# The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

Authors: V. N. Grebenev; A. N. Grishkov; M. Oberlack
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The extended symmetry of the functional of length
determined in an affine spaceK3 of the correlation vectors for
homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variable t)
of the velocity fluctuations is used to equip this space by a
family of the pseudo-Riemannian metrics dl2(t)  (Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation space K3 and expose the Lie
algebra associated with the equivalence transformation of the
above-mentioned functional for the quadratic form dlD22(t)
generated by dl2(t) which is similar to the Lie algebra
constructed by Grebenev et al. (2012). Then, using the properties of this
Lie algebra, we show that there exists a nontrivial central
extension wherein the central charge is defined by the same
bilinear skew-symmetric form c as for the Witt algebra which
measures the number of internal degrees of freedom of the system.
For the applications in turbulence, as the main result, we
establish the asymptotic expansion of the transversal correlation
function for large correlation distances in the frame of
dlD22(t).


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## Body

## 1. Introduction

This paper is a continuation of [1–3] wherein we investigated both the geometry and the group of transformations of an affine space K3 of the correlation vectors. In [1], we used the two-point velocity-correlation tensor to equip the correlation space K3 by the structure of a pseudo-Riemannian manifold of a variable signature and gave the geometric realization of the two-point velocity-correlation tensor which presents a metric tensor in the case of homogeneous isotropic turbulence. This construction presents the template for embedding the couple (K3,dl2(t)) into the Euclidean space ℝ3 with the standard metric. The Lagrangian system in the extended space K3×R+ was introduced in [3] that allowed us to attract common concept and technics of the Lagrangian mechanics for the application in turbulence. Dynamics in time of a singled out fluid volume equipped with a family of pseudo-Riemannian metrics was described in the frame of the geometry generated by dl2(t) whose components are the correlation functions that evolve according to the von Kármán-Howarth equation [4]. Notice here that the first integrals of the equations of geodesic curves form the “kinematic” conservation laws; see, for more details, [3]. In [2], we considered the functional of length and studied the infinitesimal transformations admitted by this functional. We extended the variational symmetries up to the equivalence transformations which generate an infinite-dimensional Lie algebra (the so-called extended symmetries algebra). The properties obtained of this algebra were discussed both for the signature (++) and for the signature (+-) of the above-mentioned pseudo-Riemannian metrics.The main result of the paper is the derivation of the asymptotic expansion of the transversal correlation function for large correlation distances. Notice that this expansion is obtained in the case when the correlation spaceK3 is equipped with the metric generated by the two-point velocity-correlation tensor. This derivation is based on the algebraic constructions as in Conformal Field Theory (CFT). Also we present a nontrivial extension of the above-mentioned infinite-dimensional Lie algebra.Central extensions of groups and Lie algebras occur in CFT in the investigations of conformal symmetries where the Virasoro algebra is used as the fundamental set of infinitesimal symmetries. This implies the existence of an infinite number of independent constraints, which yields the exceptional feature of the two-dimensional conformal theory. This algebra is the Lie algebra of the central extension of the group of diffeomorphisms of the circle whose basic elements{Ln},n∈Z, satisfy the following commutation relations:
(1)[Ln,Lm]=(n-m)Ln+m+c12n(n2-1)δn+m,
where δk=1 for k=0 and δk=0 for k≠0. The quantity c is known as the central charge, and its value in general depends on the particular theory under consideration. For the application in turbulence in [5, 6], the central charge c is expressed via the diffusion coefficient κ of SLE: the Shram-Lövner evolution, of random curves in planar domains; see [5, 6]. Here, the diffusion coefficient κ allows us to classify the conformally invariant random curves into classes denoted by SLEκ. The formula c=(8-3κ)(κ-6)/2κ classifies the models, for instance, SLE6 and SLE8/3, corresponds to c=0, while SLE4 corresponds to c=1. The isolines of vorticity for the 2D Euler equations and the temperature isolines in the SQG model (surface of quasi-geostrophic model) belong to the class SLE, more exactly, SLEκ1 and SLEκ1, correspondingly, where κ1≈6 and κ2≈4. The energy-momentum tensor was calculated for SLE in [7].The paper is organized as follows. Section1 contains the results obtained (see [1–3] for more details) in a compressed form about the geometry of the correlation space K3 and the extended group of the transformations admitted by the Lagrangian system. In Section 2, we consider the energy-momentum tensor associated with the metric dlD22(t) and show that the components T(z) and T¯(z-) of this tensor are holomorphic functions. Then, in the full analogy of CFT, we expose the asymptotic expansion both for T(z) and T¯(z-) as z→∞ and z-→∞. As a consequence of this result, we get the asymptotic expansion of the transversal correlation function for the large values of the correlation distances. Appendix to this paper is devoted to the central extension of the corresponding infinite-dimensional Lie algebra. We show that this nontrivial central extension can be determined by the same bilinear skew-symmetric form c (the central charge) as for the central extension of the Witt algebra.


## 2. Geometry and Group Transformations of the Correlation Space

In this section, we demonstrate in the compressed form the results obtained in [1] about a geometric realization of the two-point velocity-correlation tensor Bij and present the extended symmetries of the functional of length based on the calculation of the equivalence transformations of the eikonal equation (see, for more details, [2]).


