June  2018, 10(2): 189-208. doi: 10.3934/jgm.2018007

Vortex pairs on a triaxial ellipsoid and Kimura's conjecture

1. 

Departamento de Matemática, Universidade Federal Rural de Pernambuco, Recife, PE CEP 52171-900 Brazil

2. 

Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE CEP 50740-540 Brazil

3. 

Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG CEP 36036-900 Brazil

* Corresponding author

Received  October 2016 Revised  December 2017 Published  May 2018

We consider the problem of point vortices moving on the surface of a triaxial ellipsoid. Following Hally's approach, we obtain the equations of motion by constructing a conformal map from the ellipsoid into the sphere and composing with stereographic projection. We focus on the case of a pair of opposite vortices. Our approach is validated by testing a prediction by Kimura that a (infinitesimally close) vortex dipole follows the geodesic flow. Poincaré sections suggest that the global flow is non-integrable.

Citation: Adriano Regis Rodrigues, César Castilho, Jair Koiller. Vortex pairs on a triaxial ellipsoid and Kimura's conjecture. Journal of Geometric Mechanics, 2018, 10 (2) : 189-208. doi: 10.3934/jgm.2018007
References:
[1]

S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, 185–237, Fields Inst. Commun., 73, Springer, New York, 2015. doi: 10.1007/978-1-4939-2441-7_10.  Google Scholar

[2]

V. A. Bogomolov, The dynamics of vorticity on a sphere, (Russian), Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza, 6 (1977), 57-65.   Google Scholar

[3]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.  Google Scholar

[4]

A. V. BolsinovV. S. Matveev and A. T. Fomenko, Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry, Sb. Math., 189 (1998), 1441-1466.  doi: 10.1070/SM1998v189n10ABEH000346.  Google Scholar

[5]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Berlin: Springer-Verlag, 1954.  Google Scholar

[6]

B. C. Carlson, Computing elliptic integrals by duplication, Numerische Mathematik, 33 (1979), 1-16.  doi: 10.1007/BF01396491.  Google Scholar

[7]

C. Castilho and H. Machado, The N-vortex problem on a symmetric ellipsoid: A perturbation approach, J. Math. Phys., 49 (2008), 022703, 12pp. doi: 10.1063/1.2863515.  Google Scholar

[8]

T. Craig, Orthomophic Projection of an Ellipsoid upon a sphere, Amer. J. Math., 3 (1880), 114-127.  doi: 10.2307/2369466.  Google Scholar

[9]

D. G. CrowdyE. H. KropfC. C. Green and M. M. S. Nasser, The Schottky-Klein prime function: A theoretical and computational tool for applications, IMA J. Applied Math., 81 (2016), 589-628.  doi: 10.1093/imamat/hxw028.  Google Scholar

[10]

D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25pp. doi: 10.1098/rspa.2014.0890.  Google Scholar

[11]

I. S. Gradshteyn, J. M. Ryzhik and A. Jeffrey, Table of Integrals, Series, and Products, 4. ed. New York: Academic Press, 1965.  Google Scholar

[12]

F. Goes, L. Beibei, M. Budninskiy, Y. Tong and M. Desbrun, Discrete 2-Tensor Fields on Triangulations, Eurographics Symposium on Geometry Processing 33:5, editors Thomas Funkhouser and Shi-Min Hu, 2014. Google Scholar

[13]

I. Gromeka, Sobranie Socinenii (Russian) (Collected works), Izdat. Akad. Nauk SSSR, Moscow, 1952.  Google Scholar

[14]

D. Hally, Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys., 21 (1980), 211-217.  doi: 10.1063/1.524322.  Google Scholar

[15]

H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, (German), J. Reine Angew. Math., 55 (1858), 25-55.  doi: 10.1515/crll.1858.55.25.  Google Scholar

[16]

M. Henon, On the numerical computation of Poincaré maps, Physica D, 5 (1982), 412-414.  doi: 10.1016/0167-2789(82)90034-3.  Google Scholar

[17]

C. G. J. Jacobi, Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution], Jour. Crelle, 19 (1839), 309-313.  doi: 10.1515/crll.1839.19.309.  Google Scholar

[18]

