Renormalization group in the problem of active scalar advection

N. V. Antonov   ; M. M. Kostenko

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 26, Tome 487 (2019), pp. 5-27

Résumé

The field theoretic renormalization group (RG) is applied to the model of a near-equilibrium fluid coupled to a scalar field (like temperature or density of an impurity) which is active, that is, influencing the dynamics of the fluid itself. It is shown that the only possible nontrivial infrared asymptotic regimes are governed by “passive” fixed points of the RG equations, where the back reaction is irrelevant. This result reminds of that obtained in [Nandy and Bhattacharjee, J. Phys. A: Math. Gen. 31, 2621 (1998)] in a model describing active convection by fully developed turbulence. Furthermore, we establish the existence of “exotic” fixed points with negative and complex effective couplings and transport coefficients that may suggest possible directions for future studies.

Bibliographie

[1] D. Forster, D. R. Nelson, M. J. Stephen, “Large-distance and long-time properties of a randomly stirred fluid”, Phys. Rev., 16 (1977), 732 | DOI | MR

[2] D. Forster, D. R. Nelson, M. J. Stephen, “Long-time tails and the large-eddy behavior of a randomly stirred fluid”, Phys. Rev. Lett., 36 (1976), 867 | DOI

[3] M. Kardar, G. Parisi, Y.-C. Zhang, “Dynamic Scaling of Growing Interfaces”, Phys. Rev. Lett., 56 (1986), 889 | DOI | Zbl

[4] S. Grossmann, E. Schnedler, “Fluctuation corrections of the turbulence spectrum by renormalization group methods”, Z. Phys., 26 (1977), 307–317

[5] P. C. Martin, C. De Dominicis, “The long distance behavior of randomly stirred fluids”, Progr. Theor. Phys. Suppl., 64 (1978), 108–123 | DOI

[6] C. De Dominicis, P.C. Martin, “Energy spectra of certain randomly stirred fluid”, Phys. Rev. A, 19 (1979), 419 | DOI

[7] P. L. Sulem, J.-D. Fournier, A. Pouquet, “Fully developed turbulence and renormalization group”, Lecture Notes in Physics, 104, Springer, Berlin, 1979, 91

[8] T. Nakano, F. Tanaka, “Effects of large-scale fluctuations in fully developed turbulence”, Progr. Theor. Phys., 65 (1981), 120–139 | DOI

[9] J.-D. Fournier, P.L. Sulem, A. Pouquet, “Infrared properties of forced magnetohydrodynamic turbulence”, J. Phys. A: Math. Gen., 15 (1982), 1393–1420 | DOI | Zbl

[10] J.-D. Fournier, U. Frisch, “Remarks on renormalization group in statistical fluid dynamics”, Phys. Rev. A, 28 (1983), 1000–1002 | DOI

[11] L. Ts. Adzhemyan, A. N. Vasil'ev, Yu. M. Pis'mak, “Renormalization-group approach in the theory of turbulence: The dimensions of composite operators”, Theor. Math. Phys., 57 (1983), 1131–1141 | DOI | MR | Zbl

[12] L. Canet, B. Delamotte, N. Wschebor, “Fully developed isotropic turbulence: Symmetries and exact identities”, Phys. Rev. E, 91 (2015), 053004 ; “Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixed-point solution”, Phys. Rev. E, 93 (2016), 063101 | DOI | MR | DOI

[13] C. Mejía-Monasterio, P. Muratore Ginanneschi, “Nonperturbative renormalization group study of the stochastic Navier-Stokes equation”, Phys. Rev. E, 86 (2012), 016315 | DOI

[14] A. A. Fedorenko, P. Le Doussal, K. J. Wiese, “Functional renormalization-group approach to decaying turbulence”, J. Stat. Mech., 2013, P04014 | DOI | MR

[15] S. L. Ogarkov, On Functional and Holographic Renormalization Group Methods in Stochastic Theory of Turbulence, 2016, arXiv: 1605.07560 [hep-th]

[16] M. Tarpin, L. Canet, C. Pagani, N. Wschebor, Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach, 2018, arXiv: 1809.00909 [cond-mat.stat-mech] | MR

[17] L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil'ev, The Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach, London, 1999 | MR | Zbl

[18] A. N. Vasiliev, The Field Theoretic Renormalization Group in Critical Behaviour Theory and Stochastic Dynamics, Chapman Hall/CRC, Boca Raton, 2004, 681 pp. | MR