### 2.1. Geometric Realization of the Two-Point Velocity-Correlation Tensor

We recall only the elementary information about the structure of the two-point velocity-correlation tensor of the velocity fluctuations for homogeneous isotropic flows. The modern theory of the properties and structure of second-order (Cartesian) correlation tensors is given in [8].The statistical description of fluid turbulence employ the Reynolds decomposition to separate the fluid velocityu→ at a point x→ into its mean and fluctuating components as u→=u-→+u→′. Here, (·-) is the mean velocity, while u→ is the corresponding fluctuating quantity, usually interpreted as representing turbulence. The two-point correlation tensor is defined by the following:
(2)Bij(x,x′,t)=(ui′(x,t))(uj′(x′,t))¯,
where t∈ℝ+ and x,x′ are the points of a three-dimensional space filled by turbulent fluid.  Formula (2) is rewritten as follows:
(3)Bij(x,r→,t)=(ui′(x,t))(uj′(x+r→,t))¯,
where the vector r→ is determined by the pair (x,x′), where x and x′ are the starting point and endpoint, correspondingly, or r→=x→′-x→. Therefore, we will consider an affine space K3 with the adjoined vector space of the correlation vectors K→3≡{r→=(r1,r2,r3)}. The assumption of homogeneity and isotropy of turbulent flow (invariance with respect to rotation, reflection, and translation) implies that this tensor depends only on the length of the correlation vector r→ and time t; that is,
(4)Bij(x,r→,t)=[BLL(|r→|,t)-BNN(|r→|,t)]rirj|r→|2+BNN(|r→|,t)δij.
Moreover, for isotropic turbulence, Bij(|r→|,t) is a symmetric tensor and the correlations Bij can be expressed by using only the longitudinal correlational function BLL(|r→|,t) and the transversal correlation function BNN(|r→|,t) [9]; that is, the correlation tensor Bij takes the diagonal form with the components B11=BLL and B22≡B33=BNN in a suitable system of the coordinates of the adjoined vector space. Further instead of directly employing the correlation functions BLL and BNN, we use their normalized representations f and g where BLL=u′2(t)¯f(|r→|,t), BNN=u′2(t)¯g(|r→|,t) with the turbulence intensity equals u′2(t)¯=BLL(0,t). Then, the corresponding quadratic form (or linear element) takes the following form:
(5)dl2(t)=u′2(t)¯f(|r→|,t)dr12+u′2(t)¯g(|r→|,t)(dr22+dr32),
where dl2(t) is an indefinite quadratic form in view of the properties g(|r→|,t), see below. The normalized transversal correlation function g satisfies the following relation (taken from the continuity) [10]:
(6)g(|r→|,t)=f(|r→|,t)+r2∂∂rf(|r→|,t).
The property that f decays faster than |r→|-2 on infinity together with (6) yields [10]
(7)∫0∞‍|r→|g(|r→|,t)d|r→|=0.
Hence, g(|r→|,t) is an alternative sign function. Typical forms of experimentally measured functions f and g are given in Figure 1. we use the data presented to determine the qualitative behaviors of f and g, in particular, the algebraic properties of these correlation functions. Thus, we will assume that f is a positive everywhere function and g changes sign only in interval (-ε+|r→*|,|r→*|+ε), ε>0. g is a positive function on [0,±|r→*|) and therefore g<0 outside of [-|r→*|,|r→*|]. The change sign of g means that the quadratic forms dl2(t) have a variable signature. The normalized longitudinal correlational function f(|r→|,t) is dynamically evolved due to the von Kármán-Howarth equation [4]:
(8)∂u′2(t)¯f(|r|,t)∂t=1r4∂∂rr4(u′2(t)¯3/2h(|r|,t)+2ν∂∂ru′2(t)¯f(|r|,t)).h is the normalized triple-correlation function and u′2(t)¯ is the turbulence intensity (a positive everywhere function that vanishes on infinity) or the velocity scale for the turbulent kinetic energy; u′2(t)¯3/2 determines the scale for the turbulence transfer. This single equation directly follows from the Navier-Stokes equation (see, for example, [9]) and contains two unknowns, f and h, with the turbulence intensity u′2(t)¯ which cannot be defined from (8) without the use of additional hypothesis.Figure 1
Typical forms of the normalized longitudinal and transversal correlation functions.If we consider in the correlation spaceK3 with K→3 equipped by the standard Euclidean scalar product (i.e., in ℝ3) an infinite cylindrical domain (a singled out fluid tube at some fixed time), then the metric (induced by the quadratic form Bij) of the surface which bounds this domain takes the following form:
