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We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4 n ̂ point vortices of identical strengths, n ̂ ∈ N. The quantized point vortices are stationary in the sense that they are equilibria of a "one-way interaction" model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes. [ABSTRACT FROM AUTHOR]

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 formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green's function. We derive an explicit formula for the hydrodynamic Green's function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N , we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N = 1 , 2 , 3 , 4 vortices per fundamental lattice. • Explicit formula for hydrodynamic Green's function in a doubly-periodic domain. • Hamiltonian equations for point vortices in a rectangular fundamental domain. • Constant background vorticity and non-zero sum of point vortex strengths. • Conformal mapping approach in a complex plane using the Schottky–Klein prime function. • New equilibria with inhomogeneous vortex strengths and lattice structure with defects. [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]

This is a review article of recent research developments on the motion of a polygonal ring configuration of vortex structures with singular vorticity distributions in incompressible and inviscid flows on a non-rotating sphere. Numerical computation of a single vortex sheet reveals that the Kelvin-Helmholtz instability gives rise to the formation of a polygonal ring arrangement of rolling-up spirals. An application of methods of Hamiltonian dynamics to the N-vortex problem on the sphere shows that the motion of the ring configuration of homogeneous point vortices, which is a simple model for the rolling-up spirals, becomes chaotic after a long time evolution. Some remarks on an extension of the present research and a generic non-self-similar collapse are also provided. [ABSTRACT FROM AUTHOR]

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5 n or 6 n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic. [ABSTRACT FROM AUTHOR]

We consider the motion of N vortex points on sphere, called the N-vortex problem, which is a Hamiltonian dynamical system. The three-vortex problem is integrable and its motion has already been resolved. On the other hand, when the moment of vorticity vector, which consists of weighed sums of the vortex positions, is zero at the initial moment, the four-vortex problem is integrable, but it has not been investigated yet. The present paper gives a description of the integrable four-vortex problem with the reduction method to a three-vortex problem used by Aref and Stremler. Moreover, we examine whether the vortex points collide self-similarly in finite time. The four-vortex collapse is proved to be impossible. We consider if it is possible for not all but part of the vortex points to collapse self-similarly. Moreover, we discuss the topological structure of periodic orbits obtained in the present problem. [ABSTRACT FROM AUTHOR]

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston–Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system. [ABSTRACT FROM AUTHOR]

We consider the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices fixed on north and south poles. Analytic and numerical research revealed that a vortex sheet in two-dimensional space has the following three properties. First, the vortex sheet is linearly unstable due to Kelvin–Helmholtz instability. Second, the curvature of the vortex sheet diverges in finite time. Last, the vortex sheet evolves into a rolling-up doubly branched spiral, when the equation of motion is regularized by the vortex method. The purpose of this paper is to investigate how the curvature of the sphere and the presence of the pole vortices affect these three properties mathematically and numerically. We show that some low spectra of disturbance become linearly stable due to the pole vortices and thus the singularity formation tends to be delayed. On the other hand, however, the vortex sheet, which is regularized by the vortex method, acquires complex structure of many rolling-up spirals. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We consider an axisymmetric, swirling vortex sheet in an inviscid and incompressible flow. Caflisch et al. pointed out that the vortex sheet acquired a singularity in finite time, but the property of the singularity was not revealed. In the present paper we show convincing numerical evidences of the singularity formation by applying the same numerical methods as what was used in the study of a two-dimensional (2D) vortex sheet. We find that the radial and axial components of the axisymmetric vortex sheet behave like the 2D singularity that has been observed in many vortex-sheet motions, while the azimuthal component of the sheet behaves differently. Furthermore, the singularity appears along the vortex line and the first derivative of the vortex sheet strength forms a cusp, while the known singularities are associated with the curvature along curves perpendicular to the vortex lines and the sheet strength has a cusp. [ABSTRACT FROM AUTHOR]

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure. [ABSTRACT FROM AUTHOR]