Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T01:26:28.436Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability and energetics of corotating uniform vortices

Published online by Cambridge University Press:  20 April 2006

David G. Dritschel
David G. Dritschel
Affiliation:
Geophysical Fluid Dynamics Program, Princeton University. New Jersey 08540
Get access Rights & Permissions [Opens in a new window]

Abstract

Equilibrium shapes of two-dimensional rotating configurations of uniform vortices are numerically calculated for two to eight corotating vortices. Additionally, a perturbation series is developed which approximately describes the vortex shapes. The equilibrium configurations are subjected to a linear stability analysis. This analysis both confirms existing results regarding point vortices and shows that finite vortices may destabilize via a new form of instability derived from boundary deformations. Finally, we examine the energetics of the equilibrium configurations. We introduce a new energy quantity called ‘excess energy’, which is particularly useful in understanding the constraints on the evolution of unstable near-equilibrium configurations. This theory offers a first glance at nonlinear stability. As an example, the theory explains some features of the merger of two vortices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 157 , August 1985 , pp. 95 - 134
Copyright
© 1985 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.)

References

Arnol'D V.1965 Conditions for the nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978. [English transl. Sov. Maths 6, 331–334.]Google Scholar
Christiansen, J. P. & Zabusky N. J.1973 Instability, coalescence and fission of finite-area vortex structures. J. Fluid Mech. 61, 219–243.Google Scholar
Dieudonné J.1969 Foundations of Modern Analysis, pp. 112–113. Academic.
Dritschel D. G.1985 The nonlinear stability of corotating uniform vortices. Submitted to J. Fluid Mech.Google Scholar
Havelock T. H.1931 The stability of motion of rectilinear vortices in ring formation. Phil. Mag. 11, 617.Google Scholar
Lamb H. H.1932 Hydrodynamics. Dover.
Love A. E. H.1893 Proc. Lond. Math. Soc. 35, 18.
Morikawa, G. K. & Swenson E. V.1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058–1073.Google Scholar
Overman, E. A. & Zabusky N. J.1982 Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297–1305.Google Scholar
Pierrehumbert R. T.1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
Pierrehumbert, R. T. & Widnall S. E.1981 The structure of organized vortices in a free shear layer. J. Fluid Mech. 102, 301–313.Google Scholar
Poincaré H.1893 Théorie des Tourbillons (ed. G. Carré), chap. 4. Deslis Frères.
Saffman, P. G. & Schatzman J. C.1982 Stability of a vortex street of finite vortices. J. Fluid Mech. 117, 171–185.Google Scholar
Saffman, P. G. & Szeto R.1980 Equilibrium shapes of a pair of uniform vortices. Phys. Fluids 23, 2339–2342.Google Scholar
Snow J. T.1978 On inertial instability as related to the multiple-vortex phenomenon. J. Atmos. Sci. 35, 1660–1667.Google Scholar
Thomson J. J.1883 A Treatise on the Motion of Vortex Rings, pp. 94–108. Macmillan.
Ting L.1983 On the application of the integral invariants and decay laws of vorticity distributions. J. Fluid Mech. 17, 497–506.Google Scholar
Wu H. M., Overman, E. A. & Zabusky N. J.1982 Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases. I. Numerical algorithms and results. Univ. Pittsburgh Tech. Rep. ICMA-82-44.Google Scholar