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Instability of coupled geostrophic density fronts and its nonlinear evolution

Published online by Cambridge University Press:  01 October 2008

EMILIE SCHERER  and
VLADIMIR ZEITLIN
EMILIE SCHERER
Affiliation:
Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure / Université Paris VI, 24 rue Lhomond, 75231 Paris Cedex 5, Francezeitlin@lmd.ens.fr
VLADIMIR ZEITLIN
Affiliation:
Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure / Université Paris VI, 24 rue Lhomond, 75231 Paris Cedex 5, Francezeitlin@lmd.ens.fr
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Abstract

Instability of coupled density fronts, and its fully nonlinear evolution are studied within the idealized reduced-gravity rotating shallow-water model. By using the collocation method, we benchmark the classical stability results on zero potential vorticity (PV) fronts and generalize them to non-zero PV fronts. In both cases, we find a series of instability zones intertwined with the stability regions along the along-front wavenumber axis, the most unstable modes being long wave. We then study the nonlinear evolution of the unstable modes with the help of a high-resolution well-balanced finite-volume numerical scheme by initializing it with the unstable modes found from the linear stability analysis. The most unstable long-wave mode evolves as follows: after a couple of inertial periods, the coupled fronts are pinched at some location and a series of weakly connected co-rotating elliptic anticyclonic vortices is formed, thus totally changing the character of the flow. The characteristics of these vortices are close to known rodon lens solutions. The shorter-wave unstable modes from the next instability zones are strongly concentrated in the frontal regions, have sharp gradients, and are saturated owing to dissipation without qualitatively changing the flow pattern.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 613 , 25 October 2008 , pp. 309 - 327
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. & Perthame, B. 2004 A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065.CrossRefGoogle Scholar
Bouchut, F. 2004 Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Birkhauser.CrossRefGoogle Scholar
Bouchut, F. 2007 Efficient numerical finite-volume schemes for shallow water models. In Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (ed. Zeitlin, V.), Elsevier, pp. 191–258.Google Scholar
Bouchut, F., Le Sommer, J. & Zeitlin, V. 2004 Frontal geostrophic adjustment and non-linear-wave phenomena in one dimensional rotating shallow water. Part 2. High resolution numerical simulations. J. Fluid Mech. 514, 35–63.CrossRefGoogle Scholar
Bouchut, F., Le Sommer, J. & Zeitlin, V. 2005 Breaking of balanced and unbalanced equatorial waves. Chaos 15, 13 503–13 521.CrossRefGoogle ScholarPubMed
Cushman-Roisin, B. 1985 Linear stability of large, elliptical warm-core rings. J. Phys. Oceanogr. 16, 1158–1164.2.0.CO;2>CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377.CrossRefGoogle Scholar
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flow in shallow water. J. Fluid Mech. 184, 477–504.CrossRefGoogle Scholar
Houghton, R. W., Schlitz, R., Beardsley, R. C., Butman, B. & Chamberlin, J. L. 1982 The middle atlantic bight cold pool: evolution of the temperature structure during summer 1979. J. Phys. Oceanogr. 12, 1019–1029.2.0.CO;2>CrossRefGoogle Scholar
Killworth, P. D 1983 a Long-wave instability of an isolated front. Geophys. Astrophys. Fluid Dyn. 25, 235–258.CrossRefGoogle Scholar
Killworth, P. D 1983 b On the motion of isolated lenses on a beta plane. J. Phys. Oceanogr. 13, 146–167.2.0.CO;2>CrossRefGoogle Scholar
Knessl, C. & Keller, J. B. 1995 Stability of linear shear flows in shallow water. J. Fluid Mech. 303, 203–214.CrossRefGoogle Scholar
Le Sommer, J., Scherer, E. & Zeitlin, V. 2006 Inertial motions during the transient adjustment of a density anomaly in the equatorial ocean: application to the western Pacific warm pool. J. Phys. Oceanogr. 36, 2270–2282.CrossRefGoogle Scholar
Paldor, N. & Ghil, M. 1990 Finite-wavelength instabilities of a coupled density front. J. Phys. Oceanogr. 20, 114–123.2.0.CO;2>CrossRefGoogle Scholar
Poulin, F. J. & Flierl, G. R. 2003 The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173–2192.2.0.CO;2>CrossRefGoogle Scholar
Reznik, G. M., Zeitlin, V. & Ben Jelloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow water model. J. Fluid Mech. 445, 93–120.CrossRefGoogle Scholar
Ripa, P. 1983 General stability conditions for zonal flows in a one-layer model on the β plane or the sphere. J. Fluid Mech. 126, 463–489.CrossRefGoogle Scholar
Ripa, P. 1987 On the stability of elliptical vortex solutions of the shallow-water equations. J. Fluid Mech. 183, 343–363.CrossRefGoogle Scholar
Rubino, A., Dotsenko, S. & Brandt, P. 2003 Near-inertial oscillations of geophysical surface frontal currents. J. Phys. Oceanogr. 33, 1990–1999.2.0.CO;2>CrossRefGoogle Scholar
Sakai, S. 1989 Rossby–Kelvin instablility: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.CrossRefGoogle Scholar
Smith, P. C. 1976 Baroclinic instability in the Denmark strait overflow. J. Phys. Oceanogr. 6, 355–371.2.0.CO;2>CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in Matlab. SIAM.CrossRefGoogle Scholar
Vanneste, J. 1998 A nonlinear critical layer generated by the interaction of free Rossby waves. J. Fluid Mech. 371, 319–344.CrossRefGoogle Scholar
Zeitlin, V. 2007 Nonlinear wave phenomena in rotating shallow water with applications to geostrophic adjustment. In Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (ed. Zeitlin, V.), Elsevier, pp. 257–321.CrossRefGoogle Scholar