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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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).

[19]

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

[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.

[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.

[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.

[23]

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

[24]

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

[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).

[26]

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

[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.

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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).

[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.

[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.

[38]

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

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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).

[19]

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

[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.

[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.

[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.

[23]

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

[24]

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

[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).

[26]

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

[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.

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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).

[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.

[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.

[38]

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

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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