Instantons in a Lagrangian model of turbulence
Leonardo S. Grigorio, Freddy Bouchet, Rodrigo M. Pereira, Laurent Chevillard
aa r X i v : . [ phy s i c s . f l u - dyn ] A ug Instantons in a Lagrangian model of turbulence
L S Grigorio , , F Bouchet , R M Pereira , and L Chevillard Univ. Lyon, École Normale Supérieure de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France, Centro Federal de Educação Tecnológica Celso Suckow da Fonseca,Av. Governador Roberto Silveira 1900, Nova Friburgo, 28635-000, Brazil and CAPES Foundation, Ministry of Education of Brazil, Brasília/DF 70040-020, Brazil
Abstract
The role of instantons is investigated in the Lagrangian model for the velocity gradient evolutionknown as the Recent Fluid Deformation (RFD) approximation. After recasting the model into the path-integral formalism, the probability distribution function (pdf) is computed along with the most probablepath in the weak noise limit through the saddle-point approximation. Evaluation of the instanton solutionis implemented numerically by means of the iteratively Chernykh-Stepanov method. In the case of thelongitudinal velocity gradient statistics, due to symmetry reasons, the number of degrees of freedom canbe reduced to one, allowing the pdf to be evaluated analytically as well, thereby enabling a prediction ofthe scaling of the moments as a function of Reynolds number. It is also shown that the instanton solutionlies in the Vieillefosse line concerning the RQ -plane. We illustrate how instantons can be unveiled in thestochastic dynamics performing a conditional statistics. . INTRODUCTION This paper aims at obtaining the stationary probability distribution function and large fl uc-tuations of a stochastic model of turbulence proposed by Chevillard and Meneveau [1]. Themodel, known as Recent Fluid Deformation (RFD) approximation, consists in a set of stochasticdi ff erential equations describing the evolution of the eight degrees of freedom of the velocitygradient tensor of a fl uid particle along its Lagrangian trajectory in an incompressible fl ow.Large deviations of the velocity gradient in turbulent fl ows are associated with high dissipationrates and enstrophy and are crucial to the understanding of intermittency phenomena - a topicof intense research in turbulence. In order to evaluate the pdfs of the velocity gradients (andalso the probability of large fl uctuations) we made use of the iterative numerical procedure ofChernykh-Stepanov [2] which amounts to solving the saddle-point equations that minimize theaction, providing this way the most probable path leading to a given fl uctuation, which will berefered to as the instanton. In the case of the longitudinal velocity gradient, due to symmetryreasons, the number of degrees of freedom can be reduced to one, allowing the pdf to be ob-tained analytically as well. These analytical probability distribution functions (pdfs) obtainedare in excellent agreement with the numerical ones obtained by numerical integration of thestochastic di ff erential equations. Another result is that the instanton lies along the Vieillefosseline in the so-called RQ -plane. For the longitudinal velocity gradient, this instanton approachgives unprecedented prediction for the pdf tail and for the dynamics of the optimal path, alongwith a prediction of how the moments scale with the Reynolds number. That gives us a theoret-ical approach to the dynamics leading to these rare events, and thus intermittency. This point isthe main originality of this work.Of central interest in turbulence is the behavior of small scales statistics. More speci fi cally,scaling and universality at small scales of motion in turbulent fl ows is a long standing problem[3]. Due to the intense fl uctuations within small scales, large deviations of the velocity fi elddi ff erences are very pronounced for high values of Reynolds number. These large excursions ofthe velocity gradient are apparent in the pdf, where drastic departures from gaussian behaviorare manifest - and also termed intermittency.It is clear that a theory of turbulence capable of explaining intermittency and the scaling ofhigh order structure functions must rely on a deep understanding of the dynamics of the smallscales. A natural candidate to probe such scales is the velocity gradient tensor. Nevertheless,2btaining the statistics of the velocity gradient tensor is a di ffi cult task. A common approach toaddress the evolution of the velocity gradient is the Lagrangian