Boffetta, Guido

Musacchio, S.

English

Institutional Research Information System University of Turin

PHYSICS OF FLUIDS VOLUME 13, NUMBER 4 APRIL 2001Predictability of the inverse energy cascade in 2D turbulenceG. Boffetta and S. MusacchioDipartimento di Fisica Generale, Universita` di Torino, Via Pietro Giuria 1, 10125 Torino, Italyand INFM Sezione di Torino Universita`, Torino, Italy~Received 23 June 2000; accepted 5 January 2001!The predictability problem in the inverse energy cascade of two-dimensional turbulence isaddressed by means of high resolution direct numerical simulations. The growth rate as a functionof the error level is determined by means of a finite size extension of the Lyapunov exponent. Forerrors within the inertial range, the linear growth of the error energy, predicted by dimensionalargument, is verified with great accuracy. Our numerical findings quantitatively confirm the resultsof the classical TFM closure approximation. © 2001 American Institute of Physics.@DOI: 10.1063/1.1350877#BRIEF COMMUNICATIONSThe purpose of this Brief Communications section is to present important research results of more limited scope than regulararticles appearing in Physics of Fluids. Submission of material of a peripheral or cursory nature is strongly discouraged. BriefCommunications cannot exceed four printed pages in length, including space allowed for title, figures, tables, references, and anabstract limited to about 100 words.Unpredictability is an essential property of turbulentflows. Turbulence is characterized by a large number of de-grees of freedom interacting with nonlinear dynamics. Thusturbulence is chaotic ~and hence unpredictable!, but the stan-dard approach of dynamical system theory is not sufficient tocharacterize predictability in turbulence.1In fully developed turbulence, the maximum Lyapunovexponent diverges, with the Reynolds number thus beingvery large for typical turbulent flows.2 Nevertheless, a largevalue of the Lyapunov exponent does not imply automati-cally short time predictability. A familiar example is the at-mosphere dynamics: convective motions in the atmospheremake the small scale features unpredictable after 1 h or less,but large scale dynamics can be predicted for several days, asit is demonstrated by weather forecasting. This effect, whichcan be called ‘‘strong chaos with weak butterfly effect,’’arises in systems possessing many characteristic scales andtimes. From this point of view, turbulence probably repre-sents the example most extensively studied.The first attempts at the study of predictability in turbu-lence date back to the pioneering work of Lorenz1 and to theKraichnan and Leith papers.3,4 On the basis of closure ap-proximations, it was possible to obtain quantitative predic-tions on the evolution of the error in different turbulent situ-ations, both in two and three dimensions.A more recent approach to the problem is based on dy-namical system theory. Chaotic properties and predictabilityof turbulent flow have been extensively investigated in sim-plified models of turbulence, called shell models, with par-ticular emphasis on the relations with intermittency.5–9 Be-cause predictability experiments in fully developedturbulence are numerically rather expensive, a similar study1070-6631/2001/13(4)/1060/3/$18.00 106Downloaded 20 Mar 2001 to 192.84.137.11. Redistribution subjeon direct numerical simulations of Navier–Stokes equationsis still lacking.In this communication we address the predictabilityproblem for two-dimensional turbulence by means of highresolution direct numerical simulations. Turbulence is gener-ated in the inverse cascade regime where a robust energycascade is observed.10 The absence of intermittency correc-tions makes the problem simpler than in the three-dimensional case: velocity statistics ~energy spectrum, struc-ture functions! are found to be in close agreement with self-similar theory a` la Kolmogorov.The