Vortices on closed surfaces. (English) Zbl 1402.76034

Boatto, Stefanella; Koiller, Jair

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].

MSC:

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

Citations:

Zbl 0148.45301; Zbl 0211.57401; Zbl 0576.58008; Zbl 0063.03560; Zbl 0966.53046; Zbl 1144.53055

Software:

SKPrime

References: