The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence
The extended symmetry of the functional of length determined in an affine space of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variable ) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metrics (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 and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic form generated by 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 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 .
This paper is a continuation of [1–3] wherein we investigated both the geometry and the group of transformations of an affine space of the correlation vectors. In , we used the two-point velocity-correlation tensor to equip the correlation space 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 into the Euclidean space with the standard metric. The Lagrangian system in the extended space was introduced in  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 whose components are the correlation functions that evolve according to the von Kármán-Howarth equation . Notice here that the first integrals of the equations of geodesic curves form the “kinematic” conservation laws; see, for more details, . In , 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 space 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 , satisfy the following commutation relations: where for and for . The quantity 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 is expressed via the diffusion coefficient of : the Shram-Lövner evolution, of random curves in planar domains; see [5, 6]. Here, the diffusion coefficient allows us to classify the conformally invariant random curves into classes denoted by . The formula classifies the models, for instance, and , corresponds to , while corresponds to . The isolines of vorticity for the Euler equations and the temperature isolines in the model (surface of quasi-geostrophic model) belong to the class , more exactly, and , correspondingly, where and . The energy-momentum tensor was calculated for in .
The paper is organized as follows. Section 1 contains the results obtained (see [1–3] for more details) in a compressed form about the geometry of the correlation space 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 and show that the components and of this tensor are holomorphic functions. Then, in the full analogy of CFT, we expose the asymptotic expansion both for and as and . 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 (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  about a geometric realization of the two-point velocity-correlation tensor 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, ).
2.1. 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 .
The statistical description of fluid turbulence employ the Reynolds decomposition to separate the fluid velocity at a point into its mean and fluctuating components as . Here, is the mean velocity, while is the corresponding fluctuating quantity, usually interpreted as representing turbulence. The two-point correlation tensor is defined by the following: where and are the points of a three-dimensional space filled by turbulent fluid. Formula (2) is rewritten as follows: where the vector is determined by the pair , where and are the starting point and endpoint, correspondingly, or . Therefore, we will consider an affine space with the adjoined vector space of the correlation vectors . 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 and time ; that is, Moreover, for isotropic turbulence, is a symmetric tensor and the correlations can be expressed by using only the longitudinal correlational function and the transversal correlation function ; that is, the correlation tensor takes the diagonal form with the components and in a suitable system of the coordinates of the adjoined vector space. Further instead of directly employing the correlation functions and , we use their normalized representations and where , with the turbulence intensity equals . Then, the corresponding quadratic form (or linear element) takes the following form: where is an indefinite quadratic form in view of the properties , see below. The normalized transversal correlation function satisfies the following relation (taken from the continuity) : The property that decays faster than on infinity together with (6) yields  Hence, is an alternative sign function. Typical forms of experimentally measured functions and are given in Figure 1. we use the data presented to determine the qualitative behaviors of and , in particular, the algebraic properties of these correlation functions. Thus, we will assume that is a positive everywhere function and changes sign only in interval , . is a positive function on and therefore outside of . The change sign of means that the quadratic forms have a variable signature. The normalized longitudinal correlational function is dynamically evolved due to the von Kármán-Howarth equation : is the normalized triple-correlation function and is the turbulence intensity (a positive everywhere function that vanishes on infinity) or the velocity scale for the turbulent kinetic energy; determines the scale for the turbulence transfer. This single equation directly follows from the Navier-Stokes equation (see, for example, ) and contains two unknowns, and , with the turbulence intensity which cannot be defined from (8) without the use of additional hypothesis.
If we consider in the correlation space with equipped by the standard Euclidean scalar product (i.e., in ) an infinite cylindrical domain (a singled out fluid tube at some fixed time), then the metric (induced by the quadratic form ) of the surface which bounds this domain takes the following form: where denotes the Euclidean radius of the cross-section of the surface . We can account that and identify this manifold with . The functions and are nondimensional with and physically is a positive function such that () as tends to infinity. Moreover, and are bounded even functions such that , , and goes faster to zero than , when tends to infinity. Physically such behavior of is acceptable  and the map acts as , , where is determined by the following: Now, we rewrite the metric in the frame of the variable :
The metric (12) admits a one-parametric group of (isometric) motion , of the following form: with the generator The scalar product of the generator equals for each time .
A point is called the pole  of a (pseudo-) Riemannian manifold if is a fixed point of a group of diffeomorphisms , which acts on .
