Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-18T19:01:34.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations

Published online by Cambridge University Press:  26 April 2006

Darryl D. Holm
Darryl D. Holm
Affiliation:
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 393 - 406
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

With an Appendix by D. David and T. K. Ohsumi.

References

Ball, F. K. 1963 Some general theorems concerning the finite motion of a shallow liquid lying on a paraboloid. J. Fluid Mech. 17, 240–256.Google Scholar
Ball, F. K. 1965 The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J. Fluid Mech. 22, 529–545.Google Scholar
Basset, A. B. 1880 A Treatise on Hydrodynamics, vol. II Chap. XV. Cambridge: Deighton-Bell. (Reprinted by Dover 1961.)
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.
Cushman-Roisin, B. 1987 Exact analytical solutions for elliptical vortices of the shallow-water equations. Tellus 39A, 235–244.Google Scholar
Cushman-Roisin, B., Hell, W. H. & Nof, D. 1985 Oscillations and rotations of elliptical warmcore rings. J. Geophys. Res. 90, 11756–11764.Google Scholar
Dedekind, R. 1860 Zusatz zu der vorstehenden Abhandlung. J. Reine Angew. Math. 58, 217–228.Google Scholar
Dirichlet, G. L. 1860 Untersuchungen über ein Problem der Hydrodynamik. J. Reine Angew. Math. 58, 181–216.Google Scholar
Dyson, F. J. 1968 Dynamics of a spinning gas cloud. J. Math. Mech. 18, 91–101.Google Scholar
Goldsbrough, G. R. 1930 The tidal oscillations in an elliptic basin of variable depth.. Proc. R. Soc. Lond. A 130, 157–167.Google Scholar
Greenhill, A. G. 1879 On the rotation of a liquid ellipsoid about its mean axis. Proc. Camb. Phil. Soc. 3, 233–246.Google Scholar
Holm, D. D. 1982 Gyroscopic analog for magnetohydrodynamics. In Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems (ed. M. Tabor & Y. Treve), pp. 73–84. American Institute of Physics.
Holm, D. D. 1983 Magnetic tornadoes: three-dimensional affine motions in ideal magnetohydrodyamics.. Physica D 8, 170–182.Google Scholar
Holm, D. D. 1986 Gyroscopic analog for collective motion of a stratified fluid. J. Math. Anal. Appl. 117, 57–80.Google Scholar
Holm, D. D. & Long, B. 1989 Lyapunov stability of ideal stratified fluid equilibria in hydrostatic balance. Nonlinearity 2, 23–35.Google Scholar
Kirchhoff, G. 1876 Vorlesungen über Mathematische Physik, vol. 1. Leipzig: Teubner.
Kirwan, A. D. & Liu, J. 1991 The shallow water equations on an F plane In Proc. Intl School of Physics Enrico Fermi, Varenna, Italy.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press. (Reprinted by Dover 1945.)
Riemann, B. 1860 Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides. Abh. d. Königl. Gesell. der Wis. zu Göttingen 9, 3–36.Google Scholar
Ripa, P. 1987 On the stability of elliptical vortex solutions of the shallow-water equations. J. Fluid Mech. 183, 343–363.Google Scholar
Rogers, C. 1989 Elliptic warm-core theory: the pulsrodon.. Phys. Lett. A 138, 267–273.Google Scholar
Rogers, C. & Ames, W. F. 1989 Nonlinear Boundary Value Problems in Science and Engineering, 4.12. Academic.
Thacker, W. C. 1981 Some exact solutions to the nonlinear shallow water wave equations. J. Fluid Mech. 107, 499–508.Google Scholar
Young, W. R. 1986 Elliptical vortices in shallow water. J. Fluid Mech. 171, 101–119.Google Scholar