Nonlinear Interaction of Kelvin and the Eastward Branch of Yanai Waves in the Equatorial Waveguide

Here we present another theoretical mechanism through which the Kelvin wave can excite and exchange energy with other equatorial wave types. This mechanism is based on nonlinear resonance between two wave packets: Kelvin and the short wavelength eastward branch of Yanai wave. In fact, according to the linear equatorial wave theory, for higher zonal wavenumbers the eastward branch of Yanai wave becomes nondispersive and its group velocity asymptotically matches the Kelvin wave speed. Thus, the short wavelength eastward propagating Yanai and Kelvin wave packets resonate with each other through the nonlinear advective terms of the equatorial beta-plane shallow-water equations, and we have analyzed here the dynamics of this resonant interaction. We have shown that this resonant nonlinear interaction is described by a system of two coupled hyperbolic PDEs governing the long-time scale evolution of the two wave amplitudes along the characteristics. This system of hyperbolic PDEs has a quadratic conservation law related to the time invariance of the sum of the total energy of the two wave types. The reduced asymptotic equations also allow wave breaking and shock formation that dissipate energy from the smooth part of the solution. Consequently, numerical solutions of the reduced equations obtained through both conservative and dissipative schemes are considered and their results are compared. The numerical experiments with the reduced asymptotic equations have shown that, for periodic initial conditions, a breaking nonlinear Kelvin wave can amplify a pre-existing Yanai wave, even if the pre-existing Yanai mode has infinitesimal amplitude. Furthermore, this amplification is strongly dependent on the relative phase between the two wave packets and is maximized when the two waves are 90-degree out of phase with each other. In addition, in this situation when the Yanai wave amplification is maximum, the strong gradient associated with the breaking Kelvin wave acts to focusing the excited Yanai wave around the shock region. Alternatively, the numerical solutions of the reduced system explored here demonstrate that a single Yanai wave can excite a nonlinear breaking Kelvin wave, which in turn makes the Yanai mode to converge to a step function as time evolves and shock dissipates energy. An important feature regarding the applicability of our theory is that, although it describes Kelvin wave interaction with high zonal wavenumber Yanai modes, near resonant interaction might occur for Kelvin and planetary to intermediate scale Yanai modes. For example, for zonal wavenumber k = 3, the difference between Kelvin and the eastward propagating Yanai wave speeds is less than 10%. Thus, Kelvin and planetary to intermediate scale eastward Yanai modes may exhibit a strong interaction with each other as well. Therefore, we argue here that a large part of energy of a breaking nonlinear Kelvin wave in the atmosphere can leak to the eastward branch of Yanai mode and this process may contribute to explain the strong spectral peak band in intermediate scale eastward propagating Yanai modes found in observational studies.