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Destabilization of mixed Rossby gravity waves and the formation of equatorial zonal jets

Published online by Cambridge University Press:  08 August 2008

Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Laboratoire de Physique des Océans, IFREMER, BP 70, 29280 Plouzané, France
Earth Simulator Center, Yokohama, Japan


The stability of mixed Rossby gravity (MRG) waves has been investigated numerically using three-dimensionally consistent high-resolution simulations of the continuously stratified primitive equations. For short enough zonal wavelength, the westward phase propagating MRG wave is strongly destabilized by barotropic shear instability leading to the formation of zonal jets. The large-scale instability of the zonally short wave generates zonal jets because it consists primarily of sheared meridional motions, as shown recently for the short barotropic Rossby wave problem.

Simulations were done in a variety of domain geometries: a periodic re-entrant channel, a basin with a short MRG wave forced in its western part and a very long channel initialized with a zonally localized MRG wave. The characteristics of the zonal jets vary with the geometry. In the periodic re-entrant channel, barotropic zonal jets dominate the total flow response at the equator and its immediate vicinity. In the other cases, the destabilization leads to zonal jets with quite different characteristics, especially in the eastward group propagating part of the signal. The most striking result concerns the formation of zonal jets at the equator, alternating in sign in the vertical, with vertical scale short compared to the scale of the forcing or initial conditions.

A stability analysis of a simplified perturbation vorticity equation is formulated to explain the spatial scale selection and growth rate of the zonal jets as functions of the characteristics of the basic state MRG wave. For both types of zonal jets, the model predicts that their meridional scales are comparable to the zonal scale of the MRG wave basic state, while their growth rates scale as μ ∝ Fr |k|, where Fr is the Froude number of the meridional velocity component of the basic state and k its non-dimensional zonal wavenumber. The vertical scale of the baroclinic zonal jets corresponds to the dominant harmonic ppeak of the basic state in the fastest growing mode, given by ppeak≈0.55k2. Thus, the shorter the zonal wavelength of the basic state MRG wave, the narrower the meridional scale of the zonal jets, both barotropic and baroclinic, with the vertical scale of the baroclinic jets being tied to their meridional scale through the equatorial radius of deformation, which decreases as the square root of the vertical wavenumber. The predictions of the spatial scales are in both qualitative and quantitative agreement with the numerical simulations, where shorter vertical scale baroclinic zonal jets are favoured by shorter-wavelength longer-period MRG wave basic states, with the vertical mode number increasing as the square of the MRG wave period.

An Appendix deals with the case of zonally long and intermediate wavelength MRG waves, where a weak instability regime causes a moderate adjustment involving resonant triad interactions without leading to jet formation. For eastward phase propagating waves, adjustment does not lead to significant angular momentum redistribution.

