Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Gravity and the Shape of Turbulence Yaron Oz (Tel-Aviv University) February 2010 (CERN) With B. Keren-Zur, C. Eling, G. Falkovich, I. Fouxon, X. Liu, Y. Neiman PRL 101 (2008) 261602, JHEP 0903, 120 (2009), PLB 680, 496 (2009), arXiv:0906.4999 (JHEP), arXiv:0909.3574, JFM 644, 465 (2010), arXiv:1002.0804 and to appear. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Outline Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence Turba is a Latin word for crowd. Turbulence originally refers to the disorderly motion of a crowd. Scientifically it refers to a complex and unpredictable motions of a fluid. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence • The dynamics of fluids is a long standing challenge that remained as an unsolved problem for centuries. • Understanding its main features, chaos and turbulence, is likely to provide an understanding of the principles and non-linear dynamics of a large class of systems far from equilibrium. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Black Hole Dynamics • We consider a conceptually new viewpoint to study these features using black hole dynamics. • The existence of horizon is crucial: fields can fall into the black hole but cannot emerge, this breaks time reversal symmetry and allows Einstein equations to describe dissipative effects. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Black Hole Dynamics Since the gravitational field is characterized by a curved geometry, the gravity variables provide a geometrical framework for studying the dynamics of fluids: A geometrization of turbulence. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Non-Relativistic Fluid: The Navier-Stokes Equations • The fundamental formulation of the nonlinear dynamics of non-relativistic fluids is given by the incompressible Navier-Stokes (NS) equations ∂t vi + vj ∂j vi = −∂i P + ν∂jj vi + fi (1) • vi (x, t), i = 1, ..., d, (d ≥ 2) obeying ∂i vi = 0 is the velocity vector field, P(x, t) is the fluid pressure divided by the density, ν is the (kinematic) viscosity and fi (x, t) are the components of an externally applied (random) force. • The equations can be studied mathematically in any space dimensionality d, with two and three space dimensions having an experimental realization. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Reynolds Number and Turbulence • An important dimensionless parameter is the Reynolds number LV Re = (2) ν where L and V are, respectively, a characteristic scale and velocity of the flow. • Experimental and numerical analysis data show that for Re ≪ 1, the flows are regular (Laminar). For a Reynolds number in the range between 1 and 100 the flow exhibits a complicated (chaotic) structure, while for Re ≫ 100 the flow is highly irregular (turbulent) with a complex spatio-temporal pattern formed by the turbulent velocity field. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence in Nature • Most flows in nature are turbulent. This is simple to see by −6 m2 noting that the viscosity of water is ν ≃ 10 sec , while that −5 m2 of air is ν ≃ 1.5 × 10 sec . Thus, a medium size river has 7 a Reynolds number Re ∼ 10 . Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Statistical Approach • In the turbulent regime the velocity field exhibits a highly complex spatial and temporal structure and appears to be a random process. • A single realization of a solution to the NS equations is considered to be unpredictable. • Thus, although the NS equation is deterministic (without a random force), a statistical approach to the turbulent flows seems to provide an appropriate description. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Statistical Approach • For instance, instead of making attempts to understand the whole picture of a single-time velocity field, one considers the statistics of velocity difference between points separated by a fixed distance. The statistics is defined by averaging over space. V(x) V(y) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Energy Cascade In the energy cascade picture, energy is transmitted to the fluid by large eddies at a scale L, that transmit the energy to smaller scales by breaking to smaller eddies due to instability, until the viscous scale l is reached, where the energy dissipates due to friction. L 3 K 41Theory : ∼ R 4 l e Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Scaling • There is experimental and numerical evidence that in the range of distance scales l ≪ r ≪ L (the inertial range) the flows exhibit a universal behavior, e.g. the space-averaged equal-time correlators of velocity differences in the inertial range are characterized by critical exponents. • For instance, the longitudinal n-point functions scale as (r ≡ x − y) n r ξn Sn(r) ≡ h (v(x) − v(y)) · i∼ r (3) r The 1941 exact scaling result of Kolmogorov ξ3 = 1 (anomaly in time-reversal symmetry), agrees well with the experimental data, while the other exponents are measurable real numbers. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook K41 Theory • Longitudinal n-point functions r n n Sn(r) ≡ h (v(x) − v(y)) · i∼ r 3 (4) r • Energy spectrum − E(k) ∼ k 5/3 (5) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Intermittency A central assumption of K41 theory is the self-similarity of the velocity field at the inertial range. This misses intermittency. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Exponents from Experiment A major open problem is to calculate the anomalous exponents. While in two space dimensions the anomalous exponents ξn seem to follow the Kolmogorov linear scaling and are given by rational numbers, this is not the case in three space dimensions. Anomalous Exponents n/3 ξn 1 1 2 3 n Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Exponents: Analytic Treatment? • The highly non-linear interactions at all scales and the absence of a small expansion parameter make the analytical treatment a formidable task. • Clearly, new ideas are desperately needed. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook New Predictions • Recently, we showed that the Kolmogorov relation follows not from energy conservation but from the ability to write the equations in conservation form in terms of currents and sources. • This allows us to derive an analog for a compressible case, and in particular to relativistic hydrodynamics, where the energy cascade picture does not exist. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Incompressible Turbulence: A New Experimental Prediction (with Falkovich and Fouxon) • In non-relativistic incompressible turbulence: Dr hv(r)p(r)v 2(0)i = d Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook A Large d Conjecture • The anomalous exponents ξn of incompressible Navier-Stokes turbulence approach unity in the limit of infinite number of space dimensions d as in Burgers Turbulence ∂t v + v∂x v − ν∂xx v = f • This may provide a starting point for a large d expansion around Burgers statistics rather than Gaussian statistics as has been attempted before. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Third Millennium Problem • The NS and Euler equations are nonlinear partial differential evolution equations. A major open problem posed by these equations is the understanding and control of their solutions. • Of particular importance is the short distance behavior and existence of singularities in the solutions, i.e. starting from smooth initial data, with a bounded energy condition and a smooth external force, can the solutions develop a finite-time singularity. • This is known not to be the case in two space dimensions, but is not known in three space dimensions ("The Third Millennium Problem" announced by the Clay Mathematics Institute). Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Shock Wave Turbulence and Black