The paper presents calculated spectra of El Nino Southern oscillation (ENSO) indices. The ENSO spectra have components with periods that are multiples of the Earth’s free (1.2 years) and forced (18.6 years) nutation periods. Analysis of a 41-year series of exciting functions for the atmospheric angular momentum confirms the existence of such periodicity. Nutation waves responsible for the El Nino phenomena in the ocean, the Southern oscillation in the atmosphere, and the presence of subharmonics of the Chandler period (1.2 years) and superharmonics of the lunar period (18.6 years) in the ENSO spectra are described. A model for the nonlinear nutation of the Earth-ocean-atmosphere system is constructed. In this model, the ENSO, acting at frequencies of combinational resonances, excites the Chandler wobble of the Earth’s poles. At the same time, this wobble interacts with the nutation motions of the atmosphere and World Ocean.