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 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 generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

“…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.…”

“…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.…”

Steady point vortex pair in a field of Stuart-type vorticity

“…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.…”

Geophysics and Stuart vortices on a sphere meet differential geometry

“…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).…”