The Anomalous Scaling Exponents of Turbulence in General Dimension from Random Geometry

We propose an exact analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation, and is valid in any number of space dimensions. It incorporates intermittency by gravitationally dressing the Kolmogorov linear scaling via a coupling to a random geometry. The formula has one real parameter $\gamma$ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree with the experimental and numerical data in two and three space dimensions, and with numerical data in four space dimensions. Intermittency increases with $\gamma$, and in the infinite $\gamma$ limit the scaling exponents approach the value one, as in Burgers turbulence. At large $n$ the $n$th order exponent scales as $\sqrt{n}$. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.


I. INTRODUCTION
The remarkable phenomenon of fluid turbulence is one of the major unsolved problems of physics [1]. Most fluid motions in nature at all scales are turbulent. Aircraft motions, river flows, atmospheric phenomena, astrophysical flows and even blood flows are some examples of set-ups where turbulent flows occur. Despite centuries of research, we still lack an analytical description and understanding of fluid flows in the non-linear regime. Insights to turbulence hold a key to understanding the principles and dynamics of non-linear systems with a large number of strongly interacting degrees of freedom far from equilibrium. In addition to being a major challenge to basic science, understanding turbulence is likely to have an important impact on diverse practical problems ranging from environmental issues such as pollution and concentration of chemicals to cardiovascular physiology.
In this paper we will mainly consider incompressible fluid flows in d ≥ 2 space dimensions.
They are the relevant flows when the velocities are much smaller than the speed of sound.
The incompressible Navier-Stokes (NS) equations provide a mathematical formulation of the fluid flow evolution. They read where v i is the fluid velocity and p is the fluid pressure. An important dimensionless parameter in the study of fluid flows is the Reynolds number R e = lv ν , where l is a characteristic length scale, v is the velocity difference at that scale, and ν is the kinematic viscosity. The Reynolds number quantifies the relative strength of the non-linear interaction compared to the viscous term. When the Reynolds number is of order a thousand or more, one observes numerically and experimentally a turbulent structure of the flow. This phenomenological observation is general, and fluid details are of no importance. The turbulent velocity field exhibits highly complex spatial and temporal structures and appears to be a random process. Thus, even though the NS equations are deterministic (in the absence of a random force), a single realization of a solution to the NS equations is unpredictable.
Instead of studying individual solutions to the NS equations, one is led to consider the statistics of the solutions. The statistics can be defined in various ways. One can use an ensemble average by averaging over initial conditions. Turbulence that is reached in this way is a decaying one. Alternatively, one can introduce a random force. This allows reaching a sustained steady state turbulence with an energy source and a viscous sink. The statistical properties of turbulent flows are remarkable. Numerical and experimental data show that the statistical average properties exhibit a universal structure shared by all turbulent flows, independently of the details of the flow excitations. One defines the inertial range to be the range of distance scales L V r L F , where the scales L V and L F are determined by the viscosity and forcing, respectively. Turbulence at the inertial range of scales reaches a steady state that exhibits statistical homogeneity and isotropy.
One defines the longitudinal velocity difference between points separated by a fixed dis- The structure functions S n (r) = (δv(r)) n exhibit in the inertial range a scaling The exponents ξ n in (3) are universal, and depend only on the number of space dimensions.
In 1941 Kolmogorov [2] argued that in three space dimensions the incompressible nonrelativistic fluid dynamics in the inertial range follows a cascade breaking of large eddies to smaller eddies, called a direct cascade, where energy flux is being transferred from large eddies to small eddies without dissipation. He further assumed scale invariant statistics, that is where, P (δv(r)) is the probability density function, and h is a real parameter. Using that he deduced a linear scaling of the exponents ξ n = n/3.
All direct cascades are known numerically and experimentally to break scale invariance and do not simply follow Kolomogorov scaling. Note, that in two space dimensions the energy cascade is inverse, that is the energy flux is instead transferred to large scales. Kolmogorov's assumption that the random velocity field is self-similar is incorrect in direct cascades, but it seems to hold in the inverse cascade. The self-similarity assumption misses the intermittency of the turbulent flows. Thus, in order to calculate the scaling exponents one has quantify the inertial range intermittency effects.
The calculation of the anomalous exponents and their deviation from the Kolmogorov scaling is a major open problem. A complication in calculating the anomalous exponents is the large number of strongly interacting degrees of freedom that are involved in transferring the excitations from the injection (forcing) scale L F to the viscous scale L V throughout the inertial range. Another complication is the lack of a physical principle in non-equilibrium dynamics, analogous to the Gibbs measure in equilibrium statistical mechanics.
Our proposal in this paper is that intermittency can be taken into account by a gravitational dressing of Kolmogorov scaling, i.e. by a coupling to a random geometry. As we will explain, this idea is inspired by the relation of fluid dynamics and black hole horizon dynamics in one higher space dimension [3]. We propose an exact analytical formula for the scaling exponents of incompressible fluid turbulence in any number of space dimensions where γ(d) is a numerical real parameter that depends on the number of space dimensions d. The formula is a KPZ (Knizhnik-Polyakov-Zamolodchikov)-type relation [4].
A major part of the paper will be devoted to checking our proposed formula (5). We will first verify that the scaling exponents ξ n obtained from (5) satisfy the convexity inequality, and the supersonic bound constraint [1]. We will then show that they agree with the experimental and numerical data in two and three space dimensions, and with the numerical data in four space dimensions. Intermittency increases with γ, and in the infinite γ limit the scaling exponents approach the value one, as in Burgers turbulence. At large n the nth order exponent scales as √ n.
We will show that the formula does not apply to the Kraichnan model for passive scalar advection by a random velocity field [5,6]. This is expected since the relation between fluid flows and black hole geometry that inspired our proposal does not include the passive scalar system.
The paper is organized as follows. In section II we will explain the coupling to a random geometry, discuss the proposed formula and its properties and perform analytical checks and comparison to experimental and numerical data. While we will establish certain properties of the function γ(d), we will not calculate its precise form in the paper. In section III we will apply the formula to the passive scalar model. In section IV we will discuss the relation between fluid flows and black hole geometry that inspired our proposal. Section V is devoted to a discussion and open problems.

