Chaos and predictability of homogeneous-isotropic turbulence
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Chaos and predictability of homogeneous-isotropic turbulence
G. Boffetta , and S. Musacchio Department of Physics and INFN, Universit`a di Torino, via P. Giuria 1, Torino, Italy Institute of Atmospheric Sciences and Climate (CNR), Torino, Italy Universit´e Cˆote d’Azur, CNRS, LJAD, Nice, France
We study the chaoticity and the predictability of a turbulent flow on the basis of high-resolutiondirect numerical simulations at different Reynolds numbers. We find that the Lyapunov exponent ofturbulence, which measures the exponential separation of two initially close solution of the Navier-Stokes equations, grows with the Reynolds number of the flow, with an anomalous scaling exponent,larger than the one obtained on dimensional grounds. For large perturbations, the error is transferredto larger, slower scales where it grows algebraically generating an “inverse cascade” of perturbationsin the inertial range. In this regime our simulations confirm the classical predictions based on closuremodels of turbulence. We show how to link chaoticity and predictability of a turbulent flow in termsof a finite size extension of the Lyapunov exponent.
The strong chaoticity of turbulence does not spoil com-pletely its predictability. Such apparent paradox is re-lated to the hierarchy of timescales in the dynamics ofturbulence which ranges from the fastest Kolmogorovtime to the slowest integral time.Ruelle argued many years ago that the growth of in-finitesimal perturbations in turbulence is ruled by thefastest timescale [1]. This leads to the prediction thatthe Lyapunov exponent is proportional to the inverse ofthe Kolmogorov time, and hence it increases with theReynolds number. Turbulent flows at high Re are there-fore strongly chaotic [2]. Nonetheless, the time that ittakes for a small perturbation to affect significantly thedynamics of the large scales is expected to be of the or-der of the slow integral time [3]. The ratio between theseextreme timescales increases with the Reynolds numberand therefore allows a finite predictability time to coexistwith strong chaos [4]. This is evident from everyday ex-perience: while the Kolmogorov time of the atmosphere(in the planetary boundary layer) is a fraction of a second[5] the weather is predictable for days.The study of the predictability problem in turbulencedates back to the pioneering works of Lorenz [3] and ofLeith and Kraichnan [6, 7]. The main idea of those stud-ies is that a finite perturbation at a given scale in theinertial range of turbulence grows with the characteris-tic time at that scale. Therefore, while an infinitesimalperturbation is expected to grow exponentially fast, finiteperturbations grow only algebraically in time, making thepredictability of the flow much longer. These ideas wereapplied to the predictability of decaying turbulence [8],two-dimensional turbulence [9, 10] and three-dimensionalturbulence at moderate Reynolds numbers [11].In this letter we investigate, on the basis ofhigh-resolution direct numerical simulations, chaos inhomogeneous-isotropic turbulence by measuring thegrowth of the separation between two realizations start-ing from very close initial conditions. In the limit of in-finitesimal separation we compute the leading Lyapunovexponent of the flow (the rate of exponential growth of the separation [12]) and we find that it increaseswith the Reynolds number, but surprisingly faster thanwhat predicted on dimensional grounds [1] and what ob-served in low-dimensional models of turbulence [13]. Forlarger separation we observe the transition to an alge-braic growth of the error, in agreement with the predic-tions of closure models [7]. Finally, we discuss the re-lation between chaoticity and the predictability time ofturbulence (defined as the average time for the perturba-tion to reach a given threshold) in terms of the finite-sizegeneralization of the Lyapunov exponents.We consider the dynamics of an incompressible velocityfield u ( x , t ) given by the Navier-Stokes equations ∂ t u + u · ∇ u = − ∇ P + ν ∆ u + f , (1)where P is the pressure field and ν is the kinematic vis-cosity of the fluid. The term f represents a mechanicalforcing needed to sustain the flow. In the following wewill present results in which f is a deterministic forcingwith imposed energy input [14, 15]. The Navier-Stokesis solved numerically by a fully parallel pseudo-spectralcode in a cubic box of size L at resolution N with pe-riodic boundary conditions in the three directions. Themain parameters of the simulations are reported in Ta-ble I and further details are found in the SupplementaryMaterial .In presence of forcing and dissipation, the turbulentflow reaches a statistically steady state in which the en-ergy dissipation rate ε = ν h ( ∂ α u β ) i is equal to theinput of energy provided by the forcing (brackets indi-cate average over the physical space). The turbulentstate is characterized by a Kolmogorov energy spectrum E ( k ) = Cε / k − / . The kinetic energy E = R E ( k ) dk =(1 / h| u | i fluctuates around a constant mean value,which defines the typical intensity of the large scale flow U = (2 E/ / . The integral time is defined as T = E/ε and the integral scale is L = U T .We performed a series of simulations at increasingReynolds number Re = U L/ν . In order to ensure thatthe viscous range is resolved with the same accuracy inall the simulations, the increase of Re as been achievedby increasing the resolution N and reducing the viscosityin order to keep fixed k max η = 1 .
7, where k max = N/ η = ( ν /ε ) / isthe Kolmogorov scale. N Re E U L η τ η λ .
700 0 .
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063 2 . .
678 0 .
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01 0 .
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665 0 .
666 4 .
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1, and the box size is L = 2 π . N is the grid resolution, Re = UL/ν the Reynolds number, E the kinetic energy, U = (2 E/ / is the large-scale velocity, L = UE/ε the integral scale, η = ( ν /ε ) / the Kolmogorovscale, τ η = ( ν/ε ) / the Kolmogorov time and λ the Lyapunovexponent. For the study of chaos and predictability we are in-terested in measuring the growth of an uncertainty inthe velocity field. Starting from an initial velocity field u ( x ,
0) in the stationary turbulent state, we generate aperturbed velocity field u ( x , O (10 − )). We consider verysmall initial perturbations in order to guarantee that theseparation between the two realizations is along the mostunstable direction in phase space when the error entersin the non-linear stage and therefore we do not considerthe effect of the distribution of the initial error on thepredictability of the flow [16]. The two realizations ofthe velocity field are then simultaneously evolved in timeaccording to (1). For each resolution, we performed anaverage over several independent realizations.A natural measure of the uncertainty is the error en-ergy E ∆ ( t ) and the error energy spectrum E ∆ ( k, t ), de-fined on the basis of the error field δ u ≡ ( u − u ) / √ E ∆ ( t ) = Z ∞ E ∆ ( k, t ) dk = 12 h| δ u ( x , t ) | i . (2)With the normalization coefficient 1 / √ E ∆ = E for completely uncorrelated fields.Figure 1 shows the time evolution of the error en-ergy E ∆ for the simulation at the highest Re , aver-aged over an ensemble of 10 independent realizations.In the initial stage the error grows exponentially as E ∆ ( t ) = E ∆ (0) exp( L t ) (see inset of Fig. 1) where L is the generalized Lyapunov exponent of order 2 [17].At later times we observe a regime of linear growth ofthe error E ∆ ( t ) ≃ εt . The growth rate displays largefluctuations as the error approaches its saturation value E ∆ ( t ) ≃ E . This is due to the fluctuations of the kinetic E ∆ ( t ) / E t/T -8 -6 -4 -2 E ∆ ( t ) / E t/T FIG. 1: (Color online) Error energy E ∆ ( t ) growth for thesimulation at N = 1024. The error energy is averaged over10 different realizations (black line). The fluctuations of theerror energy within one standard deviation from the mean arerepresented by the shaded area. Inset: The initial exponentialgrowth of the error. energy which occur on the same time scale of the satu-ration of the error and are associated to the dynamics ofthe large scales. It is worth to notice that the late regimeof saturation of the error might display a non-universalbehavior with respect to the forcing mechanism. As anexample, the deterministic force used in our study is pro-portional to the large-scale velocity. At late times, whenthe error has significantly affected the large scales, theforce acting on the two fields u and u becomes differ-ent. This could induce a faster saturation of the errorwith respect to other forcing mechanism which enforcelarge-scale correlations.During the initial stage of exponential growth the er-ror energy spectrum E ∆ ( k, t ) is peaked at wavenumbersaround the dissipation range k ≃ k η ≃ /η and growsexponentially in a self similar way, as shown in Fig. 2.At later times, the error propagates to lower wavenum-bers and the error spectrum develops a scaling range E ∆ ( k ) ∼ k − / (see Fig. 3). At each time it is possi-ble to identify the error wavenumber k E ( t ) at which theerror energy spectrum has reached a given fraction α ≃ E ∆ ( k E , t ) /E ( k E ) = α . The twovelocity fields u and u can be then assumed to be com-pletely decorrelated at scales smaller than 1 /k E and stillcorrelated at larger scales.The transition from the exponential growth to the lin-ear growth of E ∆ occurs when the two fields are com-pletely decorrelated on the dissipative scales, that is when k E ≃ k η . The idea, originally proposed by Lorenz [3],is that the time that it takes to decorrelate completelythe two fields at a given scale ℓ ≃ /k within the in-ertial range is proportional to the turnover time of theeddies at that scale τ ℓ ∼ ε − / ℓ / [18]. This leads to the -14 -12 -10 -8 -6 -4 E ∆ ( k ) k -10 -8 -6 -4 E ∆ t/T FIG. 2: The spectrum of the error E ∆ ( k, t ) at times t/T =0 . , . , . , . , .