(9)ds2(t)=u′2(t)¯f(|r→|,t)dr12+u′2(t)¯g(|r→|,t)ρ2dϕ2,
where ρ denotes the Euclidean radius of the cross-section {a}×S1(ρ) of the surface ℝ×S1(ρ),a∈ℝ. We can account that ρ=1 and identify this manifold with ℝ×S1(1). The functions f and g are nondimensional with f(0,t)=g(0,t)=1 and physically f is a positive function such that f→0 (g→0) as |r→| tends to infinity. Moreover, f and g are bounded even functions such that f≤1, |g|≤1, and f goes faster to zero than |r→|-2, when |r→| tends to infinity. Physically such behavior of f is acceptable [10] and the map
(10)q(a,t)=∫0a‍f(|r→|,t)dr1,a∈ℝ,
acts as (-∞,∞)↦[-L*(t),L*(t)], L*(t)=L(t)/2u′2(t)¯, where L(t) is determined by the following:
(11)L(t)=2∫0∞‍u′2(t)¯f(|r→|,t)dr1.
Now, we rewrite the metric ds2(t) in the frame of the variable q:
(12)ds2(t)=u′2(t)¯{dq2+G(q,t)dϕ2},G(q,t)=g(|r|,t).The metric (12) admits a one-parametric group of (isometric) motion gτ(p→)≡g(p→,a1), p→=(q,ϕ) of the following form:
(13)gτ:(q,ϕ)↦(q,ϕ+χτ),χ=const,
with the generator
(14)X=ξi∂∂pi≡χ∂∂ϕ.
The scalar product of the generator X equals
(15)X2=〈X,X〉=〈χ∂∂ϕ,χ∂∂ϕ〉≡χ2〈∂∂ϕ,∂∂ϕ〉=u′2(t)¯χ2G(q,t)
for each time t.A pointp0 is called the pole [11] of a (pseudo-) Riemannian manifold M if p0 is a fixed point of a group of diffeomorphisms g(x→,a→), a→=(a1,…,ar) which acts on M.We note that ifp=p0 (p0 is the pole of gτ), then X2(p0)=0 and due to (15) p0 coincides with the roots of the equation G(q,·)=0. Therefore, the points q*∈[-L*(t),L*(t)] wherein G vanishes are the poles of gτ. In view of our assumption on g(|r|,t), the equation G(q,·)=0 has only 4 roots qi*, i=1,…,4 such that |q1*|=q4*=L*(t) and |q2*|=q3*. Thus, the metric (12) has the different signature for q∈I1=(q2*,q3*), q∈I2=(-L*(t),q2*), and, q∈I3=(q3*,L*(t)), respectively, where qi* depends on t. This metric determines for q∈I1 the element of length of the surface of revolution in ℝ3 and the radius-vector R→=(q,ϕ,·) of this surface is given by the following:
(16)R→(q,ϕ,·)=(q,{u′2(t)¯G}1/2(q,·)cosϕ,{u′2(t)¯G}1/2(q,·)sinϕ).
Therefore, the model manifold defined by (12) for q∈I1 is a cylindrical-type surface MI1t=(q2*,q3*)×S1(1) wherein the radius of the cross-section {q}×S1(1) equals G1/2(q,t). For the q∈Ii, i=2,3 where G is negative, the positive defined metric
(17)ds2(t)=u′2(t)¯{dq2-|G(q,t)|dϕ2},G(q,t)=g(|r→|,t),
can be realized (see, for details, [1]) as a surface of revolution (for each fixed time) in the Minkowski space ℝ1,23 with the element of length
(18)dρ2=dx12-dx22-dx32
when the form ds2(t) is of a fixed sign [12]. Here, the rotation presents the motion along the pseudocircle of the radius |G(q,·)|1/2, q∈Ii. Indeed, let us fix the point pa=(qa,ϕa) on the cross-section {qa}×S1(1) and consider the action of the group gτ on pa; that is, the orbit Opa:τ↦gτ(pa). This action is a motion along {qa}×S1(1) and if pa does not coincide with the poles gτ, then Opa is a not compact set [11]. In particular, Opa⊆{(x1,x2):x12-x22=u′2(t)¯|G(qa,t)|} for each fixed time t which coincides with the so-called pseudocircle under the embedding MI3t (MI2t) into the Minkowski space ℝ1,23. Moreover, the poles are saddle points of a negative index for the orbits Op, p∈MI3t (MI2t). The cross-sections {qa}×S1(1) of MI3t (MI2t) for qa∈{q3*,L*(t)} (resp., qa∈{-L*(t),q2*}) are the pseudocircles of zero radius and consist of the isotropic rays with the initial points q3* and L*(t) (resp., -L*(t) and q2*). The action of gτ on the point p is a motion along these piecewise linear isotropic curves when p∈{-L*(t),q2*,q3*,L*(t)}. We can identify MI3t (MI2t) with the foliation space of orbits MIjt=⋃p‍Op and associate the modulus of the transversal correlation function G(q,t) with the length of the velocity vector ξ→(p) of the orbit Op by |ξ→(p)|=u′2(t)¯χ|G(q,t)|. The length of displacement of the point p (or the length of arch), with respect to the vector field generated by gτ(p), is determined by the formula
(19)λ(ϕa,ϕb)=∫ϕaϕb‍χu′2(t)¯|G(q,t)|dϕ≡χu′2(t)¯|G(q,tc)|(ϕb-ϕa),χ=const,
for each fixed time that defines the following length scale along the orbit Op:
(20)λOp=χu′2(t)¯|G(q,t)|2πϕ,q∈(q3*,L*(t)),χ=1.
The constant χ can be fixed by normalizing the velocity vector ξ→(p).