model equation is the two-dimensional Navier–Stokes equation written for the scalar vorticity v(r,t)52Dc(r,t) with generalized dissipation and linear friction] tv1J~v ,c!5~21 !p11npDpv2av2 f , ~1!where J represents the Jacobian with the stream function cand the velocity is u5(]yc ,2]xc). p is the order of thedissipation; p51 for ordinary dissipation, p.1 for hyper-viscosity. As it is customary in numerical simulations, weuse hyperviscous dissipation (p58) in order to extend theinertial range. Although this can affect the small scale fea-tures of the vorticity field,11 in our simulations dissipation isnot involved in the cascade and has simply the role of re-moving entrophy at small scales. The friction term in ~1!removes energy at large scales: it is necessary in order toavoid Bose–Einstein condensation on the gravest mode12and to obtain a stationary state. Energy is injected into thesystem by a random forcing d-correlated in time which isactive on a shell of wavenumbers around k f only. Numericalintegration of ~1! is performed by a standard pseudo-spectral0 © 2001 American Institute of Physicsct to AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html1061Phys. Fluids, Vol. 13, No. 4, April 2001 Predictability of the energy cascadecode fully dealiased with second-order Adams–Bashforthtime stepping on a doubly periodic square domain with reso-lution N51024.Stationary turbulent flow is obtained after a very longsimulation starting from a zero initial vorticity field. At sta-tionarity one observes a wide inertial range with a well de-veloped Kolmogorov energy spectrum E(k)5Ce2/3k25/3~Fig. 1!. Structure functions in physical space are found inagreement with the self-similar Kolmogorov theory.10Starting from a configuration of the velocity fieldu1(r,0) in the stationary state, one generates a second ~per-turbed! configurationu2~r,0!5u1~r,0!1A2du~r,0!, ~2!in which the initial error du(r,0) is very small ~the factor A2is only for normalization convenience!. The two configura-tions are integrated in time according to ~1! and the evolutionof the error du(r,t) is computed according to ~2!. Of course,because we are interested in studying the error growth in-duced by the turbulent dynamics, we use the same realizationof random forcing in both simulations.From ~2! one defines the error energy and the error en-ergy spectrum as4,13ED~ t !5E0‘ED~k ,t ! dk512E udu~r,t !u2d2r . ~3!Normalization in ~2! ensures that ED(k ,t)→E(k) for un-correlated fields ~i.e., for t→‘). Assuming that the initialerror can be considered infinitesimal, the magnitude of thedifference field starts growing exponentially and ED(t).ED(0)exp(2lt) where l is the maximum Lyapunov expo-nent of the system.14 The error growth in this stage is con-fined at the faster scales in the inertial range, correspondingin our model to the scales close to the forcing wavenumberk f , while at larger scales the two fields remain correlated~see Fig. 1!. At larger times, when ED(k f ,t) becomes com-parable with E(k f), the exponential growth terminates, be-cause the two fields are completely decorrelated at smallscales (k<k f). The error growth continues at larger scales inthe inertial range, where the two fields are still correlated,and an algebraic regime sets in. The dimensional predictionFIG. 1. Stationary energy spectrum E(k) ~thick line! and error spectrumED(k ,t) at time t50.1,0.2,0.4,0.8,1.6. k f5320 is the forcing wavenumber.In the inset we plot the compensated spectrum e22/3k5/3E(k).Downloaded 20 Mar 2001 to 192.84.137.11. Redistribution subjeproposed by Lorenz1 assumes that the time it takes for theerror to induce a complete uncertainty at wavenumber k isproportional to the characteristic time at that scale, t}t(k).Within the Kolmogorov framework, t(k).e21/3k22/3. Re-verting this dimensional expression one can say that at fixedtime the error have reached the scale kE(t).e21/2t23/2. Atlarger scales the error is still very small in comparison withthe typical energy, while at smaller scale the two fields arecompletely decorrelated. Thus at each time we have a char-acteristic scale kE(t) which divide uncorrelated scales fromcorrelated ones:ED~k ,t !5H 0 if k,kE~ t !,E~k ! if k.kE~ t !. ~4!By inserting ~4! in ~3!, using the Kolmogorov spectrum forE(k) and assuming the dimensional expression for kE(t) oneends with the prediction1,9ED~ t !5Get . ~5!The numerical constant G in ~5! can be obtained only byrepeating the argument more formally within a closureframework.3,4,15 The physical meaning of G is the ratio of theFIG. 2. Average error energy ^ED(t)& growth. Dashed line represents clo-sure prediction ~5!, dotted line is the saturation value E. The initial expo-nential growth is emphasized by the lin-log plot in the inset.FIG. 3. Finite size Lyapunov exponent l(d) as a function of velocity un-certainty d . The asymptotic constant value for d→0 is the maximumLyapunov exponent of the turbulent flow. Dashed line represent prediction~7!. In the inset we show in the compensated plot l(d)d2/e . The line rep-resent the fit to the constant A.3.9.ct to AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html1062 Phys. Fluids, Vol. 13, No. 4, April 2001 G. Boffetta and S. Musacchiorate of uncorrelated energy production to the rate of energyinjected by the forcing and transferred to large scales e .In Fig. 2 we plot the time evolution of the error energy^ED(t)& obtained from direct numerical simulations aver-aged over 20 realizations. The exponential regime is clearlyvisible at small time, while the linear regime ~5! is barelyobservable, making the precise determination of G difficult.The dimensional predictability argument given abovecan be rephrased in a language more close to dynamical sys-tems by introducing the finite size Lyapunov exponent~FSLE! analysis. FSLE is a generalization of the Lyapunovexponent to finite size errors, which was recently proposedfor the analysis of systems with many characteristic scales.8In a nutshell, one computes the ‘‘error doubling time’’Tr(d), i.e., the time it takes for an error of size d to grow afactor r ~for r52 we have actually a doubling time!. TheFSLE is defined in terms of the average doubling time asl~d!51^Tr~d!&lnr . ~6!It is easy to show that definition ~6! reduces to the standardLyapunov exponent l in the infinitesimal error limit d→0.8For finite error, the FSLE measures the effective errorgrowth rate at error size d . Let us remark that taking aver-ages at fixed time, as in ~5!, is not the same as averaging atfixed error size, as in ~6!. This is particularly true in the caseof intermittent systems, in which strong fluctuations of theerror in different realizations can hide scaling laws like ~5!.8From a numerical point of view, the computation of l(d) isnot more expensive than the computation of the Lyapunovexponent with a standard algorithm.The same dimensional argument leading to ~5! can beused for predicting the behavior of the FSLE in the inertialrange. Taking d5AED as error, one easily obtainsl~d!5Aed22 . ~7!The constant A, which again is not determined by dimen-sional arguments, relates the energy flux in the cascade to therate of error growth. In the absence of intermittency and forr.1, the two constants in ~5! and ~7! are related by A5(r21)/logr G.The scaling ~7!, which can be shown to be unaffected bypossible intermittency corrections ~as in 3D turbulence8!, isvalid within the inertial range u(k f),d,U where u(k f) isthe typical velocity fluctuation at forcing wavenumber andU.A2E is the typical large scale velocity. At large errorsd.U , we expect error saturation, ED→E and thus l(d)→0.Figure 3 shows the FSLE computed from our simula-tions. For small errors d,u(k f) @corresponding to an errorspectrum ED(k f ,t)!E(k f)] we observe the convergence ofl(d) to the leading Lyapunov exponent. Its value is essen-tially the inverse of the smallest characteristic time in