Fibonacci turbulence
FFibonacci turbulence
N. Vladimirova , , M. Shavit and G. Falkovich Weizmann Institute of Science, Rehovot 76100 IsraelBrown University, Providence, RI 02912, USA (Dated: January 27, 2021)Never is the difference between thermal equilibrium and turbulence so dramatic, as when aquadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuatingmodes. That happens in two very different yet deeply connected classes of systems: incompressiblehydrodynamics and resonantly interacting waves. This work presents the first case of a detailedinformation-theoretic analysis of turbulence in such strongly interacting systems. The analysis elu-cidates the fundamental roles of space and time in setting the cascade direction and the changesof the statistics along it. We introduce a beautifully simple yet rich family of discrete models withneighboring triplet interactions and show that it has families of quadratic conservation laws definedby the Fibonacci numbers. Depending on the single model parameter, three types of turbulence werefound: single direct cascade, double cascade, and the first ever case of a single inverse cascade. Wedescribe quantitatively how deviation from thermal equilibrium all the way to turbulent cascadesmakes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probabil-ity distribution. We reveal where the information (entropy deficit) is encoded and disentangle thecommunication channels between modes, as quantified by the mutual information in pairs and theinteraction information inside triplets.
I. INTRODUCTION
Existence of quadratic invariants and Gaussianity ofequilibrium in a strongly interacting system may seemexceptional. And yet two very distinct wide classes ofphysical systems share that property. The first class isthe family of hydrodynamic models, starting from thecelebrated hydrodynamic Euler equation and includingmany equations for geophysical, astrophysical and mag-netohydrodynamic flows. The second class, as will bedescribed in this paper, contains systems of resonantlyinteracting waves. In particular, we show that the dis-cretized models of the first class exactly correspond tothe second one. We shall consider one particular (ar-guably the simplest) family of such models and describefar-from equilibrium (turbulent) states of such systems.In 1966 V. Arnold noticed the group-theoretic anal-ogy (whose big time in turbulence research may yet tocome) between the ordinary differential Euler equationsdescribing solid body dynamics and the partial differ-ential Euler equation describing fluid flows: both canbe described as a geodesic equation on the relevant Liegroup with respect to a one sided (left or right) invari-ant metric [1]. On the one hand, the moment of mo-mentum M of a solid body in the rotating referenceframe satisfies the Euler equation d M /dt = M × Ω ,where Ω i = I − ij M j and I ij is the inertia tensor. Onthe other hand, in an isentropic flow the velocity curl, ω = ∇ × v , satisfies another equation bearing the nameof Euler: ∂ω/∂t = ∇ × ( v × ω ). Both Euler equa-tions, for solids and for fluids, have quadratic nonlinear-ity and possess quadratic invariants. The same is true forthe rich family of two-dimensional hydrodynamic models,where a scalar field a (vorticity, temperature, potential)is linearly related to the stream function ψ , which deter-mines the velocity carrying the field: ∂a/∂t = − ( v · ∇ ) a , v = ( ∂ψ/∂y, − ∂ψ/∂x ), ψ ( r ) = (cid:82) d r (cid:48) | r − r (cid:48) | m − a ( r (cid:48) ). Forthe 2D Euler equation, m = 2. Other cases include sur-face geostrophic ( m = 1), rotating shallow fluid or mag-netized plasma ( m = − ∂a k ∂t = (cid:88) q k × q q m a q a k − q . (1)In 1969 A. Obukhov suggested to model fluid turbulenceby the chains of ODEs having these properties [2]:˙ u i = Γ ijl u j u l , Γ iil = 0 = Γ i,jl + Γ l,ji + Γ j,li . (2)The indices in (2) are lowered using the metric definedby the quadratic invariant 2 E = g ij u i u j . The position ofthe indices is irrelevant in what follows.Here we add the class of models describing resonantlyinteracting waves with the general Hamiltonian, H w = (cid:88) i ω i | b i | + (cid:88) ijl (cid:0) V l,ij b ∗ i b ∗ j b l + V ∗ l,ij b i b j b ∗ l (cid:1) , (3)where V l,ij (cid:54) = 0 only if ω i + ω j = ω l . By the gauge trans-formation, a i = b i exp( ıω i t ), we can turn the equations ofmotion, ı ˙ b i = ∂ H w /∂b ∗ i into a system of the type (1,2): ı ˙ a i = (cid:88) jl (cid:0) V ∗ i,jl a j a l + 2 V l,ij a ∗ j a l (cid:1) . (4)This means that quadratic and cubic parts of the Hamil-tonian are conserved separately. If such a system isbrought into contact with thermostat, it is straightfor-ward to show that the statistics is Gaussian: ln P{ a i } ∝− (cid:80) i ω i | a i | . Two-mode and three-mode systems are in-tegrable, multi-mode systems are generally dynamicallychaotic due to resonance overlap; islands of regular be-havior shrink exponentially