Bottom-topography effect on the instability of flows around a circular island

Abstract

Instabilities of a two-dimensional quasigeostrophic circular flow around a rigid circular wall (island) with radial offshore bottom slope are studied analytically. The basic flow is composed of two concentric, uniform potential-vorticity (PV) rings with zero net vorticity attached to the island. Linear stability analysis for perturbations in the form of azimuthal modes leads to a transcendental eigenvalue equation. The non-dimensional governing parameters are beta (associated with the steepness of the bottom slope, hence taken to be negative), the PV in the inner ring and the radii of the inner and outer rings. This setting up of the problem allows us to derive analytically the eigenvalue equation. We first analyse this equation for weak slopes to understand the asymptotic first-order corrections to the flat-bottom case. For azimuthal modes 1 and 2, it is found that the conical topographic beta effect stabilizes the counterclockwise flows, but destabilizes clockwise flows. For a clockwise flow, the beta effect gives rise to the mode-1 instability, contrary to the flat-bottom case where this mode is always stable. Moreover, however small the slope steepness (beta) is, it leads to the mode-1 instability in a large region in the parameter space. For steep slopes, the beta term in the PV expression may dominate the relative vorticity term, causing stabilization of the flow, as compared to the flat-bottom case, for both directions of the basic flow. When the flow is counterclockwise and the slope steepness is increased, mode 2 turns out to be entirely stable and modes 3, 4 and 5 enlarge their stability regions. In a clockwise flow, when the slope steepness is increased, mode 1 regains its stability in the entire parameter space, and mode 2 becomes more stable than mode 3. The bifurcation of mode 1 from stability to instability is discussed in terms of the Rossby waves at the contours of discontinuity of the basic PV and outside the uniform-PV rings.