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On the instability of geostrophic vortices

Published online by Cambridge University Press:  21 April 2006

Glenn R. Flierl
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases, the other two kinds of instability are nonlinearly destabilized.

Type
Research Article
Copyright
© 1988 Cambridge University Press

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References

Benny, D. J. & Maslowe, S. A.1975 The evolution in space and time of nonlinear waves in parallel shear flows. Stud. Appl. Maths 54, 181205.Google Scholar
Biebuyck, G. L.1986 Self propagation of a barotropic circular eddy. In Summer Study Program in Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution Tech. Rep. WHOI-86–45, pp. 193197.
Childress, S.1984 A vortex-tube model of eddies in the inertial range. Geophys. Astrophys. Fluid Dyn. 29, 2964.Google Scholar
Dritschel, D. G.1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157182.Google Scholar
Flierl, G. R.1978 Models of vertical structure and the calibration of two-layer models. Dyn. Atmos. Oceans 2, 341381.Google Scholar
Flierl, G. R.1984 The emergence of dipoles from instabilities on the f and beta planes. In Summer Study Program in Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution Tech. Rep. WHOI-84–44, pp. 104110.
Flierl, G. R.1987 Isolated eddy models in geophysics. Ann. Rev. Fluid Mech. 19, 493530.Google Scholar
Fu, L. L. & Flierl, G. R.1980 Energy and enstrophy transfers in a realistically stratified ocean. Dyn. Atmos. Oceans 5, 141.Google Scholar
Gent, P. R. & McWilliams, J. C.1986 The instability of circular vortices. Geophys. Astrophys. Fluid Dyn. 35, 209233.Google Scholar
Griffiths, R. W. & Linden, P. F.1981 The stability of vortices in a rotating stratified fluid. J. Fluid Mech. 117, 343377.Google Scholar
Hart, J. E.1974 On the mixed stability problem for quasi-geostrophic ocean currents. J. Phys. Oceanogr. 4, 349356.Google Scholar
Helfrich, K. R. & Send, U.1988 Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech. 197, 331348.Google Scholar
Hogg, N. G. & Stommel, H. M.1985 The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning heat flow. Proc. R. Soc. Lond. A 397, 120.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W.1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Met. Soc. 111, 877949.Google Scholar
Howard, L. N. & Gupta, A. S.1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463476.Google Scholar
Ikeda, M.1981 Instability and splitting of mesoscale rings using a two-layer quasi-geostrophic model on an f-plane. J. Phys. Oceanogr. 11, 987998.Google Scholar
Joyce, T.1984 Velocity and hydrographic structure of a Gulf Stream warm core ring. J. Phys. Oceanogr. 14, 936947.Google Scholar
Lindzen, R. S. & Tung, K. K.1978 Wave overreflection and shear instability. J. Atmos. Sci. 35, 16261632.Google Scholar
Malanotte—Rizzoli, P. 1982 Planetary solitary waves in geophysical flows. Adv. Geophys. 24, 147224.Google Scholar
Michalke, A. & Timme, A.1967 On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647666.Google Scholar
Olson, D. B., Schmitt, R., Kennelly, M. & Joyce, T. 1985 Two-layer diagnostic model of the long-term physical evolution of warm-core ring 82B. J. Geophys. Res. 90, 88138822.Google Scholar
Pedlosky, J.1985 The instability of continuous heton clouds. J. Atmos. Sci. 42, 14771486.Google Scholar
Polvani, L. M., Zabusky, N. J. & Flierl, G. R.1988 Two-layer geostrophic V-states and merger. 1. Constant potential vorticity lower layer. Submitted to J. Fluid Mech.Google Scholar
Saunders, P. M.1973 The instability of a baroclinic vortex. J. Phys. Oceanogr. 3, 6165.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, 96106.Google Scholar