Advection by Compressible Turbulent Flows: Renormalization Group Study of Vector and Tracer Admixture
N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, T. Lučivjanský
aa r X i v : . [ n li n . C D ] A p r Advection by Compressible Turbulent Flows:Renormalization Group Study of Vector and TracerAdmixture
N. V. Antonov , N. M. Gulitskiy , M. M. Kostenko , T. Luˇcivjansk´y April 19, 2019 Department of Physics, Saint Petersburg State University,7/9 Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia, Faculty of Sciences, P.J. ˇSaf´arik University, Moyzesova 16, 040 01 Koˇsice, Slovakia.
Abstract
Advection-diffusion problems of magnetic field and tracer field are analyzed us-ing the field theoretic perturbative renormalization group. Both advected fields areconsidered to be passive, i.e., without any influence on the turbulent environment,and advecting velocity field is generated by compressible version of stochastic Navier-Stokes equation. The model is considered in the vicinity of space dimension d = 4and is a continuation of previous work [N.V. Antonov et al., Phys. Rev. E , 033120(2017)]. The perturbation theory near the special dimension d = 4 is constructedwithin a double expansion scheme in y (which describes scaling behavior of the ran-dom force that enters a stochastic equation for the velocity field) and ε = 4 − d .We show that up to one-loop approximation both types of advected fields exhibitsimilar universal scaling behavior. In particular, we demonstrate this statement onthe inertial range asymptotic behavior of the correlation functions of advected fields.The critical dimensions of tensor composite operators are calculated in the leadingorder of ( y, ε ) expansion. Keywords fully developed turbulence, magnetohydrodynamics, advection-diffusion problem, field-theoretic renormalization group, anomalous scaling
Many natural phenomena involve broad range of spatial or time scales. For instance,continuous phase transitions, diffusion-driven systems, population dynamics or turbulentflows provide famous examples [1, 2, 3]. Both from theoretical and practical point of viewturbulence plays a distinguished role. Due to a relatively low value of air viscosity [4, 5, 6],1urbulent flows are much more easily generated than is commonly believed. Despite a vastamount of efforts that has been put into, the fundamental understanding of turbulenceremains unsolved and it is widely regarded as a last unsolved classical problem. Arguably,turbulence exhibits many interesting features, such as scaling behavior, prominent inter-mittency, coherent structures and others [4, 6, 7]. A distinctive quantitative aspect isknown as anomalous scaling, i.e., singular power-law dependence of outer scale L of somestatistical quantities in the inertial-convective range [4, 5]. Its proper investigation requiresa lot of thorough analysis. The general aim of theory is to predict possible macroscopicbehavior of a turbulent fluid and to give a quantitative prediction about characteristicquantities (correlation and structure functions).In astrophysics turbulence plays probably even more important role than in terrestrialevents [8, 9, 10]. Being a mechanism for an explanation of many effects: magnetic dynamoin interior of planets, convective processes in stars, outbreaks of prominences on Sun’ssurface, galaxy formations and others [11, 12, 13, 14, 15, 16], it is clear that mutual interplaybetween turbulent flow and additional advected field is quite common in nature. Well-known model for a theoretical description of magnetohydrodynamic (MHD) is so-calledKazantsev-Kraichnan kinematic model [17]. Its basic premise is to assume that a magneticvector field (later in this article referred just as a vector field) is passively advected byturbulent velocity field with no backward influence to the velocity field (for a generalintroduction to magnetohydrodynamic see, e.g., [18]). Thus, the Kazantzev-Kraichnanmodel can be viewed as a simplified version to the full MHD problem, in which the Lorentzforce is neglected. There are many studies [19, 20] devoted to this problem, mainly becauseit provides a mechanism for a generation of turbulent dynamo [8, 18]. The main pointof criticism on Kazantsev-Kraichnan model is an assumption of the velocity field, whichaccording to this model is simply given by a Gaussian random variable. More appropriateapproach would consider velocity field to be generated by some dynamical mechanism.As a rule, in astrophysical context we are dealing with a compressible fluid ratherthan incompressible [9]. Here, we therefore employ a compressible version of Navier-Stokesequation (NS) for a generation of velocity fluctuations [21, 22] and study its effect on anadvection of magnetic field. Such (compressible) MHD models witnessed a considerablescientific activity in recent years [23, 24, 25, 26, 27, 28, 29, 30].This work is motivated by the previous studies [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]of the incompressible case. Besides advection problem of vector quantity, we consider inthis paper also a problem of passive scalar. In particular, we have in mind tracer fieldadvected by the aforementioned compressible version of stochastic Navier-Stokes eqaution.The reason is that as concrete calculations shows, results for vector and tracer case sharesimilarities. As we will see, concrete expressions for universal quantities will be the sameup to some factor.The investigation of such behavior as anomalous scaling requires a lot of thorough anal-ysis to be carried out. The phenomenon manifests itself in a singular intermittent behaviorof some statistical quantities (correlation and response functions, structure functions, etc.)in the inertial-convective range in the fully developed turbulence regime [4, 6, 7]. As hasbeen mentioned previously, turbulent flows are accompanied by a lack of typical scale.This shares a somewhat formal similarity with a physics related to critical phenomena. Avery useful and computationally effective approach to the problems with many interactingdegrees of freedom on different scales is the field-theoretic renormalization group (RG) ap-2roach which can be subsequently accompanied by the operator product expansion (OPE);see the monographs [1, 2, 42].It is a difficult problem to investigate both the Navier-Stokes equation for a compressiblefluid and passive advection problems by this velocity ensemble. The first relevant discussionand analysis of passive advection emerged a few decades ago for the Kraichnan’s velocityensemble which modelled advection of impurity by incompressible fluid [43, 44, 