2019
DOI: 10.1098/rspa.2018.0666
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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 T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Lio… Show more

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Cited by 8 publications
(6 citation statements)
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“…The original solutions of Stuart (1967) are singly-periodic solutions and we can ask if there are singly periodic hybrid equilibria. Stuart vortex solutions have been extended to a sphere (Crowdy 2004) and a torus (Sakajo 2019). The existence of more general hybrid equilibria on these compact surfaces, which consist of greater numbers of point vortices, is an open question.…”
Section: Summary and Future Directionsmentioning
confidence: 99%
See 1 more Smart Citation
“…The original solutions of Stuart (1967) are singly-periodic solutions and we can ask if there are singly periodic hybrid equilibria. Stuart vortex solutions have been extended to a sphere (Crowdy 2004) and a torus (Sakajo 2019). The existence of more general hybrid equilibria on these compact surfaces, which consist of greater numbers of point vortices, is an open question.…”
Section: Summary and Future Directionsmentioning
confidence: 99%
“…Generalising the planar Stuart vortices to the case of a non-rotating sphere, Crowdy (2004) found analytical solutions for everywhere smooth vorticity on the surface of a sphere except for point vortices at the north and south poles. These ideas can be extended to obtain Stuart vortex solutions on a torus (Sakajo 2019) and on a hyperbolic sphere (Yoon, Yim & Kim 2020). The introduction in Krishnamurthy et al.…”
Section: Introductionmentioning
confidence: 99%
“…The latter solutions, which are also expressible in closed form, describe smooth "cat's-eye" rings of N 2 vortices around a latitude circle with steady point vortices at the two spherical poles. Those solutions have recently been generalized to a toroidal surface by Sakajo (2019). An analogous construction in the planar case (Crowdy 2003) yields a single point vortex at the origin surrounded by an N -polygonal ring of smooth vortices.…”
Section: Introductionmentioning
confidence: 99%
“…In both cases Aubin set k = 3 and took f i = F i , i.e. the basis of spherical harmonics of degree 1 given in (18). In these corollaries the primary candidate curvature h is by virtue of the variational principle replaced by h−F (h), where F (h) is an unspecified spherical harmonics of degree 1 (note that the notation in [7] is different).…”
mentioning
confidence: 99%