Abstract

Abstract

The linear instability of circular vortices over isolated topography in a homogeneous and inviscid fluid is examined for the shallow-water and quasi-geostrophic models in the $f$ -plane. The eigenvalue problem associated with azimuthal disturbances is derived for arbitrary axisymmetric topographies, either submarine mountains or valleys. Amended Rayleigh and Fjørtoft theorems with topographic effects are given for barotropic instability, obtaining necessary criteria for instability when the potential vorticity gradient is zero somewhere in the domain. The onset of centrifugal instability is also discussed by deriving the Rayleigh circulation theorem with topography. The barotropic instability theorems are applied to a wide family of nonlinear, quasi-geostrophic solutions of circular vortices over axisymmetric topographic features. Flow instability depends mainly on the vortex/topography configuration, as well as on the vortex size in comparison with the width of the topography. It is found that anticyclones/mountains and cyclones/valleys may be unstable. In contrast, cyclone/mountain and anticyclone/valley configurations are stable. These statements are validated with two numerical methods. First, the generalised eigenvalue problem is solved to obtain the wavenumber of the fastest-growing perturbations. Second, the evolution of the vortices is simulated numerically to detect the development of linear perturbations. The numerical results show that for unstable vortices over narrow topographies, the fastest growth rate corresponds to mode $1$ , which subsequently forms asymmetric dipolar structures. Over wide topographies, the fastest perturbations are mainly modes $1$ and $2$ , depending on the topographic features.

This paper determines analytically the velocity field induced by a configuration of linked vortices, with the aim of finding the translational velocity of such a configuration. The linked-vortex configuration consists of two vortices; the vortices are thin tubes of circular cross section lying on the surface of an immaterial torus of small aspect ratio r1r0 (where r1 is the cross section radius of a torus and r0 is its centerline radius). The induced velocity field has been found based on the vector potential associated with the Biot-Savart law by using a multipolar expansion; the comparison of this field with the material condition on the surface of the vortices allows one to calculate the translational velocity. The solution obtained retains effects to first order at the multipolar expansion, which corresponds to the effects the vortex curvature has at the vorticity distribution on its cross section. This solution agrees with the numerical results in the range of 2%. The method presented is generalized to the case of n linked vortices.