45]. Furtherstudies developed its more realistic generalizations [32, 33, 34, 35, 36, 37, 38, 39, 40, 41,46, 47, 48, 49, 50, 51] and, in particular, to the compressible case [52, 53, 54, 55, 56, 57,58, 59, 60, 61, 62, 63, 64, 65, 66, 67].As we will see, studied models of compressible fluid reveal intriguing behavior near thespecific space dimension d = 4. Usually, d plays a passive role in advection problems, butsometimes may affect the RG procedure: consideration of compressible fluid near d = 4is very close to analysis of incompressible fluid near special space dimension d = 2. Inthis case an additional divergence appears in the 1-irreducible Green function h v ′ v ′ i ,see [68, 69, 70]. This feature allows us to employ a double expansion scheme, in whichthe formal expansion parameters are y , which describes the scaling behavior of a randomforce, and ε = 4 − d , i.e., a deviation from the space dimension d = 4.The paper is organized as follows. First, we begin with a description of compressiblefluid dynamics in Section 2. Then, in Section 3 we proceed to a description of advectionproblem of passive tracer quantity and magnetic (vector) field, respectively. In Section 4we reformulate studied models into a field-theoretic formalism, which is subsequently an-alyzed in Section 5. Discussion of the fixed points’ structure and related scaling regimesis presented in Section 6. In Section 7 the renormalization of a certain composite fields isconsidered and anomalous exponents are calculated. In Section 8 OPE is applied to thevarious correlation functions. The concluding Section 9 is devoted to the brief discussion. A quantitative parameter that describes intensity of turbulent motion is so-calledReynolds number Re which represents a ratio between inertial and dissipative forces [4, 6,21]. For high enough values of Re ≫ L (input) to microscopic l (dissipative) scales take place. Oneof the microscopic models used for a description of fully developed turbulence in inertialinterval is based on a stochastic version of Navier-Stokes equation [1, 42]. According to itthe dynamics of a compressible fluid is governed by the following equation [21, 61] ρ ∇ t v i = ν [ δ ik ∂ − ∂ i ∂ k ] v k + µ ∂ i ∂ k v k − ∂ i p + f vi , (1)where the operator ∇ t denotes Lagrangian convective derivative ∇ t = ∂ t + v k ∂ k , ρ = ρ ( t, x ) is a fluid density field, v i = v i ( t, x ) is the velocity field, p = p ( t, x ) is the pressurefield, and f vi is the external force, ∂ t = ∂/∂t is a time derivative, ∂ i = ∂/∂x i is i -thcomponent gradient, and ∂ = ∂ i ∂ i is the Laplace operator. Two parameters ν and µ in Eq. (1) are two viscosity coefficients [21]. In this work we use a shorthand notationin which summations over repeated vector indices (Einstein summation convention) arealways implied. In subsequent sections we employ RG method, in which it is necessary todistinguish between unrenormalized (bare) and renormalized parameters. The former wedenote by a subscript “0.” 3et us note two important remarks regarding the physical interpretation of Eq. (1).First, velocity field v i should be regarded as a fluctuating part of the total velocity field. Inother words, it is implicitly assumed that the mean (regular) part of the velocity field hasbeen already subtracted [4, 6]. This point of view reflects philosophy behind the theoryof critical phenomena, where order parameter fluctuates around certain mean value aswell [1, 71]. Second, the random force f vi accounts for two underlying physical processes:a) continuous input of energy, which is needed in order to compensate losses of energy dueto viscous terms in Eq. (1), and b) interactions between fluctuating part of the velocityand the regular mean flow [6, 42].To conclude the theoretical setup of velocity field, Eq. (1) has to be supplemented bytwo equations: a continuity equation ∂ t ρ + ∂ i ( ρv i ) = 0 (2)and an additional relation coming from thermodynamic considerations δp = c δρ. (3)Here, δp = p − p and δρ = ρ − ρ give deviations from the equilibrium (mean) values ofpressure field p and density field ρ , a parameter c is the adiabatic speed of sound.Viscous terms in Eq. (1) proportional to parameters ν and µ characterize dissipativeprocesses, which are predominantly relevant at small spatial scales. Without a continuousinput of energy turbulent processes necessarily fade away and the flow eventually becomesregular. There are several possibilities for theoretical description of energy input [42, 72].It is advantageous to define properties of the random force f vi in frequency-momentumrepresentation h f vi ( t, x ) f vj ( t ′ , x ′ ) = δ ( t − t ′ )(2 π ) d Z k>m d d k D vij ( k )e i k · ( x − x ′ ) , (4)where the delta function ensures Galilean invariance of the model [42]. The integral is in-frared (IR) regularized with a parameter m ∼ L − v , where L v denotes outer scale, i.e., scaleof the largest turbulent eddies [42, 73]. Parameter d denotes dimensionality of space. Inwhat follows d will be considered as a continuous parameter in a dimensional regularization,therefore we write it explicitly and do not immediately insert its realistic three-dimensionalvalue. The kernel function D vij ( k ) reads D vij ( k ) = g ν k − d − y (cid:26) P ij ( k ) + αQ ij ( k ) (cid:27) + g ν δ ij . (5)The non-local term proportional to the charge g is chosen in a power law form thatfacilitates application of RG method. Dimensionless parameter α measures an intensitywith which energy flows into a system via longitudinal modes [16]. Scaling exponent y measures a deviation from a logarithmic behavior achieved for y = 0. Moreover, itis possible to obtain a perturbative expansion in formally small y [1, 73]. Stochastictheory of turbulence is mainly interested in the limiting case y → P ij and Q ij in the momentum space read P ij ( k ) = δ ij − k i k j /k , Q ij ( k ) = k i k j /k ; (6) k = | k | is the wave number. As we will see in Section 5 the presence of local term in (5)is imposed by the renormalizability considerations [74, 75, 76, 77]. Effectively, presence oftwo charges leads to a double expansion scheme in ( y, ε ), where y has been introduced inEq. (5), and ε = 4 − d , i.e., ε gives a deviation from the space dimension d = 4 [51]. In this section we briefly describe differential equations that govern advection of impu-rity fields by some velocity flow: time evolution of magnetic field in so-called Kazantsev-Kraichnan model and dynamics of simple tracer admixture.The inclusion of magnetic field in Kazantsev-Kraichnan model follows a simple physicalconsiderations called magnetohydrodynamic limit [8, 9]. We assume that the mediumis completely neutral at macroscopic scale and that free path of the particles is muchsmaller than Debye length. Therefore, we may neglect the displacement current, which isresponsible for bulk motion of the ions and electrons, and describe our system in the bulkvariables of density, pressure, and mean velocity fields only. From the technical point ofview and RG principles displacement current is IR irrelevant and, therefore, we do not needto preserve it in our model. Taking into account Faraday’s law ∂ t B = − ∇ × E togetherwith a generalized Ohm’s law for a conducting fluid in motion J = σ ( E + v × B ) one getsadvection-diffusion equation ∂ t B − ∇ × ( v × B ) = κ ∇ B . In a similar philosophy toSec. 