Vortices on closed surfaces. (English) Zbl 1402.76034

Chang, Dong Eui (ed.) et al., Geometry, mechanics, and dynamics. The legacy of Jerry Marsden. Selected papers presented at a focus program, Fields Institute for Research in Mathematical Sciences, Toronto, Canada, July 2012. New York, NY: Springer (ISBN 978-1-4939-2440-0/hbk; 978-1-4939-2441-7/ebook). Fields Institute Communications 73, 185-237 (2015).
Summary: It was recognized, since the seminal papers of V. I. Arnol’d [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] and D. G. Ebin and J. Marsden [Ann. Math. (2) 92, 102–163 (1970; Zbl 0211.57401)], that Euler’s equations are the right reduction of the geodesic flow in the group of volume preserving diffeomorphisms. In 1983 J. Marsden and A. Weinstein [Physica D 7, 305–323 (1983; Zbl 0576.58008)] went one step further, pointing out that vorticity evolves on a coadjoint orbit on the dual of the infinite dimensional Lie algebra consisting of divergence free vector fields.
Here we pursue a suggestion of that paper, namely, to present an intrinsic Hamiltonian formulation for a special coadjoint orbit, which contains the motion of \(N\) point vortices on a closed two dimensional surface \(S\) with Riemannian metric \(g\). Our main results reformulate the problem on the plane, mainly C. C. Lin’s works [Proc. Natl. Acad. Sci. USA 27, 570–575, 575–577 (1941; Zbl 0063.03560)] about vortex motion on multiply connected planar domains. Our main tool is the Green function \(G_g(s,s_0)\) for the Laplace-Beltrami operator of \((S,g)\), interpreted as the stream function produced by a unit point vortex at \(s_ 0\in S\). Since the surface has no boundary, the vorticity distribution \(\omega\) has to satisfy the global condition \(\iint_S\omega\,\Omega=0\), where \(\Omega\) is the area form. Thus the Green function equation has to include a background of uniform counter-vorticity. As a consequence, vortex dynamics is affected by global geometry. Our formulation satisfies Kimura’s requirement [Y. Kimura, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1981, 245–259 (1999; Zbl 0966.53046)] that a vortex dipole describes geodesic motion. A single vortex drifts on the surface, with Hamiltonian given by Robin’s function, which in the case of topological spheres is related to the Gaussian curvature [J. Steiner, Duke Math. J. 129, No. 1, 63–86 (2005; Zbl 1144.53055)]. Results on numerical simulations on flat tori, the catenoid and in the triaxial ellipsoid are depicted. We present a number of questions, intending to connect point vortex streams on surfaces with questions from the mathematical mainstream. 219 References.
For the entire collection see [Zbl 1317.53004].


76B47 Vortex flows for incompressible inviscid fluids
53D20 Momentum maps; symplectic reduction


Full Text: DOI arXiv


This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.