We note that if ( is the pole of ), then and due to (15) coincides with the roots of the equation . Therefore, the points wherein vanishes are the poles of . In view of our assumption on , the equation has only roots , such that and . Thus, the metric (12) has the different signature for , , and, , respectively, where depends on . This metric determines for the element of length of the surface of revolution in and the radius-vector of this surface is given by the following: Therefore, the model manifold defined by (12) for is a cylindrical-type surface wherein the radius of the cross-section equals . For the , where is negative, the positive defined metric can be realized (see, for details, ) as a surface of revolution (for each fixed time) in the Minkowski space with the element of length when the form is of a fixed sign . Here, the rotation presents the motion along the pseudocircle of the radius , . Indeed, let us fix the point on the cross-section and consider the action of the group on ; that is, the orbit . This action is a motion along and if does not coincide with the poles , then is a not compact set . In particular, for each fixed time which coincides with the so-called pseudocircle under the embedding () into the Minkowski space . Moreover, the poles are saddle points of a negative index for the orbits , (). The cross-sections of () for (resp., ) are the pseudocircles of zero radius and consist of the isotropic rays with the initial points and (resp., and ). The action of on the point is a motion along these piecewise linear isotropic curves when . We can identify () with the foliation space of orbits and associate the modulus of the transversal correlation function with the length of the velocity vector of the orbit by . The length of displacement of the point (or the length of arch), with respect to the vector field generated by , is determined by the formula for each fixed time that defines the following length scale along the orbit : The constant can be fixed by normalizing the velocity vector .
2.2. Extended Symmetry Transformations
Let us consider again in the correlation space with equipped by the standard Euclidean scalar product (i.e., in ) an infinite cylindrical domain (a singled out fluid tube at some fixed time). Fix the cross-section of this domain where is a two-dimensional disk. Then the quadratic form induces on the above-mentioned cross-section the metric where . Therefore, the metric (21) is a conformal form-type metric. First, we study the case of positive values of . Consider the set of piecewise smooth curves with fixed endpoints and . Let be the paper (the simple action) for each fixed time . Then the formula defines the function of distance on the cross-section. We can account that is the so-called natural parameter; that is, Therefore, where the symbol denotes length of the curve . Consider the infinitesimal transformations of the variables and In order to investigate the invariance of 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 , we consider the unit covector defined by the formulas (for brevity we omitted the index for ): Then (23) is transformed to This equation is the eikonal-type equation. Therefore, in order to find symmetries of the functional , we can consider symmetries admitted by (27) that leaves 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 . 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 space where . Infinitesimally, we look for an operator in the following form [13, 14]: where the coefficients are defined due to the following: Here, denotes the first prolongation of . The infinitesimal operator reads as follows: Its Lie (infinite-dimensional) subalgebra is of the following form: and (≡τ) is a scalar invariant of . Therefore, is a symmetry operator admitted by the functional . Here, the functions and satisfy the Cauchy-Riemann differential equations and . To get a fine structure of the equivalence transformation generated by the infinitesimal , we consider the complex coordinates and . Then or more exactly The operator takes the following form: Here, and , and we use that and for the holomorphic function . The tangent space is spanned by the following: For small perturbations and , we find the representation of the operator . Infinitesimal holomorphic transformations of the variables and read as follows: which are generated by the vector field Using the Laurent series where () are infinitesimal small numbers. We can look at , as the harmonics of decomposition of with respect to the basis functions (). Each harmonics generate the transformations and the corresponding infinitesimal generators: , presents the basis of the infinite dimensional conformal Lie algebra Therefore, the basis of the operator is , The factor is transformed into where under the change of variables , and , correspondingly.
The liner hull of the over is called the Witt algebra. The Witt algebra is a dense subalgebra of the Lie algebra of holomorphic vector fields on . Therefore, we can define the algebra generated by the infinitesimal operator as This is the linear hull of the basis elements . Here, , where . Correspondingly for , we have , where . with the Lie bracket actually become Lie algebras over for that we determine the Lie bracket of the , and correspondingly of the , . The direct calculations demonstrate that The last relation follows from the following: The commutation relations (44) and (45) will be significant for us to prove that central extension for is defined by the same skew-symmetric bilinear form as for the Witt algebra.
The case of negative values of means that defines the function of imaginary distance and instead of (23) we have to consider The calculations of equivalence transformations of (47) are similar as for (23) where the variable is replaced on .
3. Asymptotic Expansion of the Transversal Correlation Function
First, using the functional of action of the trajectory which in the complex coordinates takes the form we demonstrate (in the full analogy with CFT) the representation of the operators and in terms of the components of the energy-momentum tensor obtained. Notice that this tensor has only two independent components and which can be expressed by the transversal correlation function. and are holomorphic functions of the variables and whose variations under the infinitesimal transformations are of the form where coincides with the central charge for the corresponding nontrivial extensions. According to , these formulas lead to the commutation relations for the field representations of and which coincide with the commutation relations of nontrivial central extensions of the algebra . 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 trajectory and write this in the form We note that the extremals of this functional coincide with the geodesic curves and the length of the geodesic curve coincides with for the corresponding index . admits the same infinite-dimensional Lie algebra .
