Linear Stability of Self-Similar Motions of Three Point Vortices in a Generalized Two-Dimensional Fluid System

This study performs theoretical and numerical analyses of the linear stability of the self-similar motions of three vortices in a point vortex model of a generalized two-dimensional (2D) fluid system. In this system, the point vortex induces an azimuthal velocity of the form rα−3 around the vortex, where r is the distance from the position of the point vortex and α is a real number less than or equal to 3. When α = 2, this model reduces to the point vortex model of an ideal incompressible 2D fluid, the so-called 2D Euler system. We focus on the case 0 < α ≤ 3. Linearized evolution equations for perturbations applied to the self-similar motions of three point vortices are reduced to a 6 × 6 eigenvalue problem. Numerical and theoretical analyses are combined to evaluate eigenvalues and the corresponding eigenvectors in order to determine the linear stability of the self-similar motions. It has long been well known that for the 2D Euler system, i.e., when α = 2, the collapse and expansion motions are linearly unstable and stable, respectively. Even when α ≠ 2, the collapse and expansion motions are found to be linearly unstable and stable, respectively. One of the peculiar differences between systems with α = 2 and α ≠ 2 is that point vortices induce a rotational flow in systems with α ≠ 2. The effects of vorticity on the stability of the self-similar motions of three point vortices are discussed.