with the growing number ofmodes. a r X i v : . [ n li n . C D ] J a n Our interest in resonances is connected to that in non-equilibrium. Thermal equilibrium does not distinguishbetween resonant and non-resonant interactions becauseof the detailed balance: whatever correlations can bebuilt over time between resonantly interacting modes, thereverse process destroying these correlations is equallyprobable. This is not so away from thermal equilibrium,especially in turbulence.Neglecting non-resonant and accounting only resonantinteractions is the standard approach to weakly inter-acting systems, even though the weak nonlinearity as-sumption breaks for resonant modes. Weak turbulencetheory gets around this by considering continuous distri-bution and integrating over resonances to get the mas-ter equation or kinetic wave equation [3–6]. There is astrong temptation in theoretical statistical physics to re-strict consideration to two opposite limits: either treatfew modes or infinitely many. The temptation is evenstronger in the studies on non-equilibrium. And yetnot only most of the real-world phenomena fall in be-tween these limits, but, as we show here, one learnssome fundamental lessons comparing equilibrium andnon-equilibrium states of systems with a finite numberof degrees of freedom, where phase coherence can playa prominent role. A similar lesson condensed matterphysics taught us by discovering the world of mesoscopicphenomena.The previous treatment of mode discreteness was fo-cused on the sparseness of resonances for the particularcases when resonant surfaces did not pass through inte-ger lattice [6, 7]. Yet in many cases resonance surfaceslay in the lattice. For example, in a quite generic caseof quadratic dispersion relation, ω k ∝ k , Pythagoreantheorem makes the resonance surface for three-wave in-teractions just perpendicular to any wavevector, so thatin any rectangular box resonantly interacting triads fillthe lattice of its modes.Class of models (1,2,4) is ideally suited for the com-parative analysis of thermal equilibrium and turbulence.We show here that such analysis sheds light on the mostfundamental aspects of turbulence, particularly the rolesof spatial and temporal scales in determining cascade di-rections and build-up of intermittency. We shall use boththe traditional viewpoint of cascades and the relativelyrecent viewpoint of information theory, that is we addressboth energy and entropy of turbulence. We consider theparticular sub-class of models that allow only neighbor-ing interactions, and find it the most versatile tool to dateto study turbulence as an ultimate far-from-equilibriumstate. We carry here such detailed study of the knowntypes of direct-only and double cascades with unprece-dented numerical resolution. Even more important, ourmodels allow for an inverse-only cascade never encoun-tered before. II. FIBONACCI TURBULENCE
We consider a sub-class of the models (1,2,4) which isHamiltonian with a local interaction: H = (cid:88) i V i (cid:0) a ∗ i a ∗ i +1 a i +2 + a i a i +1 a ∗ i +2 (cid:1) . (5)The equations of motion ı ˙ a i = ∂ H /∂a ∗ i are as follows: ı ˙ a i = V i − a i − a i − + V i − a ∗ i − a i +1 + V i a ∗ i +1 a i +2 . (6)This model can have numerous classical and quantum ap-plications, since i can be denoting real-space sites, spec-tral modes, masses of particles, number of monomers ina polymers, etc. The Hamiltonian describes, in particu-lar, decay and coalescence of waves or quantum particles,breakdown and coagulation of particles or polymerizationof polymers, etc, when interactions of comparable enti-ties are dominant. In particular, the model describes theresonant interaction of waves whose frequencies are theFibonacci numbers F i = { , , , , . . . } defined by theidentity F i + F i +1 = F i +2 with F = 0. Indeed, suchwaves are described by the Hamiltonian H = (cid:88) i (cid:2) F i | a i | + V i (cid:0) a ∗ i a ∗ i +1 a i +2 + a i a i +1 a ∗ i +2 (cid:1)(cid:3) . (7)The first term corresponds to the linear terms in theequations of motion, while the second term representsthe only possible resonant interactions, since no non-consecutive Fibonacci numbers sum into another Fi-bonacci number (Zeckendorf theorem). For any real t ,the Hamiltonian (7) is invariant under the U (1) × U (1)transformation a i → a i e ıF i t due to F i + F i +1 = F i +2 .The transformation (to the wave envelopes) reduces theequation of motion ˙ a i = ∂ H /∂a ∗ i to (6).If i are spectral parameters, they are usually under-stood as shell numbers with wave numbers k = F i =[ φ i − ( − φ ) − i ] / √ φ is the golden mean: φ − φ =1), so that power-law dependencies on k mean exponen-tial dependencies on i . The model (6) thus belongs tothe class of the so-called shell models [8], that is (2) withneighboring interactions. Coefficients of shell models arechosen to have one or two quadratic integrals of motion.In particular, the Sabra shell model [9, 