Advertisement

Intermittency and Transport Barriers in Fluids and Plasmas

  • Emanuel V. Chimanski
  • Caroline G. L. Martins
  • Roman Chertovskih
  • Erico L. Rempel
  • Marisa Roberto
  • Iberê L. Caldas
  • Abraham C.-L. Chian
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 22)

Abstract

Leaking chaotic systems represent physical situations in which a hole or leak is introduced in a closed chaotic system. When such a hole is present, trajectories can escape from a trapping region of the phase space and wander for some time, before they return to the first region or settle to a different attractor. In the first case, the system displays intermittency, whereas in the second case, transient chaos is observed. The presence of transport barriers can prevent the leaking of trajectories between regions of the phase space. In the present study, transport barriers and intermittency are investigated in two dynamical systems. First, the topology of the phase space for symplectic maps is analyzed when a control parameter is varied, where a robust torus may or not be present. The patterns obtained are compared and the effect of the robust torus on the dynamical transport is described. In a second example, Raleigh-Bénard convection is studied in three-dimensional direct numerical simulations. By varying the magnitude of the Rayleigh number, a route to hyperchaos is reported, where an interior crisis leads to intermittency between quasiperiodic and hyperchaotic states.

Notes

Acknowledgements

EVC, RC, and ELR acknowledge the financial support from FAPESP (grants 2016/07398-8, 2013/01242-8, and 2013/26258-4, respectively). ELR also acknowledges, financial support from CNPq (grant 305540/2014-9) and CAPES (grant 88881.068051/2014-01). RC was also partially supported by the project POCI-01-0145-FEDER-006933/SYSTEC financed by ERDF (European Regional Development Fund) through COMPETE 2020 (Programa Operacional Competitividade e Internacionalização), and by FCT (Fundação para a Ciência e a Tecnologia, Portugal).