framework, which can drasticallyreduce the degrees of freedom and lead to a simpli fi ed picture of the small scales. Turbulencein the Lagrangian frame has some di ff erent features compared to Eulerian turbulence, such asa shorter correlation time of the velocity gradient. This property inspired the Recent Fluid De-formation [1] approximation, which is a model where the shape of a fl uid particle following thelocal velocity fi eld has a short memory. This closure was studied in the last years [4], [5], [6] andextended to account for passive scalar transport and MHD [7] and was dealt with analyticallyby an e ff ective action approach, based on noise renormalisation [8].In order to study large deviations of the velocity gradients in this model the path integralframework is used, which is very suitable to investigate large fl uctuations. The reason lies inthe fact that for weak noise driven systems, the probability is dominated by the action minima.The trajectory which minimizes the action is called instanton. This approach is equivalent to theFreidlin-Wentzell theory of large fl uctuations [10] and provides a proper way to fi nd which is themost probable evolution leading to a large event. It can be, therefore, a valuable approach to dealwith an important question in hydrodynamic turbulence, that is what are the common structuresfound at small scale turbulence. Structures related to large values of velocity gradient are ofvital importance in the study of turbulence, since they are responsible for most of dissipationthat takes place at the smallest scales of fl uid motion, typically associated with large strainand vorticity. Many works have devoted a long e ff ort on the identi fi cation of such objects. Inparticular the use of instanton techniques to achieve this goal in Burgers turbulence can be foundin references [2], [19], [20], [21]. Reference [8] applies the path-integral approach to evaluatethe pdf in the RFD model. However, the set of saddle-point equations was linearized to obtain anapproximate instanton solution. To correct this truncated saddle-point equations, a perturbativemethod was carried out.In this work we determine the instanton of the RFD addressing what is the most probableevolution of a Lagrangian particle and also calculate its contribution to the pdf in the weaknoise limit by solving the full set of non-linear saddle-point equations. For the case of a diagonal(longitudinal) component of the velocity gradient, analytical results can be computed for its pdf.The paper is organized as follows. In section II the Recent Fluid Deformation equations arereviewed. Section III is devoted to the results and is divided in four parts. Part A presentsthe model dressed in the Martin-Siggia-Rose/Janssen/de Dominicis path integral formalism [15],316], [17] and how this approach can be used to address large deviations. Part B displays thetransverse velocity gradient statistics after solving the instanton equations by means of the nu-merical Chernykh-Stepanov [2] algorithm. In part C it is shown that the longitudinal velocitygradient is subject to an analytical solution in addition to the numerical one. In the sequel, partD presents how instantons are uncovered by performing a conditioned statistics with respect tothe stochastic dynamics, which are confronted with the previously obtained instantons. Finalremarks close the paper in section IV. II. THE RFD LAGRANGIAN STOCHASTIC MODELA. Recent Fluid Deformation for Lagrangian turbulence
Proposed in [1], the Recent Fluid Deformation (RFD) is a scheme for modelling the evolutionof velocity gradient of a fl uid particle along its trajectory in the Lagrangian frame. By taking thegradient of the Navier-Stokes equation, we write dA ij dt = − A ik A kj − ∂ p∂x i ∂x j + ν ∂ A ij ∂x m ∂x m , (2.1)where d/dt is the convective derivative, p stands for pressure divided by fl uid density and ν cor-responds to the kinematical viscosity. In equation (2.1), A ij = ∂ j u i is the velocity gradient tensorin cartesian components. The di ffi culty in obtaining statistics from the velocity gradient Navier-Stokes is that the pressure Hessian and the viscous term are not closed in terms of a Lagrangiantrajectory. A review of di ff erent attempts of closures can be found at [9]. The simplest closureis achieved by neglecting dissipation and nonlocal e ff ects of the pressure Hessian. Although, asolution is available, it can be shown that it develops a divergence at fi nite time [11], [12]. TheRFD has the merit