Integral-equation approach to resonances in circular two-layer flows around an island with bottom topography

Abstract

This paper presents an integral-equation approach to the linear instability problem of two-layer quasi-geostrophic flows around circular islands with radial offshore bottom slope. The flows are composed of concentric uniform potential-vorticity (PV) rings in each layer, with the PV of each ring being opposite in sign. The study extends an earlier similar barotropic model and focuses on the degree to which the topographic waves resonate with the deformation waves at the rings’ peripheries. The integral approach poses the instability problem in a physically elucidating way, whereby the resonating waves in the system are directly identified. Four types of instabilities are identified: instability caused by the resonance of waves at the liquid contours at the edge of each PV ring, instability caused by the resonance of the wave at the upper-layer contour and the topographic waves outside the lower-layer contour, a similar resonance of the lower-layer contour with the topographic waves, and a resonance between one of the eigenmodes of the contour subsystem with the topographic waves. The three latter resonances lead to critical layer instabilities and can be identified as resonances between the contour waves and a collection of singular topographic modes with a critical layer. The PV perturbations in the outer region can be represented asymptotically (far from the origin) as a combination of barotropic and baroclinic modes. Usually, the asymptotically barotropic mode is the mode in resonance with the contours, but, for small growth rates, the asymptotically baroclinic mode may be the dominant mode.