Advertisement

Anomalous scaling in the Kazantsev-Kraichnan model with finite time correlations: two-loop renormalization group analysis of relevant composite operators

  • Eva Jurčišinová
  • Marian JurčišinEmail author
  • Martin Menkyna
Regular Article
  • 36 Downloads

Abstract

The field theoretic renormalization group technique together with the operator product expansion in the second order of the perturbation theory (in the two-loop approximation) is used for the investigation of the influence of the finite time correlations of the velocity field on the anomalous dimensions of the leading set of composite operators, which drive the anomalous scaling of correlation functions of a weak magnetic field in the framework of the kinematic Kazantsev–Kraichnan model in the presence of a large scale anisotropy. The system of possible scaling regimes of the model is found and two important special limits of the model are briefly discussed. The general two-loop expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the spatial dimension d and of the renormalization group fixed point value of the parameter u, which drives the presence of the finite time correlations of the velocity field in the model. The anisotropic hierarchies among various anomalous dimensions are investigated and it is shown that, regardless of the fixed point value of the parameter u as well as regardless of the spatial dimension of the system, the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The properties of the anomalous dimensions of the leading composite operators in the Kazantsev–Kraichnan model with finite time correlations of the velocity field are compared to the properties of the corresponding anomalous dimensions of the composite operators relevant in the framework of the Kraichnan model of passively advected scalar field with finite time correlations. It is shown that, regardless of the fixed point value of the parameter u, the two-loop corrections to the anomalous dimensions are much more important in the framework of the Kazantsev–Kraichnan vector model than in the Kraichnan model of a passive scalar advection. At the same time, again regardless of the strength of time correlations, the two-loop values of the leading anomalous dimensions in the Kazantsev–Kraichnan model of the passive magnetic field are significantly more negative than the corresponding two-loop values of the relevant anomalous dimensions in the framework of the Kraichnan model. It means that the anomalous scaling of the correlation functions of the passive magnetic field, deep inside the inertial interval of the turbulent environment with finite time correlations of the velocity field, must be much more pronounced than in the case of the correlation functions of the passively advected scalar field.

Graphical abstract

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 301 (1941) [reprinted in Proc. R. Soc. Lond. A 434, 9 1991] ADSGoogle Scholar
  2. 2.
    A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 31, 538 (1941) Google Scholar
  3. 3.
    A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 32, 16 (1941) [reprinted in Proc. R. Soc. Lond. A 434, 15 1991] ADSGoogle Scholar
  4. 4.
    A.S. Monin, A.M. Yaglom, in Statistical Fluid Mechanics (MIT Press, Cambridge, MA, 1975), Vol. 2 Google Scholar
  5. 5.
    W.D. McComb, The Physics of Fluid Turbulence (Clarendon, Oxford, 1990) Google Scholar
  6. 6.
    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995) Google Scholar
  7. 7.
    K.R. Sreenivasan, R.A. Antonia, Ann. Rev. Fluid Mech. 29, 435 (1997) ADSCrossRefGoogle Scholar
  8. 8.
    G. Falkovich, K. Gawȩdzki, M. Vergassola, Rev. Mod. Phys. 73, 913 (2001) ADSCrossRefGoogle Scholar
  9. 9.
    L.Ts. Adzhemyan, N.V. Antonov, A.N. Vasil’ev, The Field Theoretic Renormalization Group in Fully Developed Turbulence (Gordon & Breach, London, 1999) Google Scholar
  10. 10.
    R.A. Antonia, B.R. Satyaprakash, A.K.F. Hussain, J. Fluid Mech. 119, 55 (1982) ADSCrossRefGoogle Scholar
  11. 11.
    F. Anselmet, Y. Gagne, E. Hopfinger, R.A. Antonia, J. Fluid Mech. 140, 63 (1984) ADSCrossRefGoogle Scholar
  12. 12.
    C. Meneveau, K.R. Sreenivasan, Phys. Rev. A 41, 2246 (1990) ADSCrossRefGoogle Scholar
  13. 13.