II. EXACT FORMULA FOR THE SCALING EXPONENTS
A. Coupling to a Random Geometry By coupling to a random geometry we mean changing the Euclidean measure dx on a R d to a random measure dµ γ (x) = e γφ(x)− γ 2 2 dx, where the random field φ(x) has zero mean and the covariance φ(x)φ(x) ∼ − log |x − y| when |x − y| is small (but still in the inertial range).
Consider a set of scaling exponents ξ 0 with respect to the Euclidean measure. Denote the same set of exponents, but now with respect to the random measure, by ξ. Then ξ and ξ 0 are related by the KPZ relation Our proposal (5) is that one can incorporate the effect of intermittency at the inertial of range of scales by coupling to a random geometry and evaluating the Kolmogorov linear scaling exponents ξ 0 = n 3 with respect to the random measure. Mathematically, this is a known method to obtain a multifractal structure from a fractal one (for a review see [7] and references therein). Physically, it is highly nontrivial that the steady state statistics of turbulence can be viewed as such a combination of the scale invariant statistics and intermittency. Note, that intermittent features appear at short length scales, and this is when the effects of the random field φ are prominent.
Let us make a few comments on the mathematical structure of coupling to a random geometry. First, note that there are numerical factors that depend on the number of space dimensions, between γ appearing in the random measure and γ in (6) [7]. Since what is relevant for us is the formula (6), we will keep for simplicity the notation where γ 2 appears in (6).
The KPZ relation has been first derived by coupling a two-dimensional CFT to gravity and analyzing the effect of quantum gravity on the scaling dimensions of the CFT [4]. This has been dubbed "gravitational dressing". In two space-time dimensions the correlations functions of a CFT (matter) coupled to gravity (Liouville) factorize as a tensor product of the matter part and the Liouville part (this is no longer true for a non-CFT matter).
The KPZ relation has been generalized in various directions. First, to an arbitrary number of dimensions without reference to a conformal field theory structure [7]. Second, to a more general random field than the log correlated one [8,9]. We will not use the latter generalization in this paper, but it may be valuable in the study of steady state statistics of other non-linear dynamical systems out of equilibrium.
We will consider the formula (6), where γ takes values in the range [0, ∞). However, when γ > 1, the mathematical construction of the random measure changes. In the twodimensional quantum gravity language, γ is related to the central charge of the matter system c, and the critical value γ = 1 is the c = 1 barrier. The regime γ > 1 is a different phase of the theory, dubbed a "dual phase". There may be a duality relation between two phases parametrized by γ and γ that satisfy γγ = 1. This could have an interesting impact on the study of turbulence in diverse dimensions.