10] for a particu-lar choice of coefficients (non-surprisingly, connected bythe golden ratio) coincides with (6), which is Hamiltonianand has the cubic integral of motion (5).It is straightforward to show that for arbitrary V i , thedynamical equations (6) conserve a one-parameter familyof quadratic invariants F k = (cid:88) i =1 F i + k − | a i | , (8)where k could be of either sign if we define negative Fi-bonacci numbers: F − j = ( − j +1 F j . All invariants canbe obtained as linear combinations of any two of them.For example, the first two integrals are positive, indepen-dent, and in involution: F = (cid:88) i =1 F i | a i | , F = (cid:88) i =1 F i +1 | a i | . (9)In a closed system, the microcanonical equilibrium is P = δ ( H − C ) (cid:81) k δ ( F k − C k ). We now add dissipationand white-in-time pumping:˙ a i = − ı∂ H /∂a ∗ i + ξ i − γ i a i . (10)Here (cid:104) ξ i a ∗ i (cid:105) = P i /
2. Denoting H i = 2Re( a ∗ i a i − a i − ), itis straightforward to show, also in a general case (3,4),that (cid:80) i d (cid:104)H i (cid:105) /dt = − (cid:80) i ( γ i + γ i − + γ i − ) (cid:104)H i (cid:105) , whichmust be zero in a steady state. At least when all sums γ i + γ i − + γ i − are the same, (cid:80) i (cid:104)H i (cid:105) = (cid:104)H(cid:105) = 0 (one canprobably imagine exotic cases where separate (cid:104)H i (cid:105) (cid:54) = 0but we shall not consider them). If pumping and damp-ing are in a detailed balance, so that (cid:80) k α k F i + k = γ i /P i for every i , the thermal equilibrium distribution is Gaus-sian: P = exp( − (cid:80) k α k F k ) — it is a steady solution ofthe Fokker-Planck equation realizing maximum entropy.The distribution is exactly Gaussian despite the systembeing described by a cubic Hamiltonian and thus stronglyinteracting. The only restriction on the numbers α k isnormalization. In particular, when only α = 1 /T isnonzero, we get the equilibrium equipartition with theoccupation numbers n i ≡ (cid:104)| a i | (cid:105) = P i /γ i = T /F i .In a turbulent cascade, the fluxes of the quadratic in-variants can be expressed via the third cumulant. Gaugeinvariance and Zeckendorf theorem ensure that the triplecumulants are nonzero only for consecutive modes in theinertial range: J i ≡ Im (cid:104) a ∗ i a i − a i − (cid:105) , (11) F i + k − d (cid:104)| a i | (cid:105) dt = 2 F i + k − ( V i − J i − V i − J i +1 − V i J i +2 )= Π k ( i − − Π k ( i ) = − ∂ i Π k ( i ) . (12)The right hand side is the discrete divergence of the fluxΠ k ( m ) ≡ − m (cid:88) i =1 F i + k − d (cid:104)| a i | (cid:105) dt = 2 F m + k V m − J m +1 + 2 F m + k − V m J m +2 . (13)The 3rd order cumulants are zero in equilibrium, butin turbulence they are nonzero to carry the flux. In theinertial interval, the flux must be constant and its diver-gence zero. Indeed, we find the steady solution where allthe fluxes are non-zero constants: V i − J i = CF M − i +1 , Π k ( m ) = CF M + k − . (14)In computing fluxes, one uses the Cassini identities: F m F m +2 − F m +1 = ( − m +1 and F m F n + F m − F n − = F m + n − . All the fluxes have the same sign, that is allthe integrals F k flow in the same direction. This ex-plains the failure of the attempts to get a double cascade(and model 2D turbulence) in shell models [11, 12]. We shall show in the next section what kind of fine-tuning isneeded to get a double cascade when both cascades carrythe same integrals. In [9], the (quadric) spectral flux ofthe (cubic) Hamiltonian was also defined, but pumpingdoes not produce it, so that (cid:104)H(cid:105) = 0 in a steady turbulentstate, as well as in thermal equilibrium. III. CASCADE DIRECTION
To get an analytic insight into our turbulence, par-ticularly, to understand the flux direction, consider aninvariant sub-space of solutions with purely imaginary a k = iρ k for all k : ∂ρ i ∂t = V i − ρ i − ρ i − − V i − ρ i − ρ i +1 − V i ρ i +1 ρ i +2 (15)In this case, H ≡
0. The invariant subspace owes its ex-istence to the invariance of (6) with respect to the sym-metry a → − a ∗ .Consider the chain running between some integers M and N , either positive or negative, and assume V i /V i − = φ α . Then for ρ i = Aφ iβ and M +1 < i < N − ∂ρ i ∂t = A V i − φ iβ (cid:0) φ − β − φ α − φ α +3 β (cid:1) . (16)The right hand side of (16) turns into zero for β = − (1 + α ) /
3, which defines a steady solution ρ i = φ − (1+ α ) / (also with the replacement φ → − /φ ). This solutioncan describe either direct or inverse cascade, since thesymmetry ρ → − ρ , t → − t means that one reverses theflux by changing the sign of ρ in this case. Indeed, con-sider the evolution from the initial state where all ampli-tudes are zero except the first two ρ M , ρ M +1 . The firstterm in (15) then will produce ρ M +2 of the same sign as V M ρ M ρ M +1 , which makes the flux positive, as it shouldbe for a direct cascade. Alternately, by pumping the lasttwo modes, the last term of (15) produces a negativeflux. Which cascade can be realized in reality: direct,inverse or both? Physically it is clear that the sign of theflux must be determined by the only parameter α , thatis by how mode interaction depends on the mode num-ber. Indeed, for α = 1 /
