Nonlinear coupling of three zonal flow modes in the Charney–Hasegawa–Mima system

We derive a set of equations describing nonlinear coupling of three zonal flows (ZFs) from the Charney–Hasegawa–Mima equation. While ZFs cannot directly interact with one another, we show that their mutual interaction is possible being mediated via the nonlinear couplings involving two drift waves and one ZF. Its basic properties are studied and contrasted with those of the well-known three drift wave system. We show that the shortest wavelength ZF can decay into two ZFs with longer wavelengths satisfying the wavenumber matching condition, resulting in a nonlinear saturation. In addition, we propose a simple model based on the successive decay instabilities of ZFs resulting in the spectral transfer to longer wavelength region. In the long-term asymptotic state, most fluctuation energy is predicted to condensate at the three longest wavelength ZF modes allowed by the system.