B. An Exact Formula
We propose that the scaling exponents of incompressible fluid turbulence ξ n in any number of space dimensions d satisfy the KPZ-type relation (5). Solving for ξ n we get where in choosing the branch we required finite exponents ξ n . γ(d) is a numerical real parameter that depends on the number of space dimensions d. It can be determined from any moment, for instance, from the energy spectrum.
There are several immediate properties of the formula (7) that we can see. First, using n = 3 in (7) one gets the exponent ξ 3 = 1 in any dimension, an exact result derived by Kolmogorov which agrees with numerical simulations and experiments. In [10] this scaling was derived without employing the cascade picture.
Second, the scaling exponent ξ 2 is a monotonically increasing function of γ, while the exponents ξ n , n > 3 are monotonically decreasing functions of γ. Third, in the limit n → 0 we get that ξ n → 0, as expected. Fourth, in the limit γ → 0 we have ξ n → n 3 , that is scale invariant statistics with no intermittency. Fifth, in the limit γ → ∞, we have ξ n → 1, as in Burgers turbulence. The scaling exponents take values in the range 2 3 ≤ ξ 2 ≤ 1, and 1 ≤ ξ n ≤ n 3 for n ≥ 3. We will propose that the limit γ → ∞, is the limit of infinite number of space dimensions d. The subleading correction, relevant for developing a systematic 1 d expansion reads Sixth, in the limit n → ∞ for fixed γ, we have thus growing as √ n. Seventh, at the "critical" value γ = 1 we get C. Analytical Constraints on the Scaling Exponents • Absence of supersonic mode: If there exist two consecutive even numbers 2n and 2n+2 such that ξ 2n > ξ 2n+2 , then the velocity of the flow cannot be bounded. Using (7) it is straightforward to show that ξ 2n ≤ ξ 2n+2 for any γ, thus (7) satisfies the absence of supersonic velocity requirement.
Using (7) it is straightforward to show that the inequality (11) holds. Equality is achieved when γ = 0, when γ → ∞ and when n i = n j for some i = j and arbitrary γ.

D. Comparison to Experimental and Numerical Data
The anomalous scaling exponents (7) depend on the parameter γ, which is a function of d. We do not know the exact expression of γ, but it can be calculated knowing one of the structure functions, such as the energy spectrum With this knowledge we can then make an infinite number of predictions. In the following we will compare the analytical expression (7) to the available numerical and experimental data in various dimensions. The best fit value of the free parameter γ 2 is about 0.161. The error on the data is about ±1 percent.

Two Space Dimensions
In two space dimensions the energy cascade is an inverse cascade, where the energy flux flows to scales larger than the injection scale. In this case, one has the energy spectrum agreeing with the Kolmogorov scaling ξ 2 = 2 3 . Using (7), this implies that γ(2) = 0, and that all the other scaling exponents follow the Kolmogorov scaling ξ n = n 3 .

Three Space Dimensions
We first use the data for the anomalous scaling exponents quoted in [11] from wind tunnel experiments at Reynolds number ∼ 10 4 . This experimental data is consistent with numerical data from simulations of the Navier-Stokes equations, see e.g. [12]. Fitting (5) to this data, we see in Figure 1 an excellent agreement. We find that γ 2 is about 0.161.
Next, we consider the numerical results for low order structure function exponents and non-integer n given in [13]. The numerical data is consistent with experiment at Reynolds number 10 4 . For this data, the fitted value of γ 2 is about 0.159 and in Figure 2 we again see excellent agreement.   (5) to the 4d exponents given in [14]. The solid line is the 4d fit with γ 2 about 0.278.

Four Space Dimensions
In four space dimensions, numerical simulations of the Navier-Stokes equations were performed in [14]. The authors found an increase in intermittency, i.e. ξ n for n > 3. We took the data for the structure function exponents in 4d given in [14] and performed a fit to (5). This is shown in Figure 3. Although taken at a relatively low Reynolds number, the results are in agreement with a simple increase in the γ 2 parameter in our formula (5). The value of γ 2 in four space dimensions is fitted to about 0.278. Note that their numerical data for same simulation in three space dimensions predicts γ 2 about 0.188, which is higher than the experimental data above. This could be related to the relatively low Reynolds numbers involved.

E. Intermittency and the Large d Limit
In order to observe intermittency one has to study the short distance statistical properties of the fluid flow. There are various measures for intermittency, such as F n (r) are expected to grow as a power-law in the limit r → 0, while staying in the inertial range of scales.
We can analyze the properties of F n (r) using (7). They scale as ∼ r α , where α is a decreasing function of γ. In the limit γ → 0 one has α → 0 and no intermittency, while as γ → ∞ we get the maximal intermittency α = 2−n 2 . Numerically, one sees in [14] a clear growth of F n (r), n ≥ 4 in the limit r → 0, when as we increase the number of space dimensions in the simulation. The data is not accurate enough to observe the growth when n = 3.
Another exponent that is used to quantify the intermittency is Experimentally in three space dimensions it has been measured in the range 0.2 to 0.25 (see e.g. [15]). Using (7) with γ 2 = 0.161 we get µ = 0.222. Expanding around γ = 0 (d = 2) we have µ = 2γ 2 + o(γ 4 ), while expanding around infinite γ we have µ = 1 − 1 γ 2 + o( 1 γ 4 ). In [10], (also see [14]) it was conjectured that in the limit of infinite d all the exponents ξ n approach the same value, one, as in Burgers turbulence [16,17]. With our formula (5) this means that γ goes to infinity in the limit of infinite d, and therefore ξ n = 1 for any n.
This suggests the interesting possibility of having a systematic 1 d expansion (8).

F. The Energy Spectrum
The structure function S 2 (r) ∼ r ξ 2 gives the energy spectrum of the fluid. Using (7) we see that ξ 2 is a monotonic function of γ that takes values in the range 2 3 ≤ ξ 2 ≤ 1 when γ goes from zero to infinity. In momentum space a deviation from the Kolmogorov spectrum for small γ (small d) reads For large γ (large d) we have

III. PASSIVE SCALAR TURBULENCE
It is natural to ask whether our proposed exact formula for the scaling exponents of incompressible fluid turbulence is applicable for other systems that exhibit turbulent structure. In the following we will consider the Kraichnan model for passive scalar advection by a random Gaussian field of velocities v i , which is white-in-time [5,6]. The statistics of the velocities is determined by a zero mean v i (t, r) = 0, and by the covariance In the inertial range D ij ( r) − D ij (0) ∼ | r| ζ , where ζ takes values between 0 and 2.
Examples of passive scalar systems are smoke in the air, salinity in the water and temperature when one can neglect thermal convection. The evolution equation describes a passively-advected scalar field T driven by the velocity field where κ is the molecular diffusivity of T and f is an external force.
Here the scale invariant statistics is Gaussian with ξ 2n = n(2 − ζ). We can now attempt to include the intermittency by the random geometry dressing and the KPZ-type equation Solving for ξ 2n we get This formula is similar, but not exactly the one proposed by Kraichnan [6]. In the limit n → ∞ for fixed γ, we have ξ 2n growing as √ n. This is not the expected behaviour, rather ξ 2n should approach a constant [17]. Indeed, the fluid/gravity relation that inspired our proposal does not apply to the passive scalar case.

