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Shell model intermittency is the hidden self-similarity

Alexei A. Mailybaev*

  • Instituto de Matemática Pura e Aplicada, Rio de Janeiro 22460-320, Brazil

  • *alexei@impa.br
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Phys. Rev. Fluids 7, 034604 – Published 30 March, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.034604

Abstract

We show that the intermittent dynamics observed in the inertial interval of the Sabra shell model of turbulence can be rigorously related to the property of scaling self-similarity. In this connection, the space-time scaling symmetries [like in the Kolmogorov 1941 (K41) theory] are replaced by the hidden scaling symmetry, which is an exact symmetry of inviscid dynamics represented in special rescaled coordinates and times. We derive formulas expressing the anomalous scaling exponents in terms of Perron-Frobenius eigenvalues of linear operators based on the self-similar statistics. Theoretical conclusions are verified by extensive numerical simulations.

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References (25)

  1. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  2. E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Doklady 18, 216 (1973).
  3. K. Ohkitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys. 81, 329 (1989).
  4. V. S. L'vov, E. Podivilov, A. Pomyalov, I. Procaccia, and D. Vandembroucq, Improved shell model of turbulence, Phys. Rev. E 58, 1811 (1998).
  5. L. Biferale, Shell models of energy cascade in turbulence, Annu. Rev. Fluid Mech. 35, 441 (2003).
  6. L. Kadanoff, D. Lohse, J. Wang, and R. Benzi, Scaling and dissipation in the GOY shell model, Phys. Fluids 7, 617 (1995).
  7. R. Benzi, L. Biferale, M. Sbragaglia, and F. Toschi, Intermittency in turbulence: Computing the scaling exponents in shell models, Phys. Rev. E 68, 046304 (2003).
  8. V. S. L'vov and I. Procaccia, Analytic calculation of the anomalous exponents in turbulence: Using the fusion rules to flush out a small parameter, Phys. Rev. E 62, 8037 (2000).
  9. G. L. Eyink, Lagrangian field theory, multifractals, and universal scaling in turbulence, Phys. Lett. A 172, 355 (1993).
  10. V. L'vov and I. Procaccia, Fusion Rules in Turbulent Systems with Flux Equilibrium, Phys. Rev. Lett. 76, 2898 (1996).
  11. A. A. Mailybaev, Hidden spatiotemporal symmetries and intermittency in turbulence, arXiv:2010.13089.
  12. A. A. Mailybaev, Hidden scale invariance of intermittent turbulence in a shell model, Phys. Rev. Fluids 6, L012601 (2021).
  13. A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13, 82 (1962).
  14. R. Benzi, L. Biferale, and G. Parisi, On intermittency in a cascade model for turbulence, Phys. D (Amsterdam, Neth.) 65, 163 (1993).
  15. G. L. Eyink, S. Chen, and Q. Chen, Gibbsian hypothesis in turbulence, J. Stat. Phys. 113, 719 (2003).
  16. N. Vladimirova, M. Shavit, and G. Falkovich, Fibonacci Turbulence, Phys. Rev. X 11, 021063 (2021).
  17. G. Parisi and U. Frisch, On the singularity structure of fully developed turbulence, in Predictability in Geophysical Fluid Dynamics, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985), pp. 84–87.
  18. Z.-S. She and E. Leveque, Universal Scaling Laws in Fully Developed Turbulence, Phys. Rev. Lett. 72, 336 (1994).
  19. A. A. Mailybaev, Solvable intermittent shell model of turbulence, Commun. Math. Phys. 388, 469 (2021).
  20. A. A. Mailybaev, Spontaneously stochastic solutions in one-dimensional inviscid systems, Nonlinearity 29, 2238 (2016).
  21. P. D. Lax, Linear Algebra and Its Applications (Wiley, Hoboken, NJ, 2007), Chap. 16.
  22. K. Deimling, Nonlinear Functional Analysis (Courier Corporation, New York, 2010), Sec. 19.5.
  23. T. Tao, An Introduction to Measure Theory (American Mathematical Society, Providence, RI, 2011).
  24. A. A. Mailybaev and S. Thalabard, Hidden scale invariance in Navier-Stokes intermittency, Philos. Trans. R. Soc. A. 380, 20210098 (2022).
  25. W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math. 2, 53 (2002).

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Phys. Rev. Fluids 7, 034604– Published 30 March, 2022
  • Received 15 January 2022
  • Accepted 16 March 2022
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