Renormalization group analysis of models of advection of a vector admixture and a tracer field by a compressible turbulent flow. (English. Russian original) Zbl 1434.76051
Theor. Math. Phys. 200, No. 3, 1294-1312 (2019); translation from Teor. Mat. Fiz. 200, No. 3, 429-451 (2019).
Summary: Using a quantum field theory renormalization group, we consider models of advection of a vector field and a tracer field by a compressible turbulent flow. Both advected fields are considered passive, i.e., they do not have a backward influence on the fluid dynamics. The velocity field is generated by the stochastic Navier-Stokes equation. We consider the model in the vicinity of the special space dimension \(d = 4\). Analysis of the model in the vicinity of this dimension allows constructing a double expansion in the parameters \(y\) (related to the correlator of the random force for the velocity field) and \(\epsilon = 4 - d\). We show that in the framework of the one-loop approximation, the two models have similar scaling behavior, i.e., similar behavior of the correlation and structure functions in the inertial range. We calculate all critical dimensions, in particular, of tensor composite operators, in the leading order of the double expansion in \(y\) and \(\epsilon \).
MSC:
| 76F30 | Renormalization and other field-theoretical methods for turbulence |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
Keywords:
developed turbulence; magnetohydrodynamics; turbulent advection; renormalization group; anomalous scalingReferences:
| [1] | A.N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics [in Russian], Petersburg Inst. Nucl. Phys. Press, St. Petersburg (1998); English transl., CRC, Boca Raton, Fla. (2004). |
| [2] | U. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, Cambridge Univ. Press, New York (2014). |
| [3] | J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, Oxford (2002). · Zbl 1033.81006 · doi:10.1093/acprof:oso/9780198509233.001.0001 |
| [4] | U. Frisch, Turbulence: The legacy of A. N. Kolmogorov, Cambridge Univ. Press, Cambridge (1995). · Zbl 0832.76001 · doi:10.1017/CBO9781139170666 |
| [5] | A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics [in Russian] (Part 2), Nauka, Moscow (1967); English transl.: Statistical Fluid Mechanics: Mechanics of Turbulence, Dover, New York (2007). · Zbl 1140.76004 |
| [6] | P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers, Oxford Univ. Press, Oxford (2015). · Zbl 1315.76001 · doi:10.1093/acprof:oso/9780198722588.001.0001 |
| [7] | G. Falkovich, K. Gawȩdzki, and M. Vergassola, “Particles and fields in fluid turbulence,” Rev. Modern Phys., 73, 913-975 (2001). · Zbl 1205.76133 · doi:10.1103/RevModPhys.73.913 |
| [8] | D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge Univ. Press, Cambridge (2003). · Zbl 1145.76300 · doi:10.1017/CBO9780511535222 |
| [9] | S. N. Shore, Astrophysical Hydrodynamics: An Introduction, Wiley-VCH, Weinheim (2007). · doi:10.1002/9783527619054 |
| [10] | E. Priest, Magnetohydrodynamics of the Sun, Cambridge Univ. Press, Cambridge (2014). |
| [11] | A. Pouquet, U. Frisch, and J. Léorat, “Strong MHD helical turbulence and the nonlinear dynamo effect,” J. Fluid Mech., 77, 321-354 (1976). · Zbl 0336.76019 · doi:10.1017/S0022112076002140 |
| [12] | C.-Y. Tu and E. Marsch, “MHD structures, waves, and turbulence in the solar wind: Observations and theories,” Space Sci. Rev., 73, 1-210 (1995). · doi:10.1007/BF00748891 |
