Equatorial inertial-parametric instability of

This study revisits the problem of the zonally symmetric instability on the equatorial β-plane. Rather than treating the classical problem of a steady basic flow, it treats a sequence of problems of increasing complexity in which the basic flow is oscillatory in time with a frequency ω0. First, for the case of a homogeneous fluid, a time-oscillating barotropic shear forcing may excite a subharmonic parametric resonance of inertial oscillations. Because of the continuous distribution of inertial oscillation frequencies, this resonance occurs at critical inertial latitudes yc such that βyc =±ω0/2. Next the effects of stratification, characterized by Brunt–Väisäla ̈ frequency N, are taken into account. It is shown analytically (in the asymptotic limit of a weak shear) that the forced temporal oscil-lation leads to an inertial-parametric instability, when a resonance condition between the basic flow frequency and the sum of two inertio-gravity free-mode frequencies is met. This inertial-parametric instability has a well-defined inviscid vertical scale selection favouring the high-vertical mode mc ∼ 7.45m0, where m0 =βN/ω20 is the equatorial vertical mode characteristic of frequency ω0. The viscous critical shear o