2 stochastic version then takes the following form ∂ t θ i + ∂ k ( v k θ i − v i θ k ) = κ ∂ θ i + f θi , (7)where θ i is a fluctuating component of total magnetic field, κ is the magnetic diffusion,and we have added stochastic term f θi on the right hand side being the random componentof the curl of current and stemming from intrinsic stochasticity of the magnetic field [42].Detailed exposition of the MHD equation can be found in the literature [8, 9, 18]. Let usnote that in stochastic approach to MHD Eq. (7) should be understood as an equationfor the fluctuating part θ i = θ i ( t, x ) of the total magnetic field B i , i.e., B i = B ( n i + θ i )with B i = B n i and n being a constant background field and a constant unit vector,respectively [19, 20, 78, 79].Random force f θi in Eq. (7) is assumed to be a Gaussian random variable with zeromean and given covariance, h f θi ( t, x ) f θj ( t ′ , x ′ ) i = δ ( t − t ′ ) C ij ( r /L θ ) , r = x − x ′ , (8)where C ij ( r /L θ ) is a function, whose precise functional form is unimportant. It has afinite limit at ( r /L θ ) → r /L θ ) → ∞ . Magnetic field θ i isdivergence-free, which yields an equality between terms ∂ k ( v i θ k ) and ( θ k ∂ k ) v i .In more realistic scenarios there should be an additional Lorentz term in Eq. (1), whichcorresponds to the active advection of magnetic field. This would require presence of theLorentz term v × B ∼ J ∼ ( ∇ × B ) × B , which would affect dynamics of velocity field5nd the resulting model would contain two interconnected stochastic differential equations.However, this is beyond the scope of the present paper. Moreover, it was found that insome special cases the only IR attractive fixed point in full model corresponds to passive(not active) advection of impurity fields [19, 20, 80].Thus, model (7) corresponds to a model of passive advection of magnetic field, whichwe later refer to as a vector model. Related problem can be considered for a case ofscalar quantity θ = θ ( t, x ) which represents the density of some pollutant, temperaturefield, concentration, etc. There are two permissible kinds of passive scalar fields in na-ture: the density field (density of some pollutant) and the tracer field which describes thetemperature or entropy [21]. The advection of a density field is governed by equation ∂ t θ + ∂ i ( v i θ ) = κ ∂ θ + f θ , (9)whereas advection of a tracer field is governed by equation ∂ t θ + ( v i ∂ i ) θ = κ ∂ θ + f θ ; (10)here in both equations κ is the corresponding molecular diffusivity coefficient and f = f ( t, x ) is again a Gaussian random variable with zero mean and given covariance, h f ( t, x ) f ( t ′ , x ′ ) i = δ ( t − t ′ ) C ( r /L ) , r = x − x ′ . (11)The function C in Eq. (11) meets same criteria as function C ij in Eq. (8). For the incom-pressible fluid the density and tracer advection problems are identical since transversalitycondition ∂ i v i = 0 makes expressions ∂ i ( v i θ ) and ( v i ∂ i ) θ equal, but for the compressibleflows differences might appear [81]. The case of density advection was considered earlierin [74, 76]; the case of tracer field is considered here together with vector model. The main aim of this study is to investigate the scaling behavior of various statisticalquantities (Green functions) of the theory near the special space dimension d = 4. In sta-tistical physics we are interested in the macroscopic large-scale behavior that correspondsto the IR range. Our main theoretical tool is the renormalization group theory, which al-lows us to identify scaling regimes and analyse certain composite operators. An importantdifference of the present study with the traditional approaches is a special role of the spacedimension d = 4.Fortunately, despite the obvious differences between the stochastic formulations forvector and tracer fields [compare Eqs. (7) and (10)], there exist some similarities whichallows us to perform their RG analysis at once. In order to derive renormalizable fieldtheory, the stochastic equation (1) has to be divided by ρ , and fluctuations in viscousterms have to be neglected [82]. Further, by using the expressions (2) and (3) the problemcan be recast in the form of two coupled equations: ∇ t v i = ν [ δ ik ∂ − ∂ i ∂ k ] v k + µ ∂ i ∂ k v k − ∂ i φ + f i , (12) ∇ t φ = − c ∂ i v i . (13)Here, a new field φ = φ ( t, x ) has been introduced and it is related to the density fluctuationsvia the relation φ = c ln( ρ/ρ ) [74, 82], and f i = f i ( t, x ) is the external force normalizedper unit mass. 6ccording to the general theorem [1, 3], stochastic problems summarized by Eqs. (7),(12), (13) and Eqs. (10), (12), (13), respectively, are equivalent to the field theoretic modelswith a doubled set of fields and certain De Dominicis-Janssen action functional [83, 84, 85].In the case of Kazantsev-Kraichnan model it is given by a sum of two terms S = S vel + S mag , (14)where S vel describes a velocity part S vel = v ′ i D vij v ′ j v ′ i (cid:20) −∇ t v i + ν ( δ ij ∂ − ∂ i ∂ j ) v j + u ν ∂ i ∂ j v j − ∂ i φ (cid:21) + φ ′ [ −∇ t φ + v ν ∂ φ − c ( ∂ i v i )] (15)with D vij being the correlation function (5). Note that we have introduced a new di-mensionless parameter u = µ /ν > v ν φ ′ ∂ φ with another positivedimensionless parameter v , which is needed to ensure multiplicative renormalizability.Also we employ a condensed notation, in which integrals over the spatial variable x andthe time variable t are implicitly assumed, for instance φ ′ ∂ t φ = R d t R d d x φ ′ ( t, x ) ∂ t φ ( t, x ).The term S mag in the action (14) takes form S mag = 12 θ ′ i D θij θ ′ j + θ ′ k [ − ∂ t θ k − ( v i ∂ i ) θ k + ( θ i ∂ i ) v k + ν w ∂ θ k ] , (16)where for convenience we have introduced new dimensionless parameter w via κ = ν w ,and D θij denotes correlation function (8). On the other hand, advection of the tracer fieldcorresponds to the field-theoretic