of incorporating pressure and viscous e ff ects preventing divergences in A . Itmay be compared to the tetrad model [13], though instead of dealing with an equation for theevolution of fl uid deformation, it is strongly modelled. The rationale goes as follows. Write thepressure Hessian as ∂ p∂x i ∂x j ≈ ∂X m ∂x i ∂X n ∂x j ∂ p∂X m ∂X n (2.2)where ∂X j /∂x i denotes the Jacobian of the change of coordinates from Eulerian to Lagrangiancoordinates. In (2.2), spatial derivatives of the Jacobian were neglected. The Cauchy-Green4ensor, de fi ned by C ij = ∂x i ∂X k ∂x j ∂X k (2.3)is assumed to have the form C = exp[ τ A ] exp[ τ A T ] , (2.4)where τ corresponds to a short time associated to the correlation time of the velocity gradient inthe Lagrangian frame, assumed to be of the order of the Kolmogorov time scale. The idea behindthe RFD approximation is that after a short period of time ( ∼ τ ) the shape of a Lagrangianparticle is uncorrelated with its initial shape. Therefore, it is possible to assume an isotropicshape for a fl uid particle at initial time, which implies an isotropic pressure Hessian ∂ p∂X m ∂X n = δ mn ∂ p∂X l ∂X l . Taking it into account, (2.2) turns to ∂ p∂x i ∂x j ≈ C − ij C − qq A mn A nm . (2.5)Similar reasoning can be applied to model the viscous term, yielding ν ∂ A ij ∂x m ∂x m ≈ ∂X k ∂x m ∂X l ∂x m ∂ A ij ∂X k ∂X l ≈ − T C − qq A ij (2.6)where T stands for the integral time scale, which comes from dimensional arguments as ν/ ( ∂X ) ≈ /T , considering that ∂X is on the order of a typical distance travelled by aparticle during time τ , which scales with the Taylor microscale length. Therefore, substitutingeqs. (2.5) and (2.6) in (2.1), the RFD model equation is given by ˙ A = − A + C − Tr ( A ) Tr ( C − ) − Tr ( C − )3 T A + g F , (2.7)where a random forcing was supplemented to provide stationary statistics. In (2.7), g is thestrength of the stochastic force, related to energy injection rate, and will play an important rolein the discussion. F is a zero average white noise tensor such that h F ij ( t ) F kl ( t ′ ) i = G ijkl δ ( t − t ′ ) , (2.8)with G ijkl = 2 δ ik δ jl − δ il δ jk − δ ij δ kl . (2.9)The force correlator G ijkl is the general th -order tensor which respects isotropy and also ensuresincompressibility, i.e. , Tr A = 0 . It can be shown that G jjkl = 0 and G ijkl = G klij , which followimmediately from equation (2.8). 5 II. RESULTSA. Instantons in the Martin-Siggia-Rose path integral
As in many applications of large deviations, it is customary to evaluate the probability toreach a fi nal state A ( t ) = A at time t = t starting from time t , with A ( t ) = A . The initialcon fi guration A is usually taken to be at, or close to, an attractor of the deterministic dynamics,whilst the initial time is assumed to be t = −∞ , such that the stationary transition probabilitywill depend solely on A ( t ) . In this work, we want to evaluate the probability of fi nding a largevalue of one component A αβ ( t ) , either longitudinal, or transverse, which can be accomplishedwith the auxiliary of the Martin-Siggia-Rose/Janseen/de Dominics path integral [15], [16], [17].Therefore, denoting the referred transition probability by ρ αβ ( a ) = ρ ( A αβ ( t ) = a | A ( t ) = 0) with α and β prescribed ( ρ αβ should not be understood as a tensor, the indices simply referto the transition probability of the component A αβ to the value a at a fi nal time t ), the pathintegral formalism leads to ρ αβ ( a ) = h δ ( A αβ ( t ) − a ) i = Z D [ A ] exp (cid:20) − Z t t dt L OM [ A ( t ) , ˙ A ( t )] (cid:21) δ ( A αβ ( t ) − a ) , (3.1)where the angular brackets stand for the averaging over force realisations, which can be ac-counted for, alternatively, by performing a sum over all possible paths A ( t ) starting from A andarriving at A . The fi nal condition is enforced by the Dirac delta functional, and the Onsager-Machlup Lagrangian L OM [ A ( t ) , ˙ A ( t )] [18] reads L OM [ A , ˙ A ] = 12 g (cid:18) ( ˙ A ij − V ij ) Q − ijkl ( ˙ A kl − V kl ) − Tr [ ˙ A − V ] (cid:19) , (3.2)with Q − ijkl = (8 / δ ik δ jl + ( − / δ il δ jk such that G ijkl = Q ijkl − Q ijmm Q klnn /Q ppqq . Equiv-alently, the probability transition can be written in terms of the Martin-Siggia-Rose Lagrangian L MSR [ A ( t ) , ˆ A ( t )] as ρ αβ ( a ) = h δ ( A αβ ( t ) − a ) i = Z D [ A ] D [ ˆ A ] exp (cid:20) − Z t t dt L MSR [ A ( t ) , ˆ A ( t )] (cid:21) δ ( A αβ ( t ) − a ) , (3.3)with L MSR [ A , ˆ A ] = g A ij G ijkl ˆ A kl − i Tr [ ˆ A T ( ˙ A − V )] . (3.4)The relationship between the two Lagrangians is made clearer by noting that from (3.4) theconjugated momentum reads P = ∂ L /∂ ˙ A = − i ˆ A , so the Lagrangians are related by a Legendre6ransform. In order to obtain the instanton equations we have, thus, to derive the stationaryaction (3.4) with the endpoint A αβ (0) = a imposed by the Dirac delta functional. We are goingto clarify how this constraint turns to a fi nal condition for the auxiliary variable P ( t ) , since thispoint is not usually discussed in the literature. Many authors consider the physical reasoning ofGuraire and Migdal [19] based on the negative viscosity sign. The limitation of this argument isthat it applies only to fl uid systems. The discussion below encompasses more general cases.Starting from the Onsager-Machlup Lagrangian we calculate the action variation with respectto the path A ( t ) with initial point fi xed, that is, δ A ( t ) = 0 , yielding δS = Z t t dt (cid:26) Tr (cid:20) ∂ L ∂ A δ A T ( t ) + ∂ L ∂ ˙ A δ ˙ A T ( t ) (cid:21) + λδ ( t − t ) δA αβ ( t )] (cid:27) (3.5) = Z t t dt Tr (cid:20) ∂ L ∂ A δ A T ( t ) + ddt (cid:18) ∂ L ∂ ˙ A δ A T ( t ) (cid:19) − ddt ∂ L ∂ ˙ A δ A T ( t ) (cid:21) ++ lim ǫ → Z t + ǫt dt λ δ ( t − t ) δA αβ ( t ) (3.6) = Z t t dt Tr (cid:20)(cid:18) ∂ L ∂ A − ddt ∂ L ∂ ˙ A (cid:19) δ A T ( t ) (cid:21) + Tr (cid:18) ∂ L ∂ ˙ A δ A T ( t ) (cid:19) t t + λ δA αβ ( t )] (3.7)The last term in (3.5) is due to writing the Dirac delta in terms of its Fourier representation. Bydemanding the action variation to be stationary with respect to the path A ( t ) we arrive at ∂ L ∂ A − ddt ∂ L ∂ ˙ A = 0 (3.8)Tr [ P ( t ) δ A T ( t )] − Tr [ P ( t ) δ A T ( t )] | {z } =0 + λ δA αβ ( t )] = 0 , (3.9)where we used the de fi nition P ( t ) ≡ ∂ L /∂ ˙ A ( t ) . Equation (3.8) is the Euler-Lagrange equationwhich gives the evolution with time, while (3.9) implies P ij ( t ) = − δ iα δ jβ λ . This completes ourderivation relating the fi nal point condition of A with P ( t ) . Note that in this case, the endpointis not fi xed as usual. The Dirac delta relaxed the endpoint, allowing it to have non vanishingvariation ( δ A ( t ) = 0 ).Therefore, since a fi nal condition for the canonical momentum is obtained, it is more conve-nient to minimize the MSR action rather than minimising the OM Lagrangian, since the formeris fi rst order in time. Hence, substituting − i ˆ A ( t ) by P ( t ) in (3.4), we are led to solve the set ofsaddle-point equations ˙ A ij = V ij ( A ) + g G ijkl P kl (3.10) ˙ P ij = − P kl ∇ ij V kl ( A ) , (3.11)7ith endpoint condition P ij ( t ) = − λ δ iα δ jβ . The solution of equations (3.10) and (3.11) mini-mize the MSR action (3.4), (3.3) subject to the endpoint constraint A αβ ( t ) = a .Thus, we end up with a system of mixed initial- fi nal condition which naturally suggests that P should be integrated backwards in time whereas A is integrated forwards in time. This kind ofproblem was tackled numerically by Chernyk and Stepanov [2] and by [20], [21] in the contextof the Burgers equation. There they were seeking large values of the velocity gradient in onepoint. It was found that the instantons turned out to be the shocks which are present in theunderlying dynamics of the system. See also [22] for a review of applications of this approach,including the study of instantons in the stochastic Navier-Stokes equation.By scaling the auxiliary variable ˜ P = g P the action changes as S [ P , A ] → ˜ S [˜ P , A ] /g ,yielding for the conditional probability distribution ρ αβ ( a ) = Z D [ P ] D [ A ] δ [ A αβ (0) − a ] exp ( − ˜ S [˜ P , A ] g ) (3.12)where ˜ S [˜ P , A ] is independent of g . In the weak noise limit g → , the probability ρ αβ ( a ) willbe dominated by the contribution from the action minimizer. This is in accordance with theFreidlin-Wentzell theory of large deviations [10], which states that − lim g → g ln ρ αβ ( a ) = I = min ˜ S (3.13)where the action minima min ˜ S , is evaluated at the optimal path satisfying A αβ ( t = 0) = a and A ( t = −∞ ) = 0 . The rate function I , independent of g , controls the behavior of thetransition probability for asymptotically vanishing g . It contains information not just aboutsmall fl uctuations around the attractor of A ( A = 0 for the dynamics considered) but also aboutlarge fl uctuations.For the sake of clarity, we split the cases where the fi xed fi nal value of the velocity gradientis either one of the diagonal (longitudinal) or o ff -diagonal (transverse) components. B. Transverse gradient statistics