35 (from bottom to top) in the linearphase for the simulation at N = 1024 averaged over 10 inde-pendent realizations. Inset: The error energy E ∆ as a func-tion of time in semilogarithmic plot. -6 -4 -2 E ( k ) k k E L t/T FIG. 3: The spectrum of the error E ∆ ( k, t ) at times t/T =0 . , . , . , . , . , . , . , . E ( k ) (solid line) for simulations at N = 1024 averaged over 10independent realizations. The dotted line represents the Kol-mogorov scaling k − / . Inset: The error wavenumber k E asa function of time (crosses), compared with the dimensionalscaling k E ∼ t / (dotted line). dimensional prediction k E ( t ) ≃ ε − / t − / (3)for the evolution of the error wavenumber, which is con-firmed by our numerical finding (see inset of Fig. 2).Equation (3) provides an estimation of the predictabil-ity time T P that an infinitesimal error takes to contami-nate a given wavenumber k , T p ( k ) = Aε − / k − / [7, 19]where the dimensionless coefficient A depends on thethreshold α (and possibly on the Reynolds number). Inour simulation at Re = 8516 we measure A = 12 for α = 0 . A = 10 obtainedfrom early studies with closure models in the limit ofinfinite Re [7].Integrating the error spectrum with the ansatz E ∆ ( k, t ) = 0 for k < k E ( t ) E ∆ ( k, t ) = E ( k ) for k > k E ( t )and using the dimensional scaling (3), one obtains theprediction for the linear growth of the error energy: E ∆ ( t ) = Gεt . (4)The value of the dimensionless constant G measured inthe simulation at Re = 8516 is G = 0 . ± .
05, notfar from that obtained by the test field model closure G = 0 .
23 [7].As already discussed, in the early stage the perturba-tion can be considered infinitesimal and therefore growsexponentially as shown in the inset of Fig. 1. This is thesignature of the chaotic nature of the flow and the pre-dictability is characterized by the Lyapunov exponent λ .On dimensional grounds the Lyapunov exponent can beassumed to be proportional to the inverse of the fastesttime-scale of the flow, i.e., the Kolmogorov timescale τ η = ( ν/ε ) / [1]. Since the ratio between τ η and theintegral timescale T increases with the Reynolds numberas T /τ η ∼ Re / one has the prediction that the Lya-punov exponent is proportional to the square root of theReynolds number: λ ≃ τ − η ≃ T − Re / . (5)Therefore the predictability time T P for infinitesimal per-turbations vanishes in the limit of large Re .The dimensional prediction (5) is obtained under theassumption of self-similarity of the velocity field withKolmogorov scaling exponent h = 1 / h ∈ (0 : 1) one has λ ≃ τ − η ≃ T − Re β with β = (1 − h ) / (1 + h ). Averaging over the multifractalspectrum D ( h ) allows to take into account intermittencycorrections and this gives β = 0 .
459 [13, 20].We have computed the Lyapunov exponent λ by mea-suring the average rate of logarithmic divergence of twoclose realizations, a standard method in the study of dy-namical systems [17, 21, 22], for the simulations at dif-ferent Reynolds numbers (see Table I). Interestingly, wefind that the Lyapunov exponent increases with Re fasterthan the dimensional prediction (5), as shown in Fig. 4.Fitting the measured values with a power law λT ≃ Re β gives the exponent β = 0 . ± .