2, the scaling of the flux steadysolution coincides with that of the thermal equilibrium: (cid:104) ρ i (cid:105) = 0, (cid:104) ρ i ρ j (cid:105) = δ ij F − i ∝ φ − i , which, for instance,can be excited by an imaginary pumping acting on everymode in detailed balance with dissipation. Physical com-mon sense suggests that the cascade must carry the con-served quantity (cid:80) i F i ρ i from excess to scarcity [4, 13].For α > / ρ i = φ − α ) i/ decayswith i faster than the equipartition ρ i ∝ /F i ∝ φ − i ,so that it must correspond to a direct cascade. By thesame token, we must have an inverse cascade for α < / α = 1 / ρ i ( t ) = φ iβ [1 + w i ( t )], satisfiesthe linear equation, which turns into a differential advec-tion equation in the region i (cid:29)
1, assuming w i is a slowfunction of i there: φ i (1 − α ) / ∂w i ∂t ≡ ∂w i ∂τ ≈ A V M φ − α [ w i ( φ − − φ − ) − φ − α ( φ + φ − ) ∂w i ∂i ] = − v ∂w i ∂i . (17)This equation describes propagation of perturbationsalong the characteristics: w i ( t ) = f ( i − vτ ) = f ( ξ ). Thecharacteristics are described by i − vτ = i − φ − α (3 φ +3 φ − + 1) tφ i (2 α − / =const. We see that perturbationsrun in the direction of slowing down (also true in weakturbulence [4]): to the left for α > / α < /
2. Since the perturbations are produced bypumping, we conclude that the stable cascade must beestablished in the direction opposite to the direction ofpropagation of perturbations. For α > /
2, the stablecascade is the direct one, which indeed is steeper thanthermal equilibrium. On the contrary, for α < / α = 1 /
2, the perturbations of the power-law distributionrun to the right, but the true cascades in this case arenot pure power laws, the occupation numbers decreaserelative to equilibrium in both directions from pumping,see Figure 1, which is discussed below in more detail.In a general complex case, arguing that the cascadechanges direction when α crosses 1 / F k is equal to Π = P F p + k − where p is the position of the pumping. The input ratemust be equal to the dissipation rate Π = 2 γ d F d + k − n d for any choice of γ d taken at the dissipation position d .In order for n d to smoothly match the cascade, one mustchoose γ d comparable to the nonlinear interaction time: γ d (cid:39) V d J / d (cid:39) V d (Π /V d F d ) / . This gives an order-of-magnitude estimate n d (cid:39) (Π /V d F d ) / . Such reasoningcan be applied to every i , which in turn gives the estimatefor the spectrum of occupation numbers: n i (cid:39) (Π /V i F i ) / . (18)Since the direction of the flux is toward the occupa-tion numbers that are lower than thermal equilibrium, n i ∝ F − i , then again we see that the flux changes di-rection when V i ∝ F / i . The dimensionless degree of non-Gaussianity on such a spectrum, ξ ≡ J i n / i (cid:39) Π V i F i n / i (cid:39) P F p V i F i n / i , (19)must be independent of i . For the spectrum close toequilibrium, ξ ∝ F / i /V i F i = F / i /V i .Figures 1 and 2 confirm these predictions. We placethe pumping at a single mode, i = p , between two dissi-pation regions on the ends, letting the system to choosethe cascade direction. The system (10) with pumpingand damping has been evolved numerically using LSODE solver [14]. At each step, random Gaussian noise of power P is applied to the pumping-connected mode injectingflux Π p = P F p . Damping with γ L and γ R is appliedto the two left-most and two right-most modes respec-tively. For α = 1 / V i = √ F i ), the system is weaklydistorted from equilibrium, with a constant flux on eachside of the pumping. For α (cid:54) = 1 / α > / V i = F i ), we have a thermal equilibriumto the left of pumping and the direct cascade (18) witha constant ξ to the right. In the opposite case ( α < / V i =const), we find an inverse cascade (18) with constant ξ to the left and equilibrium equipartition to the right ofpumping. In both cases, the damping on the flux side iscarefully selected to avoid build-up in the spectrum (thedamping on the equilibrium side can be then set to zeroto establish cleaner scaling). We have chosen V i = F i and V i = const because they qualitatively correspond tothe Kolmogorov scaling of the direct energy cascade inincompressible turbulence and to the inverse wave actioncascade in deep water turbulence respectively.Thermal equilibrium at the scales exceeding the pump-ing scale has been predicted and observed for direct-onlycascades [15]. Nobody has seen before an inverse-onlycascade, neither in hydrodynamic-type systems nor inwave turbulence or shell models. In all known cases in-verse cascades appear in systems with at least two con-served quantities that scale differently. All our conservedquantities (8) scale the same in the limit i (cid:29)
