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Asymptotic analysis of transition to turbulence and chaotic advection in shear zonal flows on a beta-plane

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Abstract

The generation of narrow-band Rossby wave packets and the modulated vortex chains induced by them in a weakly-dissipative zonal flow on the beta-plane with a velocity profile in the form of a shear layer is studied. The analysis is performed within the framework of the asymptotic approach based on the distinguishing a thin critical layer inside of which the vortex chains are formed. The evolution equations, describing the simultaneous development of a wave packet envelope and vorticity perturbations in a nonlinear critical layer, are derived for a weakly supercritical flow. A transition to the complex dynamics of a wave packet (low-mode turbulence) is studied within the framework of a numerical solution of the derived equations and its mechanism is revealed. The onset of chaotic advection and anomalous diffusion of passive scalar in the critical layer is considered, and the exponent of the diffusion law is calculated.

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References

  1. A. M. Oboukhov, Turbulence and Dynamics of the Atmosphere (Gidrometeoizdat, Leningrad, 1988) [in Russian].

    Google Scholar 

  2. R. N. Ferreira and W. H. Schubert, “Barotropic Aspects of ITCZ Breakdown,” J. Atmos. Sci. 54, 261–285 (1997).

    Article  Google Scholar 

  3. C. W. Hughes, “The Antarctic Circumpolar Current as a Waveguide for Rossby Waves,” J. Phys. Oceanogr. 26, 1375–1387 (1996).

    Article  Google Scholar 

  4. P. S. Marcus, “Jupiter’s Great Red Spot and Other Vortices,” Annu. Rev. Astron. Astrophys. 31, 523–573 (1993).

    Article  Google Scholar 

  5. A. R. Vasavada and A. P. Showman, “Jovian Atmospheric Dynamics: An Update after Galileo and Cassini,” Rep. Progr. Phys. 68, 1935–1996 (2005).

    Article  Google Scholar 

  6. T. H. Solomon, W. J. Holloway, and H. L. Swinney, “Shear Flow Instabilities and Rossby Waves in Barotropic Flow in a Rotating Annulus,” Phys. Fluids A5, 1971–1982 (1993).

    Google Scholar 

  7. F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, “Stability and Eddy Structures of Quasidimeric Shear Flows,” Usp. Fiz. Nauk 160(7), 1–47 (1990).

    Google Scholar 

  8. M. V. Nezlin and E. N. Snezhkin, Rossby Eddies and Helical Structures: Astrophysics and Plasma Physics in Shallow-Water Experiments (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  9. V. V. Alekseev, S. V. Kiseleva, and S. S. Lappo, Laboratory Models of Physical Processes in Atmosphere and Ocean (Nauka, Moscow, 2005) [in Russian].

    Google Scholar 

  10. J. A. Van de Konijnenberg, A. H. Nielsen, J. J. Rasmussen, et al., “Shear Flow Instability in a Rotating Fluid,” J. Fluid Mech. 387, 177–204 (1999).

    Article  Google Scholar 

  11. H. J. Kwon and M. Mak, “On the Equilibration in Nonlinear Barotropic Instability,” J. Atmos. Sci. 45, 294–308 (1988).

    Article  Google Scholar 

  12. G. R. Flierl, P. Malanotte-Rizzoli, and N. J. Zabusky, “Nonlinear Waves and Coherent Vortex Structures in Barotropic β-Plane Jets,” J. Phys. Oceanogr. 17(9), 1408–1438 (1987).

    Article  Google Scholar 

  13. P. S. Marcus and C. Lee, “A Model for Eastward and Westward Jets in Laboratory Experiments and Planetary Atmospheres,” Phys. Fluids 10(6), 1474–1489 (1998).

    Article  Google Scholar 

  14. M. R. Shoeberl and D. L. Hartmann, “The Dynamics of the Stratospheric Polar Vortex and Its Relation to Springtime Ozone Depletions,” Science 251, 46–52 (1991).