### 2.2. Extended Symmetry Transformations

Let us consider again in the correlation spaceK3 with K→3 equipped by the standard Euclidean scalar product (i.e., in ℝ3) an infinite cylindrical domain (a singled out fluid tube at some fixed time). Fix the cross-section {0}×D2 of this domain where D2 is a two-dimensional disk. Then the quadratic form dl2(t) induces on the above-mentioned cross-section the metric
(21)dlD22(t)=u′2(t)¯g(|r→|,t)(dr22+dr32),
where |r→|=r2+r3. Therefore, the metric (21) is a conformal form-type metric. First, we study the case of positive values of g(|r→|,t). Consider the set Ω(D2,v,w) of piecewise smooth curves γ:J→D2 with fixed endpoints γ(0)=v and γ(1)=w. Let Lγ:Ω(D2,v,w)→ℝ be the paper (the simple action) for each fixed time t. Then the formula
(22)dt(v,w)=minγ∈Ω(D2,v,w)Lγt(γ),Lγt=∫γ‍u′2(t)¯g(|r→|,t)(rτ22+rτ32)dτ,
defines the function of distance dt:({0}×D2)×({0}×D2)→ℝ on the cross-section. We can account  that τ is the so-called natural parameter; that is,
(23)u′2(t)¯g(|r→|,t)(rτ22+rτ32)=1alongthecurveγ.
Therefore,
(24)Lγt=∫γ‍1·dτ=τγ,
where the symbol τγ denotes length of the curve γ. Consider the infinitesimal transformations of the variables r2 and r3(25)r2*=r2+δr2,r3*=r3+δr3.
In order to investigate the invariance of dt under the action of infinitesimal transformations, it is sufficient to prove that the length of arch of extremals of the functional above is invariant. Instead of the vector (r2,r3), we consider the unit covector (τr2,τr3) defined by the formulas (for brevity we omitted the index γ for τ):
(26)rτ2=τr2u′2(t)¯g(|r→|,t),rτ3=τr3u′2(t)¯g(|r→|,t).
Then (23) is transformed to
(27)τr22+τr32=u′2(t)¯g(|r→|,t).
This equation is the eikonal-type equation. Therefore, in order to find symmetries of the functional Lγt, we can consider symmetries admitted by (27) that leaves τ invariant. The restatement of the variational symmetry in the terms of symmetry of partial differential equations (27) enables us to extend the class of symmetry transformations admitted by the functional Lγt. More exactly, we extend the variational symmetry up to the equivalence transformations. Recall that equivalence transformations for a differential equation in a given class is a change of variables which maps the equation to another equation in the same class. An equivalence transformation admitted by (27) is a point transformation given on (r2,r3,u1,u2) space where u1=τ,u2=u′2(t)¯g(|r→|,t). Infinitesimally, we look for an operator in the following form [13, 14]:
(28)Y=μ(r2,r3,u1,u2)∂∂r2+ν(r2,r3,u1,u2)∂∂r3+λi(r2,r3,u1,u2)∂∂ui,
where the coefficients are defined due to the following:
(29)Y1|(27)=0.
Here, Y1 denotes the first prolongation of Y. The infinitesimal operator Y reads as follows:
(30)Y=Φ(r2,r3)∂∂r2+Ψ(r2,r3)∂∂r2+γ(u1)∂∂u1+2(dγdu1-Φr2(r2,r3))u2∂∂u2.
Its Lie (infinite-dimensional) subalgebra is of the following form:
(31)X=Φ(r2,r3)∂∂r2+Ψ(r2,r3)∂∂r3-2Φr2(r2,r3)u2∂∂u2,
and u1(≡τ) is a scalar invariant of X. Therefore, X is a symmetry operator admitted by the functional Lγt. Here, the functions Φ(r2,r3) and Ψ(r2,r3) satisfy the Cauchy-Riemann differential equations Φr2=Ψr3 and Φr3=-Ψr2. To get a fine structure of the equivalence transformation generated by the infinitesimal X, we consider  the complex coordinates z=r2+ir3 and z-=r2-ir3. Then
(32)dlD22=u′2(t)¯g(zz-)dzdz-,
or more exactly
(33)dlD22=u′2(t)¯g(zz-)2(dz⊗dz-+dz-⊗dz).
The operator X1 takes the following form:
(34)X1=F(z)ddz+F¯(z-)ddz--Fz(z)u2d∂u2-F¯z-(z-)u2(zz-)d∂u2.
Here, F=Φ+iΨ and F¯=Φ-iΨ, and we use that F(z,z-)≡F(z) and F¯(z,z-)≡F¯(z-) for the holomorphic function F. The tangent space is spanned by the following:
(35)ddz=12(∂∂r2-i∂∂r3),ddz-=12(∂∂r2+i∂∂r3).
For small perturbations z↦z+ϵ(z) and z-↦z-+ε-(z-), we find the representation of the operator X1. Infinitesimal holomorphic transformations of the variables z and z- read as follows:
(36)z*=z+ϵ(z)=z+F(z)δs,z-*=z-+ϵ-(z-)=z-+F¯(z-)δs,
which are generated by the vector field
(37)(F(z)ddz,F¯(z-)ddz-).
Using the Laurent series
(38)ϵ(z)=-∑n=-∞∞‍εnzn+1,ϵ-(z-)=-∑n=-∞∞‍ε-nz-n+1,
where εn (ε-n) are infinitesimal small numbers. We can look at εnzn+1, ε-nz-n+1 as the harmonics of decomposition of ϵ(z),ϵ-(z-) with respect to the basis functions {zn+1} ({z-n+1}). Each harmonics generate the transformations z→z′≡z-εnzn+1,z-→z-′≡z--ε-nz-n+1 and the corresponding infinitesimal generators:  ln=-zn+1(d/dz),l-n=-z-n+1(d/dz-), presents the basis of the infinite dimensional conformal Lie algebra
(39)Z=F(z)ddz+F¯(z-)ddz-.
Therefore, the basis of the operator X(40)X=F(z)ddz+F¯(z-)ddz--Fz(z)u2(zz-)ddu2-F¯z-(z-)u2(zz-)ddu2
is kn⊕k-n, n∈ℤ(41)kn=-zn+1ddz-(n+1)znu2ddu2,k-n=-z-n+1ddz--(n+1)z-nu2ddu2.
The factor u2 is transformed into
(42)u2*=u2(zz*)2,forkn,u2*=u2(z-z-*)2,fork-n,
where (zz*)2=(z-z-*)2 under the change of variables z↦z*, and z-↦z-*, correspondingly.The liner hull of theln over ℂ is called the Witt algebra. The Witt algebra W is a dense subalgebra of the Lie algebra of holomorphic vector fields on ℂ∖{0}. Therefore, we can define the algebra generated by the infinitesimal operator X as
(43)W2=ℂ{kn,n∈Z}⊕ℂ{k-n,n∈Z}.
This is the linear hull of the basis elements kn⊕k-n. Here, kn:f(z,x)↦-zn+1fz(z,x)-(n+1)znfx(z,x), where z∈ℂ∖{0},x∈ℝ+. Correspondingly for k-n, we have k-n:f(z-,x)↦-z-n+1fz-(z-,x)-(n+1)z-nfx(z-,x), where z-∈ℂ∖{0},x∈ℝ+. W2 with the Lie bracket actually become Lie algebras over ℂ for that we determine the Lie bracket of the kn, km and correspondingly of the k-n, k-m. The direct calculations demonstrate that
(44)[kn,km]f=knkmf-kmknf=(n-m)kn+m,(45)[k-n,k-m]f=(n-m)k-n+mf,[kn,k-m]f=0.
The last relation follows from the following:
(46)ddz-zn=0,ddzz-n=0.
The commutation relations (44) and (45) will be significant for us to prove that central extension for W2 is defined by the same skew-symmetric bilinear form as for the Witt algebra.The case of negative values ofg(zz-) means that Lγt defines the function of imaginary distance and instead of (23) we have to consider
(47)u′2(t)¯i2|g(|r→|,t)|(rτ22+rτ32)=1alongthecurveγ,i2=-1.
The calculations of equivalence transformations of (47) are similar as for (23) where the variable u2=u′2(t)¯g(|r→|,t) is replaced on u2=u′2(t)¯i2|g(|r→|,t)|.