1. It isinteresting if one can find another class of systems with asingle conservation law and the turbulent spectrum lesssteep than equilibrium. In weak wave turbulence, this re-quires the sum of the space dimensionality and the scalingexponent of the three-wave interaction to be less than thefrequency scaling exponent [4]. We do not know such aphysical system, nor we aware of any fundamental lawthat forbids its existence. Remark that the connectionbetween the cascade direction, its stability and steepnessrelative to equipartition has been firmly established inthe weak turbulence theory [4, 13]. In all known exam-ples, the formal turbulent solution with a wrong flux signis not realized; the system chooses instead to stay closeto equipartition with a slight deviation that provides forthe flux in the right direction [4, 16]. Similarly, when weplace pumping and damping at the “wrong” ends of a fi-nite chain, our system heats up, staying close to thermal n i F i P p - / i -1-0.5 0 0.5 1 0 10 20 30 40 P / P p ip = 5p = 10p = 20p = 30p = 36 | x | i i-1, m-i~ 1 / i FIG. 1. Compensated spectra, fluxes and skewness for α = 1 / p = 5, 10, 20, 30, and 36 onthe 40 mode interval. Pumping rate is selected to provide the same flux in all cases, Π p = 67 .
65. In all cases damping ratesare γ L = γ R = 1. Inset reproduces the longer arm of the cascades in log-log scale.
0 20 40 60 n i F i i ~ F i1/3 p = 20 of 40p = 30 of 40p = 50 of 60 n i F i i~ F i-1/3 p = 20 of 40p = 10 of 40p = 10 of 60 | x | i | x | i
0 20 40 60-1-0.5 0 P / P p P / P p FIG. 2. Compensated spectra, fluxes, and the dimensionlessskewness for α = 0 (left) and α = 1 (right) for systems withdifferent location of pumping. In all cases Π p = 67 .
65. For α = 0 damping rates are γ L = 1 . γ R = 0; for α = 1damping rates are γ L = 0 and γ R = 140 at mode 40 and γ R = 3500 at mode 60. equilibrium. As far as shell models are concerned, ourresults resolve the old puzzle of the inability to establishan inverse cascade there — apparently what was neededis to use a system where the interaction time increaseswith the mode number.It is important that our system is a one-dimensionalchain, as well as shell models, so that there is nospace and consequently no distinction in the phase vol-ume (number of modes) between infrared and ultravi- olet parts of the spectrum. The directions along thechain are only distinguished temporally, i.e. in termsof growth/decay of the typical interaction time. Thesame combination V i /F i ∝ φ α − determines the i -dependence of the inverse interaction time both for theequilibrium, V i b / i = V i F − / i T / , and for a cascade, V i (Π /V i F i ) / = ( V i /F i ) / Π / . As the above stabilityconsideration shows, the steepness and stability are con-nected for this class of models, so that the cascade canonly proceed from slow modes to fast modes in Fibonacciturbulence. Similarly in shell models [11, 12, 17] (albeitwith parameters and conservation laws distinct from ourmodel), a cascade proceeding from fast modes to slowmodes was never observed. It was argued that this isbecause the fast modes act like thermal noise on the slowones, which must lead to equilibrium [11]. That this can-not be generally true follows from the existence of theinverse energy cascade in 2D incompressible turbulenceand from numerous examples in weak wave turbulencewhere non-linear interaction time either grows or decaysalong the cascade. Moreover, the formation of the cas-cade spectrum proceeds from fast to slow modes (and notnecessarily from pumping to damping), according to theinformation-theory argument [18].Why is the flux direction unambiguously related to thecascade acceleration in shell models in general and in ourmodel in particular, in distinction from other cases? Theargument can be made by considering capacity, a mea-sure that tells at which end the conserved quantity isstored — perturbations are known to run towards thatend [4]. For example, the power-law energy density spec-trum (cid:15) k ∝ k − s in d dimensions has the total energy (cid:82) (cid:15) k d d k — at which end it diverges is determined by thesign of d − s . This is generally unrelated to the directionof the energy cascade, determined by the sign of s , whichtells whether the spectrum is more or less steep thanthe equipartition. However, in shell models the exponen-tial character of i -dependencies makes the total energy (cid:80) i F i | a i | determined by either the last or the first termof the sum, which solely depends on whether F i | a i | issteeper than equipartition or not, that is by the sign ofthe flux.Which direction then the cascade goes in the symmet-ric case, V i = √ F i ? Now the naive cascade solution (18)coincides with thermal equipartition, F i n i =const, andthe interaction time is independent of the mode numberfor such n i . If we start from thermal equilibrium andapply pumping to