    Article  Google Scholar 

  15. M. R. Shoeberl, L. R. Lait, P. A. Newman, et al., “The Structure of the Polar Vortex,” J. Geophys. Res. 97(D8), 7859–7882 (1992).

    Google Scholar 

  16. K. Ishioka and S. Yoden, “Non-Linear Aspects of a Barotropically Unstable Polar Vortex in a Forced Dissipative System: Flow Regimes and Tracer Transport,” J. Meteor. Soc. Japan 73, 201–212 (1995).

    Google Scholar 

  17. L. J. Hickernell, “Time-Dependent Critical Layers in Shear Flows on the Beta-Plane,” J. Fluid Mech. 142, 431–449 (1984).

    Article  Google Scholar 

  18. S. M. Churilov, “The Nonlinear Stabilization of a Zonal Shear Flow Instability,” Geophys. Astrophys. Fluid Dyn. 46(3), 159–175 (1989).

    Article  Google Scholar 

  19. S. M. Churilov, “The Influence of Ekman Dissipation on the Development of Perturbations in a Zonal Shear Flow,” Geophys. Astrophys. Fluid Dyn. 46(3), 177–190 (1989).

    Article  Google Scholar 

  20. S. M. Churilov and I. G. Shukhman, “Critical Layer and Nonlinear Evolution of Disturbances in Weakly Supercritical Shear Flows,” Izv. Akad. Nauk, Fiz. Atm. Okeana 31(4), 557–569 (1995).

    Google Scholar 

  21. N. J. Balmforth and C. Piccolo, “The Onset of Meandering in a Barotropic Jet,” J. Fluid Mech. 449, 85–114 (2001).

    Article  Google Scholar 

  22. K. P. Bowman, “Rossby Wave Phase Speeds and Mixing Barriers in the Stratosphere. Pt I: Observations,” J. Atmos. Sci. 53(6), 905–916 (1996).

    Article  Google Scholar 

  23. H. L. Kuo, “Dynamics of Quasigeostrophic Flows and Instability Theory,” Adv. Appl. Mech. 13, 247–330 (1973).

    Article  Google Scholar 

  24. N. P. Shakina, Hydrodynamic Instability in the Atmosphere (Gidrometeoizdat, Leningrad, 1990) [in Russian].

    Google Scholar 

  25. Y. Tomikawa, M. Yoshiki, and K. Sato, “A Neutral Wave Observed in the Antarctic Polar Vortex,” J. Meteor. Soc. Japan 84(1), 97–113 (2006).

    Article  Google Scholar 

  26. D. del-Castillo-Negrete, “Asymmetric Transport and Non-Gaussian Statistics of Passive Scalars in Vortices in Shear,” Phys. Fluids 10(3), 576–594 (1998).

    Article  Google Scholar 

  27. L. J. Pratt, M. S. Lozier, and N. Beliakova, “Parcel Trajectories in Quasigeostrophic Jets: Neutral Modes,” J. Phys. Oceanogr. 25, 1451–1466 (1995).

    Article  Google Scholar 

  28. K. V. Koshel’ and S. V. Prants, Chaotic Advection in the Ocean (NITs Regulyar. Khaot. Dinam., Izhevsk, 2008) [in Russian].

    Google Scholar 

  29. K. Ngan and T. G. Shepherd, “Chaotic Mixing and Transport in Rossby-Wave Critical Layers,” J. Fluid Mech. 334, 315–351 (1997).

    Article  Google Scholar 

  30. R. Mizuta and S. Yoden, “Chaotic Mixing and Transport Barriers in an Idealized Stratospheric Polar Vortex,” J. Atmos. Sci. 58, 2616–2629 (2001).