| [13] | S. A. Balbus and J. F. Hawley, “Instability, turbulence, and enhanced transport in accretion disks,” Rev. Modern Phys., 70, 1-53 (1998). · doi:10.1103/RevModPhys.70.1 |
| [14] | G. Chabrier, “Galactic stellar and substellar initial mass function,” Publ. Astron. Soc. Pac., 115, 763-795 (2003). · doi:10.1086/376392 |
| [15] | B. G. Elemegreen and J. Scalo, “Interstellar turbulence I: Observations and processes,” Ann. Rev. Astron. Astrophys., 42, 211-273 (2004). · doi:10.1146/annurev.astro.41.011802.094859 |
| [16] | C. Federrath, “On the universality of supersonic turbulence,” Mon. Not. R. Astron. Soc., 436, 1245-1257 (2013); arXiv:1306.3989v4 [astro-ph.SR] (2013). · doi:10.1093/mnras/stt1644 |
| [17] | N. V. Antonov, “Renormalization group, operator product expansion, and anomalous scaling in models of turbulent advection,” J. Phys. A: Math. Gen., 39, 7825-7865 (2006). · Zbl 1117.76030 · doi:10.1088/0305-4470/39/25/S04 |
| [18] | H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Univ. Press, Cambridge (1978). |
| [19] | J. D. Fournier, P. L. Sulem, and A. Pouquet, “Infrared properties of forced magnetohydrodynamic turbulence,” J. Phys. A: Math. Gen., 15, 1393-1420 (1982). · Zbl 0498.76097 · doi:10.1088/0305-4470/15/4/037 |
| [20] | L. Ts. Adzhemyan, A. N. Vasil’ev, and M. Gnatich, “Renormalization-group approach to the theory of turbulence: Inclusion of a passive admixture,” Theor. Math. Phys., 58, 47-51 (1984). · Zbl 0556.76042 · doi:10.1007/BF01031034 |
| [21] | Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics [in Russian] (1986), Moscow |
| [22] | P. Sagaut and C. Cambon, Homogeneous Turbulence Dynamics, Cambridge Univ. Press, Cambridge (2008). · Zbl 1154.76003 · doi:10.1017/CBO9780511546099 |
| [23] | J. Kim and D. Ryu, “Density power spectrum of compressible hydrodynamic turbulent flows,” Astrophys. J., 630, L45-L48 (2005); arXiv:astro-ph/0507591v1 (2005). · doi:10.1086/491600 |
| [24] | V. Carbone, R. Marino, L. Sorriso-Valvo, A. Noullez, and R. Bruno, “Scaling laws of turbulence and heating of fast solar wind: The role of density fluctuations,” Phys. Rev. Lett., 103, 061102 (2009). · doi:10.1103/PhysRevLett.103.061102 |
| [25] | F. Sahraoui, M. L. Goldstein, P. Robert, and Yu. V. Khotyainstsev, “Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale,” Phys.Rev.Lett., 102, 231102 (2009). · doi:10.1103/PhysRevLett.102.231102 |
| [26] | H. Aluie and G. L. Eyink, “Scale locality of magnetohydrodynamic turbulence,” Phys.Rev.Lett., 104, 081101 (2010); arXiv:0912.3752v1 [astro-ph.SR] (2009). · doi:10.1103/PhysRevLett.104.081101 |
| [27] | S. Galtier and S. Banerjee, “Exact relation for correlation functions in compressible isothermal turbulence,” Phys. Rev. Lett., 107, 134501 (2011); arXiv:1108.4529v1 [astro-ph.SR] (2011). · doi:10.1103/PhysRevLett.107.134501 |
| [28] | S. Banerjee and S. Galtier, “Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence,” Phys.Rev.E, 87, 013019 (2013); arXiv:1301.2470v1 [physics.flu-dyn] (2013). · doi:10.1103/PhysRevE.87.013019 |
| [29] | S. Banerjee, L. Z. Hadid, F. Sahraoui, and S. Galtier, “Scaling of compressible magnetohydrodynamic turbulence in the fast solar wind,” Astrophys. J. Lett., 829, L27 (2016). · doi:10.3847/2041-8205/829/2/L27 |
| [30] | L. Z. Hadid, F. Sahraoui, and S. Galtier, “Energy cascade rate in compressible fast and slow solar wind turbulence,” Astrophys. J., 838, 9 (2017); arXiv:1612.02150v1 [astro-ph.SR] (2016). · doi:10.3847/1538-4357/aa603f |
