On the Motion of Vortices in Two Dimensions: I. Existence of the Kirchhoff-Routh Function

Helicoids and vortices

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a ‘vortex crystal’, an equilibrium of point vortices in two-dimensional fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such rotors are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either driven rotors in a viscous membrane, or ideal vortices with minute perturbations, spontaneously self assemble into a distinct arrangement. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in prominent structural states. We find that the rotationally invariant interactions isotropically suppress long-wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.

Rotor-like dynamics is observed in many natural systems, from the rotor proteins in cellular membranes to atmospheric models. Here, the authors uncover geometrical conservation laws that limit distribution of driven rotors in a membrane or a soap film and allow to predict their structural states.

The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

Giant vortex clusters in a two-dimensional quantum fluid

Adding energy to a system through transient stirring usually leads to more disorder. In contrast, point-like vortices in a bounded two-dimensional fluid are predicted to reorder above a certain energy, forming persistent vortex clusters. In this study, we experimentally realize these vortex clusters in a planar superfluid: a 87Rb Bose-Einstein condensate confined to an elliptical geometry. We demonstrate that the clusters persist for long time periods, maintaining the superfluid system in a high-energy state far from global equilibrium. Our experiments explore a regime of vortex matter at negative absolute temperatures and have relevance for the dynamics of topological defects, two-dimensional turbulence, and systems such as helium films, nonlinear optical materials, fermion superfluids, and quark-gluon plasmas.

Point vortex interactions on a toroidal surface

Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation for N-point vortices from Green's function associated with the Laplace-Beltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance.

The motion of point vortices on closed surfaces

We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech.157, 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech.255, 35–64 (doi:10.1017/S0022112093002381)).

Statistical mechanics of a neutral point-vortex gas at low energy

The statistics of a neutral point-vortex gas in an arbitrary two-dimensional simply connected and bounded container are investigated in the framework of the microcanonical ensemble, following the cumulant expansion method of Pointin and Lundgren [Phys. Fluids 19, 1459 (1976)]. The equation for vorticity fluctuations, obtained when a thermodynamic scaling limit is taken, is solved explicitly. The solution depends on an infinite sequence of negative "domain inverse temperatures," determined by the domain shape, which are obtained from solutions of a "vorticity mode" eigenvalue problem. An explicit expression for the thermodynamic curve relating inverse temperature and energy is found and is shown to depend on the geometry and not on the scale of the domain. Explicit formulas are then obtained for the time variance of the projection of the vorticity field onto each vorticity mode. The results are verified by two methods. First, for a chosen single-parameter family of domains, direct sampling of the microcanonical ensemble is used to demonstrate the accuracy of the formula for the thermodynamic curve. Second, direct numerical simulations are used to verify the formulas for the variance of the projections of the vorticity field, with convincing results.

Mass of a spin vortex in a Bose-Einstein condensate

In contrast with charge vortices, spin vortices in a two-dimensional ferromagnetic condensate move inertially (if the condensate has zero magnetization along an axis). The Magnus force, which prevents the inertial motion of the charge vortices, cancels for spin vortices, because they are composed of two oppositely rotating vortices. The inertial mass of spin vortices varies inversely with the strength of spin-dependent interactions and directly with the width of the condensate layer, and can be measured as a part of experiments on how spin vortices orbit one another. For Rb87 in a 1 microm thick trap, mv approximately 10(-21) kg.

Equation of motion for point vortices in multiply connected circular domains

The paper gives the equation of motion for N point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail.

Interacting vortices and spin-up in two-dimensional turbulence

In simulations of decaying two-dimensional turbulence, Clercx and co-workers discovered the unexpected phenomenon of "spin-up." (This is the spontaneous acquisition of angular momentum by a turbulent two-dimensional fluid in a rigid container.) Here we show that this phenomenon can readily be interpreted in terms of statistical models of two-dimensional turbulence. When the net vorticity is zero in a bounded system, there are two distinct types of statistical equilibrium. The first has the expected property that its angular momentum is zero. However, the second type has a nonzero angular momentum even though its circulation vanishes. The relative probability of the two types of equilibrium depends strongly on the shape of the boundary and weakly on the energy. The angular momentum predicted for the second type of equilibrium is in good agreement with that found in simulations at high Reynolds number.

Calculating the lift on a finite stack of cylindrical aerofoils

The classic exact solution due to Lagally (Lagally, M. 1929 Die reibungslose strömung im aussengebiet zweier kreise. Z. Angew. Math. Mech . 9 , 299–305.) for streaming flow past two cylindrical aerofoils (or obstacles) is generalized to the case of an arbitrary finite number of cylindrical aerofoils. Given the geometry of the aerofoils, the speed and direction of the oncoming uniform flow and the individual round-aerofoil circulations, the complex potential associated with the flow is found in analytical form in a parametric pre-image region that can be conformally mapped to the fluid region. A complete determination of the flow then follows from knowledge of the conformal mapping between the two regions. In the special case where the aerofoils are all circular, the conformal mapping from the parametric pre-image region to the fluid domain is a Möbius mapping. The solution for the complex potential in such a case can then be used, in combination with the Blasius theorem, to compute the distribution of hydrodynamic forces on the multi-aerofoil configuration.

Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains

Explicit formulae for the Kirchhoff–Routh path functions (or Hamiltonians) governing the motion of N -point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero. The method uses the Schottky–Klein prime function to find representations of the hydrodynamic Green's function in multiply connected circular domains. The Green's function is then used to construct the associated Kirchhoff–Routh path function. The path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains. Illustrative examples are presented for the case of single vortex motion in multiply connected domains.

Chaotic capture of vortices by a moving body. I. The single point vortex case

The study of the dynamical properties of vortex systems is an important and topical research area, and is becoming of ever increasing usefulness to a variety of physical applications. In this paper, we present a study of a model of a rotational singularity which obeys a logarithmic potential interacting with a bluff body in a uniform inviscid laminar flow, e.g., a line vortex interacting with a cylinder in three dimensions or a point vortex with a circular boundary in two dimensions. We show that this system is Hamiltonian and simple enough to be solved analytically for the stagnation points and separatrices of the flow, and a bifurcation diagram for the relevant parameters and classification of the various types of motion is given. We also show that, by introducing a periodic perturbation to the body, chaotic motion of the vortex can be readily generated, and we present analytic criteria for the generation of chaos using the Poincare-Melnikov-Arnold method. This leads to an important dynamical effect for the model, i.e., that the possibility exists for the vortex to be chaotically captured around the body for periods of time which are extremely sensitive to initial conditions. The basic mechanism for this capture is due to the chaotic dynamics and is similar to that of other chaotic scattering phenomena. We show numerically that cases exist where the vortex can be captured around an elliptic point external to (and possibly far from) the body, and the existence of other very complicated motions are also demonstrated. Finally, generalizations of the problem of the vortex-body interaction are indicated, and some possible applications are postulated such as the interaction of line vortices with aircraft wings.