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Modulational instability of Rossby and drift waves and generation of zonal jets
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Connaughton, Colm, Nadiga, Balasubramanya T., Nazarenko, Sergey and Quinn, Brenda E.. (2010) Modulational instability of Rossby and drift waves and generation of zonal jets. Journal of Fluid Mechanics, Vol.654 . pp. 207231. ISSN 00221120

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Official URL: http://dx.doi.org/10.1017/S0022112010000510
Abstract
We study the modulational instability of geophysical Rossby and plasma drill waves within the CharneyHasegawaMima (CH M) model both theoretically, using truncated (fourmode and threemode) 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 offzonal inclined modulations that are close to being in threewave 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 (Pit vs. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Karmanlike 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 twodimensional vortex streets appear to be more stable than purely onedimensional zonal jets, and their zonalaveraged speed can reach amplitudes much stronger than is allowed by the RayleighKuo instability criterion for the onedimensional case. In the long term, the system transitions to turbulence helped by the vortexpairing instability (for strong waves) and the resonant wave wave interactions (for weak waves).
Item Type:  Journal Article 

Subjects:  Q Science > QC Physics 
Divisions:  Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Rossby waves, Drift waves 
Journal or Publication Title:  Journal of Fluid Mechanics 
Publisher:  Cambridge University Press 
ISSN:  00221120 
Date:  2010 
Volume:  Vol.654 
Number of Pages:  25 
Page Range:  pp. 207231 
Identification Number:  10.1017/S0022112010000510 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Open Access 
Related URLs:  
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. Balk, A. M. 1991 A new invariant for Rossby wave systems. Phys. Lett. A 155, 20–24. Balk, A. M. 1997 New conservation laws for the interaction of nonlinear waves. SIAM Rev. 39 (1), 68–94. Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1990a Nonlocal drift wave turbulence. Sov. Phys. JETP 71, 249–260. Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1990b On the nonlocal turbulence of drift type waves. Phys. Lett. A 146, 217–221. Balk, A. M., Nazarenko, S. V. & Zakharov, V. E. 1991 A new invariant for drift turbulence. Phys. Lett. A 152, 276–280. Benjamin, T. & Feir, J. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417. Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 A mechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech. 628, 395. Bustamante, M. D. & Kartashova, E. 2009 Effect of the dynamical phases on the nonlinear amplitudes’ evolution. Europhys. Lett. 85 (3), 34002. Charney, J. G. 1949 On a physical basis for numerical prediction of largescale motions in the atmosphere. J. Meteorol. 6, 371–85. 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. 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. J. Denier & J. S. Frederiksen), World Scientific Lecture Notes in Complex Systems, vol. 6, pp. 415–430. World Scientific. 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. Dorland, W., Hammett, G. W., Chen, L., Park, W., Cowley, S. C., Hamaguchi, S. & Horton, W. 1990 Numerical simulations of nonlinear 3D ITG fluid turbulence with an improved Landau damping model. Bull. Am. Phys. Soc. 35, 2005. Dritschel, D. G. & McIntyre, M. E. 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddytransport barriers. J. Atmos. Sci. 65, 855–874. 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. Gill, A. E. 1974 The stability on planetary waves on an infinite betaplane. Geophys. Fluid Dyn. 6, 29–47. Hasegawa, A. & Mima, K. 1978 Pseudothreedimensional turbulence in magnetized nonuniform plasma. Phys. Fluids 21, 87. Horton, W. & Ichikawa, Y.H. 1996 Chaos and Structures in Nonlinear Plasmas. World Scientific. James, I. N. 1987 Suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci. 44 (24), 3710. Janssen, P. A. E. M. 2003 Nonlinear fourwave interactions and freak waves. J. Phys. Ocean. 33, 863–884. Kartashova, E. & L’vov, V. S. 2007 Model of intraseasonal oscillations in Earth’s atmosphere. Phys. Rev. Lett. 98 (19), 198501. Kraichnan, R. H. 1967 Inertial ranges in twodimensional turbulence. Phys. Fluids 10, 1417–1423. Kuo, H. L. 1949 Dynamic instability of twodimensional nondivergent flow in a barotropic atmosphere. J. Meteorol. 6, 105–122. Lewis, J. M. 1988 Clarifying the dynamics of the general circulation: Phillips’s 1956 experiment. Bull. Am. Meteorol. Soc. 79 (1), 39–60. Lorentz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258–269. Mahanti, A. C. 1981 The oscillation between Rossby wave and zonal flow in a barotropic fluid. Arch. Met. Geoph. Biokl. A 30, 211–225. Manfredi, G., Roach, C. M. & Dendy, R. O. 2001 Zonal flow and streamer generation in drift turbulence. Plasma Phys. Control. Fusion 43, 825–837. Manin, D. Yu. & Nazarenko, S. V. 1994 Nonlinear interaction of smallscale Rossby waves with an intense largescale zonal flow. Phys. Fluids 6 (3), 1158–1167. Maximenko, N. A., Melnichenko, O. V., Niiler, P. P. & Sasaki, H. 2008 Stationary mesoscale jetlike features in the ocean. Geophys. Res. Lett. 35, L08603. McWilliams, J. C. 2006 Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press. Mima, K. & Lee, Y. C. 1980 Modulational instability of strongly dispersive drift waves and formation of convective cells. Phys. Fluids 23, 105. Nadiga, B. T. 2006 On zonal jets in oceans. Geophys. Res. Lett. 33 (10), L10601. Nazarenko, S. & Quinn, B. 2009 Triple cascade behaviour in quasigeostrophic and drift turbulence and generation of zonal jets. Phys. Rev. Lett. 103 (11), 118501. 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. Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86, 5831. Rhines, P. 1975 Waves and turbulence on a betaplane. J. Fluid Mech. 69, 417–443. Rudakov, L. I. & Sagdeev, R. Z. 1961 On the instability of a nonuniform rarefied plasma in a strong magnetic field. Sov. Phys. Dokl. 6, 415. Sagdeev, Z. & Galeev, A. A. 1969 Nonlinear Plasma Theory. Benjamin. SanchezLavega, A., Rojas, J. F. & Sada, P. V. 2000 Saturn’s zonal winds at cloud level. Icarus 147, 405. Simon, A. A. 1999 The structure and temporal stability of Jupiter’s zonal winds: a study of the north tropical region. Icarus 141, 29. 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. 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., Kl¨uber, O., Kornherr, M., Lackner, K., Lisitano, G., Lister, G. G., Mayer, H. M., Meisel, D., M¨uller, E. R., Murmann, H., Niedermeyer, H., Poschenrieder, W., Rapp, H. & R¨ohr, H. 1982 Regime of improved confinement and high beta in neutralbeam heated divertor discharges of the ASDEX tokamak. Phys. Rev. Lett. 49 (19), 1408–1412. 
URI:  http://wrap.warwick.ac.uk/id/eprint/5549 
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