## 2.1. Geometric Realization of the Two-Point Velocity-Correlation Tensor

We recall only the elementary information about the structure of the two-point velocity-correlation tensor of the velocity fluctuations for homogeneous isotropic flows. The modern theory of the properties and structure of second-order (Cartesian) correlation tensors is given in [8].The statistical description of fluid turbulence employ the Reynolds decomposition to separate the fluid velocityu→ at a point x→ into its mean and fluctuating components as u→=u-→+u→′. Here, (·-) is the mean velocity, while u→ is the corresponding fluctuating quantity, usually interpreted as representing turbulence. The two-point correlation tensor is defined by the following:
(2)Bij(x,x′,t)=(ui′(x,t))(uj′(x′,t))¯,
where t∈ℝ+ and x,x′ are the points of a three-dimensional space filled by turbulent fluid.  Formula (2) is rewritten as follows:
(3)Bij(x,r→,t)=(ui′(x,t))(uj′(x+r→,t))¯,
where the vector r→ is determined by the pair (x,x′), where x and x′ are the starting point and endpoint, correspondingly, or r→=x→′-x→. Therefore, we will consider an affine space K3 with the adjoined vector space of the correlation vectors K→3≡{r→=(r1,r2,r3)}. The assumption of homogeneity and isotropy of turbulent flow (invariance with respect to rotation, reflection, and translation) implies that this tensor depends only on the length of the correlation vector r→ and time t; that is,
(4)Bij(x,r→,t)=[BLL(|r→|,t)-BNN(|r→|,t)]rirj|r→|2+BNN(|r→|,t)δij.
Moreover, for isotropic turbulence, Bij(|r→|,t) is a symmetric tensor and the correlations Bij can be expressed by using only the longitudinal correlational function BLL(|r→|,t) and the transversal correlation function BNN(|r→|,t) [9]; that is, the correlation tensor Bij takes the diagonal form with the components B11=BLL and B22≡B33=BNN in a suitable system of the coordinates of the adjoined vector space. Further instead of directly employing the correlation functions BLL and BNN, we use their normalized representations f and g where BLL=u′2(t)¯f(|r→|,t), BNN=u′2(t)¯g(|r→|,t) with the turbulence intensity equals u′2(t)¯=BLL(0,t). Then, the corresponding quadratic form (or linear element) takes the following form:
(5)dl2(t)=u′2(t)¯f(|r→|,t)dr12+u′2(t)¯g(|r→|,t)(dr22+dr32),
where dl2(t) is an indefinite quadratic form in view of the properties g(|r→|,t), see below. The normalized transversal correlation function g satisfies the following relation (taken from the continuity) [10]:
(6)g(|r→|,t)=f(|r→|,t)+r2∂∂rf(|r→|,t).
The property that f decays faster than |r→|-2 on infinity together with (6) yields [10]
(7)∫0∞‍|r→|g(|r→|,t)d|r→|=0.
Hence, g(|r→|,t) is an alternative sign function. Typical forms of experimentally measured functions f and g are given in Figure 1. we use the data presented to determine the qualitative behaviors of f and g, in particular, the algebraic properties of these correlation functions. Thus, we will assume that f is a positive everywhere function and g changes sign only in interval (-ε+|r→*|,|r→*|+ε), ε>0. g is a positive function on [0,±|r→*|) and therefore g<0 outside of [-|r→*|,|r→*|]. The change sign of g means that the quadratic forms dl2(t) have a variable signature. The normalized longitudinal correlational function f(|r→|,t) is dynamically evolved due to the von Kármán-Howarth equation [4]:
(8)∂u′2(t)¯f(|r|,t)∂t=1r4∂∂rr4(u′2(t)¯3/2h(|r|,t)+2ν∂∂ru′2(t)¯f(|r|,t)).h is the normalized triple-correlation function and u′2(t)¯ is the turbulence intensity (a positive everywhere function that vanishes on infinity) or the velocity scale for the turbulent kinetic energy; u′2(t)¯3/2 determines the scale for the turbulence transfer. This single equation directly follows from the Navier-Stokes equation (see, for example, [9]) and contains two unknowns, f and h, with the turbulence intensity u′2(t)¯ which cannot be defined from (8) without the use of additional hypothesis.Figure 1
Typical forms of the normalized longitudinal and transversal correlation functions.If we consider in the correlation spaceK3 with K→3 equipped by the standard Euclidean scalar product (i.e., in ℝ3) an infinite cylindrical domain (a singled out fluid tube at some fixed time), then the metric (induced by the quadratic form Bij) of the surface which bounds this domain takes the following form:
(9)ds2(t)=u′2(t)¯f(|r→|,t)dr12+u′2(t)¯g(|r→|,t)ρ2dϕ2,
where ρ denotes the Euclidean radius of the cross-section {a}×S1(ρ) of the surface ℝ×S1(ρ),a∈ℝ. We can account that ρ=1 and identify this manifold with ℝ×S1(1). The functions f and g are nondimensional with f(0,t)=g(0,t)=1 and physically f is a positive function such that f→0 (g→0) as |r→| tends to infinity. Moreover, f and g are bounded even functions such that f≤1, |g|≤1, and f goes faster to zero than |r→|-2, when |r→| tends to infinity. Physically such behavior of f is acceptable [10] and the map
(10)q(a,t)=∫0a‍f(|r→|,t)dr1,a∈ℝ,
acts as (-∞,∞)↦[-L*(t),L*(t)], L*(t)=L(t)/2u′2(t)¯, where L(t) is determined by the following:
(11)L(t)=2∫0∞‍u′2(t)¯f(|r→|,t)dr1.
Now, we rewrite the metric ds2(t) in the frame of the variable q:
(12)ds2(t)=u′2(t)¯{dq2+G(q,t)dϕ2},G(q,t)=g(|r|,t).The metric (12) admits a one-parametric group of (isometric) motion gτ(p→)≡g(p→,a1), p→=(q,ϕ) of the following form:
(13)gτ:(q,ϕ)↦(q,ϕ+χτ),χ=const,
with the generator
(14)X=ξi∂∂pi≡χ∂∂ϕ.
The scalar product of the generator X equals
(15)X2=〈X,X〉=〈χ∂∂ϕ,χ∂∂ϕ〉≡χ2〈∂∂ϕ,∂∂ϕ〉=u′2(t)¯χ2G(q,t)
for each time t.A pointp0 is called the pole [11] of a (pseudo-) Riemannian manifold M if p0 is a fixed point of a group of diffeomorphisms g(x→,a→), a→=(a1,…,ar) which acts on M.We note that ifp=p0 (p0 is the pole of gτ), then X2(p0)=0 and due to (15) p0 coincides with the roots of the equation G(q,·)=0. Therefore, the points q*∈[-L*(t),L*(t)] wherein G vanishes are the poles of gτ. In view of our assumption on g(|r|,t), the equation G(q,·)=0 has only 4 roots qi*, i=1,…,4 such that |q1*|=q4*=L*(t) and |q2*|=q3*. Thus, the metric (12) has the different signature for q∈I1=(q2*,q3*), q∈I2=(-L*(t),q2*), and, q∈I3=(q3*,L*(t)), respectively, where qi* depends on t. This metric determines for q∈I1 the element of length of the surface of revolution in ℝ3 and the radius-vector R→=(q,ϕ,·) of this surface is given by the following:
(16)R→(q,ϕ,·)=(q,{u′2(t)¯G}1/2(q,·)cosϕ,{u′2(t)¯G}1/2(q,·)sinϕ).
Therefore, the model manifold defined by (12) for q∈I1 is a cylindrical-type surface MI1t=(q2*,q3*)×S1(1) wherein the radius of the cross-section {q}×S1(1) equals G1/2(q,t). For the q∈Ii, i=2,3 where G is negative, the positive defined metric
(17)ds2(t)=u′2(t)¯{dq2-|G(q,t)|dϕ2},G(q,t)=g(|r→|,t),
can be realized (see, for details, [1]) as a surface of revolution (for each fixed time) in the Minkowski space ℝ1,23 with the element of length
(18)dρ2=dx12-dx22-dx32
when the form ds2(t) is of a fixed sign [12]. Here, the rotation presents the motion along the pseudocircle of the radius |G(q,·)|1/2, q∈Ii. Indeed, let us fix the point pa=(qa,ϕa) on the cross-section {qa}×S1(1) and consider the action of the group gτ on pa; that is, the orbit Opa:τ↦gτ(pa). This action is a motion along {qa}×S1(1) and if pa does not coincide with the poles gτ, then Opa is a not compact set [11]. In particular, Opa⊆{(x1,x2):x12-x22=u′2(t)¯|G(qa,t)|} for each fixed time t which coincides with the so-called pseudocircle under the embedding MI3t (MI2t) into the Minkowski space ℝ1,23. Moreover, the poles are saddle points of a negative index for the orbits Op, p∈MI3t (MI2t). The cross-sections {qa}×S1(1) of MI3t (MI2t) for qa∈{q3*,L*(t)} (resp., qa∈{-L*(t),q2*}) are the pseudocircles of zero radius and consist of the isotropic rays with the initial points q3* and L*(t) (resp., -L*(t) and q2*). The action of gτ on the point p is a motion along these piecewise linear isotropic curves when p∈{-L*(t),q2*,q3*,L*(t)}. We can identify MI3t (MI2t) with the foliation space of orbits MIjt=⋃p‍Op and associate the modulus of the transversal correlation function G(q,t) with the length of the velocity vector ξ→(p) of the orbit Op by |ξ→(p)|=u′2(t)¯χ|G(q,t)|. The length of displacement of the point p (or the length of arch), with respect to the vector field generated by gτ(p), is determined by the formula
(19)λ(ϕa,ϕb)=∫ϕaϕb‍χu′2(t)¯|G(q,t)|dϕ≡χu′2(t)¯|G(q,tc)|(ϕb-ϕa),χ=const,
for each fixed time that defines the following length scale along the orbit Op:
(20)λOp=χu′2(t)¯|G(q,t)|2πϕ,q∈(q3*,L*(t)),χ=1.
The constant χ can be fixed by normalizing the velocity vector ξ→(p).