action S = S vel + S tracer , (17)where S vel is given by Eq. (15) and S tracer reads S tracer = 12 θ ′ D θ θ ′ + θ ′ [ − ∂ t θ − ( v i ∂ i ) θ + ν w ∂ θ ] (18)with D θ being the correlation function (11).In a field-theoretic formulation various stochastic quantities (corresponding to Greenfunctions in quantum field theory) are calculable as functional integrals with a given weightfunctional exp S . Main benefits of such approach are transparency of a perturbation theoryin Feynman graphs and feasibility of the other powerful methods such as renormalizationgroup and operator product expansion [1, 2, 3].It is convenient to express the propagators of the theory in momentum-frequency rep-resentation h v i v ′ j i = h v ′ j v i i ∗ = P ij ( k ) ǫ − + Q ij ( k ) ǫ R − , h v j φ ′ i = h φ ′ v j i ∗ = − i k j R , (19) h v i v j i = P ij ( k ) d f | ǫ | + Q ij ( k ) d f (cid:12)(cid:12)(cid:12) ǫ R (cid:12)(cid:12)(cid:12) , h φv ′ j i = h v ′ j φ i ∗ = − i c k j R , (20) h φφ ′ i = h φ ′ φ i ∗ = ǫ R , h φφ i = c k d f | R | , (21) h v i φ i = h φv i i ∗ = i c d f ǫ k i | R | , (22)7igure 1: Graphical representation of all propagators of the models given by the quadraticpart of the actions (14) and (17).where the symbol z ∗ denotes the complex conjugate of the expression z and the followingabbreviations have been used: d f = g ν k − d − y + g ν , d f = αg ν k − d − y + g ν , ǫ = − i ω + ν k ,ǫ = − i ω + u ν k , ǫ = − i ω + v ν k , R = ǫ ǫ + c k . (23)Graphical representation of propagators is depicted in Fig. 1. Remaining propagators formagnetic admixture and tracer field take the following form, respectively h θ i θ ′ j i = h θ ′ j θ i i ∗ = P ij ( k ) − i ω + κ k , (24) h θθ ′ i = h θ ′ θ i ∗ = 1 − i ω + κ k . (25)There are two additional non-zero propagators h θ i θ j i and h θθ i , but in actual calculationsthey are in fact not needed [62, 63]. Therefore, we do not quote them here. All theremaining propagators are identically zero, i.e., h φ ′ φ ′ i = h v ′ i φ ′ i = h v ′ i v ′ j i = h θ ′ i θ ′ j i = h θ ′ θ ′ i = 0. Self-explanatory graphical representations of non-linearities in studied modelstogether with their vertex factors are depicted in Fig. 2 and Fig. 3.Since the theory of critical phenomena corresponds to limit k →
0, in accordance withgeneral theory all the terms in actional functional (and, as a consequence, in propagators)should have the same IR behavior. From the quantity R in Eqs. (23) it follows that c k ≃ ǫ ǫ . This means, that c is IR significant parameter and, moreover, c ≃ k . Thus, c → c = 0 in Feynman graphs just as a simplest way to performe calculations and giveFigure 2: Graphical representation of all interaction vertices of the model related velocitynon-linearities of the action (15). For brevity we have retained only those momentumarguments that appear in a resulting vertex factors.8igure 3: Graphical representation of all interaction vertices of the model given by theadvection terms derived from action (16) and action (18), respectively.no restriction for the model. Here, we deal with basic feature of the theory; the situationis analogous to well-known ϕ model, where parameter τ = T − T c is IR significant andrequirement τ → Theoretical models (14) and (17) are amenable to a standard loop expansion using well-known Feynman diagrammatic rules [1, 2]. However, as it is often the case, ordinary per-turbation theory is plagued by divergences and these must be properly taken care of. Thehelp comes from renormalization group method, which is considered here in dimensionalregularization within minimal subtraction (MS) scheme.From a practical point of view, theory is renormalized once all 1-particle irreducibleGreen functions Γ (further referred to as 1-irreducible functions) are finite [1, 2]. For dy-namical models two independent scales have to be introduced: the time scale T and thelength scale L . Canonical dimensions of model parameters are found from the require-ment that each term of the action functionals (14) and (17) be dimensionless quantitywith respect to both the momentum and frequency scales separately. We adopt standardnormalization conditions d kk = − d kx = 1 , d ωk = d ωx = 0 , d ωω = − d ωt = 1 , d kω = d kt = 0 . (26)Then, the overall (total) canonical dimension of any quantity F is described by two num-bers, the frequency dimension d ωF and the momentum dimension d kF , and given quantity F therefore scales as [ F ] ∼ [ T ] − d ωF [ L ] − d kF . Based on d kF and d ωF , the total canonical dimension d F = d kF + 2 d ωF can be introduced,which in the renormalization theory of dynamic models plays the same role as the conven-tional (momentum) dimension does in static problems [1, 3]. Assuming quadratic dispersionlaw ω ∼ k brought about a scaling relation between time and spatial scale which ensuresthat all the viscosity and diffusion coefficients in the model are dimensionless.The canonical dimensions for the velocity part of the model (15) are given in Tab. 1,whereas parameters of the magnetic and tracer part are given in Tab. 2. From Tabs. 1and 2 it directly follows that the coupling constants g ∼ [ L ] − y and g ∼ [ L ] − ε becomesimultaneously dimensionless at y = ε = 0 what corresponds to the logarithmic theory.Since we use MS scheme the ultraviolet (UV) divergences in the Green functions manifestthemselves as poles in y , ε and their linear combinations.9able 1: Canonical dimensions of the fields and parameters entering field-theoretic ac-tion (15) for velocity fluctuations. F v ′ i v i φ ′ φ ν , ν c , c g g u , v , u , v , g , g , αd kF d + 1 − d + 2 − − − y − d d ωF − − d F d − d − y − d θ i (magnetic field) and θ (tracer field), respectively. F θ ′ i , θ ′ θ i , θ κ , κ w , wd kF d − d ωF / − / d F d + 1 − δ Γ = d + 2 − X Φ N Φ d Φ , (27)where N Φ is the number of the given type of field entering the function Γ, d Φ is thecorresponding total canonical dimension of field Φ, and the summation runs over all typesof the fields Φ entering the 1-irreducible function Γ, see [1].Superficial UV divergences whose removal requires counterterms can be present only inthose functions Γ for which the formal index of divergence δ Γ is a non-negative integer. Adimensional analysis should be augmented by the several additional considerations. Theyare summarized in the previous works [62, 63, 74] and we do not repeat them here. Thecrucial property of studied models is Galilean invariance, whose direct consequence isthat fields v ′ i , v i , θ i and θ are not renormalized. Thus, models under considerations withthe actions (14) and (17) are renormalizable and the only graphs that are divergent andneeded to be considered are two-point Green functions. For a velocity part (15), thefollowing graphs should be analyzed: . (28)For an advected part (vector or tracer field with appropriate changes in propagators andvertices) we have one additional graph: . (29)Remaining graphs are either UV finite or the Galilean invariance prohibits their appearance.The calculation of the divergent parts of Feynman graphs proceeds in a straightforwardfashion and details can be found in [62, 63, 74].10lthough, the intermediate steps involved in calculation of the graph presented in (29)differs for vector and tracer cases, the resulting divergent part of the graph is the sameand reads D = S d d p ν (cid:26) − d w (cid:20) g y (cid:18) µm (cid:19) y + g ε (cid:18) µm (cid:19) ε (cid:21) − u − wu ( u + w ) (cid:20) αg y (cid:18) µm (cid:19) y + g ε (cid:18) µm (cid:19) ε (cid:21)(cid:27) , (30)where S d = S d / (2 π ) d with S d = 2 π d/ / Γ( d/
