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Coaxial scattering of Euler-equation translating V-states via contour dynamics

Published online by Cambridge University Press:  20 April 2006

Edward A. Overman  and
Norman J. Zabusky
Edward A. Overman
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15261
Norman J. Zabusky
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15261
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Abstract

The robustness of localized states that transport energy and mass is assessed by a numerical study of the Euler equation in two space dimensions. The localized states are the translating ‘V-states’ discovered by Deem & Zabusky. These piecewise- constant dipolar (i.e. oppositely-signed ± or ±) vorticity regions are steady translating solutions of the Euler equations. A new adaptive contour-dynamical algorithm with curvature-controlled node insertion and removal is used. The evolution of one V-state, subject to a symmetric-plus-asymmetric perturbation is examined and stable (i.e. non-divergent) fluctuations are observed. For scattering interactions, coaxial head-on (or ± on ±) and head-tail (or & on ±) arrangements are studied. The temporal variation of contour curvature and perimeter after V-states separate indicate that internal degrees of freedom have been excited. For weak interactions we observe phase shifts and the near recurrence to initial states. When two similar, equal-circulation but unequal-area V-states have a head-on interaction a new asymmetric state is created by contour ‘exchange’. There is strong evidence that this is near to a V-state. For strong interactions we observe phase shifts, ‘breaking’ (filament formation) and, for head-tail interactions, merger of like-signed vorticity regions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 125 , December 1982 , pp. 187 - 202
Copyright
© 1982 Cambridge University Press

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References

Deem, G. S. & Zabusky, N. J. 1978a Stationary. V-states: interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859.Google Scholar
Deem, G. S. & Zabusky, N. J. 1978b Stationary ‘V-states: interactions, recurrence and breaking. In Solitons in Action (ed. K. Lonngren & A. Scott), pp. 277–293, Academic.
Flierl, G. R., Larichev, V. D., McWilliams, J. C. & Reznick, G. M. 1980 The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans. 5, 1–41.Google Scholar
Love, A. E. H. 1894 On the motion of paired vortices with a common axis. Proc. Lond. Math. Soc. 25, 185.Google Scholar
Mcwilliams, J. C. 1980 An application of equivalent modons to atmospheric blocking. Dyn. Atmos. Oceans 4, 43–46.Google Scholar
Mcwilliams, J. C. & Zabusky, N. J. 1982 Interactions of isolated vortices. I. Modons colliding with modons. Geophys. Astrophys. Fluid Dyn. 19, 207–227.Google Scholar
Makhankov, V. G. 1980 Computer experiments in soliton theory. Comp. Phys. Commun. 21, 1–49.Google Scholar
Overman Ii, E. A. & Zabusky, N. J. 1982 Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297–1306.Google Scholar
Pierrehumbert, R. T. 1980 A family of steady translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
Simonov, Yu, A. & Tjon, J. A. 1980 Inelastic effects in classical field-theoretical models with confinement. Ann. Phys. (N.Y.) 129, 110–130.Google Scholar
Wu, H. M., Overman II, E. A. & Zabusky, N. J. 1982 Steady-state solutions of the Euler equations in two dimensions. Rotating and translating V-states with limiting cases. Preprint, I.C.M.A., University of Pittsburgh. Submitted to J. Comp. Phys.Google Scholar
Zabusky, N. J. 1981 Computational synergetics and mathematical innovation. J. Comp. Phys. 43, 195–249.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comp. Phys. 30, 96–106.Google Scholar
Zabusky, N. J. & Overman II, E. A. 1982 Regularization of contour dynamical algorithms. J. Comp. Phys. (to be published).Google Scholar