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Modulational instability of Rossby and drift waves and generation of zonal jets

Published online by Cambridge University Press:  05 May 2010

COLM P. CONNAUGHTON ,
BALASUBRAMANYA T. NADIGA ,
SERGEY V. NAZARENKO  and
BRENDA E. QUINN
COLM P. CONNAUGHTON*
Affiliation:
Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
BALASUBRAMANYA T. NADIGA
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
SERGEY V. NAZARENKO
Affiliation:
Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
BRENDA E. QUINN
Affiliation:
Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
*
†Email address for correspondence: connaughtonc@gmail.com
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Abstract

We study the modulational instability of geophysical Rossby and plasma drift waves within the Charney–Hasegawa–Mima (CHM) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. We review the linear theory of Gill (Geophys. Fluid Dyn., vol. 6, 1974, p. 29) and extend it to show that for strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weak waves, the maximum growth occurs for off-zonal inclined modulations that are close to being in three-wave resonance with the primary wave. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the nonlinear jet pinching predicted by Manin & Nazarenko (Phys. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Kármán-like vortex streets, and at this moment the truncated models fail. For weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave, so that the truncated description holds for longer. The two-dimensional vortex streets appear to be more stable than purely one-dimensional zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh–Kuo instability criterion for the one-dimensional case. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and the resonant wave–wave interactions (for weak waves).

