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Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus.

ABSTRACT: 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, ?TR,r2?=c?ed?+(8/d)? , where ?TR,r2 and ? denote the Laplace-Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c, d are real parameters with cd?

SUBMITTER: Sakajo T

PROVIDER: S-EPMC6501666 | biostudies-literature | 2019 Apr

REPOSITORIES: biostudies-literature

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Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus.

Sakajo Takashi T

Proceedings. Mathematical, physical, and engineering sciences 20190403 2224

A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 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 gene ...[more]

PMID: 31105449

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Project description:Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose–Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation AΓ = 0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Dataset Information

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus.

Publications

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus.

Similar Datasets

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