The Tidal Problem for Simple Circulations

Abstract

Simple wave-circulation systems on the rotating spherical surface are considered in the framework of Laplace's tidal theory. For solid body rotation of the shallow 1-layer system the comparison of tidal theory and β-plane concepts shows that Matsuno's approach provides a consistent approximation for divergent low-frequency waves, while the midlatitude β-plane captures fewer aspects of the system. Tidal theory suggests an alternative formalism for extratropical low-frequency waves. The geography of wave activity depends on internal physics of the circulation: on a mean flow with constant layer thickness low-frequency waves populate an equatorial belt, while a hydrostatically driven eastward mean flow with varying layer thickness confines divergence-free Rossby waves to the extratropics. Effects of stratification and vertical shear are considered for solid body rotation of bishallow water with constant layer thickness. Rotation and mean flow stabilize against Kelvin–Helmholtz instability and the equatorial Kelvin wave is the most unstable mode of this flow. However, observed vertical shears in the tropical ocean remain well below the instability threshold. Baroclinic instability does not occur. Wave geography and stability characteristics of this system capture major features of the oceanic large-scale circulation.