This section shows the results regarding the stationary statistics ρ ( a ) of the transversevelocity gradient. Due to numerical reasons we should use a fi nite but large initial time t . Inour implementation we chose t = − T whereas t = 0 , that is, the evolution is carried outthrough six integral time scales. It was also checked numerically that this value su ffi ces for8tationarity by examining the time series of the original SDE, integrated according to [24]. Thealgorithm is an iterative procedure to obtain the solution of the set of equations (3.10) and (3.11).Before we apply the method treating the eight independent degrees of freedom encoded in A ,it is convenient to take advantage of the symmetries of the problem in order to reduce the num-ber of degrees of freedom, lowering thus the computational cost. First, we recall the Onsager-Machlup action (3.1). After we write the Dirac delta using its Fourier representation, the end-point condition can be understood as another term in the action of the form λδ ( t − t ) A αβ ( t ) (in this section ( α, β ) = (1 , ). This additional term, which manifests itself in the equations ofmotion (3.10) and (3.11) as a fi nal condition for P , breaks the parity symmetry x i → − x i and v i → − v i for i = 1 , , therefore only the symmetry x → − x and v → − v remains. Ifthe action exhibits this symmetry so does the solution to the equations of motion, provided the fi nal/initial conditions keep the same symmetry, which is the case. Hence A must be a velocitygradient tensor with re fl ection symmetry in the x direction, whose only possible form is A ( t ) = A ( t ) A ( t ) 0 A ( t ) A ( t ) 00 0 − A ( t ) − A ( t ) . (3.14)We are left with 4 independent variables instead of eight, which simpli fi es the computationconsiderably.Now, the Chernyk-Stepanov method can be performed. The idea is to decouple P ( t ) and A ( t ) for the fi rst iteration. For instance, we set A ( t ) = 0 and solve (3.11) backwards in time for an ar-bitrarily chosen λ . In the next step, we substitute the time series of P ( t ) obtained in (3.10), whichis integrated forward in time to obtain A ( t ) . This is performed recursively until the solutionsconverge. Both equations are solved by the 4th order Runge-Kutta scheme with time step dt =10 − and a piecewise cubic interpolation is performed to obtain the intermediate time steps re-quired by the method. The criteria used for convergence is that | A (0) − A old (0) | / | A old (0) | < δ , i.e , the relative error of the obtained instanton in comparison with the (old) instanton calculatedin the previous iteration should be smaller than a quantity δ (we set δ = 10 − and δ = 10 − forthe longitudinal case). With the instanton solution, the probability of arriving at a fi nal value A = a can be computed plugging it into (3.13). Spanning a set o λ ’s we can generate the pdf ρ ( a ) , since each value of λ leads to a di ff erent fi nal value of the longitudinal velocity gradient a . Figure 1(a) displays pdfs obtained by this approach for di ff erent values of forcing amplitude.9 − − − − − − (a) (b) − − − − − − (a) (b) ρ ( A ) A g l n ρ ( A ) A FIG. 1. (a) Semilog plot of the transverse velocity gradient pdf. Dots: numerical instanton evaluation.Solid lines: pdfs from SDE (2.7). The range of A lies between 5 to 6 standard deviations. Forcing valuesare g = 0 . , . , . and . where darker colours correspond to higher values of g . (b) Rescaled pdfscorresponding to vertical axis g ln ρ ( a ) showing collapse. Pdfs from numerical integration of the SDE are also plotted for comparison, showing good agree-ment between the results. The collapse depicted in fi gure 1(b) corresponds to a rescaling of thevertical axis, g ln( ρ ( A ( t ) = a )) and shows that the pdfs calculated obey the large deviationprinciple (3.13). The curve is minus the rate function (action minima) as a function of the fi nalvalue A .A last comment on the numerical scheme concerns convergence issues that may arise. Actu-ally, in the original reference of the method [2] it was reported that, for a critical value of λ , thenumerical convergence becomes problematic. In our case, it is noticed that as | λ | increases, sodoes the number of iterations to reach convergence. In the transverse case, where the numberof degrees of freedom cannot be as reduced as in the longitudinal case (cf. next subsection),convergence may fail completely. In order to circumvent this