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December  2011, 3(4): 439-486. doi: 10.3934/jgm.2011.3.439

Point vortices on the sphere: Stability of symmetric relative equilibria

Frederic Laurent-Polz 1, , James Montaldi 2, and  Mark Roberts 3,

1. 

Institut Non Linéaire de Nice, 1361 route des Lucioles, 06560 Valbonne, France
2. 

School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
3. 

Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received  March 2011 Revised  May 2011 Published  February 2012

We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.
Keywords: bifurcations, Hamiltonian systems, stability, symmetry., Point vortices.
Mathematics Subject Classification: Primary: 37J15, 37J20, 37J25, 70H33, 76B4.
Citation: Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439
References:
[1]

H. Aref, P. Newton, M. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals. Adv. Appl. Mech., 39 (2003), 1-79. Google Scholar

[2]

V. Bogomolov, Dynamics of vorticity at a sphere, Fluid Dyn., 6 (1977), 863-870. Google Scholar

[3]

S. Boatto and H. E. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere, SIAM J. Appl. Math., 64 (2003), 216-230.  Google Scholar

[4]

P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations, in "Geometric Mechanics and Symmetry," Based on lectures by Montaldi, LMS Lecture Note Series, 306, Cambridge University Press, Cambridge, (2005), 357-402.  Google Scholar

[5]

H. E. Cabral, K. R. Meyer and D. S. Schmidt, Stability and bifurcations for the $N+1$ vortex problem on the sphere, Regular and Chaotic Dynamics, 8 (2003), 259-282.  Google Scholar

[6]

H. E. Cabral and D. S. Schmidt, Stability of relative equilibria in the problem of $N+1$ vortices,, SIAM J. Math. Anal., 31 (): 231.   Google Scholar

[7]

P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry, Arch. Ration. Mech. Anal., 163 (2002), 1-33. Correction to "Hamiltonian Hopf bifurcation with symmetry'', Arch. Ration. Mech. Anal., 167 (2002), 83-84.  Google Scholar

[8]

G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation, Nonlinearity, 15 (2002), 531-549.  Google Scholar

[9]

M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry, With an appendix by Jerrold Marsden, Physica D, 24 (1987), 391-405.  Google Scholar

[10]

H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55. English translation, On integrals of the hydrodynamical equations which express vortex motion, Phil. Mag., 33 (1867), 485-512. Google Scholar

[11]

E. Hansen, "A Table of Series and Products,'' Prentice-Hall, 1975. Google Scholar

[12]

R. Kidambi and P. Newton, Motion of three point vortices on a sphere, Physica D, 116 (1998), 143-175.  Google Scholar

[13]

G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'' Kap.\ XX, Teubner, Leipzig, 1876. Google Scholar

[14]

L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere, Chaos, 14 (2004), 592-602.  Google Scholar

[15]

F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2002), 143-171.  Google Scholar

[16]

F. Laurent-Polz, Relative periodic orbits in point vortex systems, Nonlinearity, 17 (2004), 1989-2013.  Google Scholar

[17]

F. Laurent-Polz, Point vortices on a rotating sphere, Regul. Chaotic Dyn., 10 (2005), 39-58.  Google Scholar

[18]

F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'' Ph.D Thesis, University of Nice, 2002. Google Scholar

[19]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135.  Google Scholar

[20]

J. Marsden, S. Pekarsky and S. Shkoller, Stability of relative equilibria of point vortices on a sphere and symplectic integrators, Nuovo Cimento C, 22 (1999), 793-802. Google Scholar

[21]

J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'' Lecture Notes in Mathematics, 1160, Springer-Verlag, Berlin, 1985.  Google Scholar

[22]

G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices, Phys. Fluids, 21 (1978), 1092-1095. Google Scholar

[23]

K. R. Meyer and D. S. Schmidt, Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh, Celest. Mech., 4 (1971), 99-109.  Google Scholar

[24]

K. R. Meyer and D. S. Schmidt, Bifurcations of relative equilibria in the $N$-body and Kirchhoff problems, SIAM J. Math. Anal., 19 (1988), 1295-1313.  Google Scholar

[25]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  Google Scholar

[26]

J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., ().   Google Scholar

[27]

J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, ().   Google Scholar

[28]

J. Montaldi and M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  Google Scholar

[29]

J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. London Ser. A, 325 (1988), 237-293.  Google Scholar

[30]

J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.  Google Scholar

[31]

J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder, SIAM J. on Applied Dynamical Systems, 2 (2003), 417-430.  Google Scholar

[32]

J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().   Google Scholar

[33]

P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'' Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001.  Google Scholar

[34]

J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'' Ph.D Thesis, University of California, Santa Cruz, 1998. Google Scholar

[35]

R. Palais, Principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

[36]

G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, J. Geom. Phys., 9 (1992), 111-119.  Google Scholar

[37]

G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance, Math. Z., 232 (1999), 747-788.  Google Scholar

[38]

L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech., 255 (1993), 35-64.  Google Scholar

[39]

S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria, J. Math. Phys., 39 (1998), 5894-5907.  Google Scholar

[40]

