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On the stability of elliptical vortex solutions of the shallow-water equations

Published online by Cambridge University Press:  21 April 2006

P. Ripa
P. Ripa
Affiliation:
CICESE, Ensenada, BC, México
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Abstract

The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).

Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.

The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 343 - 363
Copyright
© 1987 Cambridge University Press

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References

Andrews, D. G. 1984 On the existence of nonzonal flows satisfying sufficient conditions for stability. Geophys. Astrophys. Fluid Dyn. 28, 243–256.Google Scholar
Arnol'd, V. I. 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk. SSSR 162, 975–978. (English Transl. in Soviet Math. 6, 331–334.)Google Scholar
Ball, F. K. 1963 Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 17, 240–256.Google Scholar
Ball, F. K. 1965 The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J. Fluid Mech. 22, 529–545.Google Scholar
Cushman-Roisin, B. 1986a Frontal geostrophic dynamics. J. Phys. Oceanogr. 16, 132–143.Google Scholar
Cushman-Roisin, B. 1986b Linear stability of large, elliptical warm-core rings. J. Phys. Oceanogr. 16, 1158–1164.Google Scholar
Cushman-Roisin, B., Heil, W. H. & Nof, D. 1985 Oscillations and rotations of elliptical warm-core rings. J. Geophys. Res. 90, 11756–11764.Google Scholar
Gordin, V. A. 1984 On the stability (in the Lyapunov sense) of the simplest stationary solutions of forecasting systems. Izv. Atmos. Ocean. Phys. 20, 365–370.Google Scholar
Griffths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377.Google Scholar
Griffiths, R. W. & Pearce, A. F. 1985 Satellite images of an unstable warm eddy from the Leeuwin Current. Deep-Sea Res. 32, 1371–1380.Google Scholar
Houghton, R. W., Olson, D. B. & Celone, P. J. 1986 Observation of an anticyclonic eddy near the continental shelf break of New England. J. Phys. Oceanogr. 16, 60–71.Google Scholar
Joyce, T. M. 1984 Velocity and hydrographic structure of a Gulf Stream'warm-core ring. J. Phys. Oceanogr. 14, 936–947.Google Scholar
Joyce, M. T., Backus, R., Baker, K., Blackwelder, P., Brown, O., Cowles, T., Evans, R., Fryxell, G., Mountain, D., Olson, D., Schlitz, R., Schmitt, R., Smith, P., Smith, R. & Wiebe, P. 1985 Rapid evolution of a Gulf Stream warm-core ring. Nature 308, 837–840.Google Scholar
Killworth, P. D. 1983 On the motion of isolated lenses on a beta-plane. J. Phys. Oceanogr. 13, 368–376.Google Scholar
Love, A. E. H. 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. 25, 18–43.Google Scholar
McWilliams, J. C. 1985 Submesoscale, coherent vortices in the ocean. Rev. Geophys. 23, 165–182.Google Scholar
Miles, J. W. & Ball, P. K. 1963 On free-surface oscillations in a rotating paraboloid. J. Fluid Mech. 17, 257–266.Google Scholar
Nof, D. 1981 On the beta-induced movement of isolated baroclinic eddies. J. Phys. Oceanogr. 11, 1662–1672.Google Scholar
Nof, D. 1982 On the movements of deep mesoscale eddies in the North Atlantic. J. Mar. Res. 40, 57–74.Google Scholar
Ripa, P. 1981 Symmetries and conservation laws for internal gravity waves. In Nonlinear Properties of Internal Waves (ed. B. West), A.I.P. Proc. pp. 281–306.
Ripa, P. 1983 General stability conditions for zonal flows in a one-layer model on the beta-plane or the sphere. J. Fluid Mech. 126, 463–487.Google Scholar
Salmon, R. 1982 Hamilton's principle and Ertel's theorem. Am. Inst. Phys. Conf. Proc. 88, 127–136.Google Scholar
Thacker, W. C. 1981 Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499–508.Google Scholar
Yentsch, C. S. & Phinney, D. A. 1985 Rotary motions and convection as a means of regulating primary production in warm core rings. J. Geophys. Res. 90, 3237–3248.Google Scholar
Young, W. R. 1987 Elliptical vortices in shallow water. J. Fluid Mech. 171, 101–119.Google Scholar