some intermediate mode, the systemdevelops cascades in both directions. The left panel ofthe Figure 1 shows that the pumping at site p inside theinterval (1 , N ) generates left and right fluxes in the pro-portion Π L / Π R (cid:39) ( N − p ) /p . This seems natural as in theshorter interval the steeper spectrum falls away from thepumping, which must correspond to a larger flux. Thismeans that if we want to keep the flux constant whileincreasing p or N − p , we need to keep constant the ratio( N − p ) /p .We end this section with a general remark. FibonacciHamiltonian is not symmetric with respect to revers-ing the order of modes, it sets the preferred direction,which is physically meaningful since the frequencies oftwo lower modes sum into the frequency of a high one.Yet, as we see in the case V i F − / i =const, direct andinverse cascades are pretty symmetric. So, it is natu-ral to conclude that indeed the i -dependence of V i F − / i determines which way cascade goes. IV. ALONG THE CASCADES AND AWAYFROM EQUILIBRIUM
As we have seen, thermal equilibrium statistics isexactly Gaussian with no correlation between modes,despite strong interaction (which actually establishesequipartition). The reason for the absence of correla-tion is apparently the detailed balance that cancels them.We do not expect such cancelations in non-equilibriumstates. In all cases of strong turbulence known before, thedegree of non-Gaussianity increases along a direct cas-cade and stays constant along an inverse cascade [19, 20].As we shall show now, non-Gaussianity always increasesalong the cascades in our one-dimensional chains.We present first the symmetric case, where the systemis close to the equilibrium equipartition with the temper-ature set by pumping and slowly changing with the modenumber: n i F i ≈ ( P F p ) / f ( i ). The slow function f ( i )can be suggested by the analogy with the 2D enstrophycascade [21, 22] as f ( i ) ∝ ln / F i ∝ i / , counting fromthe damping region. This gives the dimensionless cumu-lant ξ ∝ /i . This hypothesis is supported by the rightpanel of the Figure 1, which shows that ξ grows alongboth cascades by a power law in i rather than exponen-tially. Let us stress that count always starts from the dis-sipation region, where we have the balance condition Π = γ d F d + k − n d and where γ d (cid:39) V d J / d (cid:39) V d (Π /V d F d ) / | a i | / i i - p40 modes60 modes | a i | / i i - p FIG. 3. Fourth and sixth moments for α = 1 / γ L = γ R = 3, P = 0 .
1, andin 60-mode system with γ L = γ R = 30, P = 1. -5 -4 -3 -2 -1
0 4 8 12 p r obab ili t y | a i | / n i i - p = -24-14-461626 c o m pen s a t ed p r obab ili t y | a i | / n i q i / p FIG. 4. Probability (left) and deviation of probabilityfrom equilibrium (right) for α = 1 /
2. Main panels showprobabilities of occupation numbers rescaled to their aver-ages, the inset shows the probability of phase difference, θ i = ϕ i − ϕ i − − ϕ i − . Refer to the first panel for the linecolor for different modes. Data are shown for 60-mode systemwith center pumping and γ L = γ R = 30, P = 1. according to the dynamical estimate. This sets the non-linearity parameter of order unity at the damping re-gion and decaying towards pumping; the longer the in-terval, the smaller is ξ at any fixed distance from thepumping region. The limit of long intervals may then beamenable to an analytical treatment. Indeed, Figure 3demonstrates that as the interval increases, the highercumulants remain small over longer and and longer in-tervals starting from pumping. Despite the model havingultra-local interactions (every mode participates in onlythree adjacent interacting triplets), the cascade forma-tion is very nonlocal. It is somewhat similar to thermalconduction: if we keep the flux but increase the distance,the distribution gets closer to the thermal equilibrium atevery point.Turning to asymmetric (one-cascade) cases, we see thecumulants higher than third growing with F i by a powerlaw instead of logarithmic. Rather than look for scal-ing in the mode number i , we find it more natural touse F i (playing the role of frequency); at large i onehas F i ≈ φ i , where φ is the golden mean. Traditionalstudy of turbulence in general and shell models in par-ticular was focused on the single-mode moments (analog -6 -4 -2
0 10 20 30 a = 0 p - i p r obab ili t y | a i | / n i -6 -4 -2
0 10 20 30 a = 1 i - p | a i | / n i a = 0 | a i | / n i , | a i | / n i (p + 1) - ip = 30 of 40p = 50 of 60 a = 1 i - (p + 1)p = 10 of 40p = 10 of 60 q i / p q i / p FIG. 5. Probabilities (top) and forth and sixth moments (bot-tom) for the inverse cascade, α = 0 (left), and the direct cas-cade, α = 1 (right). Probabilities for the rescaled occupationnumbers are shown in the main panels, while probabilities forthe phase difference, θ i = ϕ i − ϕ i − − ϕ i − , are shown in theinsets. The variation between P ( θ i ) for different i is minor.In all cases Π p = 67 .