Recall that the classical energy-momentum tensor is defined by where is a Lagrangian. For the functional (52) with the scalar field this tensor takes the following form: is a traceless tensor due to the equality . Denote and . Then the current is defined and has an automatically vanishing divergence due to the traceless condition on .
In the complex coordinates, the functional (52) takes the following form: Since is traceless, we have = 0; these imply and there are only two nonvanishing components of the energy-momentum tensor and . Moreover and . In the line of Conformal Field Theory, we can consider the following constructions: since and are holomorphic functions, then due to Laurent expansions where the exponent is chosen so that for the scale transformation under which we have and . Therefore, the expression (56) is finally inverted by the following relations: Recall that are defined on : Their variations read as follows: It leads to (see ) Then from (60) we get as in  that and satisfy the commutation relations
Lemma 1. Let and be the fields with satisfying the commutation relationship (61). Then for large and , it holds that
The left-hand side of (61) reads as follows: The proof of this lemma is based on the direct calculations. The first term of the right-hand side of (61) equals This integral can be rewritten as follows 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, , can be presented as the following integral: Further transformations are based on using the well-known formula from the Complex Analysis Since , then applying (68), we get that (67) equals Applying the same procedure to the first and the second terms of (66), we can see that this integral equals Combining (69) and (70), we found that the integral equals the right-hand side of (64). For the completeness, we need to add at the integral (71) a regular function on the variable since this function is free of poles at ; hence, does not contribute to the integral. Therefore, comparing (71) with (64), we derive that This relationship is called the operator product expansion in CFT. The same assertion holds for the . Notice that the quantities and coincide.
Corollary 2. Consider the following:
One applies the results obtained to find asymptotic behavior of the transversal correlation function as the length of the correlation vector tends to infinity. For this, one uses the following relationship: and the equalities and . Rewrite (74) in the form and use the formulas (73). Then substituting instead of the quantity in (73), one gets the asymptotic expansion for in the follwing form: Since , one finally obtains from (75) that The sign “−” appears in view of the negative values of as .
4. Concluding Remarks
The behavior of the correlation functions and 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 as for the geometry of the correlation space determined by the two-point velocity-correlation tensor in the case of homogeneous isotropic turbulence. The question about the asymptotic expansion of for large values of the correlation distances in the physical space 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 into the physical space that presents another topic of investigations.
In this section, we show that nontrivial central extension of the Lie algebra exists.
In order to avoid a redundant complexity of the exposition of material, we recall basic definitions and present only elementary results on the algebraic constructions concerning central extensions of the Lie algebras. We begin several definitions.
A Lie algebra is called abelian if Lie bracket of is trivial; that is, for all , .
Let be an abelian Lie algebra over and a Lie algebra over . An exact sequence of algebra homomorphisms is called a central extension of by , if ; that is, for all and .
Here, we identify with the corresponding subalgebra of . For such a central extension, the abelian Lie algebra is realized as an ideal in and the homomorphism serves to identify with .
For every central extension of the Lie algebras there is a linear map with . Here, is in general not a Lie algebra homomorphism. Consider Then the map (depending on ) always has the following properties: (a) is a skew-symmetric bilinear form;(b).
Moreover, is isometric to as vector spaces by the linear isomorphism Thus, with the Lie bracket on given by for and , the map is a Lie algebra isomorphism.
The Lie bracket on can also be written as
A map with the properties (a) and (b) is called a 2-cocycle on .
With the definitions introduced, the following results hold.
Lemma A.1. Every central extension of by comes from a cocycle . Every cocycle generates a central extension of by .
The first result follows from the comments above. Let be the vector space . The bracket for , and is a Lie bracket if and only if is a cocycle. Hence, with such Lie bracket defines a central extension of by that proves the second assertion of this lemma.
Define the following sets: is a linear subspace of . The above vector spaces are abelian groups, and is the space of 2 cocycles.
Let us consider the Lie algebra and denote the corresponding components for these algebras by , . In order to find central extension of , it is sufficient to construct a central extension for each component.
Lemma A.2. The central extension of by exists; that is, This central extension is defined by the cocycle , , where which coincides with the cocycle of the Witt algebra .
The proof of this lemma completely repeats one as we came to the Virasoro algebra which is a proper central extension of the Witt algebra (see, for more details, ). For the convince, we shortly present this proof again.
First, we establish that ; that is, we have to check that for , , . Using the commutation relations (44), the direct calculations show that That proves the inclusion . Further, we have For we get Hence, for , , .
Define a homomorphism by and let , . Then, for , , , since