| [31] | P. S. Iroshnikov, “Turbulence of a conducting fluid in a strong magnetic field,” Sov. Astron., 7, 566 (1964). |
| [32] | Antonov, N. V.; Gulitskiy, N. M.; Adam, G. (ed.); Busa, J. (ed.); Hnatič, M. (ed.), Two-loop calculation of the anomalous exponents in the Kazantsev-Kraichnan model of magnetic hydrodynamics, 128-135 (2012), Berlin · doi:10.1007/978-3-642-28212-6_11 |
| [33] | N. V. Antonov and N. M. Gulitskiy, “Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model,” Phys. Rev. E, 85, 065301 (2012); arXiv:1202.5992v3 [cond-mat.stat-mech] (2012). · doi:10.1103/PhysRevE.85.065301 |
| [34] | N. V. Antonov and N. M. Gulitskiy, “Erratum: Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model,” Phys. Rev. E, 87, 039902 (2013). · doi:10.1103/PhysRevE.87.039902 |
| [35] | E. Jurčišinova and M. Jurčišin, “Anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model,” J. Phys. A: Math. Theor., 45, 485501 (2012). · Zbl 1339.82003 · doi:10.1088/1751-8113/45/48/485501 |
| [36] | E. Jurčišinova and M. Jurčišin, “Anomalous scaling of a passive scalar advected by a turbulent velocity field with finite correlation time and uniaxial small-scale anisotropy,” Phys. Rev. E, 77, 016306 (2008); arXiv:nlin/0703063v1 (2007). · doi:10.1103/PhysRevE.77.016306 |
| [37] | E. Jurčišinova, M. Jurcišin, and R. Remecký, “Influence of anisotropy on anomalous scaling of a passive scalar advected by the Navier-Stokes velocity field,” Phys. Rev. E, 80, 046302 (2009). · Zbl 1421.76058 · doi:10.1103/PhysRevE.80.046302 |
| [38] | N. V. Antonov and N. M. Gulitskiy, “Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field,” Phys. Rev. E, 91, 013002 (2015); arXiv:1406.3808v2 [cond-mat.stat-mech] (2014). · doi:10.1103/PhysRevE.91.013002 |
| [39] | N. V. Antonov and N. M. Gulitskiy, “Passive advection of a vector field: Anisotropy, finite correlation time, exact solution, and logarithmic corrections to ordinary scaling,” Phys. Rev. E, 92, 043018 (2015); arXiv:1506.05615v2 [cond-mat.stat-mech] (2015). · doi:10.1103/PhysRevE.92.043018 |
| [40] | N. V. Antonov and N. M. Gulitskiy, “Logarithmic violation of scaling in anisotropic kinematic dynamo model,” AIP Conf. Proc, 1701, 100006 (2016). · doi:10.1063/1.4938715 |
| [41] | N. V. Antonov and N. M. Gulitskiy, “Anisotropic turbulent advection of a passive vector field: Effects of the finite correlation time,” EPJ Web Conf., 108, 02008 (2016). · doi:10.1051/epjconf/201610802008 |
| [42] | L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, Renormalization Group Method in the Theory of Developed Turbulence [in Russian], St. Petersburg Univ. Press, St. Petersburg (1998); English transl.: The Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach, London (1999). |
| [43] | R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids, 11, 945-953 (1968). · Zbl 0164.28904 · doi:10.1063/1.1692063 |
| [44] | K. Gawȩdzki and A. Kupiainen, “Anomalous scaling of the passive scalar,” Phys. Rev. Lett., 75, 3834-3837 (1995) · doi:10.1103/PhysRevLett.75.3834 |
| [45] | D. Bernard, K. Gawȩdzki, and A. Kupiainen, “Anomalous scaling in the <Emphasis Type=”Italic“>N-point functions of a passive scalar,” Phys. Rev. E, 54, 2564-2572 (1996) · doi:10.1103/PhysRevE.54.2564 |
| [46] | M. Chertkov and G. Falkovich, “Anomalous scaling exponents of a white-advected passive scalar,” Phys. Rev. Lett., 76, 2706-2709 (1996). · doi:10.1103/PhysRevLett.76.2706 |