    M.S. Borgas, Phys. Fluids A 4, 2055 (1992) ADSCrossRefGoogle Scholar
  14. 14.
    V.R. Kuznetsov, V.A. Sabel’nikov, Turbulence and Combustion (Hemisphere Publishing, New York, 1990) Google Scholar
  15. 15.
    N.V. Antonov, J. Phys. A 39, 7825 (2006) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    R.A. Antonia, E.J. Hopfinger, Y. Gagne, F. Anselmet, Phys. Rev. A 30, 2704 (1984) ADSCrossRefGoogle Scholar
  17. 17.
    K.R. Sreenivasan, Proc. R. Soc. Lond., Ser. A 434, 165 (1991) ADSCrossRefGoogle Scholar
  18. 18.
    M. Holzer, E.D. Siggia, Phys. Fluids 6, 1820 (1994) ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Pumir, Phys. Fluids 6, 2118 (1994) ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    C. Tong, Z. Warhaft, Phys. Fluids 6, 2165 (1994) ADSCrossRefGoogle Scholar
  21. 21.
    T. Elperin, N. Kleeorin, I. Rogachevskii, Phys. Rev. E 52, 2617 (1995) ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    T. Elperin, N. Kleeorin, I. Rogachevskii, Phys. Rev. Lett. 76, 224 (1996) ADSCrossRefGoogle Scholar
  23. 23.
    T. Elperin, N. Kleeorin, I. Rogachevskii, Phys. Rev. E 53, 3431 (1996) ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Z. Warhaft, Ann. Rev. Fluid Mech. 32, 203 (2000) ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    B.I. Shraiman, E. Siggia, Nature 405, 639 (2000) ADSCrossRefGoogle Scholar
  26. 26.
    F. Moisy, H. Willaime, J.S. Andersen, P. Tabeling, Phys. Rev. Lett. 86, 4827 (2001) ADSCrossRefGoogle Scholar
  27. 27.
    A. Arnèodo et al., Phys. Rev. Lett. 100, 254504 (2008) ADSCrossRefGoogle Scholar
  28. 28.
    M. Vergassola, Phys. Rev. E 53, R3021 (1996) ADSCrossRefGoogle Scholar
  29. 29.
    K. Gawȩdzki, A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995) ADSCrossRefGoogle Scholar
  30. 30.
    D. Bernard, K. Gawȩdzki, A. Kupiainen, Phys. Rev. E 54, 2564 (1996) ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Chertkov, G. Falkovich, I. Kolokolov, V. Lebedev, Phys. Rev. E 52, 4924 (1995) ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    M. Chertkov, G. Falkovich, Phys. Rev. Lett. 76, 2706 (1996) ADSCrossRefGoogle Scholar
  33. 33.
    M. Avellaneda, A. Majda, Commun. Math. Phys. 131, 381 (1990) ADSCrossRefGoogle Scholar
  34. 34.
    M. Avellaneda, A. Majda, Commun. Math. Phys. 146, 139 (1992) ADSCrossRefGoogle Scholar
  35. 35.
    A. Majda, J. Stat. Phys. 73, 515 (1993) ADSCrossRefGoogle Scholar
  36. 36.
    D. Horntrop, A. Majda, J. Math. Sci. Univ. Tokyo 1, 23 (1994) MathSciNetGoogle Scholar
  37. 37.
    Q. Zhang, J. Glimm, Commun. Math. Phys. 146, 217 (1992) ADSCrossRefGoogle Scholar
  38. 38.
    R.H. Kraichnan, Phys. Rev. Lett. 72, 1016 (1994) ADSCrossRefGoogle Scholar
  39. 39.
    R.H. Kraichnan, V. Yakhot, S. Chen, Phys. Rev. Lett. 75, 240 (1995) ADSCrossRefGoogle Scholar
  40. 40.
    B.I. Shraiman, E.D. Siggia, Phys. Rev. Lett. 77, 2463 (1996) ADSCrossRefGoogle Scholar
  41. 41.
    A. Pumir, B.I. Shraiman, E.D. Siggia, Phys. Rev. E 55, R1263 (1997) ADSCrossRefGoogle Scholar
  42. 42.