## 2.2. Extended Symmetry Transformations

Let us consider again in the correlation spaceK3 with K→3 equipped by the standard Euclidean scalar product (i.e., in ℝ3) an infinite cylindrical domain (a singled out fluid tube at some fixed time). Fix the cross-section {0}×D2 of this domain where D2 is a two-dimensional disk. Then the quadratic form dl2(t) induces on the above-mentioned cross-section the metric
(21)dlD22(t)=u′2(t)¯g(|r→|,t)(dr22+dr32),
where |r→|=r2+r3. Therefore, the metric (21) is a conformal form-type metric. First, we study the case of positive values of g(|r→|,t). Consider the set Ω(D2,v,w) of piecewise smooth curves γ:J→D2 with fixed endpoints γ(0)=v and γ(1)=w. Let Lγ:Ω(D2,v,w)→ℝ be the paper (the simple action) for each fixed time t. Then the formula
(22)dt(v,w)=minγ∈Ω(D2,v,w)Lγt(γ),Lγt=∫γ‍u′2(t)¯g(|r→|,t)(rτ22+rτ32)dτ,
defines the function of distance dt:({0}×D2)×({0}×D2)→ℝ on the cross-section. We can account  that τ is the so-called natural parameter; that is,
(23)u′2(t)¯g(|r→|,t)(rτ22+rτ32)=1alongthecurveγ.
Therefore,
(24)Lγt=∫γ‍1·dτ=τγ,
where the symbol τγ denotes length of the curve γ. Consider the infinitesimal transformations of the variables r2 and r3(25)r2*=r2+δr2,r3*=r3+δr3.
In order to investigate the invariance of dt under the action of infinitesimal transformations, it is sufficient to prove that the length of arch of extremals of the functional above is invariant. Instead of the vector (r2,r3), we consider the unit covector (τr2,τr3) defined by the formulas (for brevity we omitted the index γ for τ):
(26)rτ2=τr2u′2(t)¯g(|r→|,t),rτ3=τr3u′2(t)¯g(|r→|,t).
Then (23) is transformed to
(27)τr22+τr32=u′2(t)¯g(|r→|,t).
This equation is the eikonal-type equation. Therefore, in order to find symmetries of the functional Lγt, we can consider symmetries admitted by (27) that leaves τ invariant. The restatement of the variational symmetry in the terms of symmetry of partial differential equations (27) enables us to extend the class of symmetry transformations admitted by the functional Lγt. More exactly, we extend the variational symmetry up to the equivalence transformations. Recall that equivalence transformations for a differential equation in a given class is a change of variables which maps the equation to another equation in the same class. An equivalence transformation admitted by (27) is a point transformation given on (r2,r3,u1,u2) space where u1=τ,u2=u′2(t)¯g(|r→|,t). Infinitesimally, we look for an operator in the following form [13, 14]:
(28)Y=μ(r2,r3,u1,u2)∂∂r2+ν(r2,r3,u1,u2)∂∂r3+λi(r2,r3,u1,u2)∂∂ui,
where the coefficients are defined due to the following:
(29)Y1|(27)=0.
Here, Y1 denotes the first prolongation of Y. The infinitesimal operator Y reads as follows:
(30)Y=Φ(r2,r3)∂∂r2+Ψ(r2,r3)∂∂r2+γ(u1)∂∂u1+2(dγdu1-Φr2(r2,r3))u2∂∂u2.
Its Lie (infinite-dimensional) subalgebra is of the following form:
(31)X=Φ(r2,r3)∂∂r2+Ψ(r2,r3)∂∂r3-2Φr2(r2,r3)u2∂∂u2,
and u1(≡τ) is a scalar invariant of X. Therefore, X is a symmetry operator admitted by the functional Lγt. Here, the functions Φ(r2,r3) and Ψ(r2,r3) satisfy the Cauchy-Riemann differential equations Φr2=Ψr3 and Φr3=-Ψr2. To get a fine structure of the equivalence transformation generated by the infinitesimal X, we consider  the complex coordinates z=r2+ir3 and z-=r2-ir3. Then
(32)dlD22=u′2(t)¯g(zz-)dzdz-,
or more exactly
(33)dlD22=u′2(t)¯g(zz-)2(dz⊗dz-+dz-⊗dz).
The operator X1 takes the following form:
(34)X1=F(z)ddz+F¯(z-)ddz--Fz(z)u2d∂u2-F¯z-(z-)u2(zz-)d∂u2.
Here, F=Φ+iΨ and F¯=Φ-iΨ, and we use that F(z,z-)≡F(z) and F¯(z,z-)≡F¯(z-) for the holomorphic function F. The tangent space is spanned by the following:
(35)ddz=12(∂∂r2-i∂∂r3),ddz-=12(∂∂r2+i∂∂r3).
For small perturbations z↦z+ϵ(z) and z-↦z-+ε-(z-), we find the representation of the operator X1. Infinitesimal holomorphic transformations of the variables z and z- read as follows:
(36)z*=z+ϵ(z)=z+F(z)δs,z-*=z-+ϵ-(z-)=z-+F¯(z-)δs,
which are generated by the vector field
(37)(F(z)ddz,F¯(z-)ddz-).
Using the Laurent series
(38)ϵ(z)=-∑n=-∞∞‍εnzn+1,ϵ-(z-)=-∑n=-∞∞‍ε-nz-n+1,
where εn (ε-n) are infinitesimal small numbers. We can look at εnzn+1, ε-nz-n+1 as the harmonics of decomposition of ϵ(z),ϵ-(z-) with respect to the basis functions {zn+1} ({z-n+1}). Each harmonics generate the transformations z→z′≡z-εnzn+1,z-→z-′≡z--ε-nz-n+1 and the corresponding infinitesimal generators:  ln=-zn+1(d/dz),l-n=-z-n+1(d/dz-), presents the basis of the infinite dimensional conformal Lie algebra
(39)Z=F(z)ddz+F¯(z-)ddz-.
Therefore, the basis of the operator X(40)X=F(z)ddz+F¯(z-)ddz--Fz(z)u2(zz-)ddu2-F¯z-(z-)u2(zz-)ddu2
is kn⊕k-n, n∈ℤ(41)kn=-zn+1ddz-(n+1)znu2ddu2,k-n=-z-n+1ddz--(n+1)z-nu2ddu2.
The factor u2 is transformed into
(42)u2*=u2(zz*)2,forkn,u2*=u2(z-z-*)2,fork-n,
where (zz*)2=(z-z-*)2 under the change of variables z↦z*, and z-↦z-*, correspondingly.The liner hull of theln over ℂ is called the Witt algebra. The Witt algebra W is a dense subalgebra of the Lie algebra of holomorphic vector fields on ℂ∖{0}. Therefore, we can define the algebra generated by the infinitesimal operator X as
(43)W2=ℂ{kn,n∈Z}⊕ℂ{k-n,n∈Z}.
This is the linear hull of the basis elements kn⊕k-n. Here, kn:f(z,x)↦-zn+1fz(z,x)-(n+1)znfx(z,x), where z∈ℂ∖{0},x∈ℝ+. Correspondingly for k-n, we have k-n:f(z-,x)↦-z-n+1fz-(z-,x)-(n+1)z-nfx(z-,x), where z-∈ℂ∖{0},x∈ℝ+. W2 with the Lie bracket actually become Lie algebras over ℂ for that we determine the Lie bracket of the kn, km and correspondingly of the k-n, k-m. The direct calculations demonstrate that
(44)[kn,km]f=knkmf-kmknf=(n-m)kn+m,(45)[k-n,k-m]f=(n-m)k-n+mf,[kn,k-m]f=0.
The last relation follows from the following:
(46)ddz-zn=0,ddzz-n=0.
The commutation relations (44) and (45) will be significant for us to prove that central extension for W2 is defined by the same skew-symmetric bilinear form as for the Witt algebra.The case of negative values ofg(zz-) means that Lγt defines the function of imaginary distance and instead of (23) we have to consider
(47)u′2(t)¯i2|g(|r→|,t)|(rτ22+rτ32)=1alongthecurveγ,i2=-1.
The calculations of equivalence transformations of (47) are similar as for (23) where the variable u2=u′2(t)¯g(|r→|,t) is replaced on u2=u′2(t)¯i2|g(|r→|,t)|.


## 3. Asymptotic Expansion of the Transversal Correlation Function

First, using the functional of action of the trajectoryγ(48)Eγt=∫γ‍(u2)-1(τr22+τr32)dτ,τr22+τr32=u2,
which in the complex coordinates takes the form
(49)Eγt=4∫γ‍(u2)-1(zz-,t)∂zτ(z,z-)·∂z-τ(z,z-)dτ,
we demonstrate (in the full analogy with CFT) the representation of the operators kn and k-n in terms of the components of the energy-momentum tensor obtained. Notice that this tensor has only two independent components T(z) and T¯(z-) which can be expressed by the transversal correlation function. T(z) and T¯(z-) are holomorphic functions of the variables z and z- whose variations under the infinitesimal transformations z↦z+ϵ(z),z-↦z-+ϵ-(z) are of the form
(50)δϵT(z)=ϵ(z)T′+2ϵ′(z)T(z)+(1/12)cϵ′′′(z),δϵ-T¯(z)=ϵ-(z-)T¯′+2ϵ′-(z-)T¯(z-)+(1/12)c-ϵ-′′′(z-),
where c(c-) coincides with the central charge for the corresponding nontrivial extensions. According to [15], these formulas lead to the commutation relations for the field representations of T(z) and T¯(z-) which coincide with the commutation relations of nontrivial central extensions of the algebra W2. Then we show how on the basis of these commutation relations to find asymptotic expansion of the transversal correlation function that present the basic interest of the theory of homogeneous isotropic turbulence.


