Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.

An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.

We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.

Since the 1980s, the application of concepts and ideas from dynamical systems theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time-dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time-dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully applied to time-dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular, we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of normally hyperbolic invariant manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation $\Gamma_1$ and $\Gamma_2$ $(\Gamma_2 = -\mu\Gamma_1)$ are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.

Relations satisfied by the roots of the Loutsenko sequence of polynomials are derived. These roots are known to correspond to families of stationary and uniformly translating point vortices with two vortex strengths in ratio $-2$. The relations are analogous to those satisfied by the roots of the Adler–Moser polynomials, corresponding to equilibria with ratio $-1$. The proof uses an analysis of the differential equation that these polynomial pairs satisfy.

A vortex pair+single vortex (P+S) wake behind a transversely oscillating cylinder is investigated from the Lagrangian point of view. The Lagrangian coherent structures (LCSs) of the flow are computed to analyze formation of vortices in the wake. An asymmetric vortex street is obtained by using a dynamic mesh method. The corresponding vorticity field is found to agree well with real experiments. The LCSs are approximated by ridges of the finite-time Lyapunov exponents computed from transient velocity fields. The formation process is investigated using the vorticity field and the LCSs. It is found that details of the wake pattern are sensitive to initial oscillation conditions, and that the cylinder motion causes an early roll-up of boundary layers to form new vortex structures in the wake. Lagrangian description of the flow with the help of the LCSs provides further details about formation of vortices in the cylinder flow and helps to get a new insight into the flow structure in the wake region.

We consider the equations of motion of $n$ vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of $n$ vortices. We include numerical results for all cases, for various values of $n$, and we provide key details on the computational approach.

This paper is concerned with the problem of three vortices on a sphere $S^2$ and the Lobachevsky plane $L^2$. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.