| [47] | L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, “Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar,” Phys. Rev. E, 58, 1823-1835 (1998). · doi:10.1103/PhysRevE.58.1823 |
| [48] | N. V. Antonov, “Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field,” Phys. Rev. E, 60, 6691-6707 (1999). · Zbl 1062.76512 · doi:10.1103/PhysRevE.60.6691 |
| [49] | E. Jurčišinova and M. Jurčišin, “Anomalous scaling of the magnetic field in the helical Kazantsev-Kraichnan model,” Phys. Rev. E, 91, 063009 (2015). · Zbl 1339.82003 · doi:10.1103/PhysRevE.91.063009 |
| [50] | N. V. Antonov, A. Lanotte, and A. Mazzino, “Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence,” Phys. Rev. E, 61, 6586-6605 (2000); arXiv:nlin/0001039v1 (2000) · doi:10.1103/PhysRevE.61.6586 |
| [51] | N. V. Antonov and N. M. Gulitskii, “Anomalous scaling in statistical models of passively advected vector fields,” Theor. Math. Phys., 176, 851-860 (2013). · Zbl 1286.76136 · doi:10.1007/s11232-013-0072-7 |
| [52] | H. Arponen, “Anomalous scaling and anisotropy in models of passively advected vector fields,” Phys. Rev. E, 79, 056303 (2009); arXiv:0811.0510v2 [nlin.CD] (2008). · doi:10.1103/PhysRevE.79.056303 |
| [53] | E. Jurčišinova, M. Jurčišin, and M. Menkyna, “Anomalous scaling in the Kazantsev-Kraichnan model with finite time correlations: Two-loop renormalization group analysis of relevant composite operators,” Eur. Phys. J. B, 91, 313 (2018). · Zbl 1515.82118 · doi:10.1140/epjb/e2018-90511-0 |
| [54] | M. Hnatič, J. Honkonen, and T. Lučivjanský, “Advanced field-theoretical methods in stochastic dynamics and theory of developed turbulence,” Acta Phys. Slovaca, 66, 69-264 (2016); arXiv:1611.06741v1 [cond-mat.statmech] (2016). |
| [55] | L. Ts. Adzhemyan, N. V. Antonov, J. Honkonen, and T. L. Kim, “Anomalous scaling of a passive scalar advected by the Navier-Stokes velocity field: Two-loop approximation,” Phys. Rev. E, 71, 016303 (2005); arXiv:nlin/0408057v1 (2004). · doi:10.1103/PhysRevE.71.016303 |
| [56] | N. V. Antonov, “Scaling behavior in a stochastic self-gravitating system,” Phys. Rev. Lett., 92, 161101 (2004); arXiv:astro-ph/0308369v1 (2003). · doi:10.1103/PhysRevLett.92.161101 |
| [57] | N. V. Antonov, N. M. Gulitskiy, and A. V. Malyshev, “Stochastic Navier-Stokes equation with colored noise: Renormalization group analysis,” EPJ Web Conf., 126, 04019 (2016). · doi:10.1051/epjconf/201612604019 |
| [58] | E. Jurčišinova, M. Jurčišin, and R. Remecký, “Turbulent Prandtl number in the <Emphasis Type=”Italic“>A model of passive vector admixture,” Phys. Rev. E, 93, 033106 (2016). · doi:10.1103/PhysRevE.93.033106 |
| [59] | E. Jurčišinova, M. Jurčišin, and M. Menkyna, “Simultaneous influence of helicity and compressibility on anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model,” Phys. Rev. E, 95, 053210 (2017). · doi:10.1103/PhysRevE.95.053210 |
| [60] | M. Vergassola and A. Mazzino, “Structures and intermittency in a passive scalar model,” Phys. Rev. Lett., 79, 1849-1852 (1997). · doi:10.1103/PhysRevLett.79.1849 |
| [61] | A. Celani, A. Lanotte, and A. Mazzino, “Passive scalar intermittency in compressible flow,” Phys. Rev. E, 60, R1138-R1141 (1999). · doi:10.1103/PhysRevE.60.R1138 |
| [62] | M. Chertkov, I. Kolokolov, and M. Vergassola, “Inverse cascade and intermittency of passive scalar in one-dimensional smooth flow,” Phys. Rev. E., 56, 5483-5499 (1997). · doi:10.1103/PhysRevE.56.5483 |
| [63] | L. Ts. Adzhemyan and N. V. Antonov, “Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow,” Phys. Rev. E, 58 part A, 7381-7396 (1998). · doi:10.1103/PhysRevE.58.7381 |
