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Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger

Published online by Cambridge University Press:  26 April 2006

L. M. Polvani ,
N. J. Zabusky  and
G. R. Flierl
L. M. Polvani
Affiliation:
Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Mathematics, Room 2–339, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
N. J. Zabusky
Affiliation:
Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA
G. R. Flierl
Affiliation:
Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

We generalize the methods of two-dimensional contour dynamics to study a two-layer rotating fluid that obeys the quasi-geostrophic equations. We consider here only the case of a constant-potential-vorticity lower layer. We derive equilibrium solutions for monopolar (rotating) and dipolar (translating) geostrophic vortices in the upper layer, and compare them with the Euler case. We show that the equivalent barotropic (infinite lower layer) case is a singular limit of the two-layer system. We also investigate the effect of a finite lower layer on the merger of two regions of equal-sign potential vorticity in the upper layer. We discuss our results in the light of the recent laboratory experiments of Griffiths & Hopfinger (1986). The process of filamentation is found to be greatly suppressed for equivalent barotropic dynamics on scales larger than the radius of deformation. We show that the variation of the critical initial distance for merger as a function of the radius of deformation and the ratio of the layers at rest is closely related to the existence of vortex-pair equilibria and their geometrical properties.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 205 , August 1989 , pp. 215 - 242
Copyright
© 1989 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. Longren & A. Scott), pp. 277–293. Academic.
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Dritschel, D. G. 1988 Contour surgery: a contour dynamics method for long-time behaviour of two-dimensional, inviscid, rotational flow. J. Comput. Phys. 77, 240–266.Google Scholar
Griffiths, R. W. & Hopfinger, E. J. 1986 Experiments with baroclinic vortex pairs in a rotating fluid. J. Fluid Mech. 173, 501–518.Google Scholar
Griffiths, R. W. & Hopfinger, E. J. 1987 Coalescing geostrophic vortices. J. Fluid Mech. 178, 73–79.Google Scholar
Gryanik, V. M. 1983 Dynamics of singular geostrophic vortices in a two-level model of the atmosphere (or the ocean). Izv. Akad. Nauk. SSSR Atmos. Ocean. Phys. 19, 171–179.Google Scholar
Hogg, N. & Stommel, H. 1985 The heton, an elementary interaction between discrete baroclinic geostrophic vortices and its implications concerning eddy heat-flow. Proc. R. Soc. Lond. A 397, 1–20.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
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
McWilliams, J. C. 1989 Statistical properties of decaying geostrophic turbulence. J. Fluid Mech. 198, 199–230.Google Scholar
Melander, M. V., McWilliams, J. C. & Zabusky, N. J. 1987a Axisymmetrization and vorticity gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137–159.Google Scholar
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1987b Asymmetric vortex merger in two-dimensions: Which vortex is ‘victorious’? Phys. Fluids 30, 2610–2612.Google Scholar
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1988 Symmetric vortex merger in two-dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Melander, M. V., Zabusky, N. J. & Styczek, A. S. 1986 A moment model for vortex interactions of the two-dimensional Euler equations. Part I. Computational validation of a Hamiltonian elliptical representation. J. Fluid Mech. 167, 95–115.Google Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–431.Google Scholar
Overman, E. A. & Zabusky, N. J. 1982a Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297–1305.Google Scholar
Overman, E. A. & Zabusky, N. J. 1982b Coaxial scattering of Euler equation translating V-states via contour dynamics. J. Fluid Mech. 125, 187–202.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics, chap. 6. Springer.
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasigeostrophic model. Tellus 6, 273–286.Google Scholar
Pierrehumbert, R. T. 1980 A family of steadily translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
Polvani, L. M. 1988 Geostrophic vortex dynamics. Ph.D. Thesis, Massachusetts Institute of Technology, 1988.
Polvani, L. M., Flierl, G. R. & Zabusky, N. J. 1989 Filamentation of coherent vortex structures via separatrix crossing: a quantitative estimate of onset time. Phys. Fluids. A 1, 181–184.Google Scholar
Polvani, L. M., Zabusky, N. J. & Flierl, G. R. 1988 Applications of contour dynamics to two-layer quasi-geostrophic. Fluid Dyn. Res. 3, 422–424.Google Scholar
Pratt, L. J. & Stern, M. E. 1986 Dynamics of potential vorticity fronts and eddy detachment. J. Phys. Oceanogr. 16, 1101–1120.Google Scholar
Saffman, P. G. & Schatzman, J. C. 1982 Stabilities 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 equal uniform vortices. Phys. Fluids 23, 2339–2342.Google Scholar
Salmon, R. 1982 Geostrophic turbulence. In Turbolenza e Predicibilità nella Fluidodinamica Geofisica e la Dinamica del Clima. Scuola Internazionale di Fisica Enrico Fermi, LXXXVIII, pp. 113–158.
Williams, G. P. & Wilson, R. J. 1988 The stability and genesis of Rossby vortices. J. Atmos. Sci. 45, 207–241.Google Scholar
Wu, H. M., Overman, E. A. & Zabusky, N. J. 1984 Steady state solutions of the Euler equation in two dimensions. Rotating and translating V-states with limiting cases. I. Numerical results. J. Comput. Phys. 53, 42–71.Google Scholar
Young, W. R. 1985 Some interactions between small numbers of baroclinic geostrophic vortices. Geophys. Astrophys. Fluid Dyn. 33, 42–71.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
Zou, Q., Overman, E. A., Wu, H. M. & Zabusky, N. J. 1988 Contour dynamics for the Euler equations: curvature controlled initial node placement and accuracy. J. Comput. Phys. 78, 350–372.Google Scholar