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Two-layer geostrophic vortex dynamics. Part 2. Alignment and two-layer V-states

Published online by Cambridge University Press:  26 April 2006

L. M. Polvani
L. M. Polvani
Affiliation:
Department of Mathematics, Room 2-339, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Applied Physics, Columbia University, New York, NY 10027, USA.
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Abstract

The process of alignment, a new fundamental interaction between vortices in a stratified and rapidly rotating fluid, is defined and studied in detail in the context of the two-layer quasi-geostrophic model. Alignment occurs when two vortices in different density layers coalesce by reducing their horizontal separation. It is found that only vortices whose radii are comparable with or larger than the Rossby deformation radius can align. In the same way as the merger process (in a single two-dimensional layer) is related to the reverse energy cascade of two-dimensional turbulence, geostrophic potential vorticity alignment is related the barotropic-to-baroclinic energy cascade of geostrophic turbulence in two layers. It is also shown how alignment is intimately connected with the existence of two-layer doubly connected geostrophic potential vorticity equilibria (V-states), for which the analysis of the geometry of the stream function in the corotating frame is found to be a crucial diagnostic. The finite-area analogues of the hetons of Hogg & Stommel (1985) are also determined: they consist of a propagating pair of opposite-signed potential vorticity patches located in different layers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 241 - 270
Copyright
© 1991 Cambridge University Press

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References

Babiano, A., Basdevant, C., Legbrs, B. & Sadourny, R., 1986 Vorticity and passive scalar dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379–397.Google Scholar
Benzi, R., Paternello, S. & Santangelo, P., 1988 Self-similar coherent structures in two-dimensional decaying turbulence. J. Phys A 21, 1221–1237.Google Scholar
Charney, J. C.: 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.Google Scholar
Deem, G. S. & Zabusky, N. J., 1978 Stationary V-states: interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859.Google Scholar
Dbitschel, D. G.: 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Dritschel, D. G.: 1988a Contour surgery: a contour dynamics method for long time behavior of two-dimensional, inviscid, rotational flow. J. Comput. Phys. 77, 240–266.Google Scholar
Dbitschel, D. G.: 1988b The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511–532.Google Scholar
Fliebl, G. R.: 1988 On the instability of geostrophic vortices. J. Fluid Mech. 97, 349–388.Google Scholar
Fliebl, G. R., Larichev, V. D., McWilllams, J. C. & Reznik, G. M., 1980 The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Ocean 5, 1–41.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
Gryanick, V. M.: 1983 Dynamics of singular geostrophic vortices in a two-level model of the atmosphere (or the ocean). Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 9, 171–179.Google Scholar
Helfrich, K. & Send, U., 1988 Finite amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech. 197, 331–348.Google Scholar
Herring, J. R.: 1980 Statistical theory of quasigeostrophic turbulence. J. Atmos. Sci. 37, 969–977.Google Scholar
Hogg, N. & Stommel, H., 1985 The heton, an elementary interaction between discrete baroclinic geostrophic vortices and its implication concerning eddy heat-flow. Proc. R. Soc. Lond. A 397, 1–20.Google Scholar
Hua, L. B. & Haidvogel, D. B., 1986 Numerical simulation of the vertical structure of quasigeostrophic turbulence. J. Atmos. Sci. 43, 2923–2936.Google Scholar
Kraichnan, R. H. & Montgomery, D., 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.Google Scholar
McWilllams, 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., McWilllams, 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
Melander, M. V., Zabusky, N. J. & McWilllams, J. C., 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Polvani, L. M.: 1988 Geostrophic vortex dynamics. Ph.D. thesis, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, WHOI-88-48.
Polvani, L. M. & Carton, X. J., 1989 The tripole: a new coherent vortex structure of incompressible two-dimensional flows. Geophys. Astrophys. Fluid Dyn. 51, 87–102.Google Scholar
Polvani, L. M., Flierl, G. R. & Zabusky, N. J., 1989b 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., 1989a Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger. J. Fluid Mech. 205, 215–242.Google Scholar
Pullin, D. I., Jacobs, P. A., Grimshaw, R. H. J. & Saffman, P. G. 1989 Stability and filamentation of finite-amplitude waves on vortex layers of finite thickness. J. Fluid Mech. 209, 359–384.Google Scholar
Rhines, P. B.: 1979 Geostrophic turbulence. Ann. Rev. Fluid Mech. 11, 401–441.Google Scholar
Saffman, P. G. & Szeto, R., 1979 Equilibrium shapes of a pair equal uniform vortices. Phys. Fluids 23, 171–185.Google Scholar
Verron, J., Hopfinger, E. J. & McWilllams, J. C., 1990 Sensitivity to initial conditions in the merging of two-layer baroclinic vortices. Phys. Fluids A 2, 886–889.Google Scholar
Wu, H. M., Overman, E. A. & Zabusky, N. J., 1984 Steady state solutions of the Euler equations in two dimensions. Rotating and translating V-states with limiting cases. I. Numerical algorithms and results. J. Comput. Phys. 53, 42–71.Google Scholar
Zabusky, N. J.: 1981 Recent developments in contour dynamics for the Euler equations. Ann. NY Acad. Sci. 373, 160–170.Google Scholar