2) being the surface area of the unit sphere inthe d -dimensional space; Γ( x ) is Euler’s Gamma function. Parameter µ is a renormalizationmass employed in minimal subtraction scheme [1, 3]. For vector admixture expression (30)should be multiplied by a projection operator P ij ( p ) due to a vector nature of magneticfield θ i .From the result (30) we easily derive renormalization constant Z κ for the parameter κ related to the advected fields [see Eq. (16)] and the anomalous dimension γ κ : Z κ = 1 − g dwy (cid:20) d −
11 + w + α ( u − w ) u ( u + w ) (cid:21) − g dwε (cid:20) d −
11 + w + ( u − w ) u ( u + w ) (cid:21) , (31) γ κ = g dw (cid:20) d −
11 + w + α ( u − w ) u ( u + w ) (cid:21) + g dw (cid:20) d −
11 + w + ( u − w ) u ( u + w ) (cid:21) . (32) Underlying idea of RG approach [1, 2, 3] is embodied in a relation between the initial andrenormalized action functionals S (Φ , e ) = S R ( Z Φ Φ , e, µ ), where e denotes the completeset of bare parameters, e is the set of their renormalized counterparts, and Φ stands fora complete set of fields { ϕ } and their response counterparts { ϕ ′ } . This relation can beconverted into a differential form (cid:26) D RG + N ϕ γ ϕ + N ϕ ′ γ ϕ ′ (cid:27) G R ( e, µ, . . . ) = 0 , (33)where G = h Φ · · · Φ i is an arbitrary Green function of the theory; N ϕ and N ϕ ′ are thenumbers of entering fields ϕ and ϕ ′ in G , the ellipsis commonly denotes remaining argu-ments of G (such as spatial and time variables), D RG is the operation e D µ expressed in therenormalized variables, and e D µ is the differential operation µ∂ µ at fixed bare parameters e . For the present model it takes the form D RG = D µ + X β g ∂ g − γ ν D ν − γ c D c , (34)where the sum runs over the set of all charges { g , g , u, v, w } , ν and c are viscosity andadiabatic speed of sound, respectively, and differential operator D x = x∂ x has been intro-duced. The γ and β -functions are defined as γ F = e D µ ln Z F and β g = e D µ g . The latter onescan be expressed in terms of anomalous dimensions: β g = g ( − y − γ g ) , β g = g ( − ε − γ g ) , β u = − uγ u , β v = − vγ v , β w = w ( γ ν − γ κ ) . (35)The last expression follows from the introduced definition of the charge w in Eq. (16).11acroscopic scaling regimes are naturally identified with the IR attractive (“stable”)fixed points g ∗ ≡ { g ∗ , g ∗ , u ∗ , v ∗ , w ∗ } , whose coordinates are found from the conditions [1, 3] β g ( g ∗ ) = β g ( g ∗ ) = β u ( g ∗ ) = β v ( g ∗ ) = β w ( g ∗ ) = 0 . (36)This statement is a direct consequence of the invariant couplings behavior. Let us considera set of invariant couplings g i = g i ( s, g ) with the initial data g i | s =1 = g i , where s = k/µ .IR (macroscopic) asymptotic behavior is obtained in the limit s →
0. An evolution ofinvariant couplings satisfies flow equations D s g i = β i ( g j ) , and in a limit s → g i ( s, g ∗ ) ∼ = g ∗ + const × s ω i . A set { ω i } contains all eigenvalues of thematrix Ω ij = ∂β i /∂g j | g = g ∗ . (37)The existence of IR attractive solutions of the RG equations corresponds to such fixedpoints for which the matrix (37) is positive definite. These fixed points are then naturalcandidates for macroscopically observable scaling regimes.In the double expansion approach we have used, the character of the IR behaviordepends on the mutual relation between y and ε . In work [74] the velocity part of thesystem (35), which don’t include β w , was thoroughly analyzed. The net result of theanalysis is a prediction of three IR attractive fixed points. The fixed point FPI (the trivialor Gaussian point) is stable if y , ε < g ∗ = 0 , g ∗ = 0 , both u ∗ and v ∗ are arbitrary . (38)The fixed point FPII, which is stable if ε > y < ε/
2, has the coordinates g ∗ = 0 , g ∗ = 8 ε , u ∗ = v ∗ = 1 . (39)The fixed point FPIII, which is stable if y > y > ε/
2, has the coordinates g ∗ = 16 y (2 y − ε )9[ y (2 + α ) − ε ] , g ∗ = 16 αy y (2 + α ) − ε ] , u ∗ = v ∗ = 1 . (40)The crossover between the two nontrivial points (39) and (40) takes place across the line y = 3 ε/
2, which is in accordance with results of [53].Substituting obtained values of u ∗ and v ∗ together with d = 4 we obtain for the charge w the following beta function β w = w − w (1 + w ) (cid:20) g (6 + 2 α + 9 w + 3 w ) + g (3 w + 9 w + 8) (cid:21) . (41)Note that this result agrees with previous works for the passive scalar case [74] and vectorcase [87] as well. The expression in the square brackets in Eq. (41) is always positivefor physically permissible values g > , g > , w > α >
0. Therefore, only onenontrivial solution for the fixed point w ∗ = 1 exists. It is straightforward to prove that ∂ w β w > w charge. Once scaling regimes are found, the critical dimensions for variousquantities F can be calculated via the relation∆[ F ] = d kF + ∆ ω d ωF + γ ∗ F , (42)12here d ωF is the canonical frequency dimension, d kF is the momentum dimension, γ ∗ F isthe anomalous dimension at the fixed point (FPII or FPIII), and ∆ ω = 2 − γ ∗ ν is thecritical dimension of frequency. Using Eq. (42) the critical dimensions of advected fieldsare obtained for the fixed points FPII and FPIII:∆ θ i = − ε/ , ∆ θ ′ i = d + 1 − ε/ , (43)∆ θ i = − y/ , ∆ θ ′ i = d + 1 − y/ . (44) From experimental point of view much more relevant than critical dimensions are quan-tities known as correlation and structure functions. In field-theoretic framework they areusually represented by certain composite operators. A local composite operator is a mono-mial or polynomial constructed from the primary fields and their finite-order derivatives ata single space-time point. In the Green functions with such objects, new UV divergencesarise