| [64] | N. V. Antonov, M. Yu. Nalimov, and A. A. Udalov, “Renormalization group in the problem of the fully developed turbulence of a compresible fluid,” Theor. Math. Phys., 110, 305-315 (1997). · Zbl 0918.76030 · doi:10.1007/BF02630456 |
| [65] | N. V. Antonov and M. M. Kostenko, “Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: Effects of strong compressibility and large-scale anisotropy,” Phys. Rev. E, 90, 063016 (2014); arXiv:1410.1262v1 [cond-mat.stat-mech] (2014). · doi:10.1103/PhysRevE.90.063016 |
| [66] | N. V. Antonov and M. M. Kostenko, “Anomalous scaling in magnetohydrodynamic turbulence: Effects of anisotropy and compressibility in the kinematic approximation,” Phys. Rev. E, 92, 053013 (2015); arXiv: 1507.08516v1 [cond-mat.stat-mech] (2015). · doi:10.1103/PhysRevE.92.053013 |
| [67] | M. Hnatich, E. Jurčišinova, M. Jurčišin, and M. Repašan, “Compressible advection of a passive scalar: Two-loop scaling regimes,” J. Phys. A: Math. Gen., 39, 8007-8021 (2006). · Zbl 1122.76042 · doi:10.1088/0305-4470/39/25/S14 |
| [68] | V. S. L’vov and A. V. Mikhailov, “Toward a nonlinear theory of sonic and hydrodynamic turbulence of a compressible liquid [in Russian],” Preprint No. 54, Institute of Automation and Electrometry, Novosibirsk (1977). |
| [69] | I. Staroselsky, V. Yakhot, S. Kida, and S. A. Orszag, “Long-time, large-scale properties of a randomly stirred compressible fluid,” Phys. Rev. Lett., 65, 171-174 (1990). · doi:10.1103/PhysRevLett.65.171 |
| [70] | S. S. Moiseev, A. V. Tur, and V. V. Yanovskii, “Spectra and excitation methods of turbulence in a compressible fluid,” Sov. Phys. JETP, 44, 556-561 (1976). |
| [71] | J. Honkonen and M. Yu. Nalimov, “Two-parameter expansion in the renormalization-group analysis of turbulence,” Z. Phys. B, 99, 297-303 (1996). · doi:10.1007/s002570050040 |
| [72] | L. Ts. Adzhemyan, J. Honkonen, M. V. Kompaniets, and A. N. Vasil’ev, “Improved <Emphasis Type=”Italic“>ε expansion for three-dimensional turbulence: Two-loop renormalization near two dimensions,” Phys. Rev. E, 71, 036305 (2005); arXiv:nlin/0407067v1 (2004). · doi:10.1103/PhysRevE.71.036305 |
| [73] | L. Ts. Adzhemyan, M. Hnatich, and J. Honkonen, “Improved <Emphasis Type=”Italic“>ε expansion in the theory of turbulence: Summation of nearest singularities by inclusion of an infrared irrelevant operator,” Eur. Phys. J B, 73, 275-285 (2010). · Zbl 1188.76219 · doi:10.1140/epjb/e2009-00432-8 |
| [74] | A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory of Phase Transitions [in Russian], Nauka, Moscow (1982); English transl. prev. ed., Pergamon, Oxford (1979). |
| [75] | D. Forster, D. R. Nelson, and M. J. Stephen, “Large-distance and long-time properties of a randomly stirred fluid,” Phys. Rev. A, 16, 732-740 (1977). · doi:10.1103/PhysRevA.16.732 |
| [76] | L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, “Infrared divergences and the renormalization group in the theory of fully developed turbulence,” Sov. Phys. JETP, 68, 742 (1989). |
| [77] | N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. Lučivjanskš, “Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields,” Phys. Rev. E, 95, 033120 (2017); arXiv:1611.00327v2 [cond-mat.stat-mech] (2016). · doi:10.1103/PhysRevE.95.033120 |