65. For α = 0, the damping rates are γ L = 1 . γ R = 0; for α = 1 the damping rates are γ L = 0and γ R = 140 at i = 40 and γ R = 3500 at i = 60. of structure functions), (cid:104)| a i | q (cid:105) ∝ F − ζ q i , whose anoma-lous scaling exponents, ∆( q ) = qζ / − ζ q give particularmeasures of how non-Gaussianity grows along the cas-cade. For V i = F αi , the flux law gives J i ∝ Π /V i F i , thatis ζ = α + 1. The anomalous scaling is observable innumerics for the single-cascade cases α = 0 and α = 1,as shown in the right panel of the Figure 6. This seemsto be the first case of an anomalous scaling in an inversecascade, with the anomalous dimensions having the op-posite signs to those in direct cascades. The exponentsstart fairly small but grow fast with q . The anomalousexponents, ∆( q ), can be related to the statistical La-grangian conservation laws [23, 24] in fluid turbulence;no comparable physical picture was developed for shellmodels. Without physical guiding, the set of the anoma-lous exponents is not very informative, all the more thatthey characterize only one-mode distribution.Here we suggest a complementary set of threeinformation-theoretic measures, which shed a new lighton the turbulent statistics emerging along the cascade.The main distinction of any non-equilibrium state is thatit has lower entropy than the thermal equilibrium at thesame energy. Turbulence has the entropy that is muchlower, which means that a lot of information is processedto excite the turbulence state. We pose the question:where is the information that distinguishes turbulence -1-0.5 0 0.5 0 4 8 12 D ( q ) q a = 0 a = 1 -0.25-0.2-0.15-0.1-0.05 0 0 20 40 60 S i i a = 0 a = 1~ 0.003 i~ (-0.003) i -0.04 0 0.04 0 1 2 3 4 FIG. 6. Left panel: Anomalous exponents ∆( q ) = qζ / − ζ q = q (1 + α ) − ζ q . Right panel: Down the cascade decay ofthe entropy of the one-mode complex amplitude normalizedby √ n i . The dashed lines S i − S p ≈ − . | i − p | have theslopes equal to ∆ (cid:48) (0) computed from the left panel. Directcascade - blue, inverse cascade - red. from equilibrium encoded? V. WHERE IS THE INFORMATIONENCODED?
First, the information is encoded in a single-modestatistics, which is getting more non-Gaussian deeper inthe cascade. This must be reflected in the decay of theone-mode entropy, S i = S ( x i ) = S ( | a i | / √ n i ), with thegrowth of | i − p | . This can be computed using the multi-fractal formalism: the moments (cid:104) x qi (cid:105) ∝ F − ζ q + qζ / i inthe limit of large | i − p | correspond to the multi-fractaldistribution, P ( x i ) ∝ (cid:90) g ( x i /F hi ) x − i exp[ f ( h ) ln F i ] dh , (20)where f ( h ) = min q ( ζ q − qζ / − qh ), that is f ( h ) is theLegendre transform of ζ ( q ). The entropy is then S i = − (cid:90) dx P ln P ∝ [∆ (cid:48) (0) − ∆ /
2] ln F i . This decay is logarithmic in frequency F i , that is linear in i , as indeed can be seen in Figure 6, where i is countedfrom pumping. We estimate ∆ (cid:48) (0) ≈ ∆ , and observethat the dashed lines in the right panel of Figure 6 withthe slopes (∆ − ∆ /
2) log φ well represent the entropydecay in the inertial interval in both direct and inversecascades.Second, the information is encoded in the correlationsof different modes. It is natural to assume that corre-lations are strongest for modes in interacting triplets, a i , a i +1 , a i +2 . Disentangling of information encoded canbe done by using structured groupings [25–27]: (cid:80) ni =1 S ( a i ) − (cid:80) ij S ( a i , a j ) + (cid:80) ijk S ( a i , a j , a k ) (21) − (cid:80) ijkl S ( a i , a j , a k , a l ) + . . . + ( − n +1 S ( a , . . . , a n ) . For n = 1, this gives the one-mode entropy S i whichmeasures the total amount of information one can obtainby measuring or computing one-mode statistics. Whilethe entropy itself depends on the units or parametriza-tion, all the quantities (22) for n > n = 2, wehave the widely used mutual information, I ij = S ( a i ) + S ( a j ) − S ( a i , a j ) , which measures the amount of information one can learnabout one mode by measuring another, that is character-izes the correlation between two modes. It is interestingthat all pairs in the triplet have comparable mutual in-formation in the direct cascade ( V i = F i ), while I i,i +1 ex-ceeds noticeably I i,i +2 in the inverse cascade ( V i = 1), seethe upper right panel in Figure 8. One can also define thetotal (multi-mode) mutual information as the relative en-tropy between the true joint distribution and the productdistribution: I ( a , . . . , a k ) = (cid:80) ki =1 S ( a i ) − S ( a , . . . , a k ).It is positive and monotonically decreases upon averagingover any of its arguments. As we see from Figure 8, thechanges along the cascade in one-mode entropy and intwo-mode and three-mode mutual information are com-parable, that is one obtains