Copyright © Cambridge University Press 2008

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Baldwin, M. P., Rhines, P. B., Huang, H.-P. & McIntyre, M. E. 2007 The jet-stream conundrum. Science 315, 467468.CrossRefGoogle Scholar
Bouchut, F., Le Sommer, J. & Zeitlin, V. 2005 Transport and mixing during the breaking of balanced and unbalanced equatorial waves. Chaos 15, doi: 10.1063/1.1857171.CrossRefGoogle Scholar
Bourlès, B., Andrié, C., Gouriou, Y., Eldin, G., du Penhoat, Y., Freudenthal, S., Dewitte, B., Gallois, F., Chuchla, R., Baurand, F., Aman, A. & Kouadio, G. 2003 The deep currents in the eastern equatorial Atlantic ocean. Geophys. Res. Lett. 30, (5), 8002, doi:10.1029/2002GL015095.CrossRefGoogle Scholar
Boyd, J. 1998 Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Kluwer.CrossRefGoogle Scholar
Domaracki, A. J. & Loesch, A. Z. 1977 Nonlinear interactions among equatorial waves. J. Atmos. Sci. 34, 486498.2.0.CO;2>CrossRefGoogle Scholar
Dunkerton, T. J. 1981 On the inertial stability of the equatorial middle atmosphere. J. Atmos. Sci. 38, 23542364.2.0.CO;2>CrossRefGoogle Scholar
Firing, E. 1987 Deep zonal currents in the central equatorial Pacific. J. Mar. Res. 45, 791812.CrossRefGoogle Scholar
Frisch, U., Legras, B. & Villone, B. 1994 Large-scale Kolmogorov flow on the β-plane and resonant wave interactions. Physica D 94, 3656.CrossRefGoogle Scholar
Fruman, M., Hua, B. L. & Schopp, R. 2008 Equatorial zonal jet formation through the barotropic instability of a low-frequency mixed Rossby-gravity wave, equilibration by inertial instability, and transition to super-rotation. J. Atmos. Sci. (submitted).Google Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta-plane. Geophys. Astrophys. Fluid Dyn. 6, 2947.CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Gouriou, Y., Andriè, C., Bourlès, B., Freudenthal, S., Arnault, S., Aman, A., Eldin, G., du Penhoat, Y., Baurand, F., Gallois, F. & Chuchla, R. 2001 Deep circulation in the equatorial Atlantic ocean. Geophys. Res. Lett. 28, 819822.CrossRefGoogle Scholar
Griffiths, S. D. 2003 a Nonlinear vertical scale selection in equatorial inertial instability. J. Atmos. Sci. 60, 977990.2.0.CO;2>CrossRefGoogle Scholar
Griffiths, S. D. 2003 b The nonlinear evolution of zonally symmetric equatorial inertial instability. J. Fluid Mech. 474, 245273.CrossRefGoogle Scholar
Hua, B. L., Moore, D. W. & Le Gentil, S. 1997 Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345371.CrossRefGoogle Scholar
Iga, K. 1995 Transition modes of rotating shallow water waves in a channel. J. Fluid Mech. 294, 367390.CrossRefGoogle Scholar
Lee, S. Y. & Smith, L. 2003 Stability of Rossby waves in the β-plane approximation. Physica D. 179, 5391.CrossRefGoogle Scholar
Majda, A. 2003 Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Institute Lecture Notes in Mathematics, American Mathematical Society.Google Scholar
Majda, A. & Klein, R. 2003 Systematic multi-scale models for the tropics. J. Atmos. Sci. 60, 393408.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. J. & Young, W. R. 1999 Slow evolution of zonal jets on the beta-plane. J. Atmos. Sci. 56, 784800.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. J. & Young, W. R. 2002 Stability of β-plane Kolmogorov flow. Physica D 162, 208232.CrossRefGoogle Scholar
Matsuno, T. 1966 Quasi-geostrophic motions in the equatorial area. J. Met. Soc. Japan. 44, 2543.CrossRefGoogle Scholar
Meshalkin, L. D. & Sinai, I. G. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Appl. Math. Mech. 25, 11401143.Google Scholar
Natarov, A., Richards, K. & McCreary, J. P. 2008 Two-dimensional instabilities of time-dependent zonal flows: linear shear. J. Fluid Mech. 599, 2950.CrossRefGoogle Scholar
d'Orgeville, M. & Hua, B. L. 2005 Equatorial inertial-parametric instability of zonally symmetric oscillating shear flows. J. Fluid Mech. 531, 261291.CrossRefGoogle Scholar
d'Orgeville, M., Hua, B. L., Schopp, R. & Bunge, L. 2004 Extended deep equatorial layering as an imprint of inertial instability. Geoph. Res. Lett. 31, L22303.CrossRefGoogle Scholar
d'Orgeville, M., Hua, B. L. & Sasaki, H. 2007 Equatorial deep jets triggered by a large vertical scale variability within the western boundary layer. J. Mar. Res. 65, 125.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1984 Local and global barodinic instability of a zonally varying flow. J. Atmos. Sci. 41, 21412162.2.0.CO;2>CrossRefGoogle Scholar
Reznik, G. & Zeitlin, V. 2007 Interaction of free Rossby waves with semi-transparent equatorial waveguide. Part 1. Wave triads. Physica D 226, 5579.CrossRefGoogle Scholar
Rhines, P. B. 1994 Jets. Chaos 4, 313339.CrossRefGoogle ScholarPubMed
Richards, K., Maximenko, N., Bryan, F. & Sasaki, H. 2006 Zonal jets in the Pacific. Geophys. Res. Lett. 33, L03605.CrossRefGoogle Scholar
Ripa, P. 1983 Weak interactions of equatorial waves in a one-layer model. Part I: general properties. J. Phys. Oceanogr. 13, 12081215.2.0.CO;2>CrossRefGoogle Scholar
Shchepetkin, A. & McWilliams, J. 2005 The regional oceanic modeling system: a split-explicit, free-surface, topography-following-coordinate ocean model. Ocean Modelling 9, 891902.CrossRefGoogle Scholar
Sivashinsky, G. 1985 Weak turbulence in periodic flows. Physica D 17, 243255.CrossRefGoogle Scholar
Vallis, G. & Maltrud, M. 1993 Generation of mean flows and jets on a beta plane and over topography. J. Phys. Oceanogr. 23, 13461362.2.0.CO;2>CrossRefGoogle Scholar