    A. Pumir, Europhys. Lett. 34, 25 (1996) ADSCrossRefGoogle Scholar
  43. 43.
    A. Pumir, Europhys. Lett. 37, 529 (1997) ADSCrossRefGoogle Scholar
  44. 44.
    A. Pumir, Phys. Rev. E 57, 2914 (1998) ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    I. Rogachevskii, N. Kleeorin, Phys. Rev. E 56, 417 (1997) ADSCrossRefGoogle Scholar
  46. 46.
    A. Lanotte, A. Mazzino, Phys. Rev. E 60, R3483 (1999) ADSCrossRefGoogle Scholar
  47. 47.
    I. Arad, L. Biferale, I. Procaccia, Phys. Rev. E 61, 2654 (2000) ADSCrossRefGoogle Scholar
  48. 48.
    N.V. Antonov, A. Lanotte, A. Mazzino, Phys. Rev. E 61, 6586 (2000) ADSCrossRefGoogle Scholar
  49. 49.
    N.V. Antonov, J. Honkonen, A. Mazzino, P. Muratore-Ginanneschi, Phys. Rev. E 62, R5891 (2000) ADSCrossRefGoogle Scholar
  50. 50.
    N.V. Antonov, M. Hnatich, J. Honkonen, M. Jurčišin, Phys. Rev. E 68, 046306 (2003) ADSCrossRefGoogle Scholar
  51. 51.
    M. Chaves, K. Gawȩdzki, P. Horvai, A. Kupiainen, M. Vergassola, J. Stat. Phys. 113, 643 (2003) CrossRefGoogle Scholar
  52. 52.
    H. Arponen, Phys. Rev. E 81, 036325 (2010) ADSCrossRefGoogle Scholar
  53. 53.
    R.H. Kraichnan, Phys. Fluids 11, 945 (1968) ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    A.P. Kazantsev, Sov. Phys. JETP 26, 1031 (1968) ADSGoogle Scholar
  55. 55.
    D.J. Amit, Field Theory, Renormalization Group, and Critical Phenomena (McGraw-Hill, New York, 1978) Google Scholar
  56. 56.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon, Oxford, 1989) Google Scholar
  57. 57.
    A.N. Vasil’ev, Quantum-Field Renormalization Group in the Theory of Critical Phenomena and Stochastic Dynamics (Chapman & Hall/CRC, Boca Raton, 2004) Google Scholar
  58. 58.
    D.J. Amit, V. Martin-Mayor, Field Theory, Renormalization Group, and Critical Phenomena. Graphs to Computers, 3rd edn. (World Scientific, Singapore, 2005) Google Scholar
  59. 59.
    L.Ts. Adzhemyan, N.V. Antonov, A.N. Vasil’ev, Phys. Rev. E 58, 1823 (1998) ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    L.Ts. Adzhemyan, N.V. Antonov, A.N. Vasil’ev, Theor. Math. Phys. 120, 1074 (1999) CrossRefGoogle Scholar
  61. 61.
    L.Ts. Adzhemyan, N.V. Antonov, V.A. Barinov, Yu.S. Kabrits, A.N. Vasil’ev, Phys. Rev. E 63, 025303 (2001) ADSCrossRefGoogle Scholar
  62. 62.
    L.Ts. Adzhemyan, N.V. Antonov, V.A. Barinov, Yu.S. Kabrits, A.N. Vasil’ev, Phys. Rev. E 64, 056306 (2001) ADSCrossRefGoogle Scholar
  63. 63.
    L.Ts. Adzhemyan, N.V. Antonov, M. Hnatich, S.V. Novikov, Phys. Rev. E 63, 016309 (2000) ADSCrossRefGoogle Scholar
  64. 64.
    E. Jurčišinová, M. Jurčišin, R. Remecký, M. Scholtz, Int. J. Mod. Phys. B 22, 3589 (2008) ADSCrossRefGoogle Scholar
  65. 65.
