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Hydrodynamic Vortex on Surfaces


The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type are presented. The equations are obtained from a Green’s function on the surface. The uniqueness of the Green’s function is established under hydrodynamic conditions at the surface’s boundaries and ends. The hydrodynamic force on a point vortex is computed using a new weak formulation of Euler’s equation adapted to the point vortex context. An analogy between the hydrodynamic force on a massive point vortex and the electromagnetic force on a massive electric charge is presented as well as the equations of motion for massive vortices. Any noncompact Riemann surface admits a unique Riemannian metric such that a single vortex in the surface does not move (“Steady Vortex Metric”). Some examples of surfaces with steady vortex metric isometrically embedded in \(\mathbb {R}^3\) are presented.

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  1. In an arbitrary local coordinate system \((\xi ^1,\xi ^2)\), the Riemannian Laplacian is given by \(\Delta =\frac{1}{\sqrt{|g|}}\frac{\partial }{\partial \xi ^j}g^{jk}\sqrt{|g|} \frac{\partial }{\partial \xi ^k}\) where the sum over repeated indices is assumed, the Riemannian metric is given by \(g_{jk}d\xi ^j\otimes d\xi ^k, g^{jk}\) is the inverse of matrix \(g_{jk}\), and |g| is the absolute value of the determinant of the matrix \(g_{jk}\).

  2. In the coordinates of footnote 1, \(\mathrm{d}y=*\mathrm{d}x=-\frac{1}{\sqrt{|g|}}g_{ki}\epsilon ^{il}\frac{\partial x}{\partial \xi ^l} d\xi ^k\), where \(\epsilon ^{il}=-\epsilon ^{li}\) and \(\epsilon ^{12}=1\).

  3. The constant may be determined by the normalization \(\int _S G(q,p)\mu (q)=0\).


  • Aboudi, N.: Geodesics for the capacity metric in doubly connected domains. Complex Var. 50, 7–22 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  • Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  • Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex Crystals. Adv. Appl. Mech. 39, 1–79 (2003)

    Article  Google Scholar 

  • Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Aubin, T.: Some Nonlinear Problems in Riemannian Geometry, 3rd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  • Beardon, A.F.: A Primer on Riemann Surfaces. London Math. Soc. Lect. Notes Ser. 78, Cambridge University Press (1984)

  • Boatto, S., Koiller, J.: Vortices on closed surfaces, arXiv:0802.4313 (2008)

  • Boatto, S., Koiller, J.: Vortices on closed surfaces. In: Chang, D.E., Holm, D.D., Patrick, G., Ratiu, T. (eds.) Geometry, Mechanics, and Dynamics The Legacy of Jerry Marsden, pp. 185–237. Springer, Berlin (2013)

    Google Scholar 

  • Bogomolov, V.A.: Dynamics of vorticity at a sphere. Fluid Dyn. 12, 863–870 (1977)

    Article  MATH  Google Scholar 

  • Bolotin, S., Negrini, P.: Asymptotic solutions of Lagrangian systems with gyroscopic forces. Nonlinear Diff. Eq. Appl. NoDEA 2, 417–444 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  • Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface. Regul. Chaotic Dyn. 15, 440461 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  • Crowdy, D., Marshall, J.: Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains. Proc. R. Soc. A 461, 24772501 (2005)

    MathSciNet  MATH  Google Scholar 

  • de Rham, G.: Differentiable Manifolds. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  • Dritschel, D.G., Boatto, S.: The motion of point vortices on closed surfaces. Proc. R. Soc. A 471, 20140890 (2015)

    MathSciNet  Article  Google Scholar 

  • Flucher, M., Gustafsson, B.: Vortex motion in two-dimensional hydromechanics (Technical report, (1997)

  • Friedrichs, K.O.: Special Topics in Fluid Dynamics. Gordon and Breach, New York (1966)

    MATH  Google Scholar 

  • Grotta Ragazzo, C., Koiller, J., Oliva, W.: On the motion of two-dimensional vortices with mass. J. Nonlinear Sci. 4, 375–418 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  • Gustafsson, B.: On the motion of a vortex in two-dimensional flow of an ideal fluid in simply and multiply connected domains, (Technical Report, (1979)