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Papers
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Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 207 - 231
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Arnold, V. I. & Meshalkin, L. D. 1960 Seminar led by A. N. Kolmogorov on selected problems of analysis (1958–1959). Usp. Mat. Nauk 15, 247.Google Scholar
Balk, A. M. 1991 A new invariant for Rossby wave systems. Phys. Lett. A 155, 20–24.CrossRefGoogle Scholar
Balk, A. M. 1997 New conservation laws for the interaction of nonlinear waves. SIAM Rev. 39 (1), 68–94.CrossRefGoogle Scholar
Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1990 a Non-local drift wave turbulence. Sov. Phys. JETP 71, 249–260.Google Scholar
Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1990 b On the non-local turbulence of drift type waves. Phys. Lett. A 146, 217–221.CrossRefGoogle Scholar
Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1991 A new invariant for drift turbulence. Phys. Lett. A 152, 276–280.CrossRefGoogle Scholar
Benjamin, T. & Feir, J. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417.CrossRefGoogle Scholar
Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 A mechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech. 628, 395.CrossRefGoogle Scholar
Bustamante, M. D. & Kartashova, E. 2009 Effect of the dynamical phases on the nonlinear amplitudes' evolution. Europhys. Lett. 85 (3), 34002.CrossRefGoogle Scholar
Charney, J. G. 1949 On a physical basis for numerical prediction of large-scale motions in the atmosphere. J. Meteorol. 6, 371–85.2.0.CO;2>CrossRefGoogle Scholar
Connaughton, C., Nazarenko, S. V. & Pushkarev, A. N. 2001 Discreteness and quasi–resonances in weak turbulence of capillary waves. Phys. Rev. E 63 (4), 046306.CrossRefGoogle Scholar
Dewar, R. L. & Abdullatif, R. F. 2007 Zonal flow generation by modulational instability. In Frontiers in Turbulence and Coherent Structures: Proceedings of the CSIRO/COSNet Workshop on Turbulence and Coherent Structures (ed. Denier, J. & Frederiksen, J. S.), World Scientific Lecture Notes in Complex Systems, vol. 6, pp. 415–430. World Scientific.CrossRefGoogle Scholar
Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasma: a review. Plasma Phys. Control. Fusion 47 (5), R35–R161.CrossRefGoogle Scholar
Dorland, W., Hammett, G. W., Chen, L., Park, W., Cowley, S. C., Hamaguchi, S. & Horton, W. 1990 Numerical simulations of nonlinear 3-D ITG fluid turbulence with an improved Landau damping model. Bull. Am. Phys. Soc. 35, 2005.Google Scholar
Dritschel, D. G. & McIntyre, M. E. 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, 855–874.CrossRefGoogle Scholar
Galperin, B., Nakano, H., Huang, H.-P. & Sukoriansky, S. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and Earth's oceans. Geophys. Res. Lett. 31, L13303.CrossRefGoogle Scholar
Gill, A. E. 1974 The stability on planetary waves on an infinite beta-plane. Geophys. Fluid Dyn. 6, 29–47.CrossRefGoogle Scholar
Hasegawa, A. & Mima, K. 1978 Pseudo-three-dimensional turbulence in magnetized non-uniform plasma. Phys. Fluids 21, 87.CrossRefGoogle Scholar
Horton, W. & Ichikawa, Y.-H. 1996 Chaos and Structures in Nonlinear Plasmas. World Scientific.CrossRefGoogle Scholar
James, I. N. 1987 Suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci. 44 (24), 3710.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Ocean. 33, 863–884.2.0.CO;2>CrossRefGoogle Scholar
Kartashova, E. & L'vov, V. S. 2007 Model of intraseasonal oscillations in Earth's atmosphere. Phys. Rev. Lett. 98 (19), 198501.CrossRefGoogle ScholarPubMed
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kuo, H. L. 1949 Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteorol. 6, 105–122.2.0.CO;2>CrossRefGoogle Scholar
Lewis, J. M. 1988 Clarifying the dynamics of the general circulation: Phillips's 1956 experiment. Bull. Am. Meteorol. Soc. 79 (1), 39–60.2.0.CO;2>CrossRefGoogle Scholar
Lorentz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258–269.2.0.CO;2>CrossRefGoogle Scholar
Mahanti, A. C. 1981 The oscillation between Rossby wave and zonal flow in a barotropic fluid. Arch. Met. Geoph. Biokl. A 30, 211–225.CrossRefGoogle Scholar
Manfredi, G., Roach, C. M. & Dendy, R. O. 2001 Zonal flow and streamer generation in drift turbulence. Plasma Phys. Control. Fusion 43, 825–837.CrossRefGoogle Scholar
Manin, D. Yu. & Nazarenko, S. V. 1994 Nonlinear interaction of small-scale Rossby waves with an intense large-scale zonal flow. Phys. Fluids 6 (3), 1158–1167.CrossRefGoogle Scholar
Maximenko, N. A., Melnichenko, O. V., Niiler, P. P. & Sasaki, H. 2008 Stationary mesoscale jet-like features in the ocean. Geophys. Res. Lett. 35, L08603.CrossRefGoogle Scholar
McWilliams, J. C. 2006 Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press.Google Scholar
Mima, K. & Lee, Y. C. 1980 Modulational instability of strongly dispersive drift waves and formation of convective cells. Phys. Fluids 23, 105.CrossRefGoogle Scholar
Nadiga, B. T. 2006 On zonal jets in oceans. Geophys. Res. Lett. 33 (10), L10601.CrossRefGoogle Scholar
Nazarenko, S. & Quinn, B. 2009 Triple cascade behaviour in quasigeostrophic and drift turbulence and generation of zonal jets. Phys. Rev. Lett. 103 (11), 118501.CrossRefGoogle ScholarPubMed
Onishchenko, O. G., Pokhotelov, O. A., Sagdeev, R. Z., Shukla, P. K. & Stenflo, L. 2004 Generation of zonal flows by Rossby waves in the atmosphere. Nonlinear Proc. Geophys. 11, 241–244.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86, 5831.CrossRefGoogle ScholarPubMed
Rhines, P. 1975 Waves and turbulence on a betaplane. J. Fluid Mech. 69, 417–443.CrossRefGoogle Scholar
Rudakov, L. I. & Sagdeev, R. Z. 1961 On the instability of a non-uniform rarefied plasma in a strong magnetic field. Sov. Phys. Dokl. 6, 415.Google Scholar
Sagdeev, Z. & Galeev, A. A. 1969 Nonlinear Plasma Theory. Benjamin.Google Scholar
Sanchez-Lavega, A., Rojas, J. F. & Sada, P. V. 2000 Saturn's zonal winds at cloud level. Icarus 147, 405.CrossRefGoogle Scholar
Simon, A. A. 1999 The structure and temporal stability of Jupiter's zonal winds: a study of the north tropical region. Icarus 141, 29.CrossRefGoogle Scholar
Smolyakov, A. I., Diamond, P. H. & Shevchenko, V. I. 2000 Zonal flow generation by parametric instability in magnetized plasmas and geostrophic fluids. Phys. Plasmas 7, 1349.CrossRefGoogle Scholar
Wagner, F., Becker, G., Behringer, K., Campbell, D., Eberhagen, A., Engelhardt, W., Fussmann, G., Gehre, O., Gernhardt, J., Gierke, G. V., Haas, G., Huang, M., Karger, F., Keilhacker, M., Klüber, O., Kornherr, M., Lackner, K., Lisitano, G., Lister, G. G., Mayer, H. M., Meisel, D., Müller, E. R., Murmann, H., Niedermeyer, H., Poschenrieder, W., Rapp, H. & Röhr, H. 1982 Regime of improved confinement and high beta in neutral-beam-heated divertor discharges of the ASDEX tokamak. Phys. Rev. Lett. 49 (19), 1408–1412.CrossRefGoogle Scholar
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