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1. Shape dynamics of N point vortices on the sphere

   https://doi.org/10.1088/1361-6544/aca50e  

   Ohsawa, Tomoki (December 2022, Nonlinearity)

   Abstract We give a geometric account of the relative motion or the shape dynamics of N point vortices on the sphere exploiting the S O ( 3 ) -symmetry of the system. The main idea is to bypass the technical difficulty of the S O ( 3 ) -reduction by first lifting the dynamics from S 2 to C 2 . We then perform the U ( 2 ) -reduction using a dual pair to obtain a Lie–Poisson dynamics for the shape dynamics. This Lie–Poisson structure helps us find a family of Casimirs for the shape dynamics. We further reduce the system by T N − 1 -symmetry to obtain a Poisson structure for the shape dynamics involving fewer shape variables than those of the previous work by Borisov and Pavlov. As an application of the shape dynamics, we prove that the tetrahedron relative equilibria are stable when all of their circulations have the same sign, generalizing some existing results on tetrahedron relative equilibria of identical vortices. 

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2. A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

   https://doi.org/10.1063/5.0208538  

   Anurag, Atul; Goodman, Roy H; O'Grady, Ellison K (June 2024, Physics of Fluids)

   We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria. 

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3. Four-field Hamiltonian fluid closures of the one-dimensional Vlasov–Poisson equation

   https://doi.org/10.1063/5.0102418  

   Chandre, C.; Shadwick, B. A. (October 2022, Physics of Plasmas)

   We consider a reduced dynamics for the first four fluid moments of the one-dimensional Vlasov–Poisson equation, namely, fluid density, fluid velocity, pressure, and heat flux. This dynamics depends on an equation of state to close the system. This equation of state (closure) connects the fifth-order moment—related to the kurtosis in velocity of the Vlasov distribution—with the first four moments. By solving the Jacobi identity, we derive an equation of state, which ensures that the resulting reduced fluid model is Hamiltonian. We show that this Hamiltonian closure allows symmetric homogeneous equilibria of the reduced fluid model to be stable. 

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4. Hyperuniformity and phase enrichment in vortex and rotor assemblies

   https://doi.org/10.1038/s41467-022-28375-9  

   Oppenheimer, Naomi; Stein, David B.; Zion, Matan Yah Ben; Shelley, Michael J. (February 2022, Nature Communications)

   Abstract 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. 

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5. Islands in stable fluid equilibria

   https://doi.org/10.1090/proc/16951  

   Drivas, Theodore; Ginsberg, Daniel (November 2024, Proceedings of the American Mathematical Society)

   We prove that stable fluid equilibria with trivial homology on curved, reflection-symmetric periodic channels must posses “islands”, or cat’s eye vortices. In this way, arbitrarily small disturbances of a flat boundary cause a change of streamline topology of stable steady states. 

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