Dynamics of resonantly interacting equatorial waves

Abstract

In this paper we explore some dynamical features on the non-linear interactions among equatorial waves. The shallowwater equation model with the equatorial β-plane approximation is used for this purpose. The Galerkin method is applied to the governing equations with the basis functions given by the eigensolutions of the linear problem. From the phase space expansion of two particular integrals of motion of the system, quadratic to lowest order, some constraints are obtained which the coupling coefficients must satisfy in order to ensure the invariance of such integrals. From the numerical evaluation of the coupling coefficients, these constraints are used to determine the possible resonant triads among equatorial waves. Numerical integrations of the resonant three-wave problem show that the energy of the waves in a resonant triad evolves periodically in time, with the period and amplitude of the energy oscillations dependent on the magnitude of the initial amplitudes of the waves and the way in which the initial energy is distributed among the triad components. The high-frequency modes are found to be energetically more active than the low-frequency modes. The latter tend to act as ‘catalytic’ components in a resonant triad. Integrations of the problem of two resonant triads coupled by a single mode point out the importance of gravity waves in the intertriad energy exchanges, suggesting the significance of these modes in the redistribution of energy throughout the atmospheric motion spectrum. The results also show that the intertriad energy exchanges provided by the highest frequency mode of two triads occur in a longer time-scale than the intratriad interactions. Therefore, these results also suggest the importance of the high-frequency modes in the generation of the low-frequency variability (intraseasonal and even longer term) of the atmospheric flow.