Interacting Motion of Rectilinear Geostrophic Vortices

The kinematic motion of (N+1) geostrophic (Bessel) vortices is studied. Initially, the vortices are positioned near a uniformly rotating equilibrium configuration which consists of N vortices of equal strength γ equally spaced on a circle of radius a, and one vortex of strength γ0 at the center of the circle. One length scale κ−1 is associated with each of the vortices, the limiting case κ = 0being the classical logarithmic vortex. The linearized stability of this equilibrium configuration is studied over the range of the three parameters: N≥2, 0≤ (κa) < ∞, and − ∞ <(γ0/κ) < ∞. Over the entire range of the parameters, there are double-zero eigenvalues, which imply that the motion of the center vortex is inherently nonlinear. Numerical computations show that nonlinear effects are important in a parametric transition region embracing both the stable and unstable sides of the neutral stability curves for the circle vortices, even for small initial perturbations. The stability of the center vortex is not well represented by the neutral stability curves. In addition, the nonlinear motion away from an unstable equilibrium toward a stable equilibrium configuration is studied numerically.