[19] L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil'ev, “Quantum field renormalization group in the theory of fully developed turbulence”, Physics-Uspekhi, 39 (1193), 1193 | DOI

[20] N. V. Antonov, “Renormalization group, operator product expansion and anomalous scaling in models of turbulent advection”, J. Phys. A: Math. Gen., 39 (2006), 7825 | DOI | MR | Zbl

[21] M. Hnatič, J. Honkonen, T. Lučivjanský, “Advanced field-theoretical methods in stochastic dynamics and theory of developed turbulence”, Acta Physica Slovaca, 66 (2016), 69

[22] S. A. Orszag, “Statistical theory of turbulence in fluids dynamics”, Fluid Dynamics, Les Houches Summer School of Theoretical Physics (1973), eds. R. Balian, J.-L. Peube, Gordon and Breach, London, 1977, 235 | MR

[23] A. N. Vasiliev, Functional Methods in Quantum Field Theory and Statistical Physics, CRC Press, 1998 | MR

[24] H. W. Wyld, Jr., “Formulation of the theory of turbulence in an incompressible fluid”, Ann. Phys., 14 (1961), 143–165 | DOI | MR | Zbl

[25] V. I. Belinicher, V. S. L'vov, “Scale-invariant theory of developed hydrodynamic turbulence”, Sov. Phys. JETP, 66 (1987), 303; V. S. L'vov, “Scale invariant-theory of fully-developed hydrodynamic turbulence”, Phys. Rep., 207 (1991), 1 | DOI | MR

[26] V. S. L'vov, I. Procaccia, “Towards a nonperturbative theory of hydrodynamic turbulence: Fusion rules, exact bridge relations, and anomalous viscous scaling functions”, Phys. Rev. E, 54 (1996), 6268 | DOI | MR

[27] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[28] R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence”, Phys. Fluids, 11 (1968), 945–953 ; “Anomalous scaling of a randomly advected passive scalar”, Phys. Rev. Lett., 72 (1994), 1016 ; “Passive scalar: scaling exponents and realizability”, Phys. Rev. Lett., 78 (1997), 4922 | DOI | Zbl | DOI | DOI

[29] G. Falkovich, K. Gawȩdzki, M. Vergassola, “Particles and fields in fluid turbulence”, Rev. Mod. Phys., 73 (2001), 913 | DOI | MR | Zbl

[30] M. Chertkov, G. Falkovich, I. Kolokolov, V. Lebedev, “Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive Scalar”, Phys. Rev. E, 52 (1995), 4924 ; M. Chertkov, G. Falkovich, “Anomalous scaling exponents of a white-advected passive scalar”, Phys. Rev. Lett., 76 (1996), 2706 ; M. Chertkov, G. Falkovich, “Anomalous scaling exponents of a white-advected passive scalar”, Phys. Rev. Lett., 76 (1996), 2706 | DOI | MR | DOI

[31] K. Gawedzki, A. Kupiainen, “Anomalous scaling of the passive scalar”, Phys. Rev. Lett., 75 (1995), 3834 ; D. Bernard, K. Gaw edzki, A. Kupiainen, “Anomalous scaling in the $N$-point functions of passive Scalar”, Phys. Rev. E, 54 (1996), 2564 | DOI | DOI | MR

[32] U. Frisch, A. Mazzino, M. Vergassola, “Intermittency in passive scalar advection”, Phys. Rev. Lett., 80 (1998), 5532 ; U. Frisch, A. Mazzino, A. Noullez, M. Vergassola, “Lagrangian method for multiple correlations in passive scalar advection”, Phys. Fluids, 11 (1999), 2178 ; A. Mazzino, P. Muratore Ginanneschi, “Passive scalar turbulence in high dimensions”, Phys. Rev. E, 63 (2000), 015302 | DOI | DOI | MR | Zbl | DOI

[33] L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil'ev, Phys. Rev. E, 58 (1998), 1823 | DOI | MR

[34] D. J. Amit, Field Theory, Renormalization Group, and Critical Phenomena, 2nd edition, World Scientific, Singapore, 1984 | MR

[35] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989, 1008 pp. | MR

[36] L. Ts. Adzhemyan, N. V. Antonov, V. A. Barinov, Yu. S. Kabrits, A. N. Vasil'ev, “Anomalous exponents in the rapid-change model of the passive scalar advection in the order $\varepsilon^3$”, Phys. Rev. E, 63 (2001), 025303(R) | DOI

