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, ${\nabla }_{T_{R,r}}^2\psi =c{\text{e}}^{d\psi }+\left(8/d\right)\kappa$ , where ${\nabla }_{T_{R,r}}^2$ 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.

Keywords: Liouville equation; Stuart vortex; elliptic functions; ring torus; vortex dynamics.