IV. BLACK HOLE HORIZON DYNAMICS
In the following we will briefly review the relation between fluid flows and black hole horizon geometry (for a review, see [3] and references therein), that inspired our proposal to incorporate the intermittency at the inertial range of scales by a gravitational dressing using a random geometry.
Consider the Einstein equations with a negative cosmological constant Λ in (d + 2) spacetime dimensions where g AB is the Lorentzian metric, R AB the Ricci curvature and R = g AB R AB the Ricci where the coordinates are X A = (t, r, x i ). The function f (r) = 1 − ( 4πT (d+1) ) d+1 . The horizon is defined as the surface where f (r) vanishes. It is a (d + 1)-dimensional null hypersurface, forming a causal boundary preventing any light and particles that cross it from returning.
Hence, it effectively introduces dissipation. One can associate with the black hole horizon a temperature T (appearing in f (r)), and an entropy proportional to its cross-sectional area A. In Planck units = G N = c = 1, the relation between the area and the entropy is This structure is called black hole thermodynamics.
Black holes hydrodynamics is a generalization of black hole thermodynamics, similar to the generalization of field theory thermodynamics to hydrodynamics. While black hole thermodynamics quantifies the thermal equilibrium situation, black hole hydrodynamics describes slow derivations from equilibrium. In particular, one can allow the black hole temperature to be a slowly varying function T (t, x i )), and consider black hole itself to be moving at velocity v i (t, x) with respect to some rest frame. The perturbed solution to the Einstein equations in this setting will yield a slowly evolving curved geometry, with the gravity variables providing a geometrical framework for studying the dynamics of fluids. This can be made precise in the context of a holographic correspondence, where the fluid system lives on a (d + 1) dimensional surface of r = const.
in the (d + 2) dimensional bulk solution.
The motion of fluids translates to the evolution of the black hole horizon, and the fluid variables to its geometrical data. The normal vector n to the horizon H satisfies g AB n A n B = 0. The horizon hypersurface is defined by r = r H = const. and we have v i is the geometrical representation of the fluid velocity, it is the normal to the black hole horizon hypersurface. As we perturb the black hole and get it out of equilibrium, the horizon location changes, and up to an overall constant can be parametrized by r H ∼ 1 + p(t, x) − v 2 2 . The variable p(t, x) quantifies the deviation from equilibrium and is identified as the fluid pressure.
The set of the Einstein equations projected on the horizon describe the evolution of the perturbed horizon geometry, and are equivalent to the incompressible NS equations (1) [18,19]. Note that since the Einstein equations are relativistic, this amounts to taking the non-relativistic limit. For example, one of the equations enforces the vanishing of the fractional rate of change of horizon area at lowest order and reduces to the incompressibility condition ∂ i v i = 0.
In the gravitational framework, every fluid configuration that solves the incompressible NS equations corresponds to a particular horizon hypersurface geometry. The statistical aspects of turbulence are realized by a random hypersurface geometry. It has thus been suggested that the statistical properties of the random horizon hypersurface encode the universal statistical structure of turbulence [18]. The deviation of the horizon volume measure from equilibrium is parametrized by the pressure and the velocity. One gets that the volume measure scales like d(p − v 2 2 ), where d is the number of space dimensions. In the turbulent regime, this measure is random. It is this qualitative argument that inspired us to introduce the random measure in order to quantify the intermittency effects.
Physically, the fluid pressure is a non-local field related to the velocity gradients by a constraint ∆p = −∂ i v j ∂ j v i . It couples different space regimes, and acts as an "intermittencykiller" [20]. It is argued in [10] that in the large d limit, p ∼ √ d while v 2 ∼ d, hence the effect of the pressure diminishes as the number of space dimensions increases. It has therefore the opposite role of the random log correlated φ field that was introduced in the construction of the random measure in order to get the KPZ relation. Establishing a precise relation between p, v 2 and φ can provide means to a derivation of (6), and a calculation of γ(d).

V. DISCUSSION
We proposed an exact analytical formula for the scaling exponents of inertial range incompressible fluid turbulence in any number of space dimensions d ≥ 2. The idea is that intermittency can be taken into account by gravitationally dressing the scale invariant Kolmogorov spectrum. Mathematically, the coupling to a random geometry with a random measure based on a log correlated field, maps the fractal structure of the scaling exponents to a multifractal one.
There is one parameter that depends on the number of dimensions that we denoted by γ(d). It can be deduced knowing one moment, for instance the energy spectrum. With this knowledge one can make infinite number of predictions. Our formula passes the standard analytical consistency checks, such as the convexity inequality and the absence of a supersonic mode. Its predictions agree with experimental and numerical data in two, three and four space dimensions.
The main challenge is to determine analytically the function γ(d). It is a monotonic function of the number of space dimension d, and takes values in the range [0, ∞). In the limit γ → 0 one has a linear scaling of the exponents, and when γ → ∞ one gets the exponents of Burgers turbulence. It should take the value zero at d = 2 and infinity as d → ∞, and there is probably some sort of a duality relation expected when γ → 1 γ . One may try to calculate γ using some physical models for the anomalous scaling, such as contributions from vortex filaments [21], or statistical conservation laws [17]. We expect the framework that inspired our formula -the relation between fluid dynamics and black hole horizon geometry -to provide a clean calculational scheme.
While equilibrium statistics is characterized by the Gibbs measure, there is yet no analog of this for nonequilibrium steady state statistics. We speculate that there is a general principle that allows us to consider the steady state statistics of out of equilibrium systems as a gravitationally dressed scale invariant one. If correct, this will shed much light on out of equilibrium dynamics.
Finally, it will be interesting to use the gravitational dressing to study the intemittency effects on the anomalous scaling of the transverse structure functions and multipoint correlation functions. Also, our proposed formula is valid for the inertial range of scales, and most likely does not incorporate statistical signatures of the dissipation range of scales. It is of interest to know whether the latter can be parametrize by a random geometry, since after all the Reynolds number is finite in nature.