References

  1. 1.
    Alligood, K. T., Sauer, T. D., Yorke, J. A., & Crawford, J. D. (1997). Chaos: An introduction to dynamical systems. New York: Springer.CrossRefGoogle Scholar
  2. 2.
    Altmann, E. G., Portela, J. S. E., & Tél, T. (2013). Leaking chaotic systems. Reviews of Modern Physics, 85, 869.CrossRefGoogle Scholar
  3. 3.
    Bénard, H. (1901). Les tourbillons cellulaires dans une nappe liquide. - Méthodes optiques d’observation et d’enregistrement. Journal de Physique Théorique et Appliquée, 10(1), 254–266.CrossRefGoogle Scholar
  4. 4.
    Bergé, P., Dubois, M., Mannevillel, P., & Pomeau, Y. (1980). Intermittency in Rayleigh-Bénard convection. Journal de Physique Lettres 41(15), 341–345.CrossRefGoogle Scholar
  5. 5.
    Boyd, J. P. (2000). Chebyshev and Fourier spectral methods (2nd ed.). Mineola, NY: Dover Publications.Google Scholar
  6. 6.
    Caldas, I., Viana, R., Szezech, J., Portela, J., Fonseca, J., Roberto, M., et al. (2012). Nontwist symplectic maps in tokamaks. Communications in Nonlinear Science and Numerical Simulation 17(5), 2021–2030. (Special Issue: Mathematical Structure of Fluids and Plasmas)Google Scholar
  7. 7.
    Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. New York: Dover Publications.MATHGoogle Scholar
  8. 8.
    Chertovskih, R., Gama, S. M. A., Podvigina, O., & Zheligovsky, V. (2010). Dependence of magnetic field generation by thermal convection on the rotation rate: A case study. Physica D: Nonlinear Phenomena 239(13), 1188–1209.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chian, A. C.-L., Rempel, E. L., & Rogers, C. (2007). Crisis-induced intermittency in non-linear economic cycles. Applied Economics Letters, 14, 211.CrossRefGoogle Scholar
  10. 10.
    Chu, S., & Gascard, J. C. (1991). Deep convection and deep water formation in the oceans. Amsterdam: Elsevier.CrossRefGoogle Scholar
  11. 11.
    Clerc, M. G., & Verschueren, N. (2013). Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems. Physical Review E, 88(5), 052916.CrossRefGoogle Scholar
  12. 12.
    Cox, S. M., & Matthews, P. C. (2002). Exponential time differencing for stiff systems. Journal of Computational Physics, 176(2), 430–455.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dormy, E., & Soward, A. M. (2007). Mathematical aspects of natural dynamos. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
  14. 14.
    Gollub, J. P., & Benson, S. V. (1980). Many routes to turbulent convection. Journal of Fluid Mechanics, 100(3), 449–470.CrossRefGoogle Scholar
  15. 15.
    Grebogi, C., Ott, E., Romeiras, F., & Yorke, J. A. (1987). Critical exponents for crisis-induced intermittency. Physical Review A, 36, 5365.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Horton, W. (1999). Drift waves and transport. Reviews of Modern Physics, 71, 735–778.CrossRefGoogle Scholar
  17. 17.
    Hramov, A. E., Koronovskii, A. A., Maximenko, V. A., & Moskalenko, O. I. (2012). Computation of the spectrum of spatial Lyapunov exponents for the spatially extended beam-plasma systems and electron-wave devices. Physics of Plasmas, 19(8) 082302.CrossRefGoogle Scholar
  18. 18.
    Incropera, F. P., & DeWitt, D. P. (2007). Fundamentals of heat and mass transfer (7th ed.). Hoboken, NJ: Wiley.Google Scholar
  19. 19.
    Kamide, Y., & Chian, A. C.-L. (2007). Handbook of the solar-terrestrial environment. New York: SpringerCrossRefGoogle Scholar
  20. 20.
    Kapitaniak, T., Maistrenko, Y., & Popovych, S. (2000). Chaos-hyperchaos transition. Physical Review E, 62(2), 1972–1976.CrossRefGoogle Scholar
  21. 21.
    Klocek, D. (2011). Climate: Soul of the Earth. Great Barrington, MA: Lindisfarne Books.Google Scholar
  22. 22.
    Lai, Y.-C., & Tél, T. (2010). Transient chaos. New York: Springer.MATHGoogle Scholar
  23. 23.
    Lord Rayleigh, O. M. F. R. S. (1916). On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philosophical Magazine, 32(192), 529–546.MATHGoogle Scholar
  24. 24.
    Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130.CrossRefGoogle Scholar
  25. 25.
    Macek, W. M., & Strumik, M. (2014). Hyperchaotic intermittent convection in a magnetized viscous fluid. Physical Review Letters, 112, 074502.CrossRefGoogle Scholar
  26. 26.
    Martins, C. G. L., Egydio de Carvalho, R., Caldas, I. L., & Roberto, M. (2010). The non-twist standard map with robust tori. Journal of Physics A: Mathematical and Theoretical, 43(17), 175501.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Martins, C. G. L., Egydio de Carvalho, R., Caldas, I., & Roberto, M. (2011). Plasma confinement in tokamaks with robust torus. Physica A: Statistical Mechanics and its Applications, 390(5), 957–962.CrossRefGoogle Scholar