### 3.1. The Energy-Momentum Tensor

Consider the functional of action of the trajectoryγ(51)Eγt=∫γ‍u2(rτ22+rτ32)dτ
and write this Eγt in the form
(52)Eγt=∫γ‍(u2)-1(τr22+τr32)dτ,τr22+τr32=u2.
We note that the extremals of this functional coincide with the geodesic curves and the length of the geodesic curve γ coincides with τγ for the corresponding index γ. Eγt admits the same infinite-dimensional  Lie algebra W2.Recall that the classical energy-momentum tensor is defined by(53)Tik=gklfxiα∂L∂fαxl-gikL,
where L=L(fj,fxki) is a Lagrangian. For the functional (52) with the scalar field τ this tensor takes the following form:
(54)T11=2τr22-u2·1,T22=2τr32-u2·1,T12=T21=2τr2τr3.Tik is a traceless tensor due to the equality T11+T22=2(τr22+τr32-u2)=0. Denote ω→=(r2,r3) and δω→=(δr2,δr3). Then the current is defined jμ=Tμνδων and has an automatically vanishing divergence ∂·j=(1/2)Tμμ(∂·δω) due to the traceless condition on Tμν.In the complex coordinates, the functional (52) takes the following form:
(55)Eγt=4∫γ‍(u2)-1(zz-,t)∂zτ(z,z-)·∂z-τ(z,z-)dτ.
Since Tik is traceless, we have Tzz-=Tz-z = 0; these imply ∂z-Tzz=∂zTz-z-=0 and there are only two nonvanishing components of the energy-momentum tensor T(z)=Tzz(z)=1/4(T11-2iT12-T22) and T¯(z-)=Tz-z-(z-)=1/4(T11+2iT12-T22). Moreover Tzz=(∂zτ)2 and Tz-z-=(∂z-τ)2. In the line of Conformal Field Theory, we can consider the following constructions: since T(z) and T¯(z-) are holomorphic functions, then due to Laurent expansions
(56)T(z)=∑n‍Lnz-n-2,T¯(z-)=∑n‍L-nz--n-2,
where the exponent -n-2 is chosen so that for the scale transformation z↦λ-1z under which T(z)↦λ2T(λ-1z) we have L-n↦λnL-n and L--n↦λ-nL--n. Therefore, the expression (56) is finally inverted by the following relations:
(57)Ln=12πi∮‍T(z)zn+1dz,L-n=12πi∮‍T¯(z-)z-n+1dz-.
Recall that Ln are defined on ℂ∖{0}:
(58)Tϵ=∮‍ϵ(z)T(z)dz,T¯ϵ-=∮‍ϵ-(z-)T¯(z-)dz
Their variations read as follows:
(59)δϵT(z)=ϵ(z)T′+2ϵ′(z)T(z)+(112)cϵ′′′(z),δϵ-T¯(z)=ϵ-(z-)T¯′+2ϵ′-(z-)T¯(z-)+(112)c-ϵ-′′′(z-).
It leads to (see [15])
(60)[Tϵ,T(z)]=ϵ(z)T′(z)+2ϵ(z)T(z)+c12ϵ′′′(z),[T¯ϵ-,T¯(z-)]=ϵ-(z-)T¯′(z-)+2ϵ-(z-)T¯(z-)+c12ϵ-′′′(z-).
Then from (60) we get as in [15] that Ln and L-n satisfy the commutation relations
(61)[Ln,Lm]=(n-m)Ln+m+c12δn+mn(n2-1),(62)[L-n,L-m]=(n-m)L-n+m+c-12δn+mn(n2-1),[Ln,L-m]=0.Lemma 1.
LetT(z)=∑n‍Lnz-n-2 and T(w)=∑n‍Lnw-n-2 be the fields with {Ln} satisfying the commutation relationship (61). Then for large z and w, it holds that
(63)T(z)T(w)~c2(z-w)4+2T(w)(z-w)2+∂wT(w)z-w.The left-hand side of (61) reads as follows:
(64)[Ln,Lm]=(∮|z|>|w|‍dz2πi∮‍dw2πi-∮‍dw2πi∮|z|<|w|‍dz2πi)×zn+1T(Z)wn+1T(w).
The proof of this lemma is based on the direct calculations. The first term of the right-hand side of (61) equals
(65)(n-m)Ln+m=(n-m)∮‍dw2πi(wn+m+1T(w)).
This integral can be rewritten as follows
(66)∮‍dw2πi(2(n+1)wnwm+1T(w)+wn+1wm+1∂wT(w)).
Indeed, integrating the last term of this integral by parts and combining with the first term we get the right-hand part of (65). The second term of (61), that is, (c/12)δn+mn(n2-1), can be presented as the following integral:
(67)∮‍dw2πi(c12n(n2-1)wn-2wm).
Further transformations are based on using the well-known formula from the Complex Analysis
(68)∂ζnf(ζ)=n!∮‍dw2πif(ζ)(ζ-z)n.
Since ∂z3zn+1|z=w=n(n2-1)wn-2, then applying (68), we get that (67) equals
(69)∮‍dz2πi∮‍dw2πizn+1wm+1c2(z-w)4.
Applying the same procedure to the first and the second terms of (66), we can see that this integral equals
(70)∮‍dz2πi∮‍dw2πizn+1wm+1(2T(w)(z-w)2+∂T(w)z-w).
Combining  (69) and  (70), we found that the integral
(71)∮‍dz2πi∮‍dw2πizn+1wm+1(c2(z-w)4+2T(w)(z-w)2+∂T(w)z-w)
equals the right-hand side of  (64). For the completeness, we need to add at the integral (71) a regular function R(z,w) on the variable z-w since this function is free of poles at z=w; hence, R(z,w) does not contribute to the integral. Therefore, comparing (71) with  (64), we derive that
(72)T(z)T(w)=c2(z-w)4+2T(w)(z-w)2+∂wT(w)z-w+R(z,w).
This relationship is called the operator product expansion in CFT. The same assertion holds for the T¯(z-). Notice that the quantities c and c- coincide.Corollary 2.
Consider the following:(73)T(z)T(0)~c2z4,T¯(z-)T¯(0)~c2z-4asz,z-→∞.
One  applies the results obtained to find asymptotic behavior of the transversal correlation functionBNN=u′2(t)¯g(zz-) as the length of the correlation vector tends to infinity. For this, one uses the following relationship:
(74)4∂zτ·∂z-τ=u2,
and the equalities T(z)=Tzz=(∂zτ)2 and T¯(z-)=Tz-z-=(∂z-τ)2. Rewrite (74) in the form (∂zτ)2·(∂z-τ)2=(1/16)(u2)2 and use the formulas (73). Then substituting instead of (u2)2 the quantity u′2(t)¯2g2(|r→|,t) in (73), one gets the asymptotic expansion for BNN2(|r→|,t) in the follwing form:
(75)u′2(t)¯2g2(|r→|,t)~4c2|r→|8T(0)T¯(0),zz-=|r→|2as|r→|→∞.
Since T(0)T¯(0)=(1/16)u′2(t)¯2, one finally obtains from  (75) that
(76)BNN(|r→|,t)~-8cu′2(t)¯2|r→|4.
The sign “−” appears in view of the negative values of BNN(|r→|,t) as |r→|→∞.


