Skip to main content
SpringerLink
Log in

Structure Function and Fractal Dissipation for an Intermittent Inviscid Dyadic Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a generalization of the original tree-indexed dyadic model by Katz and Pavlović for the turbulent energy cascade of the three-dimensional Euler equation. We allow the coefficients to vary with some restrictions, thus giving the model a realistic spatial intermittency. By introducing a forcing term on the first component, the fixed point of the dynamics is well defined and some explicit computations allow us to prove the rich multifractal structure of the solution. In particular the exponent of the structure function is concave in accordance with other theoretical and experimental models. Moreover, anomalous energy dissipation happens in a fractal set of dimension strictly less than 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Andreis L., Barbato D., Collet F., Formentin M., Provenzano L.: Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model. Nonlinearity 29(3), 1156 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Anselmet F., Gagne Y., Hopfinger E., Antonia R.: High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140(1), 63–89 (1984)

    Article  ADS  Google Scholar 

  3. Arneodo A., Bacry E., Muzy J.: Random cascades on wavelet dyadic trees. J. Math. Phys. 39(8), 4142–4164 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Barbato D., Bianchi L.A., Flandoli F., Morandin F.: A dyadic model on a tree. J. Math. Phys. 54, 021507 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Barbato D., Flandoli F., Morandin F.: A theorem of uniqueness for an inviscid dyadic model. C. R. Math. Acad. Sci. Paris 348(9-10), 525–528 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbato, D., Flandoli, F., Morandin, F.: Anomalous dissipation in a stochastic inviscid dyadic model. Ann. Appl. Probab. 21(6), 2424–2446 (2011)

  7. Barbato D., Morandin F.: Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity. Nonlinear Differ. Eq. Appl. NoDEA 20(3), 1105–1123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barbato D., Morandin F., Romito M.: Smooth solutions for the dyadic model. Nonlinearity 24(11), 3083 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Barbato D., Morandin F., Romito M.: Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system. Anal. PDE 7(8), 2009–2027 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barral J., Jin X., Mandelbrot B.T.: Convergence of complex multiplicative cascades. Ann. Appl. Probab. 20(4), 1219–1252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Belin F., Tabeling P., Willaime H.: Exponents of the structure functions in a low temperature helium experiment. Phys. D: Nonlinear Phenom. 93(1), 52–63 (1996)

    Article  ADS  MATH  Google Scholar 

  12. Benzi R., Biferale L., Ciliberto S., Struglia M., Tripiccione R.: Generalized scaling in fully developed turbulence. Phys. D: Nonlinear Phenom. 96(1), 162–181 (1996)

    Article  ADS  Google Scholar 

  13. Benzi R., Biferale L., Crisanti A., Paladin G., Vergassola M., Vulpiani A.: A random process for the construction of multiaffine fields. Phys. D: Nonlinear Phenom. 65(4), 352–358 (1993)

    Article  ADS  MATH  Google Scholar 

  14. Benzi R., Biferale L., Parisi G.: On intermittency in a cascade model for turbulence. Phys. D: Nonlinear Phenom. 65(1-2), 163–171 (1993)

    Article  ADS  MATH  Google Scholar 

  15. Benzi R., Biferale L., Tripiccione R., Trovatore E.: (1+1)-dimensional turbulence. Phys. Fluids 9, 2355 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Benzi R., Paladin G., Parisi G., Vulpiani A.: On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A: Math. Gen. 17(18), 3521 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  17. Bianchi L.A.: Uniqueness for an inviscid stochastic dyadic model on a tree. Electron. Commun. Probab. 18, 1–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Biferale L.: Shell models of energy cascade in turbulence. Annu. Rev. Fluid Mech. 35, 441–468 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Buckmaster T.: Onsager’s conjecture almost everywhere in time. Commun. Math. Phys. 333(3), 1175–1198 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Buckmaster, T., De Lellis, C., Székelyhidi, L.: Dissipative Euler flows with Onsager-critical spatial regularity. Commun. Pure Appl. Math. 69, 1613–1670 (2015)

  21. Cheskidov A., Friedlander S.: The vanishing viscosity limit for a dyadic model. Physics D 238(8), 783–787 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Cheskidov A., Friedlander S., Pavlović N.: Inviscid dyadic model of turbulence: the fixed point and Onsager’s conjecture. J. Math. Phys. 48(6), 065503, 16 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Cheskidov A., Friedlander S., Pavlović N.: An inviscid dyadic model of turbulence: the global attractor. Discret. Contin. Dyn. Syst. 26(3), 781–794 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Constantin P., E W., Titi E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Deriaz E., Perrier V.: Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows. J. Turbul. 7(3), 1–37 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Desnianskii V.N., Novikov E.A.: Simulation of cascade processes in turbulent flows. Prikladnaia Matematika i Mekhanika 38, 507–513 (1974)