issue we performed the followingstrategy. Let A α and P α be the α -th step in the iteration procedure of the numerical integration.The direct approach would be to use the series A α and P α in the saddle-point equation (3.11) toobtain P α +1 and A α +1 and so on. However, when the iteration ceases to converge, we modify A α +1 by A α +1 → βA α + (1 − β ) A α +1 , with β arbitrarily chosen on the interval [0 , , that is, thenext iteration is a weighted average of the old and the new ones. Although not systematic, sincewe do not know a priori which is the optimal β value, this procedure dumps large variations in10 − − − − − − (a) (b) − − − − − − − (a) (b) ρ ( A ) A g l n ρ ( A ) A FIG. 2. (a) Semilog plot of the longitudinal velocity gradient pdf. Dots: numerical instanton evaluation.Solid lines: pdfs from SDE (2.7). Dashed lines: analytical (3.18). The range of A lies between 5 to 6standard deviations. Forcing values are g = 0 . , . , . , . and . where darker plots correspond tohigher g values. (b) Rescaled pdfs corresponding to vertical axis g ln ρ ( a ) showing collapse. each step and tends to keep iterations inside the converge radii. Values as big as β = 0 . may beneeded to capture the tail of the distributions. C. Longitudinal gradient statistics
In this section we show the results concerning the longitudinal velocity gradient. In order tocalculate the instanton we make use of the even higher degree of symmetry of this case, whichreduces the number of degrees of freedom to only one. We invoke the same rationale of theprevious section. The di ff erence is that imposing A (0) = a consequently adds to the action aterm that respects parity symmetry x i → − x i , v i → − v i in all directions and hence implies thatthe instanton velocity gradient must be diagonal. This term breaks rotation symmetry though,by selecting the x direction, but the action is still invariant under rotations around the x axis.So, the action makes no preference between the x or x directions, implying A = A forthe solution. Moreover, incompressibility leads to A = diag ( A ( t ) , − A ( t ) / , − A ( t ) / , i.e , thevelocity gradient depends on a single degree of freedom. Within this simpli fi cation the saddle-point equations become much faster and stable to be integrated numerically.Apart from the numerical solution to the saddle-point equations, the high degree of symmetryenables us to derive an analytical solution in the case of longitudinal velocity gradient. With the11elocity gradient given by a diagonal form A = diag ( A ( t ) , − A ( t ) / , − A ( t ) / , a reduced MSRaction for the single degree of freedom A ( t ) can be written as S red [ A, p ] = Z − T dt (cid:20) p (cid:16) ˙ A − b [ A ] (cid:17) − g p (cid:21) , (3.15)where A ( t ) is a scalar, equivalent to the A of the original system and b [ A ] = V [ A ] . Due tothis drastic reduction of degrees of freedom, it is possible to write b [ A ] as a gradient of a function h [ a ] b [ A ] = −∇ h [ A ] , h [ A ] = A A τ τ ) A − τ A + O ( τ ) . (3.16)In that case, instantons may be obtained as the reverse of the relaxation path from A (0) to A ( −∞ ) [23]. Nevertheless, the pdf can be computed in a more straightforward manner bysolving the corresponding Fokker-Planck equation. First, we write an e ff ective SDE which leadsto the above reduced action (3.15) ˙ A = b [ A ] + gf ( t ) , (3.17)where h f ( t ) f ( t ′ ) i = δ ( t − t ′ ) is the correlation of the reduced noise f ( t ) . A straightforwardcalculation shows that the MSR action related to the SDE (3.17) is given by (3.15). The Fokker-Planck equation can be easily derived from (3.17), whose stationary solution reads ρ ( a ) = N exp( − h [ a ] /g ) , (3.18)with h [ A ] given by (3.16) and N is normalization factor. This important result validates the nu-merical procedure, as one can see in fi gure 2(a), where a good agreement between the analyticaland numerical instanton contribution to the pdf is achieved.Once the pdf ρ ( a ) is obtained analytically, it is possible to evaluate the moments of the ve-locity gradient as a power series of the noise g along with the scaling with Reynolds number,which is another original result of this paper. A straightforward computation yields for the fi rstcentral moments of the longitudinal velocity gradient,var [ a ] = g g
96 (29 − τ (1 + τ )) , (3.19) E [( a − E [ a ]) ] var / [ a ] = − g √ g (cid:18) − √ τ √ √ τ (cid:19) , (3.20) E [( a − E [ a ]) ] var [ a ] = 3 + 116 g (19 − τ (1 + τ )) . (3.21)12 . . . . .
