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 Krmn 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 . 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.

研究了无粘性广义二维 (2D) 流体系统的双排等距点涡的线性稳定性。此系统的特征在于由关系q = - （Δ）^{α / 2} ψ有源标量之间q和流函数ψ。其中，α是不超过 3 的正实数，q称为广义涡度。二维欧拉系统的双排等距点涡的稳定性 ( α = 2) 是一个众所周知的经典问题，大约 100 年前由 Krmn 最初研究。本研究考虑了两种涡排，即涡排的对称排列和交错排列。特别注意参数α对涡列稳定性的影响。众所周知，欧拉系统的对称涡列是不稳定的，而交错涡列只有在横纵间距比k为时才是中性稳定的。不考虑α，广义二维流体系统的对称涡列是不稳定的，而交错涡列是中性稳定的。交错涡列的稳定间距比是 的递减函数α为 0.63 ≲ α，当α → 3 时接近零。相反，当α ≲ 0.63时，发现间隔比的有限稳定区域。结果表明，点涡引起的变形和普通涡度及其耦合对交错涡列的稳定性起着重要作用。