    E. Jurčišinová, M. Jurčišin, Phys. Rev. E 77, 016306 (2008) ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    N.V. Antonov, Phys. Rev. E 60, 6691 (1999) ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    L.Ts. Adzhemyan, N.V. Antonov, J. Honkonen, Phys. Rev. E 66, 036313 (2002) ADSCrossRefGoogle Scholar
  68. 68.
    L.Ts. Adzhemyan, N.V. Antonov, Phys. Rev. E 58, 7381 (1998) ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    N.V. Antonov, J. Honkonen, Phys. Rev. E 63, 036302 (2001) ADSCrossRefGoogle Scholar
  70. 70.
    O.G. Chkhetiani, M. Hnatich, E. Jurčišinová, M. Jurčišin, A. Mazzino, M. Repašan, Phys. Rev. E 74, 036310 (2006) ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    O.G. Chkhetiani, M. Hnatich, E. Jurčišinová, M. Jurčišin, A. Mazzino, M. Repašan, J. Phys. A: Math. Gen. 39, 7913 (2006) ADSCrossRefGoogle Scholar
  72. 72.
    O.G. Chkhetiani, M. Hnatich, E. Jurčišinová, M. Jurčišin, A. Mazzino, M. Repašan, Czech. J. Phys. 56 (2006) 827 ADSCrossRefGoogle Scholar
  73. 73.
    L.Ts. Adzhemyan, N.V. Antonov, J. Honkonen, T.L. Kim, Phys. Rev. E 71, 016303 (2005) ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    A.V. Gladysheva, E. Jurčišinová, M. Jurčišin, R. Remecký, Phys. Part. Nucl. 41, 1023 (2010) CrossRefGoogle Scholar
  75. 75.
    E. Jurčišinová, M. Jurčišin, R. Remecký, Phys. Rev. E 80, 046302 (2009) ADSCrossRefGoogle Scholar
  76. 76.
    N.V. Antonov, N.M. Gulitskiy, M.M. Kostenko, A.V. Malyshev, Phys. Rev. E 97, 033101 (2018) ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    L.Ts. Adzhemyan, N.V. Antonov, A.V. Runov, Phys. Rev. E 64, 046310 (2001) ADSCrossRefGoogle Scholar
  78. 78.
    M. Hnatič, M. Jurčišin, A. Mazzino, S. Šprinc, Acta Phys. Slov. 52, 559 (2002) Google Scholar
  79. 79.
    S.V. Novikov, J. Phys. A: Math. Gen. 39, 8133 (2006) ADSCrossRefGoogle Scholar
  80. 80.
    E. Jurčišinová, M. Jurčišin, R. Remecký, M. Scholtz, Phys. Part. Nucl. Lett. 5, 219 (2008) CrossRefGoogle Scholar
  81. 81.
    E. Jurčišinová, M. Jurčišin, R. Remecký, J. Phys. A: Math. Theor. 42, 275501 (2009) CrossRefGoogle Scholar
  82. 82.
    N.V. Antonov, N.M. Gulitskiy, Theor. Math. Phys. 176, 851 (2013) CrossRefGoogle Scholar
  83. 83.
    L.Ts. Adzhemyan, N.V. Antonov, P.B. Gol’din, M.V. Kompaniets, J. Phys. A: Math. Theor. 46, 135002 (2013) ADSCrossRefGoogle Scholar
  84. 84.
    N.V. Antonov, N.M. Gulitskiy, Phys. Rev. E 91, 013002 (2015) ADSCrossRefGoogle Scholar
  85. 85.
    N.V. Antonov, N.M. Gulitskiy, Phys. Rev. E 92, 043018 (2015) ADSCrossRefGoogle Scholar
  86. 86.
    N.V. Antonov, M.M. Kostenko, Phys. Rev. E 92, 053013 (2015) ADSMathSciNetCrossRefGoogle Scholar
  87. 87.
    N.V. Antonov, N.M. Gulitskiy, EPJ Web Conf. 108, 02008 (2016) CrossRefGoogle Scholar
  88. 88.
