Cascades of energy and helicity in the GOY shell model of turbulence

Cascades of energy and helicity in the GOY shell model of turbulenceP. D. DitlevsenThe Niels Bohr Institute, Department for Geophysics, University of Copenhagen, Juliane Maries Vej 30,DK-2100 Copenhagen O, DenmarkReceived 22 July 1996; accepted 16 January 1997The effect of extreme hyperviscous damping, k p n ,p is studied numerically in the GOY shellmodel of turbulence. It has resently been demonstrated Leveque and She, Phys. Rev. Lett. 75, 26901995 that the inertial range scaling in the GOY model is non-universal and dependent on theviscous damping. In the present study it is shown that the deviation from Kolmogorov scaling is dueto the cascade of the second inviscid invariant. This invariant is non-positive definite and in thissense analogous to the helicity of 3D turbulent flow. © 1997 American Institute of Physics.S1070-66319702905-XThe invicid invariants, like the energy, of the Navier–Stokes equations are important quantities determining thedynamics of turbulent flow. The major difference between2D and 3D flow originates from the existence of a secondinviscid invariant, enstrophy, in 2D—absent in 3D flow. Thisgives strong constraints on the 2D flow dynamics. Conservationof enstrophy leads to the forward cascade of enstrophyand the backward cascade of energy in 2D, giving rise tolarge-scale coherent structures in the flow. In 3D there alsoexists a second quadratic inviscid invariant, namely the helicity,defined as the integral of the scalar product of thevelocity and the vorticity. It has been proposed to be importantfor 3D turbulence. 1 It was shown by Kraichnan 2 that theinteraction between waves of opposite helicity is strongerthan interactions between waves of the same helicity. Consequently,helical flow might slow down the cascade of energyto the dissipation range. 3Since the spectral helicity density grows with a factork, the modulus of the wave-vector, over the energy densitythe helicity dissipation would, in the R→ limit, preventenergy dissipation. The situation would resemble that of 2Dturbulence where the cascade of enstrophy prevents the forwardcascade of energy. In 2D turbulence the conservation ofboth dku(k) 2 and dkk 2 u(k) 2 implies that the forwardcascade of enstrophy must be accompanied by a backwardcascade of energy. This strong constraint does, however, notapply in the case of helicity transfer since helicity is notpositive definite. So we only have the weaker assessmentmentioned above that the efficiency of energy dissipationcould be depending on the non-linear helicity transfer andthe helicity dissipation.Considerable interest has lately been given to the GOYmodel of turbulence, introduced by Gletzer and examined byYamada and Ohkitani. 4 Comprehensive lists of recent referenceson the GOY model can be found in Refs. 5 and 6. Thebehavior of the helicity in different shell models have recentlybeen investigated. 6,7The GOY model is a simplified reduced wave-numberanalog to the spectral Navier–Stokes equations. The spectraldomain is represented as shells, each of which is defined bya wavenumber k n k 0 n , where is a scaling parameterdefining the shell spacing. There are 2N degrees of freedom,where N is the number of shells, namely the generalizedcomplex shell velocities, u n for n1, N. The dynamicalequation for the shell velocities isu˙ nik nau n2* u n1* b u n1* u n1* c 2 u n1* u n2* k n p u n f n,n0 ,where the first term represents the non-linear wave interactionor advection, the second term is the dissipation, and thethird term the forcing, where n 0 is some small wavenumber.Throughout this paper the standard 3D GOY model parametervalues, 2, k 0 4 , a1, bc1/2 is used.The GOY model in this form contains no information aboutphases between waves, thus there cannot be assigned a flowfield in real space.The model has two conserved integrals, in the case of noforcing and no dissipation ( f 0) inviscid invariants.These are, E 2 1 Nn1 u n 2 and H 2 1 Nn1 (1) n k n u n 2which corresponds to the conservation of energy and a secondnon-positive definite quantity interpreted as analogous tohelicity 5 for the Navier–Stokes equation of 3D turbulence,hereafter referred to as the helicity. It should be stressed thatthis analogy is only in the sense that both quantities arenon-positive definite. In this model each shell is maximallyhelical with alternating sign, since the numerical value of thehelicity density is k times the energy density.The forcing in 1 is applied at a small wavenumber andthe dissipation dominates at large wavenumbers, so we candefine an inertial range where the non-linear energy cascadingterms dominate and the Kolmogorov scaling argumentsapply. For an energy cascade we have the ‘‘Kolmogorovscaling’’ of the shell-velocities, u 1/3 k 1/3 , and for a helicitycascade we have the ‘‘helicity scaling,’’u˜1/3 k 2/3 , where and ˜ are the mean dissipation perunit time of energy and helicity, respectively. The model hasboth the Kolmogorov scaling, u n k 1/3 n g(n) and the helicityscaling, u n (1) n k 2/3 n g(n) as unstable fixed points in theunforced and inviscid case. The function, g(n) g(n 3),isany mod3 function. The mod3 symmetry is an artifact ofthe GOY model discussed in detail in Ref. 5. It will becomeimportant in the following. The Kolmogorov fixed pointplays an important role for the behavior of the GOY model,with forcing and dissipation, in the sense that the phase space11482 Phys. Fluids 9 (5), May 1997 1070-6631/97/9(5)/1482/3/$10.00 © 1997 American Institute of PhysicsDownloaded¬02¬Sep¬2005¬to¬130.225.238.2.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pof.aip.org/pof/copyright.jsp

FIG. 1. The standard GOY model with f 0 (1i)10 3 ,n 0 4 and10 5 . The model has 20 shells and is as before the run for 3000 timeunits after a 1000 time units spinup starting from the unstable Kolmogorovfixed point. log (u n ) is shown as a function of the shell numbernlog (k n ). The line indicates the Kolmogorov scaling with a spectral slopeof 1/3.trajectory of the shell velocities seems to ‘‘curl around’’ thispoint with the average values of the velocities close to thefixed point values. It was shown in a numerical study byLeveque and She 8 that the inertial range scaling in the GOYmodel is not universal and is dependent on the form of theviscous damping. They attributed this non-universality to thereflection of energy flow from the viscous subrange back intothe inertial range in the case of hyperviscosity (p2). Bystudying the extreme case p, I suggest a different explanationfor the non-universality of the inertial range scaling,namely that the cascade of helicity blocks the cascade ofenergy and thus changes the scaling. This effect is probablyspecific to the GOY model, and a consequence of the specificodd–even asymmetry of the helicity in the GOY model.The numerical study is performed on two versions of theGOY model. First, the usual model with normal (p2) dissipationand second, with viscosity only applied on the outermostshell, corresponding to p hyperviscosity. Theforcing in 1 is in both cases taken to be f f 0 /u n0* wheref 0 is a constant. This gives a constant input of energy andhelicity per unit time.Figure 1 shows the result of a numerical calculation forf 0 (1i)10 3 ,n 0 4 and 10 5 . The model has 20shells and is run for 3000 time units after a 1000 time unitsspinup starting from the unstable Kolmogorov fixed point.The first panel shows log (u n ) as a function of the shellnumber nlog (k n ), where • denotes the temporal average.The spectral slope is close to the Kolmogorov scaling,shown by the line, thus the model shows an energy cascade.The sources and sinks for energy and helicity areĖ u n*f n k 2 n u 2 n 2nandḢ 1 n k n u n*f n 1 n k 3 n u 2 n . 