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Intermittency and Transport Barriers in Fluids and Plasmas

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A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 22))

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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.

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References

  1. Alligood, K. T., Sauer, T. D., Yorke, J. A., & Crawford, J. D. (1997). Chaos: An introduction to dynamical systems. New York: Springer.

    Book  Google Scholar 

  2. Altmann, E. G., Portela, J. S. E., & Tél, T. (2013). Leaking chaotic systems. Reviews of Modern Physics, 85, 869.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  4. Bergé, P., Dubois, M., Mannevillel, P., & Pomeau, Y. (1980). Intermittency in Rayleigh-Bénard convection. Journal de Physique Lettres 41(15), 341–345.

    Article  Google Scholar 

  5. Boyd, J. P. (2000). Chebyshev and Fourier spectral methods (2nd ed.). Mineola, NY: Dover Publications.

    Google Scholar 

  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. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. New York: Dover Publications.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  9. Chian, A. C.-L., Rempel, E. L., & Rogers, C. (2007). Crisis-induced intermittency in non-linear economic cycles. Applied Economics Letters, 14, 211.

    Article  Google Scholar 

  10. Chu, S., & Gascard, J. C. (1991). Deep convection and deep water formation in the oceans. Amsterdam: Elsevier.

    Book  Google Scholar 

  11. Clerc, M. G., & Verschueren, N. (2013). Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems. Physical Review E, 88(5), 052916.

    Article  Google Scholar 

  12. Cox, S. M., & Matthews, P. C. (2002). Exponential time differencing for stiff systems. Journal of Computational Physics, 176(2), 430–455.

    Article  MathSciNet  Google Scholar 

  13. Dormy, E., & Soward, A. M. (2007). Mathematical aspects of natural dynamos. Boca Raton, FL: CRC Press.

    Book  Google Scholar 

  14. Gollub, J. P., & Benson, S. V. (1980). Many routes to turbulent convection. Journal of Fluid Mechanics, 100(3), 449–470.

    Article  Google Scholar 

  15. Grebogi, C., Ott, E., Romeiras, F., & Yorke, J. A. (1987). Critical exponents for crisis-induced intermittency. Physical Review A, 36, 5365.

    Article  MathSciNet  Google Scholar 

  16. Horton, W. (1999). Drift waves and transport. Reviews of Modern Physics, 71, 735–778.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  18. Incropera, F. P., & DeWitt, D. P. (2007). Fundamentals of heat and mass transfer (7th ed.). Hoboken, NJ: Wiley.

    Google Scholar 

  19. Kamide, Y., & Chian, A. C.-L. (2007). Handbook of the solar-terrestrial environment. New York: Springer

    Book  Google Scholar 

  20. Kapitaniak, T., Maistrenko, Y., & Popovych, S. (2000). Chaos-hyperchaos transition. Physical Review E, 62(2), 1972–1976.

    Article  Google Scholar 

  21. Klocek, D. (2011). Climate: Soul of the Earth. Great Barrington, MA: Lindisfarne Books.

    Google Scholar 

  22. Lai, Y.-C., & Tél, T. (2010). Transient chaos. New York: Springer.

    MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  24. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130.

    Article  Google Scholar 

  25. Macek, W. M., & Strumik, M. (2014). Hyperchaotic intermittent convection in a magnetized viscous fluid. Physical Review Letters, 112, 074502.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  30. Muller, R. (1985). The fine structure of the quiet Sun. Solar Physics, 100(1), 237–255.

    Google Scholar 

  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.

    Article  Google Scholar 

  32. Parker, E. N. (1989). Solar and stellar magnetic fields and atmospheric structures: Theory (pp. 271–288). Dordrecht: Springer.

    Google Scholar 

  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.

    Article  Google Scholar 

  34. Podvigina, O. M. (2006). Magnetic field generation by convective flows in a plane layer. European Physical Journal B, 50(4), 639–652.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  39. Rüdiger, G., & Hollerbach, R. (2004). The magnetic universe: Geophysical and astrophysical dynamo theory. Weinheim: Wiley.

    Book  Google Scholar 

  40. Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid Mechanics, 38(1), 203–224.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. Voyatzis, G., & Ichtiaroglou, S. (1999). Degenerate bifurcations of resonant tori in Hamiltonian systems. International Journal of Bifurcation and Chaos, 09(05), 849–863.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  45. Yanagita, T., & Kaneko, K. (1995). Rayleigh-Bénard convection patterns, chaos, spatiotemporal chaos and turbulence. Physica D: Nonlinear Phenomena, 82(3), 288–313.

    Article  Google Scholar 

Download references

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).

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Authors and Affiliations

  1. Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo, Brazil

    Emanuel V. Chimanski, Caroline G. L. Martins & Marisa Roberto

  2. Research Center for Systems and Technologies, Faculty of Engineering, University of Porto, Porto, Portugal

    Roman Chertovskih

  3. Samara National Research University, Samara, Russian Federation

    Roman Chertovskih

  4. Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo, Brazil

    Erico L. Rempel

  5. National Institute for Space Research, São José dos Campos, São Paulo, Brazil

    Erico L. Rempel

  6. Instituto de Física, Universidade de São Paulo, São Paulo, Brazil

    Iberê L. Caldas

  7. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, Australia

    Abraham C.-L. Chian

Authors
  1. Emanuel V. Chimanski
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  2. Caroline G. L. Martins
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  3. Roman Chertovskih
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  4. Erico L. Rempel
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  6. Iberê L. Caldas
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  7. Abraham C.-L. Chian
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Correspondence to Erico L. Rempel .

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Editors and Affiliations

  1. Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos, Brazil

    Elbert E. N. Macau

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Chimanski, E.V. et al. (2019). Intermittency and Transport Barriers in Fluids and Plasmas. In: Macau, E. (eds) A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems . Nonlinear Systems and Complexity, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-78512-7_5

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  • DOI: https://doi.org/10.1007/978-3-319-78512-7_5

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