C. G. J. Jacobi, Vorlesungen über Dynamik [Lectures on Dynamics], edited by Clebsch, Reimer, Berlin, 1866; second edition edited by Weierstrass, 1884; English translation by K. Balagangadharan (Hindustan Book Agency, 2009). Google Scholar

[19]

C. F. Karney, https://geographiclib.sourceforge.io/html/jacobi.html. Google Scholar

[20]

S.-C. Kim, Latitudinal point vortex rings on the spheroid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1749-1768.  doi: 10.1098/rspa.2009.0597.  Google Scholar

[21]

R. Kidambi and P. K. Newton, Motion of three point vortices on a sphere, Physica D: Nonlinear Phenomena, 116 (1998), 143-175.  doi: 10.1016/S0167-2789(97)00236-4.  Google Scholar

[22]

Y. Kimura, Vortex motion on surfaces with constant curvature, Proc. R. Soc. Lond. A, 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.  Google Scholar

[23]

G. R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanic, Teubner, Leipzig, 1876. Google Scholar

[24]

F. Klein, On Riemann Theory of Algebraic Functions and their Integrals, Cambridge: Macmillan and Bowes (1893). Available online www.gutenberg.org/ebooks/36959 Google Scholar

[25]

J. Koiller and S. Boatto, Vortex pairs on surfaces, AIP Conference Proceedings, , 77 (2009), p1130; (Geometry and Physics: ⅩⅦ International Fall Workshop, edited by F. Etayo, M. Fioravanti, and R. Santamaria). Google Scholar

[26]

J. Koiller and K. Ehlers, Rubber rolling over a sphere, Reg. Chaotic Dyn., 12 (2007), 127-152.  doi: 10.1134/S1560354707020025.  Google Scholar

[27]

C. C. Lin, On the Motion of Vortices in Two Dimensions: Ⅰ. Existence of the Kirchhoff-Routh Function; Ⅱ. Some Further Investigations on the Kirchhoff-Routh Function, Proc. Natl.Acad. Sci. USA, 27 (1941), 575-577.   Google Scholar

[28]

A. S. Miguel, Numerical description of the motion of a point vortex pair on ovaloids, J. Phys. A: Math. Theor., 46 (2013), 115502, 21pp. doi: 10.1088/1751-8113/46/11/115502.  Google Scholar

[29]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14pp. doi: 10.1063/1.4897210.  Google Scholar

[30]

J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983.  Google Scholar

[31]

M. V. NyrtsovM. E. FliesM. M. Borisov and P. J. Stooke, Jacobi conformal projection of the triaxial ellipsoid: New projection for mapping of small celestial bodies, Cartography from Pole to Pole, (2014), 235-246.   Google Scholar

[32]

C. Ragazzo, The motion of a vortex on a closed surface: An algorithm and its application to the Bolza surface, Proc. Royal Soc. London A: Mathematical, Physical and Engineering Science, 473 (2017), 20170447, 17 pp. doi: 10.1098/rspa.2017.0447.  Google Scholar

[33]

A. Regis, Dinâmica de Vórtices Pontuais Sobre o Elipsóide Triaxial (Portuguese), Ph. D. thesis, Departamento de Matemática, Universidade Federal de Pernambuco, Brazil, 2011. Google Scholar

[34]

T. Sakajo, The motion of three point vortices on a sphere, Japan Jounal of Industrial and Applied Mathematics, 16 (1999), 321-347.  doi: 10.1007/BF03167361.  Google Scholar

[35]

E. Schering, Über Die Conforme Abbildung Des Ellipsoids Auf Der Ebene, Gesammelte Mathematische Werke, ch. Ⅲ, Mayer and Muller, Berlin (1902) (available in http://name.umdl.umich.edu/AAT1702.0001.001). Google Scholar

[36]

J. Steiner, A geometrical mass and its extremal properties for metrics on $S^2$, Duke Math. J., 129 (2005), 63-86.  doi: 10.1215/S0012-7094-04-12913-6.  Google Scholar

[37]

J. Vankerschaver and M. Leok, A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects, J. Nonlinear Science, 24 (2014), 1-37.  doi: 10.1007/s00332-013-9182-5.  Google Scholar

[38]

E. Zermelo, Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche, Z. Math. Phys., 47 (1902), 201-237.   Google Scholar

show all references

References:
[1]

S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, 185–237, Fields Inst. Commun., 73, Springer, New York, 2015. doi: 10.1007/978-1-4939-2441-7_10.  Google Scholar

[2]

V. A. Bogomolov, The dynamics of vorticity on a sphere, (Russian), Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza, 6 (1977), 57-65.   Google Scholar

[3]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.  Google Scholar

[4]

A. V. BolsinovV. S. Matveev and A. T. Fomenko, Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry, Sb. Math., 189 (1998), 1441-1466.  doi: 10.1070/SM1998v189n10ABEH000346.  Google Scholar

[5]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Berlin: Springer-Verlag, 1954.  Google Scholar

[6]

B. C. Carlson, Computing elliptic integrals by duplication, Numerische Mathematik, 33 (1979), 1-16.  doi: 10.1007/BF01396491.  Google Scholar

[7]

C. Castilho and H. Machado, The N-vortex problem on a symmetric ellipsoid: A perturbation approach, J. Math. Phys., 49 (2008), 022703, 12pp. doi: 10.1063/1.2863515.  Google Scholar

[8]

T. Craig, Orthomophic Projection of an Ellipsoid upon a sphere, Amer. J. Math., 3 (1880), 114-127.  doi: 10.2307/2369466.  Google Scholar

[9]

D. G. CrowdyE. H. KropfC. C. Green and M. M. S. Nasser, The Schottky-Klein prime function: A theoretical and computational tool for applications, IMA J. Applied Math., 81 (2016), 589-628.  doi: 10.1093/imamat/hxw028.  Google Scholar

[10]

D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25pp. doi: 10.1098/rspa.2014.0890.  Google Scholar

[11]

I. S. Gradshteyn, J. M. Ryzhik and A. Jeffrey, Table of Integrals, Series, and Products, 4. ed. New York: Academic Press, 1965.  Google Scholar

[12]

F. Goes, L. Beibei, M. Budninskiy, Y. Tong and M. Desbrun, Discrete 2-Tensor Fields on Triangulations, Eurographics Symposium on Geometry Processing 33:5, editors Thomas Funkhouser and Shi-Min Hu, 2014. Google Scholar

[13]

I. Gromeka, Sobranie Socinenii (Russian) (Collected works), Izdat. Akad. Nauk SSSR, Moscow, 1952.  Google Scholar

[14]

D. Hally, Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys., 21 (1980), 211-217.  doi: 10.1063/1.524322.  Google Scholar

[15]

H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, (German), J. Reine Angew. Math., 55 (1858), 25-55.  doi: 10.1515/crll.1858.55.25.  Google Scholar

[16]

M. Henon, On the numerical computation of Poincaré maps, Physica D, 5 (1982), 412-414.  doi: 10.1016/0167-2789(82)90034-3.  Google Scholar

[17]

C. G. J. Jacobi, Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution], Jour. Crelle, 19 (1839), 309-313.  doi: 10.1515/crll.1839.19.309.  Google Scholar

[18]

C. G. J. Jacobi, Vorlesungen über Dynamik [Lectures on Dynamics], edited by Clebsch, Reimer, Berlin, 1866; second edition edited by Weierstrass, 1884; English translation by K. Balagangadharan (Hindustan Book Agency, 2009). Google Scholar

[19]

C. F. Karney, https://geographiclib.sourceforge.io/html/jacobi.html. Google Scholar

[20]

S.-C. Kim, Latitudinal point vortex rings on the spheroid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1749-1768.  doi: 10.1098/rspa.2009.0597.  Google Scholar

[21]

R. Kidambi and P. K. Newton, Motion of three point vortices on a sphere, Physica D: Nonlinear Phenomena, 116 (1998), 143-175.  doi: 10.1016/S0167-2789(97)00236-4.  Google Scholar

[22]

Y. Kimura, Vortex motion on surfaces with constant curvature, Proc. R. Soc. Lond. A, 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.  Google Scholar

[23]

G. R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanic, Teubner, Leipzig, 1876. Google Scholar

[24]

F. Klein, On Riemann Theory of Algebraic Functions and their Integrals, Cambridge: Macmillan and Bowes (1893). Available online www.gutenberg.org/ebooks/36959 Google Scholar

[25]

J. Koiller and S. Boatto, Vortex pairs on surfaces, AIP Conference Proceedings, , 77 (2009), p1130; (Geometry and Physics: ⅩⅦ International Fall Workshop, edited by F. Etayo, M. Fioravanti, and R. Santamaria). Google Scholar

[26]

J. Koiller and K. Ehlers, Rubber rolling over a sphere, Reg. Chaotic Dyn., 12 (2007), 127-152.  doi: 10.1134/S1560354707020025.  Google Scholar

[27]

C. C. Lin, On the Motion of Vortices in Two Dimensions: Ⅰ. Existence of the Kirchhoff-Routh Function; Ⅱ. Some Further Investigations on the Kirchhoff-Routh Function, Proc. Natl.Acad. Sci. USA, 27 (1941), 575-577.   Google Scholar

[28]

A. S. Miguel, Numerical description of the motion of a point vortex pair on ovaloids, J. Phys. A: Math. Theor., 46 (2013), 115502, 21pp. doi: 10.1088/1751-8113/46/11/115502.  Google Scholar

[29]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14pp. doi: 10.1063/1.4897210.  Google Scholar

[30]

J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983.  Google Scholar

[31]

M. V. NyrtsovM. E. FliesM. M. Borisov and P. J. Stooke, Jacobi conformal projection of the triaxial ellipsoid: New projection for mapping of small celestial bodies, Cartography from Pole to Pole, (2014), 235-246.   Google Scholar

[32]

C. Ragazzo, The motion of a vortex on a closed surface: An algorithm and its application to the Bolza surface, Proc. Royal Soc. London A: Mathematical, Physical and Engineering Science, 473 (2017), 20170447, 17 pp. doi: 10.1098/rspa.2017.0447.  Google Scholar

[33]

A. Regis, Dinâmica de Vórtices Pontuais Sobre o Elipsóide Triaxial (Portuguese), Ph. D. thesis, Departamento de Matemática, Universidade Federal de Pernambuco, Brazil, 2011. Google Scholar

[34]

T. Sakajo, The motion of three point vortices on a sphere, Japan Jounal of Industrial and Applied Mathematics, 16 (1999), 321-347.  doi: 10.1007/BF03167361.  Google Scholar

[35]

E. Schering, Über Die Conforme Abbildung Des Ellipsoids Auf Der Ebene, Gesammelte Mathematische Werke, ch. Ⅲ, Mayer and Muller, Berlin (1902) (available in http://name.umdl.umich.edu/AAT1702.0001.001). Google Scholar

[36]

J. Steiner, A geometrical mass and its extremal properties for metrics on $S^2$, Duke Math. J., 129 (2005), 63-86.  doi: 10.1215/S0012-7094-04-12913-6.  Google Scholar

[37]

J. Vankerschaver and M. Leok, A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects, J. Nonlinear Science, 24 (2014), 1-37.  doi: 10.1007/s00332-013-9182-5.  Google Scholar

[38]

E. Zermelo, Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche, Z. Math. Phys., 47 (1902), 201-237.   Google Scholar

Figure 2.  Scheme for the double branched covering of the torus over the ellipsoid. See Proposition 3
Figure 1.  Lines of curvature of the triaxial ellipsoid. Cuts along the top and bottom segments joining the umbilical points results (topologically) on an open cylinder. One could as well make the cuts sidewise. Coordinates $(\lambda_1, \lambda_2)$ cannot be made global
Figure 5.  Poincaré map. Prolate, nearly symmetrical $a = 1, \ b = 1.1, \ c = 9, \ H = -40$
Figure 6.  Poincaré map. Prolate $a = 1, \ b = 2, \ c = 9, \ H = -36$
Figure 7.  Poincaré map. Prolate $a = 1, \ b = 4, \ c = 9, \ H = -60$
Figure 3.  Nearly spherical example $a = 1, \ b = 1.01, \ c = 1.02$
Figure 4.  Ellipsoid $a = 1, \ b = 6, \ c = 9.$
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