J.-P. Serre, "Représentations Linéaires des Groupes Finis,'' Third revised edition, Hermann, Paris, 1978.  Google Scholar

[41]

A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry, J. Fluid Mech., 460 (2002), 83-92.  Google Scholar

[42]

T. Tokieda, Tourbillons dansants, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943-946.  Google Scholar

show all references

References:
[1]

H. Aref, P. Newton, M. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals. Adv. Appl. Mech., 39 (2003), 1-79. Google Scholar

[2]

V. Bogomolov, Dynamics of vorticity at a sphere, Fluid Dyn., 6 (1977), 863-870. Google Scholar

[3]

S. Boatto and H. E. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere, SIAM J. Appl. Math., 64 (2003), 216-230.  Google Scholar

[4]

P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations, in "Geometric Mechanics and Symmetry," Based on lectures by Montaldi, LMS Lecture Note Series, 306, Cambridge University Press, Cambridge, (2005), 357-402.  Google Scholar

[5]

H. E. Cabral, K. R. Meyer and D. S. Schmidt, Stability and bifurcations for the $N+1$ vortex problem on the sphere, Regular and Chaotic Dynamics, 8 (2003), 259-282.  Google Scholar

[6]

H. E. Cabral and D. S. Schmidt, Stability of relative equilibria in the problem of $N+1$ vortices,, SIAM J. Math. Anal., 31 (): 231.   Google Scholar

[7]

P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry, Arch. Ration. Mech. Anal., 163 (2002), 1-33. Correction to "Hamiltonian Hopf bifurcation with symmetry'', Arch. Ration. Mech. Anal., 167 (2002), 83-84.  Google Scholar

[8]

G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation, Nonlinearity, 15 (2002), 531-549.  Google Scholar

[9]

M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry, With an appendix by Jerrold Marsden, Physica D, 24 (1987), 391-405.  Google Scholar

[10]

H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55. English translation, On integrals of the hydrodynamical equations which express vortex motion, Phil. Mag., 33 (1867), 485-512. Google Scholar

[11]

E. Hansen, "A Table of Series and Products,'' Prentice-Hall, 1975. Google Scholar

[12]

R. Kidambi and P. Newton, Motion of three point vortices on a sphere, Physica D, 116 (1998), 143-175.  Google Scholar

[13]

G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'' Kap.\ XX, Teubner, Leipzig, 1876. Google Scholar

[14]

L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere, Chaos, 14 (2004), 592-602.  Google Scholar

[15]

F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2002), 143-171.  Google Scholar

[16]

F. Laurent-Polz, Relative periodic orbits in point vortex systems, Nonlinearity, 17 (2004), 1989-2013.  Google Scholar

[17]

F. Laurent-Polz, Point vortices on a rotating sphere, Regul. Chaotic Dyn., 10 (2005), 39-58.  Google Scholar

[18]

F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'' Ph.D Thesis, University of Nice, 2002. Google Scholar

[19]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135.  Google Scholar

[20]

J. Marsden, S. Pekarsky and S. Shkoller, Stability of relative equilibria of point vortices on a sphere and symplectic integrators, Nuovo Cimento C, 22 (1999), 793-802. Google Scholar

[21]

J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'' Lecture Notes in Mathematics, 1160, Springer-Verlag, Berlin, 1985.  Google Scholar

[22]

G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices, Phys. Fluids, 21 (1978), 1092-1095. Google Scholar

[23]

K. R. Meyer and D. S. Schmidt, Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh, Celest. Mech., 4 (1971), 99-109.  Google Scholar

[24]

K. R. Meyer and D. S. Schmidt, Bifurcations of relative equilibria in the $N$-body and Kirchhoff problems, SIAM J. Math. Anal., 19 (1988), 1295-1313.  Google Scholar

[25]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  Google Scholar

[26]

J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., ().   Google Scholar

[27]

J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, ().   Google Scholar

[28]

J. Montaldi and M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  Google Scholar

[29]

J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. London Ser. A, 325 (1988), 237-293.  Google Scholar

[30]

J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.  Google Scholar

[31]

J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder, SIAM J. on Applied Dynamical Systems, 2 (2003), 417-430.  Google Scholar

[32]

J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().   Google Scholar

[33]

P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'' Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001.  Google Scholar

[34]

J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'' Ph.D Thesis, University of California, Santa Cruz, 1998. Google Scholar

[35]

R. Palais, Principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

[36]

G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, J. Geom. Phys., 9 (1992), 111-119.  Google Scholar

[37]

G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance, Math. Z., 232 (1999), 747-788.  Google Scholar

[38]

L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech., 255 (1993), 35-64.  Google Scholar

[39]

S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria, J. Math. Phys., 39 (1998), 5894-5907.  Google Scholar

[40]

J.-P. Serre, "Représentations Linéaires des Groupes Finis,'' Third revised edition, Hermann, Paris, 1978.  Google Scholar

[41]

A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry, J. Fluid Mech., 460 (2002), 83-92.  Google Scholar

[42]

T. Tokieda, Tourbillons dansants, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943-946.  Google Scholar

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