3nThis means that the helicity of opposite signs is dissipatedat every second shell and therefore total helicity isproduced for odd numbered shells in the dissipation range.FIG. 2. The modified GOY model with f 0 (1i)10 8 and 0 10 7 k 2 20 429.5. This model has likewise 20 shells and is run for3000 time units. The first panel is the same as in Fig. 1. The line in thefigure has a slope of 2/3 corresponding to the cascade of helicity. Secondpanel shows the spectrum of U n u(n1)u(n)u(n1) 1/3 . The line has aslope of 1/2 corresponding to the equipartition of helicity.The inertial range flow is dominated by the non-lineartransfer of the conserved quantities. The key assumption byKolmogorov 1941 is that the inertial range flow does notdepend on the specific form of the small scale viscous dissipation.This is not the case for the GOY model. In order toillustrate this point a study of a slightly modified GOYmodel is presented here. The dissipation is taken to be activeonly on the outermost shell, number N20; this correspondsto hyperviscosity in the limit, p⇒, with 0 k pN and 0 held constant. With this choice of dissipation the behaviorof the GOY model changes dramatically even though thenon-linear terms in the governing equation remains unchanged.Figure 2 shows the result of a numerical integration withthe modified GOY model where f 0 (1i)10 8 and 0 10 7 k 2 20 429.5. The first panel shows the velocityspectrum where the mod3 symmetry of the shell modelbecomes dominant. The scaling behavior of shells numbered10,13,16,19 is different from the rest. The line in the figurehas a slope of 2/3 corresponding to the cascade of helicityrather than energy.However, if U n u(n1)u(n)u(n1) 1/3 is consideredthe mod3 symmetry is eliminated and the inertialrange scaling approximately reemerges as an arithmeticmean of the scaling behaviors for the shells3n, 3n1, 3n2 Fig. 2, second panel. The line in thefigure has a slope of 1/2 corresponding to equipartitioningof helicity. The hyper-viscosity of the model, only pullingout helicity through the 18, 19, 20 triad, cannot maintain ahelicity cascade.The result can be interpreted as a spectral bump at theend of the spectrum with an approximate equipartition ofhelicity. Similar results have been found by Borue andPhys. Fluids, Vol. 9, No. 5, May 1997Brief Communications1483Downloaded¬02¬Sep¬2005¬to¬130.225.238.2.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pof.aip.org/pof/copyright.jsp

Orszag 9 for numerical simulations of the 3D Navier–Stokesequation. In this case the accumulation seems to be an effectof energy not being able to be transferred across the ultravioletcutoff.In the case ( f 0) the model has, discarding theboundary effects, besides the two scaling fixed points a periodicsolution. It is easy to verify that the following satisfiesthe dynamical equation 1:u 3n2 tsk 1 0 3n ,u 3n1 t 3n 1 e i 1 2 t ,u 3n t 3n 2 e i 1 2 t ,with 1 s 4 1b 2 c 31 ,45 2 s 33 1b 31 c 31 ,and s is an arbitrary constant. The scaling parameter, , isrelated to the scaling fixed point by (1)/3, wherez is a solution to 1bzcz 2 0, thus a generator ofone of the conserved quantities. 10 So there are two valuespossible for ; (2i/log())/3 corresponding to thegenerator, z2, of helicity, or the fluxless fixed point ofthe GOY model, and 1/3 corresponding to the generator,z1, of energy, or the Kolmogorov fixed point.There is in this periodic solution a complete phase lockingof all the shells. The energy and helicity fluxes are bothzero in the periodic solution, so in some sense it correspondsto the fluxless fixed point of the GOY model.The shell-velocities of shells 8, 11, 14, 17, 20 are out ofphase with those of shells 9, 12, 15, 18, while shell-velocitiesof shells 10, 13, 16, 19 are almost constant. From this itseems as if the periodic solution plays the same role for thismodified GOY model as does the Kolmogorov fixed pointfor the GOY model. This could indicate the existence of anunstable limit cycle in the modified GOY model.The dissipation of helicity is irregular, associated withthe bursts in shell number 20 and all other shells, the energydissipation is completely blocked, resulting in a steadyincrease of energy concentrated on shells 10,13,16,19. Themechanism for preventing the forward cascade of energy inthis