The N-vortex problem on a symmetric ellipsoid: A perturbation approach

Abstract

We consider the N-vortex problem on an ellipsoid of revolution. Applying standard techniques of classical perturbation theory, we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations, the equations of motion for the N-vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid's generating ellipse. First order equations are obtained explicitly. We show numerically that the truncated first order system for the three vortex system on the symmetric ellipsoid is nonintegrable.

The Motion of Point Vortex Dipole on the Ellipsoid of Revolution

“…However, as commented in [3], the perturbation approach shows that the next order computations are too complicated and does not permit exact analytic solutions. Therefore, we might think of approximating solutions by certain numerical methods.…”

“…(See [3] for derivation.) From this we derive the dynamical equations of point vortex on E 2 , (Figure 1).…”

“…This was initiated by Hally [6] who obtained a conformal coordinate formulation of vortex dynamics in the complex variables for the general simply connected surfaces. In [3], this approach is adopted and a perturbation series is calculated for the conformal factor function in the case of ellipsoid of revolution. As observed there, it is very hard to compute the conformal function in general except some highly symmetric surfaces such as a plane, a sphere, a circular cylinder, etc.…”

“…Recently the basic dynamical equations of point vortex on such ellipsoid of revolution are pursued in [3] by a perturbation technique which will be used in this paper. In addition, instead of the stereographic projection, we adopt a direct geometrical approach and incorporate the parallel translation property of geodesic.…”

“…In addition, instead of the stereographic projection, we adopt a direct geometrical approach and incorporate the parallel translation property of geodesic. More precisely, regarding the ellipsoid as imbedded in R 3 , we derive and analyze the dynamical equations of vortex directly and then check the parallel translation of the vortex dipole. The difficulty in generalizing the geodesic property of dipole paths for more general curved surfaces or even for general two-dimensional manifolds is then commented.…”