due to the coincidence of the field arguments. Their removal requires an additionalrenormalization procedure [1, 3].It is not common to consider in one paper both tracer and vector admixtures. Thereason is that the expressions for tracer case are completely analogous to the vector onesin the part connected with composite fields θ and can be easily obtained by consideringoperator ∂ i θ instead of θ i in all formulas quoted below (note, that propagators (24), (25)and vertices depicted in Fig. 3 still differs for tracer and vector cases). This is a consequenceof the fact that instead of density case for tracer admixture the operators θ n are UV finiteand, therefore, we should consider operators ∂ i θ which contain derivatives. This is why bothdensity and tracer are scalar fields, but yield different expressions. Moreover, considerationof tracer field is much more closer to the vector case rather to the density one.For brevity, hereinafter we write all expressions related to operators θ i or ∂ i θ (for vectoror tracer cases, respectively) below only for the vector case and use notation θ i for vectorfield. In the case of vector admixture we should focus on the irreducible tensor operatorsof the form F ( n,l ) i ...i l = θ i · · · θ i l ( θ j θ j ) s + . . . , (45)where l is the number of the free vector indices (the rank of the tensor) and n = l + 2 s is the total number of the fields θ i entering the operator. The ellipsis stands for thesubtractions with the Kronecker’s delta symbols that make the operator irreducible (sothat a contraction with respect to any pair of the free tensor indices vanish). For instance, F (2 , ij = θ i θ j − δ ij d ( θ k θ k ) . (46)For a theoretical analysis, it is convenient to contract the tensors (45) with an arbitraryconstant vector λ = { λ i } . The resulting scalar operator then takes the form F nl = ( λ i w i ) l ( w i w i ) s + . . . , w i ≡ θ i , (47)where the subtractions, denoted by the ellipsis, necessarily contain factors of λ = λ i λ i .In order to calculate the critical dimension of an operator, we have to renormalize it.The operators (45) can be treated as multiplicatively renormalizable, F nl = Z nl F nlR , with13ertain renormalization constants Z ( n,l ) [62, 63]. The counterterm to F nl must have thesame rank as the operator itself. It means that the terms containing λ should be excludedsince the contracted fields w i w i , which naturally appear in such terms, reduce the numberof free indices. The renormalization constants Z nl are determined by the finiteness of the1-irreducible Green function Γ nl ( x ; θ ), where x = ( t, x ) is a functional argument of theoperator θ . In the one-loop approximation we have a diagrammatic equationΓ nl ( x ; θ ) = F nl + 12 , (48)where numerical factor 1 / V ( x ; x , x ) = ∂ F nl ∂w i ∂w j δ ( x − x ) δ ( x − x ) . (49)The differentiation yields ∂ F nl /∂w i ∂w j = 2 s ( w ) s − ( λw ) l (cid:2) δ ij w + 2( s − w i w j (cid:3) + l ( l − w ) s × ( λw ) l − λ i λ j + 2 ls ( w ) s − ( λw ) l − ( w i λ j + w j λ i ) , (50)where w = w k w k , ( λw ) = λ k w k and substitution w i → θ i is assumed. Two more factors w p w r are attached to the bottom of the graph due to the derivatives stemming from thevertices θ ′ i ( v k ∂ k ) θ i .The UV divergence is logarithmic and one can set all the external frequencies andmomenta equal to zero. Since propagators (24), (25) and vertices depicted in Fig. 3 aredifferent for vector and tracer cases, the core of the graph also differs for these two cases.For tracer field it reads b D prij = Z d ω π Z k>m d d k (2 π ) d k i k j D vpr ( ω, k ) 1 ω + w ν k . (51)Here, the first factor comes from the derivatives in (50), D vpr is the velocity correlationfunction [see Eq. (5)], and the last factor comes from the two propagators h θ ′ i θ j i . Theindices i and j should be later contracted with expression (50), the indices p and r withexternal fields w p and w r denoted by “legs” of the graph.For vector field the core of the graph takes the form e D xyij = Z d ω π Z k>m d d k (2 π ) d P ai ( k ) P bj ( k ) D vpr ( ω, k ) 1 ω + w ν k V apx ( k ) V bry ( k ) , (52)where P ai ( k ) and P bj ( k ) follows from propagators (25), V apx ( k ) and V bry ( k ) are two vectorvertices (see Fig. 3), and D vpr is the velocity correlation function. For brevity we do notdraw here picture for the graph with explicitly written vector indices, but they may beeasily restored from the above written expression.14fter the integration, combining all the factors, contracting the tensor indices andexpressing the result in terms of n = l + 2 s and l , one obtains= − F nl wd ( d + 2) (cid:20) Q (1 + w ) (cid:18) g y + g ε (cid:19) + Q u ( u + w ) (cid:18) αg y + g ε (cid:19)(cid:21) , (53)where Q = − n ( n + d )( d −
1) + l ( d + 1)( d + l − ,Q = − n (3 n + d −
4) + l ( d + l −
2) (54)for tracer case and Q = − n ( n + d )( d −
1) + l ( d + 1)( d + l − ,Q = − n ( n + nd − d )( d −
1) + l ( d + l −
2) (55)for vector case . Finally, expression (48) readsΓ nl ( x ; θ ) = F nl ( x ) (cid:26) − wd ( d + 2) (cid:20) Q (1 + w ) (cid:18) g y + g ε (cid:19) + Q u ( u + w ) (cid:18) αg y + g ε (cid:19)(cid:21)(cid:27) . (56)The renormalization constants Z nl calculated in the MS scheme thus take the form Z nl = 1 − dw ( d + 2) (cid:20) Q w (cid:18) g y + g ε (cid:19) + Q u ( u + w ) (cid:18) αg y + g ε (cid:19)(cid:21) (57)and for the corresponding anomalous dimensions we get γ nl = 14 dw ( d + 2) (cid:26) Q w ( g + g ) + Q u ( u + w ) ( αg + g ) (cid:27) . (58)Now in order to evaluate the critical dimensions of the operators F nl one needs tosubstitute the coordinates of the fixed points into the expression (58) and then use therelation (42). For the fixed point FPII the critical dimension is∆ nl = n ε + Q + Q ε ; (59)for the fixed point FPIII it is∆ nl = n y + y Q ( αy + 2 y − ε ) + 3 αQ ( y − ε )9[ y (2 + α ) − ε ] . (60)Both expressions (59) and (60) might be affected by higher order corrections in y and ε . Inspection of expressions (59) and (60) reveals that increasing value of n leads to ainfinite set of operators with negative critical dimensions. Their spectra are unboundedfrom below, see Appendix A in [63]. Note that in previous study [63] there are misprints in expressions (5.18) for quantities Q and Q .Right expressions are Eq. (55) written above and Eq. (5.7) in [87]. d = 3:∆ nl = ny y ( Q + αQ )6( d − d + 2) , (61)where Q and Q coincide with those given in Eqs. (54) and (55) for tracer and vectorcases, respectively. Expression (61) at d = 4 reads∆ nl (cid:12)(cid:12)(cid:12)(cid:12) d =4 = ny y ( Q + αQ )108 . (62)Expanding the expression (60) in y at fixed value ε = 1 (which corresponds to d = 3) yields∆ nl = ny y ( Q + αQ )108 + O ( y ) . (63)From the expressions (62) and (63) it follows that the expression (60), obtained as a resultof the double y and ε expansion near d = 4, may be considered as a certain partial infiniteresummation of the ordinary y expansion. This resummation significantly improves thesituation at large α : now we do not have the pathology when the critical dimensions ∆ nl are linear in α and, therefore, grow with α without a bound. Our main interest are pair correlation functions constructed from the composite op-erators, whose unrenormalized counterparts have been defined in Eq. (45). For Galileaninvariant equal-time functions we can write the representation h F mi ( t, x ) F nj ( t, x ′ ) i ≃ µ d F ν d ωF ( µr ) − ∆ mi − ∆ nj ζ mi ; nj ( mr, c ( r )) , (64)where r = | x − x ′ | and c ( r ) is effective speed of sound. Its limiting behavior is c ( r ) = c ( µr ) ∆ c µν → ( c (0) for non-local regime FPIII; c ( ∞ ) for local regime FPII , (65)see [74].Expression (64) is valid in the asymptotic limit µr ≫
1. Further, the inertial-convectiverange corresponds to the additional restriction mr ≪
1. The behavior of the functions ζ at mr → x and x ′ with acondition x − x ′ → F mi ( t, x ) F nj ( t, x ′ ) ≃ X F C F ( mr ) F (cid:18) t, x + x ′ (cid:19) , (66)where functions C F are regular in their arguments and a given sum runs over all permissiblelocal composite operators F allowed by RG and symmetry considerations. Taken intoaccount (64) and (66) in the limit mr → ζ ( mr ) ≃ X F A F ( mr )( mr ) ∆ F . (67)16ingularities for mr → h F ( p, F ( k, i constructed from scalaroperators of the type (45), one can observe that the leading contribution to the expansionis determined by the operator F ( n, with n = p + k from the same family. Therefore, inthe inertial range these correlation functions acquire the form h F ( p, ( t, x ) F ( k, ( t, x ′ ) i ∼ r − ∆ p − ∆ k +∆ n . (68)The inequality ∆ n < ∆ p +∆ k , which follows from both explicit one-loop expressions (59)and (60), indicates, that the operators F ( n, demonstrate a multifractal behavior for bothregimes FPII and FPIII; see [88, 89].A direct substitution of d = 4 leads to the following prediction for a critical dimensions∆ nl = n ∆ θ + γ ∗ nl = ( − n + nε + ( Q + Q ) ε for FPII , − n + ny + Q y + Q αy ( y − ε )36[ y (2+ α ) − ε ] for FPIII , (69)where we have Q | d =4 = − n ( n + 4) + 5 l ( l + 2) , Q | d =4 = − n + l ( l + 2) (70)for tracer case and Q | d =4 = − n ( n + 4) + 5 l ( l + 2) , Q | d =4 = − n (5 n −
4) + l ( l + 2) (71)for vector case. From Eq. (69) and Eqs. (70), (71) it follows that for fixed n a kind of anhierarchy present with respect to the index l , which measures the “degree of anisotropy:” ∂ ∆ nl ∂l > . (72)In other words, the higher l the less important contribution. The most relevant is givenby the isotropic shell with l = 0. This is in accordance with previous studies [62, 63, 87]and hypothesis about restoration of isotropy and symmetries of turbulent motion in thestatistical sence in the depth of inertial interval [90]. In the present paper the advection problem of the vector and tracer field by the Navier-Stokes velocity ensemble have been examined. The fluid was assumed to be compressibleand the space dimension was close to d = 4. This specific case allows us to take intoconsideration one more divergent function, namely h v ′ v ′ i , and construct the doubleexpansion in y and ε = 4 − d . This leads to richer results in comparison with naiveconsideration of the system near physical dimension d = 3.Using renormalization group technique two nontrivial IR stable fixed points were iden-tified and, therefore, the critical behavior in the inertial range demonstrates two differentnontrivial regimes depending on the relation between the exponents y and ε . The ex-pressions for the critical exponents of the advected fields θ were obtained in the leadingone-loop approximation. 17n order to find the anomalous exponents of the structure functions, the compositefields (45) were renormalized. The critical dimensions of them were evaluated. Moreover,operator product expansion allowed us to derive the explicit expressions for the criticaldimensions of the structure functions.The existence of the anomalous scaling (singular power-like dependence on the integralscale L ) in the inertial-convective range was established for both possible scaling regimes.From the leading order (one-loop) calculations it follows that the main contribution into theOPE is given by the isotropic term corresponding to l = 0, where l is the rank of the tensorand serves as a degree of the anisotropy; all other terms with l ≥ α (purely potentialrandom force). In contrast to analysis near d = 3, in the present case the anomalousdimensions of the composite operators (59) and (60) do not grow with α without a bound.This is a consequence of eliminating the poles in ε near d = 4, which leads to a significantimprovement of calculated expressions for critical dimensions near physical value d = 3.Expression (60) obtained in this study may be considered as an example of infinite re-summation of ordinary y expansion. It works well at large α being not expanded in y ,and the first term of this expansion coincides with the answer presented earlier in [63]; seeexpressions (62) – (63).Regarding future tasks, it would be interesting to go beyond the one-loop approximationand to analyze the behavior more accurately. Another very important issue to be furtherinvestigated is to have a closer look at the both scalar and vector active fields, i.e., toconsider a back influence of the advected fields to the turbulent environment flow. Acknowledgments
T. L. gratefully acknowledge financial support provided by VEGA grant No. 1 / / References [1] A. N. Vasil’ev,
The Field Theoretic Renormalization Group in Critical Behavior The-ory and Stochastic Dynamics (Boca Raton, Chapman Hall/CRC, 2004).[2] U. T¨auber,
Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, New York, 2014).183] J. Zinn-Justin,
Quantum Field Theory and Critical Phenomena (4th edition, OxfordUniversity Press, Oxford, 2002).[4] U. Frisch,
Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press,Cambridge, 1995).[5] A. S. Monin, A. M. Yaglom,
Statistical Fluid Mechanics:Vol 2 (MIT Press, Cambridge,1975).[6] P. A. Davidson,
Turbulence: an introduction for scientists and engineers (2th edition,Oxford University Press, Oxford, 2015).[7] G. Falkovich, K. Gaw¸edzki and M. Vergassola, Rev. Mod. Phys. , 913 (2001).[8] D. Biskamp, Magnetohydrodynamic Turbulence (Cambridge Univ. Press, Cambridge,2003).[9] S. N. Shore,
