The linear theory of Gill for the steady response due to heating of the tropical atmosphere provides an attractive simplified model for the tropical atmosphere including Hadley cells, the Walker circulation, etc. This steady state model does not capture the speed and structure of important tropical waves such as the MJO or other faster moving waves. Here we consider the weakly nonlinear transient response to Gill heating as well as the effects of topography. A new theoretical mechanism developed by the authors for nonlinear interaction of equatorial waves emerges from this analysis. Through this mechanism, large-scale equatorial Kelvin waves can modify their speed through dispersion and interaction with other large-scale equatorial waves such as Yanai or Rossby modes through heat sources or topographic resonance. These are novel equations for applied mathematics and involve the Burgers equation coupled through large scale dispersion to a family of O.D.E.'s. This resonance mechanism can prevent the breaking of a propagating nonlinear Kelvin wave, slow down its speed, and concentrate most of its energy in large-scale zonal wave numbers while simultaneously generating large-scale Yanai or Rossby modes with specific zonal wavelengths. Simplified reduced dynamic equations for this resonant interaction are developed via novel asymptotic expansions of the equatorial shallow water equations with diabatic heating and topography. Explicit exact solutions and elementary numerical experiments are utilized to display the new mechanisms of large-scale resonance for equatorial waves. For a vertical structure corresponding to the first baroclinic mode of the equatorial troposphere, these theories predict the primary resonant interaction involves eastward propagating nonlinear Kelvin waves with westward propagating nonlinear equatorial Rossby waves with zonal wave-number four.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics