Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-19T01:22:06.248Z Has data issue: false hasContentIssue false
  • English
  • Français

The repeated filamentation of two-dimensional vorticity interfaces

Published online by Cambridge University Press:  21 April 2006

David G. Dritschel
David G. Dritschel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Undular disturbances of arbitrarily small steepness to the boundary of a circular patch of uniform vorticity are found to give way to a qualitatively different type of behaviour after a time inversely proportional to the initial wave steepness squared. Repeatedly, thin filaments are drawn out at a frequency of nearly half the vorticity jump across the interface (the intrinsic frequency of linear waves on the interface), and the interaction between the filaments and the undulating boundary generate new disturbances from which still greater numbers of filaments of increasingly complicated shape are drawn out. The vortex boundary thereby experiences an extraordinary, continual and apparently irreversible growth in complexity.

Essentially the same phenomenon occurs on all vortex-patch equilibria, e.g. the Kirchhoff elliptical vortex, with even greater complexity. The steepening of a disturbance on a non-circular vortex proceeds faster than that on a circular vortex, because of the varying mean strain and shear seen by the disturbance as it travels around the vortex boundary.

Generalizations to more than one vorticity interface, to flows on the surface of a sphere, and to sharp but not infinitely sharp vorticity gradients are also discussed. The results support the view that almost any two-dimensional, inviscid, incompressible flow with large vorticity gradients will exhibit repeated filamentation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 194 , September 1988 , pp. 511 - 547
Copyright
© 1988 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

Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary V-states, interactions, recurrence, and breaking. Phys. Rev. Lett. 40, 859–862. (Also see Stationary V-states, interactions, recurrence, and breaking. In Solitons in Action (ed. K. Longren & A. Scott), pp. 277–293. Academic.)Google Scholar
Dritschel, D. G. 1980 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Dritschel, D. G. 1988a Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows. J. Fluid Mech. 191, 575–582.Google Scholar
Dritschel, D. G. 1988b Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77 (in press).Google Scholar
Dritschel, D. G. 1988c Contour dynamics/surgery on the sphere. J. Comput. Phys. (in press).Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Haynes, P. H. 1987 On the instability of sheared disturbances. J. Fluid Mech. 175, 463–478.Google Scholar
Juckes, M. N. & McIntyre, M. E. 1987 A high-resolution one-layer model of breaking planetary waves in the stratosphere. Nature 328, 590–596.Google Scholar
Lamb, H. H. 1932 Hydrodynamics. Dover.
Love, A. E. H. 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. 25, 18–34.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1983 Breaking planetary waves in the stratosphere. Nature 305, 593–600.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1984 The ‘surf-zone’ in the stratosphere. J. Atmos. Terr. Phys. 46, 825–849.Google Scholar
McIntyre, M. E. & Palmer, T. N. 1985 A note on the general concept of wave breaking for Rossby and gravity waves. PAGEOPH 123, 964–975.Google Scholar
Melander, M. V., McWilliams, J. C. & Zabusky, N. J. 1987 Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137–159.Google Scholar
Moffatt, H. K. & Moore, D. W. 1978 The response of Hill's spherical vortex to a small axisymmetric perturbation. J. Fluid Mech. 87, 749–760.Google Scholar
Pullin, D. I. 1981 The nonlinear behaviour of a constant vorticity layer at a wall. J. Fluid Mech. 108, 401–421.Google Scholar
Shariff, K. 1988 Ph.D. Thesis in preparation, Dept. of Mech. Engng, Stanford Univ., California, USA.
Stern, M. E. 1985 Lateral wave breaking and shingle formation in large scale shear flow. J. Phys. Oceanogr. 15, 1274–1283.Google Scholar
Stern, M. E. & Pratt, L. J. 1985 Dynamics of vorticity fronts. J. Fluid Mech. 161, 513–532.Google Scholar
Thomson, W. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Wan, Y. H. & Pulvirenti, M. 1985 Nonlinear stability of a circular vortex. Commun. Math. Phys. 99, 435–450.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.Google Scholar