In this study, we investigate the stability of time-dependent zonal flows to two-dimensional (zonally symmetric) disturbances. While steady currents can only experience inertial instability (II) in this setting, unsteady ones may be destabilized in other ways. For example, time-periodic flows can be subject to parametric subharmonic instability (PSI). Motivated by observations of salinity interleaving patterns in the upper equatorial Pacific Ocean, our objective is to determine the basic properties of dominant instabilities (their generation mechanism, spatial and temporal characteristics, and finite-amplitude development) for background flows that are representative of those in the upper-equatorial ocean, yet still amenable to a computational sweep of parameter space. Our approach is to explore the stability of solutions to linear and nonlinear versions of a two-dimensional model for an idealized background flow with oscillating linear shear. To illustrate basic properties of the instabilities, the f-plane and equatorial β-plane scenarios are studied using a linear model. Stability regime diagrams show that on the f-plane there is a clear separation in dominant vertical scales between PSI- and II-dominated regimes, whereas on the equatorial β-plane the parameter space contains a region where dominant instability is a mixture of the two types. In general, PSI favours lower vertical modes than II. The finite-amplitude development of instabilities on the equatorial β-plane is explored using a nonlinear model, including cases illustrating the equilibration of pure II and the development of pure PSI and mixed instabilities. We find that unless the instabilities are weak enough to be equilibrated by viscosity at low amplitude, disturbances continue to grow until the vertical shear of their meridional velocity field becomes large enough to allow for Richardson numbers less than 1/4; as a consequence, PSI-favoured vertical modes are able to reach higher amplitudes than II-favoured modes before becoming susceptible to Kelvin–Helmholtz instability, and induce tracer intrusions of a considerably larger meridional extent.
INVESTIGATION OF THE NORMAL VIBRATION MODES STABILITY IN SOME ESSENTIALLY NONLINEAR SYSTEMS