    Article  Google Scholar 

  31. J. Pedlosky, Geophysical Fluid Dynamics (Springer-Verlag, New-York, 1979; Mir, Moscow, 1984).

    Google Scholar 

  32. F. B. Lipps, “The Stability of an Asymmetric Zonal Current in the Atmosphere,” J. Fluid Mech. 21, 225–239 (1965).

    Article  Google Scholar 

  33. V. P. Reutov, “Nonstationary Critical Layer and the Nonlinear Instability Stage in Two-Dimensional Poiseuille Flow,” Prikl. Mekh. Tekh. Fiz., No. 4, 43–54 (1982).

  34. R. K. Dodd, J. C. Eilbeck, J. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Mir, Moscow, 1988; Academic Press, New York, 1982).

    Google Scholar 

  35. V. P. Reutov, S. V. Shagalov, and G. V. Rybushkina, “The Onset of Turbulence in a Shear Flow on the beta-Plane. An Asymptotic Approach Based on the Critical Layer Theory,” in Advances in Turbulence VII, Ed. by U. Frish (Kluwer, Dordrecht, 1998), pp. 483–486.

    Google Scholar 

  36. V. P. Reutov, “The Plasma-Hydrodynamic Analogy and the Nonlinear Stage of Wind Wave Instability,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 16(12), 1266–1275 (1980).

    Google Scholar 

  37. G. M. Zaslavskii, Stochasticity of Dynamic Systems (Nauka, Moscow, 1984) [in Russian].

    Google Scholar 

  38. A. B. Mikhailovskii, The Theory of Plasma Instabilities, Vol. 1: Instabilities of Homogeneous Plasma (Atomizdat, Moscow, 1975) [in Russian].

    Google Scholar 

  39. F. Drazin, Introduction to the Theory of Hydrodynamic Stability (Fizmatlit, Moscow, 2005) [in Russian].

    Google Scholar 

  40. M. I. Rabinovich and A. B. Ezerskii, Dynamic Theory of Forming (Yanus-K, Moscow, 1998) [in Russian].

    Google Scholar 

  41. I. G. Shukhman and S. M. Churilov, “Effect of Slight Stratification on the Nonlinear Spatial Evolution of a Weakly Unstable Wave in a Free Shear Layer,” J. Fluid Mech. 343, 197–233 (1997).

    Article  Google Scholar 

  42. D. Ruelle and F. Takens, “On the Nature of Turbulence,” Commun. Math. Phys. 20 167–192 (Springer, Berlin, 1971).

    Google Scholar 

  43. V. D. Shapiro and V. I. Shevchenko, “Wave-Particle Interaction in Nonequilibrium Media,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 19(5–6), 767–791 (1976).

    Google Scholar 

  44. T. H. Solomon, E. R. Weeks, and H. L. Swinney, “Chaotic Advection in a Two-Dimensional Flow: Levy Flights and Anomalous Diffusion,” Phys. D (Amsterdam) 76, 70–84 (1994).

    Google Scholar 

  45. S. Kovalyov, “Phase Space Structure and Anomalous Diffusion in a Rotational Fluid Experiment,” Chaos 10(1), 153–165 (2000).

    Article  Google Scholar 

  46. G. M. Zaslavsky, “Fractional Kinetic Equation for Hamiltonian Chaos,” Phys. D (Amsterdam) 76, 110–122 (1994).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Physics, Russian Academy of Sciences, ul. Ulyanova 46, Nizhni Novgorod, 603950, Russia

    S. V. Shagalov, V. P. Reutov & G. V. Rybushkina

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  1. S. V. Shagalov
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  2. V. P. Reutov
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  3. G. V. Rybushkina
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Correspondence to V. P. Reutov.

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Original Russian Text © S.V. Shagalov, V.P. Reutov, G.V. Rybushkina, 2010, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2010, Vol. 46, No. 1, pp. 105–118.

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Shagalov, S.V., Reutov, V.P. & Rybushkina, G.V. Asymptotic analysis of transition to turbulence and chaotic advection in shear zonal flows on a beta-plane. Izv. Atmos. Ocean. Phys. 46, 95–108 (2010). https://doi.org/10.1134/S0001433810010135

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