Point vortices on the sphere: Stability of symmetric relative equilibria

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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

Abstract
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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.
[2]	V. Bogomolov, Dynamics of vorticity at a sphere,, Fluid Dyn., 6 (1977), 863.
[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.
[4]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations,, in, 306 (2005), 357.
[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.
[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.
[7]	P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry,, Arch. Ration. Mech. Anal., 163 (2002), 1.
[8]	G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation,, Nonlinearity, 15 (2002), 531.
[9]	M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry,, With an appendix by Jerrold Marsden, 24 (1987), 391.
[10]	H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55., English translation, 33 (1867), 485.
[11]	E. Hansen, "A Table of Series and Products,'', Prentice-Hall, (1975).
[12]	R. Kidambi and P. Newton, Motion of three point vortices on a sphere,, Physica D, 116 (1998), 143.
[13]	G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'', Kap.\ XX, (1876).
[14]	L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere,, Chaos, 14 (2004), 592.
[15]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143.
[16]	F. Laurent-Polz, Relative periodic orbits in point vortex systems,, Nonlinearity, 17 (2004), 1989.
[17]	F. Laurent-Polz, Point vortices on a rotating sphere,, Regul. Chaotic Dyn., 10 (2005), 39.
[18]	F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'', Ph.D Thesis, (2002).
[19]	C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97.
[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.
[21]	J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'', Lecture Notes in Mathematics, 1160 (1160).
[22]	G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices,, Phys. Fluids, 21 (1978), 1092.
[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.
[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.
[25]	J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449.
[26]	J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., ().
[27]	J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, ().
[28]	J. Montaldi and M. Roberts, Relative equilibria of molecules,, J. Nonlinear Sci., 9 (1999), 53.
[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.
[30]	J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems,, Nonlinearity, 3 (1990), 695.
[31]	J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder,, SIAM J. on Applied Dynamical Systems, 2 (2003), 417.
[32]	J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().
[33]	P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'', Applied Mathematical Sciences, 145 (2001).
[34]	J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'', Ph.D Thesis, (1998).
[35]	R. Palais, Principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19.
[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.
[37]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance,, Math. Z., 232 (1999), 747.
[38]	L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere,, J. Fluid Mech., 255 (1993), 35.
[39]	S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria,, J. Math. Phys., 39 (1998), 5894.
[40]	J.-P. Serre, "Représentations Linéaires des Groupes Finis,'', Third revised edition, (1978).
[41]	A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry,, J. Fluid Mech., 460 (2002), 83.
[42]	T. Tokieda, Tourbillons dansants,, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943.

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.
[2]	V. Bogomolov, Dynamics of vorticity at a sphere,, Fluid Dyn., 6 (1977), 863.
[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.
[4]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations,, in, 306 (2005), 357.
[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.
[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.
[7]	P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry,, Arch. Ration. Mech. Anal., 163 (2002), 1.
[8]	G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation,, Nonlinearity, 15 (2002), 531.
[9]	M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry,, With an appendix by Jerrold Marsden, 24 (1987), 391.
[10]	H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55., English translation, 33 (1867), 485.
[11]	E. Hansen, "A Table of Series and Products,'', Prentice-Hall, (1975).
[12]	R. Kidambi and P. Newton, Motion of three point vortices on a sphere,, Physica D, 116 (1998), 143.
[13]	G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'', Kap.\ XX, (1876).
[14]	L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere,, Chaos, 14 (2004), 592.
[15]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143.
[16]	F. Laurent-Polz, Relative periodic orbits in point vortex systems,, Nonlinearity, 17 (2004), 1989.
[17]	F. Laurent-Polz, Point vortices on a rotating sphere,, Regul. Chaotic Dyn., 10 (2005), 39.
[18]	F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'', Ph.D Thesis, (2002).
[19]	C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97.
[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.
[21]	J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'', Lecture Notes in Mathematics, 1160 (1160).
[22]	G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices,, Phys. Fluids, 21 (1978), 1092.
[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.
[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.
[25]	J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449.
[26]	J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., ().
[27]	J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, ().
[28]	J. Montaldi and M. Roberts, Relative equilibria of molecules,, J. Nonlinear Sci., 9 (1999), 53.
[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.
[30]	J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems,, Nonlinearity, 3 (1990), 695.
[31]	J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder,, SIAM J. on Applied Dynamical Systems, 2 (2003), 417.
[32]	J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().
[33]	P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'', Applied Mathematical Sciences, 145 (2001).
[34]	J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'', Ph.D Thesis, (1998).
[35]	R. Palais, Principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19.
[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.
[37]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance,, Math. Z., 232 (1999), 747.
[38]	L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere,, J. Fluid Mech., 255 (1993), 35.
[39]	S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria,, J. Math. Phys., 39 (1998), 5894.
[40]	J.-P. Serre, "Représentations Linéaires des Groupes Finis,'', Third revised edition, (1978).
[41]	A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry,, J. Fluid Mech., 460 (2002), 83.
[42]	T. Tokieda, Tourbillons dansants,, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943.

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