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A steady solution of the incompressible Euler equation on a toroidal surface TR,r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇²T R,r ψ =c edψ + (8/d)κ, where ∇²T R,r and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c, d are real parameters with cd<0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29, 417–440. (doi:10.1017/S002211206 7000941)) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498, 381–402. (doi:10.1017/S002211200 3007043)) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α =R/r. A comparison with the Stuart vortex on the flat torus is also made. [ABSTRACT FROM AUTHOR]

We carry out the linear stability analysis of a polygonal ring configuration of N point vortices, called an N-ring, along the line of latitude θ0 on a torus with the aspect ratio α. Deriving a criterion for the stability depending on the parameters N, θ0 and α, we reveal how the aspect ratio α contributes to the stability of the N-ring. While the N-ring necessarily becomes unstable when N is sufficiently large for fixed α, the stability is closely associated with the geometric property of the torus for variable α; for low aspect ratio α∼1, N=7 is a critical number determining the stability of the N-ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio α≫1. That is to say, the N-ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while the N-ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of the N-ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of the N point vortices is formulated as a Hamiltonian system with N degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable α are characterized in terms of the fundamental group of the torus. [ABSTRACT FROM AUTHOR]