thesystem and represents the growth rate of the most unstablefeatures. At larger d.1022 we clearly see the transition tothe inertial range scaling ~7!. At further large d.U.0.1,l(d) falls down to zero in correspondence of error satura-tion.Downloaded 20 Mar 2001 to 192.84.137.11. Redistribution subjeIn order to emphasize scaling ~7!, in Fig. 3 we also showthe compensation of l(d) with ed22. Prediction ~7! is veri-fied with very high accuracy which allows us to determinethe value of A.3.960.1. With the present value of r.1.12, this corresponds to a value G.4.1. The physicalpicture we obtain is that the creation of uncorrelated energyin the inertial range due to chaotic dynamics is about fourtimes faster than the energy transfer rate.Our numerical result is in remarkable agreement with theold prediction obtained within the test field model closure4which gives G54.19. At least from the point of view ofpredictability, two-dimensional turbulence thus seems to bevery well captured by low-order closure scheme. As a con-sequence we can exclude, on the basis of our numerical find-ings, the existence of intermittency effects in the inverse cas-cade of error. This is a result which is probably of moregeneral interest than the specific problem discussed in thiscommunication.ACKNOWLEDGMENTSWe are grateful to A. Celani, M. Cencini and A. Vul-piani for useful discussions. Support from INFM ‘‘PRATURBO’’ is gratefully acknowledged. Numerical simula-tions were partially performed at CINECA within the project‘‘Fully developed two-dimensional turbulence.’’1E. N. Lorenz, ‘‘The predictability of a flow which possesses many scalesof motion,’’ Tellus 21, 289 ~1969!.2D. Ruelle, ‘‘Microscopic fluctuations and turbulence,’’ Phys. Lett. A 72,81 ~1979!.3C. E. Leith, ‘‘Atmospheric predictability and two-dimensional turbu-lence,’’ J. Atmos. Sci. 28, 145 ~1971!.4C. E. Leith and R. H. Kraichnan, ‘‘Predictability of turbulent flows,’’ J.Atmos. Sci. 29, 1041 ~1972!.5K. Ohkitani and M. Yamada, ‘‘Temporal intermittency in the energy cas-cade process and local Lyapunov analysis in fully-developed model turbu-lence,’’ Prog. Theor. Phys. 81, 329 ~1989!.6M. H. Jensen, G. Paladin, and A. Vulpiani, ‘‘Intermittency in a cascademodel for three-dimensional turbulence,’’ Phys. Rev. A 43, 798 ~1991!.7E. Aurell, G. Boffetta, A. Crisanti, P. Frick, G. Paladin, and A. Vulpiani,‘‘Statistical mechanics of shell models for 2D-turbulence,’’ Phys. Rev. E50, 4705 ~1994!.8E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani, ‘‘Growthof non-infinitesimal perturbations in turbulence,’’ Phys. Rev. Lett. 77,1262 ~1996!; ‘‘Predictability in the large: An extension of the concept ofLyapunov exponent,’’ J. Phys. A 30, 1 ~1997!.9T. Bohr, M. Jensen, G. Paladin, and A. Vulpiani, Dynamical SystemsApproach to Turbulence ~Cambridge University Press, Cambridge, UK,1998!.10G. Boffetta, A. Celani, and M. Vergassola, ‘‘Inverse energy cascade intwo-dimensional turbulence: Deviations from Gaussian behavior,’’ Phys.Rev. E 61, R29 ~2000!.11V. Borue, ‘‘Spectral exponents of enstrophy cascade in stationary two-dimensional homogeneous turbulence,’’ Phys. Rev. Lett. 71, 3967 ~1993!.12L. M. Smith and V. Yakhot, ‘‘Bose condensation and small-scale structuregeneration in a random force driven 2D turbulence,’’ Phys. Rev. Lett. 71,352 ~1993!.13M. Lesieur, Turbulence in Fluids, 3rd ed. ~Kluwer, Dordrecht, 1997!.14Actually, the growth rate for the energy is the generalized Lyapunov ex-ponent L(2), but one has simply L(2)52l in nonintermittent systems~Ref. 9!.15O. Metais and M. Lesieur, ‘‘Statistical predictability of decaying turbu-lence,’’ J. Atmos. Sci. 43, 857 ~1986!.ct to AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html

Predictability of the inverse energy cascade in 2D turbulence

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Predictability of the inverse energy cascade in 2D turbulence

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