| [78] | N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. Lučivjanský, “Renormalization group analysis of a turbulent compressible fluid near <Emphasis Type=”Italic“>d = 4: Crossover between local and non-local scaling regimes,” EPJ Web Conf., 125, 05006 (2016). · doi:10.1051/epjconf/201612505006 |
| [79] | N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. Lučivjanský, “Advection of a passive scalar field by turbulent compressible fluid: Renormalization group analysis near <Emphasis Type=”Italic“>d = 4,” EPJ Web Conf., 137, 10003 (2017). · doi:10.1051/epjconf/201713710003 |
| [80] | N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. Lučivjanský, “Stochastic Navier-Stokes equation and advection of a tracer field: One-loop renormalization near <Emphasis Type=”Italic“>d = 4,” EPJ Web Conf., 164, 07044 (2017). · doi:10.1051/epjconf/201716407044 |
| [81] | N. V. Antonov, A. Lanotte, and A. Mazzino, “Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence,” Phys. Rev. E, 61, 6586-6605 (2000); arXiv:nlin/0001039v1 (2000). · doi:10.1103/PhysRevE.61.6586 |
| [82] | Y. Zhou, “Renormalization group theory for fluid and plasma turbulence,” Phys. Rep., 448, 1-49 (2010). · doi:10.1016/j.physrep.2009.04.004 |
| [83] | M. K. Nandy and J. K. Bhattacharjee, “Renormalization-group analysis for the infrared properties of a randomly stirred binary fluid,” J. Phys. A: Math. Gen., 31, 2621-2637 (1998). · Zbl 0940.76022 · doi:10.1088/0305-4470/31/11/012 |
| [84] | N. V. Antonov, “Anomalous scaling of a passive scalar advected by the synthetic compressible flow,” Phys. D, 144, 370-386 (2000). · Zbl 0986.76026 · doi:10.1016/S0167-2789(00)00089-0 |
| [85] | D. Yu. Volchenckov and M. Yu. Nalimov, “The corrections to fully developed turbulent spectra due to the compressibility of fluid,” Theor. Math. Phys., 106, 307-318 (1996). · Zbl 0888.76033 · doi:10.1007/BF02071475 |
| [86] | H. K. Janssen, “On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,” Z. Phys. B, 23, 377-380 (1976). · doi:10.1007/BF01316547 |
| [87] | C. De Dominicis, “FrTechniques de renormalisation de la theorie des champs et dynamique des phenomenes critiques,” J. Phys. Colloq. France, 37, C1-247-C1-253 (1976). · doi:10.1051/jphys:019760037010100 |
| [88] | Janssen, H. K.; Enz, C. P (ed.), Field-theoretic method applied to critical dynamics, 25-47 (1979), Heidelberg · Zbl 0508.22019 · doi:10.1007/3-540-09523-3_2 |
| [89] | L. Ts. Adzhemyan, M. Yu. Nalimov, and M. M. Stepanova, “Renormalization-group approach to the problem of the effect of compressibility on the spectral properties of developed turbulence,” Theor. Math. Phys., 104, 971-979 (1995). · Zbl 0857.76039 · doi:10.1007/BF02065977 |
| [90] | N. V. Antonov, M. Hnatich, J. Honkonen, and M. Jurčišin, “Turbulence with pressure: Anomalous scaling of a passive vector field,” Phys. Rev. E, 68, 046306 (2003); arXiv:nlin/0305024v1 (2003). · doi:10.1103/PhysRevE.68.046306 |
| [91] | B. Duplantier and A. Ludwig, “Multifractals, operator-product expansion, and field theory,” Phys. Rev. Lett., 66, 247-251 (1991). · Zbl 0968.82512 · doi:10.1103/PhysRevLett.66.247 |
| [92] | G. L. Eyink, “Lagrangian field theory, multifractals, and universal scaling in turbulence,” Phys. Lett. A, 172, 355-360 (1993). · doi:10.1016/0375-9601(93)90117-I |
| [93] | N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and A. V. Malyshev, “Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models,” Phys. Rev. E, 97, 033101 (2018); arXiv:1710.04992v2 [cond-mat.stat-mech] (2017). · doi:10.1103/PhysRevE.97.033101 |
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