Astrophysical Hydrodynamics:An Introduction (Wiley-VCH VerlagGmbH& KGaA, Weinheim, 2007).[10] E. Priest,
Magnetohydrodynamics of the sun (Cambridge University Press, 2014).[11] A. Pouquet, U. Frisch, J. L´eorat, J. Fluid Mech. , 321 (1976).[12] C.-Y. Tu, E. Marsch, Space Sci. Rev. , 1 (1995).[13] S. A. Balbus, J. F. Hawley, Rev. Mod. Phys. , 1 (1998).[14] G. Chabrier, Publ. Astron. Soc. Pac. , 763 (2003).[15] B. G. Elemegreen, J. Scalo, Annu. Rev. Astron. Astrophys. , 211 (2004).[16] C. Federrath, Mon. Notices Royal Astron. Soc. , 1245 (2013).[17] N. V. Antonov, J. Phys. A: Math. Gen. , 7825 (2006).[18] H. K. Moffatt, Magnetic field generation in electrically conducting fluids (CambridgeUniversity Press, Cambridge 1978).[19] J. D. Fournier, P. L. Sulem, A. Pouquet, J. Phys. A , 1393 (1982).[20] L. Ts. Adzhemyan, A. N. Vasil’ev, and M. Gnatich, Theor. Math. Phys. (2), 777(1985).[21] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959).[22] P. Sagaut, C. Cambon,
Homogeneous Turbulence Dynamics (Cambridge UniversityPress, 2008).[23] J. Kim, and D. Ryu, Astrophys. J. , L45 (2005).[24] V. Carbone, R. Marino, L. Sorriso-Valvo, A. Noullez, and R. Bruno, Phys. Rev. Lett. , 061102 (2009). 1925] F. Sahraoui, M. L. Goldstein, P. Robert, Yu. V. Khotyainstsev, Phys. Rev. Lett. ,231102 (2009).[26] H. Aluie, and G. L. Eyink, Phys. Rev. Lett. , 081101 (2010).[27] S. Galtier, and S. Banerjee, Phys. Rev. Lett. , 134501 (2011).[28] S. Banerjee, and S. Galtier, Phys. Rev. E , 013019 (2013).[29] S. Banerjee, L. Z. Hadid, F. Sahraoui, and S. Galtier, Astrophys. J. Lett. , L27,(2016).[30] L. Z. Hadid, F. Sahraoui, and S. Galtier, Astrophys. J. , 9 (2017).[31] P. S. Iroshnikov, Sov. Astron. , 566 (1964).[32] N. V. Antonov and N. M. Gulitskiy, Lecture Notes in Comp. Science, ,128 (2012).[33] N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E , 065301(R) (2012).[34] N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E , 039902(E) (2013).[35] E. Jurˇciˇsinova and M. Jurˇciˇsin, J. Phys. A: Math. Theor., , 485501 (2012).[36] E. Jurˇciˇsinova and M. Jurˇciˇsin, Phys. Rev. E , 016306 (2008).[37] E. Jurˇciˇsinova, M. Jurˇciˇsin and R. Remecky, Phys. Rev. E , 046302 (2009).[38] N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E , 013002 (2015).[39] N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E , 043018 (2015).[40] N. V. Antonov and N. M. Gulitskiy, AIP Conf. Proc. , 100006 (2016).[41] N. V. Antonov and N. M. Gulitskiy, EPJ Web of Conf. , 02008 (2016).[42] L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil’ev: The Field Theoretic Renormaliza-tion Group in Fully Developed Turbulence (Gordon & Breach, London, 1999).[43] R.H. Kraichnan, Phys. Fluids , 945 (1968).[44] K. Gaw¸edzki and A. Kupiainen, Phys. Rev. Lett. , 3834 (1995);D. Bernard, K. Gaw¸edzki, and A. Kupiainen, Phys. Rev. E , 2564 (1996);M. Chertkov and G. Falkovich, Phys. Rev. Lett. , 2706 (1996)[45] L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, Phys. Rev. E , 1823 (1998).[46] N. V. Antonov, Phys. Rev. E , 6691 (1999).[47] E. Jurˇciˇsinova and M. Jurˇciˇsin, Phys. Rev. E , 063009 (2015).[48] N. V. Antonov, A. Lanotte, and A. Mazzino, Phys. Rev. E , 6586 (2000);N. V. Antonov and N. M. Gulitskiy, Theor. Math. Phys., (1), 851 (2013).2049] H. Arponen, Phys. Rev. E, , 056303 (2009).[50] E. Jurˇciˇsinova, M. Jurˇciˇsin, M. Menkyna, EPJ B , 313 (2018).[51] M. Hnatiˇc, J. Honkonen, T. Luˇcivjansk´y, Acta Physica Slovaca , 69 (2016).[52] L. Ts. Adzhemyan, N. V. Antonov, J. Honkonen, and T. L. Kim, Phys. Rev. E ,016303 (2005).[53] N. V. Antonov, Phys. Rev. Lett. , 161101 (2004).[54] N. V. Antonov, N. M. Gulitskiy, and A. V. Malyshev, EPJ Web of Conf. , 04019(2016).[55] E. Jurˇciˇsinova, M. Jurˇciˇsin, R. Remecky, Phys. Rev. E , 033106 (2016).[56] E. Jurˇciˇsinova, M. Jurˇciˇsin, M. Menkyna, Phys. Rev. E , 053210 (2017).[57] M. Vergassola and A. Mazzino, Phys. Rev. Lett. , 1849 (1997).[58] A. Celani, A. Lanotte, and A. Mazzino, Phys. Rev. E R1138 (1999).[59] M. Chertkov, I. Kolokolov, and M. Vergassola, Phys. Rev. E. , 5483 (1997).[60] L. Ts. Adzhemyan, N. V. Antonov, Phys. Rev. E , 7381 (1998).[61] N. V. Antonov, M. Yu. Nalimov and A. A. Udalov, Theor. Math. Phys. , 305(1997).[62] N. V. Antonov and M. M. Kostenko, Phys. Rev. E , 063016 (2014).[63] N. V. Antonov and M. M. Kostenko, Phys. Rev. E , 053013 (2015).[64] M. Hnatich, E. Jurˇciˇsinova, M. Jurˇciˇsin, and M. Repaˇsan, J. Phys. A: Math. Gen. ,8007 (2006).[65] V. S. L’vov and A. V. Mikhailov, Preprint No. 54, Inst. Avtomat. Electron., Novosi-birsk (1977).[66] I. Staroselsky, V. Yakhot, S. Kida, and S. A. Orszag, Phys. Rev. Lett., , 171 (1990).[67] S. S. Moiseev, A. V. Tur, and V. V. Yanovskii, Sov. Phys. JETP , 556 (1976).[68] J. Honkonen and M. Yu. Nalimov, Z. Phys. B , 297 (1996).[69] L. Ts. Adzhemyan, J. Honkonen, M. V. Kompaniets, A. N. Vasil’ev, Phys. Rev. E (3), 036305 (2005).[70] L. Ts. Adzhemyan, M. Hnatich and J. Honkonen, Eur. Phys. J B , 275 (2010).[71] A. Z. Patashinskii, V. L. Pokrovskii, Fluctuation Theory of Phase Transitions (Perg-amon Press, Oxford, 1979). 2172] D. Forster, D. R. Nelson, M. J. Stephen, Phys. Rev. A , 732 (1977).[73] L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, Sov. Phys. JETP , 733(1989).[74] N.V. Antonov, N. M. Gulitskiy, M. M. Kostenko, T. Luˇcivjansk´y, Phys. Rev. E ,033120 (2017).[75] N.V. Antonov, N. M. Gulitskiy, M. M. Kostenko, T. Luˇcivjansk´y, EPJ Web of Conf. , 05006 (2016).[76] N.V. Antonov, N. M. Gulitskiy, M. M. Kostenko, T. Luˇcivjansk´y, EPJ Web of Conf. , 10003 (2017).[77] N.V. Antonov, N. M. Gulitskiy, M. M. Kostenko, T. Luˇcivjansk´y, EPJ Web of Conf. , 07044 (2017).[78] N. V. Antonov, A. Lanotte, and A. Mazzino, Phys. Rev. E , 6586 (2000).[79] Ye Zhou, Phys. Rep. , 1 (2010).[80] M. K. Nandy and J. K. Bhattacharjee, J. Phys. A: Math. Gen., , 2621 (1998).[81] N. V. Antonov, Physica D , 370 (2000).[82] D. Yu. Volchenkov and M. Yu. Nalimov, Theor. Math. Phys. (3), 307 (1996).[83] H. K. Janssen, Z. Phys. B: Condens. Matter , 377 (1976).[84] C. De Dominicis, J. Phys. Colloq. France , C1-247 (1976).[85] H. K. Janssen, Dynamical Critical Phenomena and Related Topics , Lect. Notes Phys. , (Springer, Heidelberg, 1979).[86] L. Ts. Adzhemyan, M. Yu. Nalimov, and M. M. Stepanova, Theor. Math. Phys. ,971 (1995).[87] N. V. Antonov, M. Hnatich, J. Honkonen, and M. Jurˇciˇsin, Phys. Rev. E , 046306(2003).[88] B. Duplantier and A. Ludwig, Phys. Rev. Lett. , 247 (1991).[89] G. L. Eyink, Phys. Lett. A , 355 (1993).[90] N.V. Antonov, N. M. Gulitskiy, M. M. Kostenko, A. V. Malyshev, Phys. Rev. E97