15 0 0 . . . . . (a) Variance (b) Skewness (c) Flatness − . − . − . − . . . . . . . (a) Variance (b) Skewness (c) Flatness . . . . . . . . . (a) Variance (b) Skewness (c) Flatness g g g FIG. 3. Statistical moments of the longitudinal velocity gradient as a function of the forcing g . Circles:numerical integration. Solid lines: instanton analytical results (equations 3.19-3.21). We highlight this is a novel result specially considering there are few analytical results con-cerning velocity gradient models available. Let us compare it to phenomenological expectations.The forcing g may be interpreted as the energy injection rate in the Lagrangian particle per unitarea. Since stationarity demands that energy injection equals energy dissipation, the stochasticRFD equation (2.7) leads to g ∼ ∂ ε/ ( ∂x ) ∼ ε/λ , where ε is the dissipation rate and λ isthe Taylor microscale length. On dimensional grounds one would expect the velocity gradientvariance to behave as h ( ∂u ) i ∼ ε/ν ∼ ε Re/ ( U L ) ( U is a typical integral velocity scale) whichimplies h ( ∂u ) i ∼ g λ Re/ ( U L ) ∼ g T in agreement with (3.19) at least to leading order (recallwe have set T = 1 ).Comparison with the numerical solution of the SDE, fi gure 3, shows compatibility betweenanalytical and numerical moments for small values of forcing. As g increases though, the analyt-ical result disagrees with the numerical evaluation since for fi nite g the instanton approximationis not su ffi cient to estimate the pdf. Moreover, it can be also noted that the agreement betweennumerical and analytical moments decreases for higher moments, which is expected consideringthe analytical pdfs mismatch the numerical ones in the tails (specially the right tail), fi gure 2.The skewness and fl atness, though, show an incorrect scaling with respect to Reynolds num-ber which points to a drawback of the model. This drawback appearing at high Reynolds num-bers was already recognized in Ref. [26]. The new analytical results provided in Eqs. (3.19),(3.20), (3.21) shed a new light on the numerical results obtained in this reference [26]. Indeed, itis there underlined that the variance of the gradients does not behave with the free parameter13f the model, i.e. the Reynolds number, in a consistent way with the dimensional approach ofKolmogorov. To circumvent this issue, it was proposed instead in Ref. [26] to study the relativescaling of the logarithm of higher order moments of the gradients with respect to the varianceof the gradients. To interpret the departure of the observed scalings seen in Ref. [26] from nonintermittent scalings, it is then tempting to interpret them, based on the theoretical results ofEqs. (3.19), (3.20), (3.21), as being reminiscent of the forcing. Future works will be devoted toimprove the RFD approximation in order to include genuine intermittent scalings, at the cost,perhaps, of introducing a further free parameter that quanti fi es in an appropriate way intermit-tent corrections. We leave these perspectives for future investigations.Regarding the so-called RQ plane, the velocity gradient instanton starts at A = 0 evolvingto a fi nal con fi guration such that A (0) = a . If we keep track of the trajectory on the RQ plane it is noticed that it lies entirely in the Vieillefosse line ( Q + 27 R = 0 , with Q = − Tr A / and R = − Tr A / ) [11], although this is not a consequence of the model dynamics.Actually, this is simply due to kinematics since for a velocity gradient tensor taking the form A = diag ( A ( t ) , − A ( t ) / , − A ( t ) / , which in turn is a consequence of symmetry, the Vieillefosse lineis satis fi ed identically. D. Filtering and interpretation of instanton solution
In this subsection we try to assess the relevance of instantons in a fl uid dynamical modelsharing many non trivial properties with real turbulence, as it is the case for the RFD approx-imation, following reference [20]. An ensemble with trajectories of the original SDE with-out any constraint was build. With this ensemble we perform a conditioned statistics select-ing those paths ending within a small neighborhood of a , that is, A (0) ∈ [ a − da, a + da ] ( A (0) ∈ [ a − da, a + da ] if we are looking at transverse gradients). To increase the ensemblesizes, if the searched value a is crossed by any component, we perform frame rotations over theentire trajectory so that it always corresponds to component A (in the diagonal case) or A (o ff -diagonal). What is seen is that these paths concentrates around the instanton solution andafter being averaged they tend to superpose with it as depicted in fi gure 4. Figure 4(a) shows sev-eral components of velocity gradients from conditionally averaged trajectories compared to theinstanton solution with fi nal value A (0) = − . . Figure 4(b) depicts how the unconditionedcomponent A ( t ) evolves for di ff erent fi nal values of the conditioned A (0) in comparison14 . − . − . − . . − − − − − − (a) (b) − . − . − . − . − . − − − − − − (a) (b) A i j ( t ) tA A A A A ( t ) t A (0) = 0 . A (0) = 1 . A (0) = 1 . FIG. 4. (a) Di ff erent components from conditionally averaged trajectories (symbols) compared to theinstanton (solid) for the case where A is set to − . at the endpoint. Components A , A , A , A (not all shown) are negligible, in agreement with our symmetry argument. (b) Component A both from fi ltering (dashed) and instanton (solid) for the case where A is set to . , . and . at the endpoint. Inboth fi gures g = 0 . . with instanton solution. The agreement is better as the constrained fi nal value gets larger, asexpected by instanton theory. This trend has been found in the context of Burgers equation in[20] and [21]. After all it is clearly obtained that typical trajectories of the stochastic dynamics fl uctuates around but not far from the instanton trajectory provided g is small in accordancewith the large deviation principle.Conversely, the most probable trajectory leading to a certain value of longitudinal velocitygradient is such that the velocity gradient is diagonal, as claimed in section IIIB by symmetryarguments. This statement is indeed con fi rmed by the fi ltering procedure as presented in fi gure5. Figure 5(a) shows the average behaviour of velocity gradient conditioned to A (0) = 1 . in comparison with instanton trajectories. All o ff -diagonal components vanish, as illustratedby A ( t ) and A ( t ) (others not shown). In fi gure 5(b) three di ff erent constrained values areexhibited. In contrast to the previous case the agreement does not improve for larger values of A (0) , another manifestation of the mismatch observed on the tails of the diagonal pdf ( fi gure2). 15 . − . − . . . . . − − − − − − (a) (b) . . . . . − − − − − − (a) (b) A i j ( t ) tA A A A A ( t ) t A (0) = 0 . A (0) = 1 . A (0) = 1 . FIG. 5. (a) Di ff erent components from conditionally averaged trajectories (symbols) compared to theinstanton (solid) for the case where A is set to . at the endpoint. Components A , A are negligible(as well as other o ff -diagonal components not shown), in agreement with our symmetry argument. (b)Component A both from fi ltering (dashed) and instanton (solid) for the case where A is set to . , . and . at the endpoint. In both fi gures g = 0 . . IV. CONCLUSION
The role of the rare events can be revealed by means of the Martin-Siggia-Rose path integralformulation. In this work we apply this technique to a model of Lagrangian turbulence calledthe Recent Fluid Deformation (RFD). This closure comprises a stochastic model of the velocitygradient based on short time correlations in the Lagrangian frame. Within the path integral for-malism the most probable trajectory that leads to a certain event is calculated numerically and,for the longitudinal velocity gradient case, also analytically. We showed the use of symmetriescan rule out unnecessary degrees of freedom allowing less numerical e ff ort in order to computethe instanton. Apart from the bene fi ted numerical computation, the symmetries let us evaluatean analytical approximated solution for the longitudinal velocity gradient pdf, enabling us tounveil its central moments dependence on the Reynolds number.Both longitudinal and transverse cases present the instanton satisfying the Vielleifosse line.We believe that the rationale for that lies in the dominance of the non deviatoric terms fi guringthe model equation (2.7). That is, when τ → , the RFD approximation approaches the RestrictedEuler equation.Regarding vorticity alignment, instanton solutions for transverse gradients shows a complete16lignement with the intermediate strain eigenvalue, which can be seen computing the normal-ized product of the three rate of strain eigenvalues s ∗ = − √ λ λ λ / ( λ + λ + λ ) / [27],resulting s ∗ = 1 , where λ i , i = 1 , , , are the referred eigenvalues. Since the instanton cor-responds to the most probable trajectory leading to a certain value of the velocity gradient, ourresult agrees with reference [27] which showed that the pdf of s ∗ develops a sharp peak around s ∗ = 1 .The longitudinal velocity gradient pdf has a weaker agreement in comparison with the trans-verse one as the forcing increases, showing the instanton approximation is not enough to ac-count for the full statistics even for moderately low values of g . It means that fl uctuationsaround the instanton solution may play an essential role, which could be hopefully analyzedby perturbative methods. Perturbative corrections to the instanton pdf can be dealt with thee ff ective action approach [8] and is currently under study. The issue of wether the instantonapproach su ffi ces and perturbative methods are fi t to more complex fl uid dynamical systems isan important matter and deserves further investigation.As a fi nal remark, the application of the instanton study to this Lagrangian model allowedus to understand the scaling of the statistical moments with the Reynolds numbers. This opensnew possibilities in the direction of re fi nement of the RFD approximation in order to grasp moreaspects of the phenomenology of turbulence. Moreover we expect that the use of symmetriesas in this work, which led to a reduction of the degrees of freedom, can be applied to otherstochastic systems allowing more e ffi cient optimal paths computation. V. ACKNOWLEDGMENTS
L.S.G. would like to thank the fi nancial support by CAPES scholarship program, processnumber . / − and also the warm hospitality of the Laboratoire de Physique,École Normale Supérieure de Lyon where this work was developed. R.M.P. thanks CAPES for fi nancial support through the scholarship process number 9497/13-7. [1] Chevillard L and Meneveau C 2006 PRL PRE .[3] Sreenivasan K R and Antonia R A 1997 Annu. Rev. Fluid Mech.
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