    M. Hnatich, J. Honkonen, M. Jurčišin, A. Mazzino, S. Šprinc, Phys. Rev. E 71, 066312 (2005) ADSCrossRefGoogle Scholar
  89. 89.
    N.V. Antonov, N.M. Gulitskiy, Phys. Rev. E 85, 065301 (2012) ADSCrossRefGoogle Scholar
  90. 90.
    N.V. Antonov, N.M. Gulitskiy, Lecture Notes Comput. Sci. 7125, 128 (2012) CrossRefGoogle Scholar
  91. 91.
    E. Jurčišinová, M. Jurčišin, J. Phys. A: Math. Theor. 45, 485501 (2012) CrossRefGoogle Scholar
  92. 92.
    E. Jurčišinová, M. Jurčišin, Phys. Rev. E 88, 011004 (2013)(R) ADSCrossRefGoogle Scholar
  93. 93.
    E. Jurčišinová, M. Jurčišin, Phys. Rev. E 91, 063009 (2015) ADSMathSciNetCrossRefGoogle Scholar
  94. 94.
    E. Jurčišinová, M. Jurčišin, M. Menkyna, Phys. Rev. E 95, 053210 (2017) ADSCrossRefGoogle Scholar
  95. 95.
    M. Chertkov, G. Falkovich, V. Lebedev, Phys. Rev. Lett. 76, 3707 (1996) ADSCrossRefGoogle Scholar
  96. 96.
    G. Eyink, Phys. Rev. E 54, 1497 (1996) ADSCrossRefGoogle Scholar
  97. 97.
    J.P. Bouchaud, A. Comtet, A. Georges, P. Le Doussal, J. Phys. (Paris) 48, 1445 (1987) CrossRefGoogle Scholar
  98. 98.
    J.P. Bouchaud, A. Comtet, A. Georges, P. Le Doussal, J. Phys. (Paris) 49, 369 (1988) CrossRefGoogle Scholar
  99. 99.
    J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990) ADSMathSciNetCrossRefGoogle Scholar
  100. 100.
    J. Honkonen, E. Karjalainen, J. Phys. A 21, 4217 (1988) ADSCrossRefGoogle Scholar
  101. 101.
    J. Honkonen, Yu.M. Pis’mak, A.N. Vasil’ev, J. Phys. A 21, L835 (1989) CrossRefGoogle Scholar
  102. 102.
    J. Honkonen, Yu. M. Pis’mak, J. Phys. A 22, L899 (1989) CrossRefGoogle Scholar
  103. 103.
    J.D. Fournier, P.L. Sulem, A. Pouquet, J. Phys. A 15, 1393 (1982) ADSCrossRefGoogle Scholar
  104. 104.
    L.Ts. Adzhemyan, A.N. Vasil’ev, M. Gnatich, Theor. Math. Phys. 64, 777 (1985) CrossRefGoogle Scholar
  105. 105.
    P.C. Martin, E.D. Siggia, H.A. Rose, Phys. Rev. A 8, 423 (1973) ADSCrossRefGoogle Scholar
  106. 106.
    C. De Dominicis, J. Phys. (Paris), Colloq. 37, C1–247 (1976) ADSCrossRefGoogle Scholar
  107. 107.
    H.K. Janssen, Z. Phys. B 23, 377 (1976) ADSCrossRefGoogle Scholar
  108. 108.
    R. Bausch, H.K. Janssen, H. Wagner, Z. Phys. B 24, 113 (1976) ADSCrossRefGoogle Scholar
  109. 109.
    E. Jurčišinová, M. Jurčišin, Phys. Part. Nucl. 44, 360 (2013) CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Eva Jurčišinová
    • 1
    • 2
  • Marian Jurčišin
    • 1
    • 2
    • 3
    Email author
  • Martin Menkyna
    • 1
    • 2
  1. 1.Institute of Experimental Physics, Slovak Academy of SciencesKošiceSlovakia
  2. 2.Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubna, MoscowRussia
  3. 3.Department of Theoretical Physics and AstrophysicsFaculty of Science, P.J. Šafárik UniversityKošiceSlovakia

Personalised recommendations