  28. 28.
    Miranda, R. A., Rempel, E. L., & Chian, A. C.-L. (2009). On-off intermittency and amplitude-phase synchronization in Keplerian shear flows. Monthly Notices of the Royal Astronomical Society, 448(1), 804.CrossRefGoogle Scholar
  29. 29.
    Miranda, R. A., Rempel, E. L., Chian, A. C.-L., & Borotto, F. A. (2005). Intermittent chaos in nonlinear wave-wave interactions in space plasmas. Journal of Atmospheric and Solar-Terrestrial Physics, 67(17–18), 1852.CrossRefGoogle Scholar
  30. 30.
    Muller, R. (1985). The fine structure of the quiet Sun. Solar Physics, 100(1), 237–255.Google Scholar
  31. 31.
    Niemela, J. J., Skrbek, L., Sreenivasan, K. R., & Donnelly, R. J. (2000). Turbulent convection at very high Rayleigh numbers. Nature, 404(6780), 837–840.CrossRefGoogle Scholar
  32. 32.
    Parker, E. N. (1989). Solar and stellar magnetic fields and atmospheric structures: Theory (pp. 271–288). Dordrecht: Springer.Google Scholar
  33. 33.
    Paul, S., Wahi, P., & Verma, M. K. (2011). Bifurcations and chaos in large-Prandtl number Rayleigh-Bénard convection. International Journal of Non-Linear Mechanics, 46(5), 772–781.CrossRefGoogle Scholar
  34. 34.
    Podvigina, O. M. (2006). Magnetic field generation by convective flows in a plane layer. European Physical Journal B, 50(4), 639–652.CrossRefGoogle Scholar
  35. 35.
    Podvigina, O. M. (2008). Magnetic field generation by convective flows in a plane layer: the dependence on the Prandtl numbers. Geophysical & Astrophysical Fluid Dynamics, 102(4), 409–433.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Portela, J. S., Caldas, I. L., & Viana, R. L. (2008). Tokamak magnetic field lines described by simple maps. The European Physical Journal Special Topics, 165(1), 195–210.CrossRefGoogle Scholar
  37. 37.
    Rempel, E. L., Chian, A. C. L., Macau, E. E. N., & Rosa, R. R. (2004). Analysis of chaotic saddles in low-dimensional dynamical systems: The derivative nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 199(3–4), 407–424.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rempel, E. L., Proctor, M. R. E., & Chian, A. C.-L. (2009). A novel type of intermittency in a non-linear dynamo in a compressible flow. Monthly Notices of the Royal Astronomical Society, 400(1), 509.CrossRefGoogle Scholar
  39. 39.
    Rüdiger, G., & Hollerbach, R. (2004). The magnetic universe: Geophysical and astrophysical dynamo theory. Weinheim: Wiley.CrossRefGoogle Scholar
  40. 40.
    Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid Mechanics, 38(1), 203–224.CrossRefGoogle Scholar
  41. 41.
    Szezech, J. D. J., Caldas, I. L., Lopes, S. R., Viana, R. L., & Morrison, P. J. (2009). Transport properties in nontwist area-preserving maps. Chaos: An Interdisciplinary Journal of Nonlinear Science, 19(4), 043108.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Vasiliev, A., & Frick, P. (2011). Reversals of large-scale circulation at turbulent convection in rectangular boxes. Journal of Physics: Conference Series, 318(8), 82013.Google Scholar
  43. 43.
    Voyatzis, G., & Ichtiaroglou, S. (1999). Degenerate bifurcations of resonant tori in Hamiltonian systems. International Journal of Bifurcation and Chaos, 09(05), 849–863.MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wurm, A., Apte, A., Fuchss, K., & Morrison, P. J. (2005). Meanders and reconnection–collision sequences in the standard nontwist map. Chaos: An Interdisciplinary Journal of Nonlinear Science, 15(2), 023108.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Yanagita, T., & Kaneko, K. (1995). Rayleigh-Bénard convection patterns, chaos, spatiotemporal chaos and turbulence. Physica D: Nonlinear Phenomena, 82(3), 288–313.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Emanuel V. Chimanski
    • 1
  • Caroline G. L. Martins
    • 1
  • Roman Chertovskih
    • 2
    • 3
  • Erico L. Rempel
    • 4
    • 5
  • Marisa Roberto
    • 1
  • Iberê L. Caldas
    • 6
  • Abraham C.-L. Chian
    • 7
  1. 1.Instituto Tecnológico de AeronáuticaSão José dos CamposBrazil
  2. 2.Research Center for Systems and Technologies, Faculty of EngineeringUniversity of PortoPortoPortugal
  3. 3.Samara National Research UniversitySamaraRussian Federation
  4. 4.Instituto Tecnológico de AeronáuticaSão José dos CamposBrazil
  5. 5.National Institute for Space ResearchSão José dos CamposBrazil
  6. 6.Instituto de FísicaUniversidade de São PauloSão PauloBrazil
  7. 7.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations