Linear stability analysis of double rows of point vortices for an inviscid generalized two-dimensional fluid system

Abstract

Abstract The linear stability of double rows of equidistant point vortices for an inviscid generalized two-dimensional (2D) fluid system is studied. This system is characterized by the relation q = −(Δ) α/2 ψ between the active scalar q and the stream function ψ. Here, α is a positive real number not exceeding 3, and q is referred to as the generalized vorticity. The stability of double rows of equidistant point vortices for a 2D Euler system (α = 2) is a well-known classical problem and was originally investigated by Kármán approximately 100 years ago. Two types of vortex rows, i.e., symmetrical and staggered arrangements of vortex rows, are considered in this study. Special attention is paid to the effect of the parameter α on the stability of vortex rows. As is well-known, the symmetrical vortex rows for the Euler system are unstable, whereas the staggered vortex rows are neutrally stable only when the transverse-to-longitudinal spacing ratio k is k = π − 1 cosh − 1 2 . Irrespective of α, the symmetrical vortex rows for the generalized 2D fluid system are unstable, whereas the staggered vortex rows are neutrally stable. The stable spacing ratio of the staggered vortex rows is a decreasing function of α for 0.63 ≲ α and approaches zero as α → 3. In contrast, a finite stable region of the spacing ratio is found when α ≲ 0.63. It turns out that the deformation and usual vorticity induced by the point vortices and their coupling play important roles on the stability of the staggered vortex rows.