model is analogous to the 2D case, where the forwardcascade of enstrophy governs the dynamics.The system shows no signs of approaching a statisticalequilibrium simply because the helicity can grow to arbitrarilylarge negative values dominated by shell number 19.The reason for this can be understood by examining the dissipativeenergy balance. In the case of normal viscosity itfollows from 2 that the phase space trajectory will be attractedto the hyper-ellipsoid given by k 2 n u n 2 f whichis a compact 2N1 dimensional object. In the p casethis object will be the 2N1 dimensional ‘‘hyper-cylinder’’defined by u 20 f /k 2 20 ). This is not compact and ergodicitydoes not apply, thus the p→ is a singular limit and nostatistical equilibrium can be reached. The relative change inenergy over the integration is of the order 10 2 such that thesystem is in a quasi-equilibrium state where statistical equilibriumis reestablished in the limit where the energy injectionrate goes to zero. Taking an even smaller forcing doesnot change the statistics, thus the statistical time averaging ismeaningful.If the energy cascade is effectively blocked the energyshould be pulled out of the system by a drag at the smallwavenumbers corresponding to the backward cascade in 2D.This is not seen, so an inverse energy cascade cannot beestablished. The model was also run with the usual dissipationbut only active on every second shell. The result of thisrun was essentially the same as for the modified GOY model.In conclusion we see that the non-linear transfer in theGOY model depends crucially on the dissipation propertiesfor both conserved quantities, energy and helicity. It is stillan open question how much the GOY model reflects thedynamics of the Navier–Stokes equation. These findings certainlyindicate that the model is too restricted in some senseto represent real flow. The dissipative term introduced in themodified model is certainly not very realistic, in real flowhelicity of both signs will be dissipated, which it is not in themodified model presented here. The findings from thissimple model might indicate that in the inertial range flowthe conservation of helicity could block the energy cascadeand thus alter the Kolmogorov k 5/3 scaling.ACKNOWLEDGMENTSI would like to thank A. Wiin-Nielsen, J. R. Herring, andR. M. Kerr for valuable discussions. The work was grantedby the Carlsberg foundation.1 J. C. Andre and M. Lesieur, ‘‘Influence of helicity on the evolution ofisotropic turbulence at high Reynolds number,’’ J. Fluid Mech. 81, 1871977.2 R. H. Kraichnan, ‘‘Helical turbulence and absolute equilibrium,’’ J. FluidMech. 59, 745 1973.3 W-S. Wu, D. K. Lilly, and R. M. Kerr, ‘‘Helicity and thermal convectionwith shear,’’ J. Atmos. Sci. 49, 1800 1991.4 M. Yamada and K. Ohkitani, ‘‘Lyapunov spectrum of a chaotic model ofthree-dimensional turbulence,’’ J. Phys. Soc. Jpn. 56, 4210 1987; ‘‘Theinertial subrange and non-positive Lyapunov exponents in fully-developedturbulence,’’ Progr. Theor. Phys. 79, 1265 1988; 60, 983 1988.5 L. Kadanoff, D. Lohse, J. Wang, and R. Benzi, ‘‘Scaling and dissipation inthe GOY shell model,’’ Phys. Fluids 7, 617 1995.6 R. Benzi, L. Biferale, R. M. Kerr, and E. Trovatore, ‘‘Helical shell modelsfor three dimensional turbulence,’’ Phys. Rev. E 53, 3541 1996.7 L. Biferale and R. Kerr, ‘‘On the role of inviscid invariants in shell modelsof turbulence,’’ Phys. Rev. E 52, 6113 1995.8 E. Leveque and Z-S. She, ‘‘Viscous effects on inertial range scaling in adynamical model of turbulence,’’ Phys. Rev. Lett. 75, 2690 1995.9 V. Borue and S. A. Orszag, ‘‘Self-similar decay of three-dimensional homogeneousturbulence with hyperviscosity,’’ Phys. Rev. E 51, R8561995.10 P. D. Ditlevsen and I. A. Mogensen, ‘‘Cascade and statistical equilibriumin shell models of turbulence,’’ Phys. Rev. E. 53, 4785 1996.1484 Phys. Fluids, Vol. 9, No. 5, May 1997 Brief CommunicationsDownloaded¬02¬Sep¬2005¬to¬130.225.238.2.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://pof.aip.org/pof/copyright.jsp