05. It is remarkable thatthe measured deviation from the dimensional prediction β = 0 . λτ η in-creases with Re .Since the Lyapunov exponent is an average quantity,it is interesting to investigate its fluctuations and their λ T Re λ τ η Re10 -2 -1 µ τ η Re FIG. 4: (Color online) Lyapunov exponents λ as a functionof Re (squares). The solid line represents the best fit scaling λT ≃ Re . while the dashed line is the dimensional scaling λT ≃ Re / . Lower inset: The Lyapunov exponents λ com-pensated with the Kolmogorov time scale τ η as a functionof Re . Upper inset: The Lyapunov variance µ compensatedwith the Kolmogorov time scale τ η as a function of Re . dependence on Re . We have therefore measured the vari-ance µ of the distribution of the finite-time Lyapunovexponents, a standard measure of the fluctuations in achaotic system [12, 17] (see also the Supplementary Ma-terial ). The results, plotted in Fig. 4, shows that also µτ η increases with Re and faster than the Lyapunov ex-ponent (a fit gives µT ≃ Re . although the errors hereare large).The connection between predictability and chaoticityin turbulent flows can be extended also to finite pertur-bations, of the order of the velocities of the inertial range,by means of the finite size Lyapunov exponents (FSLE)Λ( δ ). The FSLE has been introduced to measure thechaoticity of systems with many characteristic time scales[4, 20]. It is defined in terms of the average time T r ( δ )that it takes for a perturbation of size δ to grow by a fac-tor r , as Λ( δ ) = ln( r ) / h T r ( δ ) i (where the average is nowover different realizations). We remind that performingaverages at fixed times is not equivalent to averaging atfixed error size. The latter procedure was found to bemore effective in intermittent systems, in which scalinglaws can be affected by strong fluctuations of the error(as in Fig. 1).In the limit δ → δ → Λ( δ ) = λ [20].For finite errors, Λ( δ ) measures the average growth rateof the uncertainty of size δ . Following the idea of Lorenz[3] that a perturbation of size δ ∼ u ℓ within the inertialrange of turbulence grows with the local eddy turnovertime τ ℓ ∼ ε − / ℓ / ∼ ε − u ℓ , one obtains the prediction[20] Λ( δ ) ≃ εδ − . (6) -1 -3 -2 -1 Λ ( δ ) T δ /U -2 -1 -3 -2 -1 λ ( δ ) / λ δ / δ * FIG. 5: (Color online) Finite-size Lyapunov exponents Λ( δ )(FSLE) as a function of the velocity uncertainty δ for N =1024 (red squares) N = 512 (blue circles) N = 256 (purpletriangles). The values of the Lyapunov exponents λ are alsoshown (dashed lines). Black solid line represents the scalingΛ( δ ) ∼ δ − . Inset: The FSLE Λ( δ ) rescaled by the Lyapunovexponents λ as a function of the rescaled uncertainty δ/δ ∗ . In Figure 5 we show the FSLE as a function of theerror δ for three values of Re . For small δ the FSLEapproaches the constant value Λ( δ ) ≃ λ , while in the in-ertial range we observe the dimensional scaling (6). Thecrossover between the two regimes is expected to occurat δ ∗ ≃ ( ε/λ ) / . Rescaling the error δ with δ ∗ andΛ( δ ) with λ we find a good collapse of the two regimesof infinitesimal and finite errors, as shown in the insetof Fig. 5. Figure 5 also shows that the crossover rangebetween the two regimes increases with Re . One possibleexplanation for this long crossover is that the transitionbetween the two regimes involves the dynamics of ed-dies which are at the border between the inertial and thedissipative scales, in the so-called intermediate dissipa-tive range [18]. The extension of this range is known togrow with the Reynolds number, and this could cause thebroadening of the crossover regime for the FSLE.Remarkably, Figure 5 shows that in the scaling rangeΛ( δ ) ∼ δ − the error growth rate Λ becomes independentboth on the Reynolds number and on the values of theLyapunov exponent. The independence of the FSLE inthe scaling range on the value λ observed for infinitesi-mal errors provides a clear explanation of how in turbu-lent flows it is possible to observe the coexistence of longpredictability time at large scales and strong chaoticityat small scales.In conclusion, we studied the chaotic and predictabilityproperties of fully developed turbulence by simulatingtwo realizations of the velocity field initially separated bya very small perturbation. At short times the separationincreases exponentially as a consequence of the chaoticityof the flow. Finite perturbations increase linearly in time,as predicted by dimensional arguments, and the time forthe perturbation to affect a wavenumber k in the inertialrange is proportional to ε − / k − / .The Lyapunov exponent is found to grow with theReynolds number faster than what predicted by a di-mensional argument and intermittency models and, as aconsequence, the product λτ η grows with Re . This in-dicates that the strong, intermittent fluctuations of tur-bulence at small scales give diverse contributions on dif-ferent observables. In addition to the interest for manyapplications, turbulence is a prototypical example of sys-tem with many scales and characteristic times. Our re-sults on the chaoticity of turbulence and its dependenceon the number of active degrees of freedom are thereforeof general interest for the study of extended dynamicalsystems.The Authors gratefully acknowledge support from theSimons Center for Geometry and Physics, Stony BrookUniversity, where part of this work was performed. TheCOST Action MP1305, supported by COST (EuropeanCooperation in Science and Technology) is acknowledged.Numerical simulations have been performed at Cinecawithin the INFN-Cineca agreement INF17 fldturb . [1] D. Ruelle, Phys. Lett. , 81 (1979).[2] R. G. Deissler, Phys. Fluids , 1453 (1986).[3] E. N. Lorenz, Tellus , 289 (1969).[4] G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani,Phys. Rep. , 367 (2002). [5] J. R. Garratt, The atmospheric boundary layer , vol. 416(Cambridge University Press, 1992).[6] C. E. Leith, J. Atmos. Sci. , 145 (1971).[7] C. E. Leith and R. H. Kraichnan, J. Atmos. Sci. , 1041(1972).[8] O. M´etais and M. Lesieur, J. Atmos. 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