    ADS  Google Scholar 

  27. Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13(1), 249–255 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Eggleston H.: The fractional dimension of a set defined by decimal properties. Q. J. Math. 20, 31–36 (1949)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Eyink G.L.: Energy dissipation without viscosity in ideal hydrodynamics i. fourier analysis and local energy transfer. Phys. D: Nonlinear Phenom. 78(3), 222–240 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Eyink G.L.: Besov spaces and the multifractal hypothesis. J. Stat. Phys. 78(1-2), 353–375 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Friedlander S., Pavlović N.: Blowup in a three-dimensional vector model for the Euler equations. Comm. Pure Appl. Math. 57(6), 705–725 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Frisch, U.: Turbulence: The legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995)

  33. Frisch U., Sulem P.-L., Nelkin M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87(04), 719–736 (1978)

    Article  ADS  MATH  Google Scholar 

  34. Gledzer, E.: System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 18(4), 216–217 (1973)

  35. Isett, P.: Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time, volume 196 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (2017)

  36. Jaffard S.: Multifractal formalism for functions part i: results valid for all functions. SIAM J. Math. Anal. 28(4), 944–970 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Katz, N.H., Pavlović, N.: Finite time blow-up for a dyadic model of the Euler equations equations. Trans. Am. Math. Soc., 357(2), 695–708 (electronic) (2005)

  38. Kolmogorov A.N.: The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 301–305 (1941)

    ADS  Google Scholar 

  39. Kolmogorov A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Lewis G.S., Swinney H.L.: Velocity structure functions, scaling, and transitions in high-reynolds-number couette-taylor flow. Phys. Rev. E 59, 5457–5467 (1999)

    Article  ADS  Google Scholar 

  41. Mandelbrot B.B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62(2), 331–358 (1974)

    Article  ADS  MATH  Google Scholar 

  42. Meneveau C., Sreenivasan K.: The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484 (1991)

    Article  ADS  MATH  Google Scholar 

  43. Meyer, Y.: Wavelets and operators, volume 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (1992). Translated from the 1990 French original by D. H. Salinger.

  44. Muzy J.-F., Bacry E., Arneodo A.: Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method. Phys. Rev. E 47(2), 875 (1993)

    Article  ADS  Google Scholar 

  45. Obukhov A.: Some specific features of atmospheric turbulence. J. Geophys. Res. 67(8), 3011–3014 (1962)

    Article  ADS  MATH  Google Scholar 

  46. Ohkitani K., Yamada M.: Temporal intermittency in the energy cascade process and local lyapunov analysis in fully-developed model turbulence. Prog. Theor. Phys. 81(2), 329–341 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  47. Olsen L.: On the Hausdorff dimension of generalized Besicovitch-Eggleston sets of d-tuples of numbers. Indag. Math. 15(4), 535–547 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Onsager L.: Statistical hydrodynamics. Il Nuovo Cimento (1943–1954) 6, 279–287 (1949)

    Article  MathSciNet  Google Scholar 

  49. Parisi, G., Frisch, U.: On the singularity structure of fully developed turbulence. In: Turbulence and Predictability in Geophysical Fluid Dynamics. Proceedings of the International School of Physics E. Fermi, pp. 84–87. Amsterdam, The Netherlands, (1985)

  50. Perrier V., Basdevant C.: Besov norms in terms of the continuous wavelet transform. Application to structure functions. Math. Models Methods Appl. Sci. 6(05), 649–664 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  51. Riedi, R.H.: Multifractal processes. Technical report, DTIC Document, (1999)

  52. She Z.-S., Leveque E.: Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72(3), 336 (1994)

    Article  ADS  Google Scholar 

  53. Stevenson R.: Divergence-free wavelet bases on the hypercube. Appl. Comput. Harmonic Anal. 30(1), 1–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Tao, T.: Finite time blowup for an averaged three-dimensional Navier–Stokes equation. J. Am. Math. Soc. 29, 601–674 (2015)

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Luigi Amedeo Bianchi

  2. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parma, Italy

    Francesco Morandin

Authors
  1. Luigi Amedeo Bianchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Morandin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luigi Amedeo Bianchi.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bianchi, L.A., Morandin, F. Structure Function and Fractal Dissipation for an Intermittent Inviscid Dyadic Model. Commun. Math. Phys. 356, 231–260 (2017). https://doi.org/10.1007/s00220-017-2974-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2974-y

Search

Navigation