Abstract
We propose an effective conformal field theory (CFT) description of steady state incompressible fluid turbulence at the inertial range of scales in any number of spatial dimensions. We derive a KPZ-type equation for the anomalous scaling of the longitudinal velocity structure functions and relate the intermittency parameter to the boundary Euler (A-type) conformal anomaly coefficient. The proposed theory consists of a mean field CFT that exhibits Kolmogorov linear scaling (K41 theory) coupled to a dilaton. The dilaton is a Nambu-Goldstone gapless mode that arises from a spontaneous breaking due to the energy flux of the separate scale and time symmetries of the inviscid Navier-Stokes equations to a K41 scaling with a dynamical exponent \( z=\frac{2}{3} \). The dilaton acts as a random measure that dresses the K41 theory and introduces intermittency. We discuss the two, three and large number of space dimensions cases and how entanglement entropy can be used to characterize the intermittency strength.
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References
C. Eling and Y. Oz, The anomalous scaling exponents of turbulence in general dimension from random geometry, JHEP 09 (2015) 150 [arXiv:1502.03069] [INSPIRE].
A.N. Kolmogorov, On positive functionals on almost-periodic functions, Dokl. Akad. Nauk. SSSR 30 (1941) 9
A.N. Kolmogorov, Dissipation of energy in locally isotropic turbulence, Dokl. Akad. Nauk. SSSR 32 (1941) 16 [Proc. Roy. Soc. London A 434 (1991) 9].
A.N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13 (1962) 82.
A.M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13 (1962) 77.
V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal structure of 2D quantum gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
J. Distler and H. Kawai, Conformal field theory and 2D quantum gravity, Nucl. Phys. B 321 (1989) 509 [INSPIRE].
R. Rhodes and V. Vargas, Gaussian multiplicative chaos and applications: a review, Probab. Surv. 11 (2014) 315 [arXiv:1305.6221] [INSPIRE].
C. Eling, I. Fouxon and Y. Oz, Gravity and a geometrization of turbulence: an intriguing correspondence, arXiv:1004.2632 [INSPIRE].
U. Frisch, Turbulence: the legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge U.K. (1995).
G.L. Eyink and K.R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys. 78 (2006) 87 [INSPIRE].
I. Arav, I. Hason and Y. Oz, Spontaneous breaking of non-relativistic scale symmetry, JHEP 10 (2017) 063 [arXiv:1702.00690] [INSPIRE].
I. Hason, Dual scale invariance spontaneous symmetry breaking and turbulence, arXiv:1708.08294 [INSPIRE].
A.M. Polyakov, The theory of turbulence in two-dimensions, Nucl. Phys. B 396 (1993) 367 [hep-th/9212145] [INSPIRE].
V. Riva and J.L. Cardy, Scale and conformal invariance in field theory: a physical counterexample, Phys. Lett. B 622 (2005) 339 [hep-th/0504197] [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
H. Elvang et al., On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
H. Elvang and T.M. Olson, RG flows in d dimensions, the dilaton effective action and the a-theorem, JHEP 03 (2013) 034 [arXiv:1209.3424] [INSPIRE].
C.P. Herzog, K.-W. Huang and K. Jensen, Universal entanglement and boundary geometry in conformal field theory, JHEP 01 (2016) 162 [arXiv:1510.00021] [INSPIRE].
C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
C.R. Graham et al., Conformally invariant powers of the Laplacian. I: existence, J. London Math. Soc. 46 (1992) 557.
J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [INSPIRE].
J. Erdmenger, Conformally covariant differential operators: Properties and applications, Class. Quant. Grav. 14 (1997) 2061 [hep-th/9704108] [INSPIRE].
R.H. Kraichnan, Turbulent cascade and intermittency growth, Proc. Roy. Soc. Lond. A 434 (1991) 65.
D. Bernard et al., Conformal invariance in two-dimensional turbulence, Nature Phys. 2 (2006) 124.
J.L. Cardy, SLE for theoretical physicists, Annals Phys. 318 (2005) 81 [cond-mat/0503313] [INSPIRE].
G. Falkovich, I. Fouxon and Y. Oz, New relations for correlation functions in Navier-Stokes turbulence, J. Fluid Mech. 644 (2010) 465 [arXiv:0909.3404] [INSPIRE].
G. Falkovich and A. Zamolodchikov, Operator product expansion and multi-point correlations in turbulent energy cascades, J. Phys. A 48 (2015) 18FT02.
A.M. Polyakov, Turbulence without pressure, Phys. Rev. E 52 (1995) 6183 [hep-th/9506189] [INSPIRE].
V. Yakhot, Mean-field approximation and a small parameter in turbulence theory, Phys. Rev. E 63 (2001) 026307.
I. Fouxon and Y. Oz, Exact scaling relations in relativistic hydrodynamic turbulence, Phys. Lett. B 694 (2010) 261 [arXiv:0909.3574] [INSPIRE].
J.R. Westernacher-Schneider, L. Lehner and Y. Oz, Scaling relations in two-dimensional relativistic hydrodynamic turbulence, JHEP 12 (2015) 067 [arXiv:1510.00736] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985) 293.
S. Chang and J. Qing, The Zeta functional determinants on manifolds with boundary, J. Funct. Anal. 147 (1997) 327.
C.R. Graham and M. Zworski, Scattering matrix in conformal geometry, math/0109089.
T. Levy and Y. Oz, work in progress.
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Oz, Y. Spontaneous symmetry breaking, conformal anomaly and incompressible fluid turbulence. J. High Energ. Phys. 2017, 40 (2017). https://doi.org/10.1007/JHEP11(2017)040
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DOI: https://doi.org/10.1007/JHEP11(2017)040