## 3.1. The Energy-Momentum Tensor

Consider the functional of action of the trajectoryγ(51)Eγt=∫γ‍u2(rτ22+rτ32)dτ
and write this Eγt in the form
(52)Eγt=∫γ‍(u2)-1(τr22+τr32)dτ,τr22+τr32=u2.
We note that the extremals of this functional coincide with the geodesic curves and the length of the geodesic curve γ coincides with τγ for the corresponding index γ. Eγt admits the same infinite-dimensional  Lie algebra W2.Recall that the classical energy-momentum tensor is defined by(53)Tik=gklfxiα∂L∂fαxl-gikL,
where L=L(fj,fxki) is a Lagrangian. For the functional (52) with the scalar field τ this tensor takes the following form:
(54)T11=2τr22-u2·1,T22=2τr32-u2·1,T12=T21=2τr2τr3.Tik is a traceless tensor due to the equality T11+T22=2(τr22+τr32-u2)=0. Denote ω→=(r2,r3) and δω→=(δr2,δr3). Then the current is defined jμ=Tμνδων and has an automatically vanishing divergence ∂·j=(1/2)Tμμ(∂·δω) due to the traceless condition on Tμν.In the complex coordinates, the functional (52) takes the following form:
(55)Eγt=4∫γ‍(u2)-1(zz-,t)∂zτ(z,z-)·∂z-τ(z,z-)dτ.
Since Tik is traceless, we have Tzz-=Tz-z = 0; these imply ∂z-Tzz=∂zTz-z-=0 and there are only two nonvanishing components of the energy-momentum tensor T(z)=Tzz(z)=1/4(T11-2iT12-T22) and T¯(z-)=Tz-z-(z-)=1/4(T11+2iT12-T22). Moreover Tzz=(∂zτ)2 and Tz-z-=(∂z-τ)2. In the line of Conformal Field Theory, we can consider the following constructions: since T(z) and T¯(z-) are holomorphic functions, then due to Laurent expansions
(56)T(z)=∑n‍Lnz-n-2,T¯(z-)=∑n‍L-nz--n-2,
where the exponent -n-2 is chosen so that for the scale transformation z↦λ-1z under which T(z)↦λ2T(λ-1z) we have L-n↦λnL-n and L--n↦λ-nL--n. Therefore, the expression (56) is finally inverted by the following relations:
(57)Ln=12πi∮‍T(z)zn+1dz,L-n=12πi∮‍T¯(z-)z-n+1dz-.
Recall that Ln are defined on ℂ∖{0}:
(58)Tϵ=∮‍ϵ(z)T(z)dz,T¯ϵ-=∮‍ϵ-(z-)T¯(z-)dz
Their variations read as follows:
(59)δϵT(z)=ϵ(z)T′+2ϵ′(z)T(z)+(112)cϵ′′′(z),δϵ-T¯(z)=ϵ-(z-)T¯′+2ϵ′-(z-)T¯(z-)+(112)c-ϵ-′′′(z-).
It leads to (see [15])
(60)[Tϵ,T(z)]=ϵ(z)T′(z)+2ϵ(z)T(z)+c12ϵ′′′(z),[T¯ϵ-,T¯(z-)]=ϵ-(z-)T¯′(z-)+2ϵ-(z-)T¯(z-)+c12ϵ-′′′(z-).
Then from (60) we get as in [15] that Ln and L-n satisfy the commutation relations
(61)[Ln,Lm]=(n-m)Ln+m+c12δn+mn(n2-1),(62)[L-n,L-m]=(n-m)L-n+m+c-12δn+mn(n2-1),[Ln,L-m]=0.Lemma 1.
LetT(z)=∑n‍Lnz-n-2 and T(w)=∑n‍Lnw-n-2 be the fields with {Ln} satisfying the commutation relationship (61). Then for large z and w, it holds that
(63)T(z)T(w)~c2(z-w)4+2T(w)(z-w)2+∂wT(w)z-w.The left-hand side of (61) reads as follows:
(64)[Ln,Lm]=(∮|z|>|w|‍dz2πi∮‍dw2πi-∮‍dw2πi∮|z|<|w|‍dz2πi)×zn+1T(Z)wn+1T(w).
The proof of this lemma is based on the direct calculations. The first term of the right-hand side of (61) equals
(65)(n-m)Ln+m=(n-m)∮‍dw2πi(wn+m+1T(w)).
This integral can be rewritten as follows
(66)∮‍dw2πi(2(n+1)wnwm+1T(w)+wn+1wm+1∂wT(w)).
Indeed, integrating the last term of this integral by parts and combining with the first term we get the right-hand part of (65). The second term of (61), that is, (c/12)δn+mn(n2-1), can be presented as the following integral:
(67)∮‍dw2πi(c12n(n2-1)wn-2wm).
Further transformations are based on using the well-known formula from the Complex Analysis
(68)∂ζnf(ζ)=n!∮‍dw2πif(ζ)(ζ-z)n.
Since ∂z3zn+1|z=w=n(n2-1)wn-2, then applying (68), we get that (67) equals
(69)∮‍dz2πi∮‍dw2πizn+1wm+1c2(z-w)4.
Applying the same procedure to the first and the second terms of (66), we can see that this integral equals
(70)∮‍dz2πi∮‍dw2πizn+1wm+1(2T(w)(z-w)2+∂T(w)z-w).
Combining  (69) and  (70), we found that the integral
(71)∮‍dz2πi∮‍dw2πizn+1wm+1(c2(z-w)4+2T(w)(z-w)2+∂T(w)z-w)
equals the right-hand side of  (64). For the completeness, we need to add at the integral (71) a regular function R(z,w) on the variable z-w since this function is free of poles at z=w; hence, R(z,w) does not contribute to the integral. Therefore, comparing (71) with  (64), we derive that
(72)T(z)T(w)=c2(z-w)4+2T(w)(z-w)2+∂wT(w)z-w+R(z,w).
This relationship is called the operator product expansion in CFT. The same assertion holds for the T¯(z-). Notice that the quantities c and c- coincide.Corollary 2.
Consider the following:(73)T(z)T(0)~c2z4,T¯(z-)T¯(0)~c2z-4asz,z-→∞.
One  applies the results obtained to find asymptotic behavior of the transversal correlation functionBNN=u′2(t)¯g(zz-) as the length of the correlation vector tends to infinity. For this, one uses the following relationship:
(74)4∂zτ·∂z-τ=u2,
and the equalities T(z)=Tzz=(∂zτ)2 and T¯(z-)=Tz-z-=(∂z-τ)2. Rewrite (74) in the form (∂zτ)2·(∂z-τ)2=(1/16)(u2)2 and use the formulas (73). Then substituting instead of (u2)2 the quantity u′2(t)¯2g2(|r→|,t) in (73), one gets the asymptotic expansion for BNN2(|r→|,t) in the follwing form:
(75)u′2(t)¯2g2(|r→|,t)~4c2|r→|8T(0)T¯(0),zz-=|r→|2as|r→|→∞.
Since T(0)T¯(0)=(1/16)u′2(t)¯2, one finally obtains from  (75) that
(76)BNN(|r→|,t)~-8cu′2(t)¯2|r→|4.
The sign “−” appears in view of the negative values of BNN(|r→|,t) as |r→|→∞.


## 4. Concluding Remarks

The behavior of the correlation functionsBLL and BNN presents significant interests for the theory of turbulence since this leads to various types of the so-called integral invariants. The Loitsyansky and Birkhoff integrals are the most famous integral invariants. In this paper, we established the asymptotic behavior of the transversal correlation function BNN(|r→|,t) as |r→|→∞ for the geometry of the correlation space K3 determined by the two-point velocity-correlation tensor in the case of homogeneous isotropic turbulence. The question about the asymptotic expansion of BNN(|r→|,t) for large values of the correlation distances in the physical space ℝ3 with the standard Euclidian metric is still open. Nevertheless, the formula (76) can be applied for studying this problem if we construct the isometric embedding of the couple (K3,dl2(t)) into the physical space that presents another topic of investigations.



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*Source: 469654-2013-03-25.xml*