value of the zonal PV c...