  • Hally, D.: Motion of Vortex in Thin Films, PhD Thesis, The University of British Columbia, Canada (1979)

  • Hally, D.: Stability of streets of vortices on surfaces of revolution with a reflection symmetry. J. Math. Phys. 21, 211–217 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  • Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1999)

    MATH  Google Scholar 

  • He, Z.-X., Schramm, O.: Fixed points. Koebe uniformization and circle packings. Ann. Math. 137, 369–406 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  • Kimura, Y., Okamoto, H.: Vortex motion on a sphere. J. Phys. Soc. Jpn 56, 4203–4206 (1987)

    MathSciNet  Article  Google Scholar 

  • Kimura, Y.: Vortex motion on surfaces with constant curvature. Proc. R. Soc. Lond. A 455, 245259 (1999)

    MathSciNet  Article  Google Scholar 

  • Lamb, H.: Hydrodynamics, 4th edn. Cambridge University Press, Cambridge (1916)

    MATH  Google Scholar 

  • Llewellyn Smith, S.G.: How do singularities move in potential flow? Physica D 240, 16441651 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  • Li, P.: Curvature and Function Theory on Riemannian Manifolds, Survey in Differential Geometry in Honor of Atiyah, Bott, Hirzebruch and Singer, VII, International Press, Cambridge, 71–111. (2000)

  • Lin, C.C.: On the motion of vortices in two dimensions. I. Existence of the Kirchhoff-Routh function. Proc. Nat. Acad. Sci. USA 27, 570575 (1941). (see also the second part of the paper in the same volume p. 575-577)

    MathSciNet  Google Scholar 

  • Lin, C.C.: On the Motion of Vortices in Two Dimensions, University of Toronto Studies, Applied Mathematics Series. University of Toronto Press, (1943)

  • Littlejohn, R.G.: “Hamiltonian Theory of Guiding Center Motion”, PhD Thesis, Lawrence Berkeley Lab (eletronically available at: (1980)

  • Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  • Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York (1994)

    Book  MATH  Google Scholar 

  • Newton, P.: The N-vortex Problem: Analytical Techniques. Springer, New York (2001)

    Book  MATH  Google Scholar 

  • Newton, P.: Point vortex dynamics in the post-Aref era, Fluid Dyn. Res. 46 031401 (11pp) (2014)

  • Oliva, W.M.: “Geometric Mechanics”, Lect. Notes Math. 1798, Springer, Heidelberg (2002)

  • Paternain, G.P.: Geodesic Flows. Birkhas̈er, Boston (1999)

    Book  MATH  Google Scholar 

  • Richards, I.: On the classification of noncompact surfaces. Trans. AMS 106, 259–269 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  • Sakajo, T., Shimizu, Y.: Point vortex interactions on a toroidal surface. Proc. R. Soc. A 472, 20160271 (2016)

    MathSciNet  Article  Google Scholar 

  • Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 2, 3rd edn. Publish or Perish, Houston (1999)

    MATH  Google Scholar 

  • Turner, A.M., Vitelli, V., Nelson, D.R.: Vortices on curved surfaces. Rev. Modern Phys. 82, 1301–1348 (2010)

    Article  Google Scholar 

  • Viglioni, H.H.B.: Vortex Dynamics on Surfaces and applications to the problem of two vortices on the torus, Doctorate thesis (in Portuguese), Universidade de São Paulo (2013)

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The authors thank to: Jair Koiller who presented and taught them the subject, Hildeberto Cabral who invited CGR to write a review on the subject, and Björn Gustafsson for the discussions.

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Correspondence to Clodoaldo Grotta Ragazzo.

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CGR is partially supported by supported by FAPESP (Brazil) 2011/16265-8.

Communicated by George Haller.

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Ragazzo, C.G., de Barros Viglioni, H.H. Hydrodynamic Vortex on Surfaces. J Nonlinear Sci 27, 1609–1640 (2017).

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  • Point vortex dynamics
  • Special metrics
  • Robin function
  • Euler equation weak solution
  • Steady vortex metric

Mathematics Subject Classification

  • 70F99
  • 76B47
  • 53C07
  • 31A15