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Anomalous scaling in the model of turbulent advection of a vector field

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Abstract

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order.

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References

  1. A. S. Monin and A. M. Yaglom, Statistical Hydromechanics [in Russian] (2nd ed.), Vol. 2, Gidrometeoizdat, St. Petersburg (1996); English transl. prev. ed.: Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press, Cambridge (1975); U. Frish, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge Univ. Press, Cambridge (1995).

    Google Scholar 

  2. S. Grossman and E. Schnedler, Z. Phys. B, 26, 307 (1977).

    Google Scholar 

  3. E. Levich, Phys. Lett. A, 79, 171 (1980); G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Modern Phys., 73, 913 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  4. L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, Phys. Rev. E, 58, 1823 (1998).

    ADS  MathSciNet  Google Scholar 

  5. L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, Theor. Math. Phys., 120, 1074 (1999).

    MathSciNet  Google Scholar 

  6. R. H. Kraichnan, Phys. Fluids, 11, 945 (1968); Phys. Rev. Lett., 72, 1016 (1994); 78, 4922 (1997); A. P. Kazantsev, JETP, 26, 1031 (1968).

    MATH  MathSciNet  Google Scholar 

  7. K. Gawedzki and A. Kupiainen, Phys. Rev. Lett., 75, 3834 (1995); D. Bernard, K. Gawedzki, and A. Kupiainen, Phys. Rev. E, 54, 2564 (1996).

    ADS  Google Scholar 

  8. M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Phys. Rev. E, 52, 4924 (1995); M. Chertkov and G. Falkovich, Phys. Rev. Lett., 76, 2706 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  9. E. Balkovsky, M. Chertkov, I. Kolokolov, and V. Lebedev, JETP Letters, 61, 1049 (1995); chao-dyn/9508002 (1995); A. Pumir, Europhys. Lett., 34, 25 (1996); Phys. Rev. E, 57, 2914 (1998); B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett., 77, 2463 (1996); A. Pumir, B. I. Shraiman, and E. D. Siggia, Phys. Rev. E, 55, R1263 (1997).

    ADS  Google Scholar 

  10. L. Ts. Adzhemyan, N. V. Antonov, V. A. Barinov, Yu. S. Kabrits, and A. N. Vasil’ev, Phys. Rev. E, 63, R025303 (2001); 64, 019901 (2001).

  11. K. Gawedzki, Nucl. Phys. B (Proc. Suppl.), 58, 123 (1997); “Turbulence under a magnifying glass,” in: Quantum Fields and Quantum Space Time (ASI Ser., Ser. B, Phys., Vol. 364, G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora, eds.), Plenum, New York (1997), p. 123; chao-dyn/9610003 (1996).

    ADS  MATH  MathSciNet  Google Scholar 

  12. L. Ts. Adzhemyan and N. V. Antonov, Phys. Rev. E, 58, 7381 (1998).

    ADS  MathSciNet  Google Scholar 

  13. N. V. Antonov, Phys. Rev. E, 60, 6691 (1999); L. Ts. Adzhemyan, N. V. Antonov, and J. Honkonen, Phys. Rev. E, 66, 036313 (2002).

    ADS  MATH  MathSciNet  Google Scholar 

  14. L. Ts. Adzhemyan, N. V. Antonov, and A. V. Runov, Phys. Rev. E, 64, 046310 (2001).

  15. I. Arad and I. Procaccia, Phys. Rev. E, 63, 056302 (2001).

  16. K. Yoshida and Y. Kaneda, Phys. Rev. E, 63, 016308 (2000); S. V. Novikov, Vestn. S.-Petersb. Univ., Ser. 4, No. 4 (25), 77 (2002).

    Google Scholar 

  17. S. V. Novikov, Theor. Math. Phys., 136, 936 (2003).

    Article  Google Scholar 

  18. A. N. Vasil’ev, Quantum-Field Renormalization Group in the Theory of Critical Behavior and Stochastic Dynamics [in Russian], PIYaF Publ., St. Petersburg (1998); English transl.: The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Chapman and Hall/CRC, Boca Raton, Fl. (2004).

    Google Scholar 

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  1. St. Petersburg State University, St. Petersburg, Russia

    L. Ts. Adzhemyan & S. V. Novikov

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  1. L. Ts. Adzhemyan
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  2. S. V. Novikov
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.

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Adzhemyan, L.T., Novikov, S.V. Anomalous scaling in the model of turbulent advection of a vector field. Theor Math Phys 146, 393–410 (2006). https://doi.org/10.1007/s11232-006-0048-y

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  • DOI: https://doi.org/10.1007/s11232-006-0048-y

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