[37] L. Ts. Adzhemyan, N. V. Antonov, J. Honkonen, “Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: Two-loop approximation”, Phys. Rev. E, 66 (2002), 036313 | DOI | MR

[38] L. Ts. Adzhemyan, N. V. Antonov, J. Honkonen, T. L. Kim, “Anomalous scaling of a passive scalar advected by the Navier–Stokes velocity field: Two-loop approximation”, Phys. Rev. E, 71 (2005), 016303 | DOI | MR

[39] R. Ruiz, D. R. Nelson, “Turbulence in binary fluid mixtures”, Phys. Rev. A, 23 (1981), 3224 | DOI

[40] A. Aronowitz, D. R. Nelson, “Turbulence in phase-separating binary mixtures”, Phys. Rev. A, 29 (1984), 2012 | DOI

[41] M. K. Nandy, J. K. Bhattacharjee, “Renormalization-group analysis for the infrared properties of a randomly stirred binary fluid”, J. Phys. A: Math. Gen., 31 (1998), 2621–2637 | DOI | MR | Zbl

[42] A. Celani, M. Cencini, A. Mazzino, M. Vergassola, “Active versus passive scalar turbulence”, Phys. Rev. Lett., 89 (2002), 234502 | DOI

[43] A. Celani, M. Cencini, A. Mazzino, M. Vergassola, “Active and passive fields face to face”, New J. Phys., 6 (2004), 72 | DOI

[44] A. Celani, M. Vergassola, “Statistical geometry in scalar turbulence”, Phys. Rev. Lett., 86 (2001), 424 | DOI

[45] I. Arad, L. Biferale, A. Celani, I. Procaccia, M. Vergassola, “Statistical conservation laws in turbulent transport”, Phys. Rev. Lett., 87 (2001), 164502 | DOI

[46] E. S. C. Ching, Y. Cohen, T. Gilbert, I. Procaccia, “Active and passive fields in turbulent transport: The role of statistically preserved structures”, Phys. Rev. E, 67 (2003), 016304 | DOI | MR

[47] A. Kupiainen, P. Muratore-Ginanneschi, “Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection”, J. Stat. Phys., 126 (2007), 669–724 | DOI | MR | Zbl

[48] L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil'ev, “Infrared divergences and the renormalization group in the theory of fully developed turbulence”, Sov. Phys. JETP, 68 (1989), 733–742 | MR

[49] P. C. Hohenberg, B. I. Halperin, “Theory of dynamic critical phenomena”, Rev. Mod. Phys., 49 (1977), 435 | DOI

[50] R. Folk, G. Moser, “Critical dynamics: a field-theoretical approach”, J. Phys. A: Math. Gen., 39 (2006), 207–313 | DOI | MR

[51] N. V. Antonov, P. I. Kakin, “Random interface growth in random environment: Renormalization group analysis of a simple model”, Theor. Math. Phys., 185 (2015), 1391–1407 | DOI | MR | Zbl

[52] N. V. Antonov, 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 (2014), 063016 | DOI | MR

[53] V. Yakhot, “Ultraviolet dynamic renormalization group: Small-scale properties of a randomly stirred fluid”, Phys. Rev. A, 23 (1981), 1486–1497 ; V. Yakhot, “Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation”, Phys. Rev. A, 24 (1981), 642–644 ; G. Sivashinsky, V. Yakhot, “Negative viscosity effect in large-scale flows”, Physics of Fluids, 28 (1985), 1040–1042 | DOI | DOI | DOI | Zbl

[54] G. Pelletier, “Langmuir turbulence as a critical phenomenon. Part 2. Application of the dynamical renormalization group method”, J. Plasma Phys., 24 (1980), 421–443 | DOI

[55] L. Ts. Adzhemyan, A. N. Vasil'ev, M. Gnatich, Yu. M. Pis'mak, “Quantum field renormalization group in the theory of stochastic Langmuir turbulence”, Theor. Math. Phys., 78 (1989), 260–272 | DOI | MR

[56] U. C. Täuber, S. Diehl, “Perturbative field-theoretical renormalization group approach to driven-dissipative Bose-Einstein criticality”, Phys. Rev. X, 4 (2014), 021010; Weigang Liu, U. C. Täuber, “Critical initial-slip scaling for the noisy complex Ginzburg–Landau equation”, J. Phys. A: Math. Theor., 49 (2016), 434001 | DOI | MR | Zbl