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Enstrophy inertial range dynamics in generalized two-dimensional turbulence

Takahiro Iwayama*

Takeshi Watanabe†

  • Department of Planetology, Graduate School of Science, Kobe University, Kobe 657-8501, Japan

  • Department of Physical Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466-8555, Japan

  • *iwayama@kobe-u.ac.jp
  • †watanabe@nitech.ac.jp
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Phys. Rev. Fluids 1, 034403 – Published 26 July, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.034403

Abstract

We show that the transition to a k−1 spectrum in the enstrophy inertial range of generalized two-dimensional turbulence can be derived analytically using the eddy damped quasinormal Markovianized (EDQNM) closure. The governing equation for the generalized two-dimensional fluid system includes a nonlinear term with a real parameter α. This parameter controls the relationship between the stream function and generalized vorticity and the nonlocality of the dynamics. An asymptotic analysis accounting for the overwhelming dominance of nonlocal triads allows the k−1 spectrum to be derived based upon a scaling analysis. We thereby provide a detailed analytical explanation for the scaling transition that occurs in the enstrophy inertial range at α=2 in terms of the spectral dynamics of the EDQNM closure, which extends and enhances the usual phenomenological explanations.

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References (31)

  1. R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  2. C. E. Leith, Diffusion approximation for two-dimensional turbulence, Phys. Fluids 11, 671 (1968).
  3. G. K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids 12, II-233 (1969).
  4. R. H. Kraichnan, Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech. 47, 525 (1971).
  5. K. Ohkitani, Nonlocality in a forced two-dimensional turbulence, Phys. Fluids A 2, 1529 (1990).
  6. Y. Kaneda and T. Ishihara, Nonuniversal k−3 energy spectrum in stationary two-dimensional homogeneous turbulence, Phys. Fluids 13, 1431 (2001).
  7. V. Borue, Inverse energy cascade in stationary two-dimensional homogeneous turbulence, Phys. Rev. Lett. 72, 1475 (1994).
  8. T. Gotoh, Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence, Phys. Rev. E 57, 2984 (1998).
  9. T. Ishihara and Y. Kaneda, Energy spectrum in the enstrophy transfer range of two-dimensional forced turbulence, Phys. Fluids 13, 544 (2001).
  10. R. K. Scott, Nonrobustness of the two-dimensional turbulent inverse cascade, Phys. Rev. E 75, 046301 (2007).
  11. J. Fontane, D. G. Dritschel, and R. K. Scott, Vortical control of forced two-dimensional turbulence, Phys. Fluids 25, 015101 (2013).
  12. R. T. Pierrehumbert, I. M. Held, and K. L. Swanson, Spectra of local and nonlocal two-dimensional turbulence, Chaos Solitons Fractals 4, 1111 (1994).
  13. I. M. Held, R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech. 282, 1 (1995).
  14. W. Blumen, Uniform potential vorticity flow: Part I. Theory of wave interactions and two-dimensional turbulence, J. Atmos. Sci. 35, 774 (1978).
  15. T. Iwayama, T. G. Shepherd, and T. Watanabe, An ‘ideal’ form of decaying two-dimensional turbulence, J. Fluid Mech. 456, 183 (2002).
  16. K. S. Smith, G. Boccaletti, C. C. Henning, I. Marinov, C. Y. Tam, I. M. Held, and G. K. Vallis, Turbulent diffusion in the geostrophic inverse cascade, J. Fluid Mech. 469, 13 (2002).
  17. T. Iwayama, S. Murakami, and T. Watanabe, Anomalous eddy viscosity for two-dimensional turbulence, Phys. Fluids 27, 045104 (2015).
  18. N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E 61, 6572 (2000).
  19. T. Watanabe and T. Iwayama, Unified scaling theory for local and non-local transfers in generalized two-dimensional turbulence, J. Phys. Soc. Jpn. 73, 3319 (2004).
  20. T. Watanabe and T. Iwayama, Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence, Phys. Rev. E 76, 046303 (2007).
  21. S. A. Orszag, Analytical theories of turbulence, J. Fluid Mech. 41, 363 (1970).
  22. R. H. Kraichnan, An almost-Markovian Galilean-invariant turbulence model, J. Fluid Mech. 47, 513 (1971).
  23. M. Lesieur, Turbulence in Fluids, 4th ed. (Springer, Berlin, 2008).
  24. J. Herring and J. C. McWilliams, Comparison of direct numerical simulation of two-dimensional turbulence with two-point closure: the effects of intermittency, J. Fluid Mech. 153, 229 (1985).
  25. B. H. Burgess and T. G. Shepherd, Spectral nonlocality, absolute equilibria, and Kraichnan-Leith-Batchelor phenomenology in two-dimensional turbulent energy cascades, J. Fluid Mech. 725, 332 (2013).
  26. T. Iwayama and T. Watanabe, Universal spectrum in the infrared range of two-dimensional turbulent flows, Phys. Fluids 26, 025105 (2014).
  27. A. Pouquet, M. Lesieur, J. C. André, and C. Basdevant, Evolution of high Reynolds number two-dimensional turbulence, J. Fluid Mech. 72, 305 (1975).
  28. C. Basdevant, M. Lesieur, and R. Sadourny, Subgrid-scale modeling of enstrophy transfer in two-dimensional turbulence, J. Atmos. Sci. 35, 1028 (1978).
  29. W. J. Bos and J.-P. Bertoglio, Large-scale bottleneck effect in two-dimensional turbulence, J. Turbul. 10, N30 (2009).
  30. T. Iwayama and T. Watanabe, Green's function for a generalized two-dimensional fluid, Phys. Rev. E 82, 036307 (2010).
  31. B. H. Burgess, R. K. Scott, and T. G. Shepherd, Kraichnan-Leith-Batchelor similarity theory and two-dimensional inverse cascades, J. Fluid Mech. 767, 467 (2015).

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Phys. Rev. Fluids 1, 034403– Published 26 July, 2016
  • Received 12 January 2016

©2016 American Physical Society

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