comparable amount of infor-mation about turbulence from these quantities.To see how much more information one gets by measur-ing or computing the three modes simultaneously com-pared to separately by pairs, one needs to use the mea-sure of the irreducible information encoded in triplets, asgiven by the third member of the hierarchy (22): II i = S ( a i ) + S ( a i +1 ) + S ( a i +2 ) + S ( a i , a i +1 , a i +2 ) − S ( a i , a i +1 ) − S ( a i , a i +2 ) − S ( a i +1 , a i +2 )= I i,i +1 + I i,i +2 + I i +1 ,i +2 − I i,i +1 ,i +2 = I ( i, i + 1) − I ( i, i + 1 | i + 2) , (22)It is called interaction information in the classical statis-tics and topological entanglement entropy in the quan-tum statistics [25, 28]. Interaction information measuresthe influence of the third variable on the amount of infor-mation shared between the other two and could be of ei-ther sign. Positive II ( X, Y, Z ) measures the redundancyin the information about Y obtained by measuring X and Z separately, while negative one measures synergy whichis the extra information about Y received by knowing X and Z together. While we cannot prove it mathemat-ically, it seems physically plausible that systems withthree-mode interaction must demonstrate synergy. In-deed, one finds a strong synergy in weak turbulence: itwas shown that I (cid:29) I + I + I [18], so that II < V i = √ F i as seen in Figure 7. Indeed, the two-modemutual information is much smaller than both the one-mode entropy and the absolute value of the interaction -0.02-0.01 0 0.01-30 -20 -10 0 10 20 30 S , I i - p S I I I -0.01-0.005 0 0.005 0.01-30 -20 -10 0 10 20 30 I , II i - pI I II FIG. 7. Deviation of entropies from equilibrium, mutualinformation, and interaction information for α = 1 / · data point. The samevalues of entropy were obtained for a set of 2 · data point,that is S i is saturated. Both I and II show a slight decreasein absolute values with the increase of the ensemble size from2 · to 5 · . information, which is negative.Let us stress that both the mutual information andthe interaction information are symmetric, that is theymeasure the degree of correlation rather than causal re-lationship or cascade direction.We compute the entropies and mutual information asfollows. First, we obtain the probability distributionin 4D space ( x i − , x i − , x i , θ i ) and integrate it to getcorresponding 1D and 2D distributions. Here, θ i = ϕ i − ϕ i − − ϕ i − , where ϕ i is the phase of mode i ,and x i = | a i | / √ n i , while n i = (cid:104)| a i | (cid:105) is the direct av-erage. Mutual information and information interactionare computed directly from entropies, S = − Σ P log P ,obtained for these distributions, since all normalizationfactors cancel out in subtraction. The entropy for anindividual mode, however, is presented relative to theGaussian entropy based on the average occupation num-ber obtained for the binned, staircase distribution for x i .We use the bin sizes ∆ x i = 1 for α = 0 and α = 1, and∆ x i = 1 / α = 1 /
2. In all cases ∆ θ = 2 π/ -0.4-0.3-0.2-0.1 0 0 20 40 60 S | i - p |a = 0, 80 modes a = 1, 60 modes I | i - p |a = 0, I a = 0, I a = 1, I a = 1, I I | i - p | a = 0 a = 1 -0.6-0.4-0.2 0 0.2 0 20 40 60 II / I | i - p | a = 0 a = 1 FIG. 8. Deviation of the entropy from equilibrium, themutual information, and the interaction information (all inbits) for α = 0 and α = 1 and center pumping. Number ofdata points 2 · for α = 0, 80 modes, 6 · for α = 0,60 modes, and 10 for α = 1. For the bin size selected, allquantities agree with those obtained in a half-reduced dataset. direct cascade than in the inverse cascade. Since therequirements on statistics grow exponentially with thedimensionality, the suggestion that one can get most ofinformation (or at least a large part of it) from lower-dimensional probability distributions is great news forturbulence measurements and modeling. To put it sim-ply, comparable amounts of information can be broughtfrom one-mode and from three-mode measurements in di-rect and inverse cascades; most of that information canbe inferred from two-mode measurements. It remains tobe seen to what degree this property of small (asymptoti-cally zero?) interaction information is a universal featureof strong turbulence.Insets in the Figures 4,5 show the probability distribu-tion of the relative phase, θ i , which is closely related tothe flux (skewness), proportional to (cid:104)| a i a i − a i − | sin θ i (cid:105) .The probability maximum is then at ± π/ i -dependence ofthe phase distributions is in accordance with the changesin skewness along i . In the two-cascade symmetric case,the distribution is flat (the phases are random) near thepumping, and the phase correlations appear along thecascades, as can be seen comparing the last panel of Fig-ure 1 with the inset in the right panel of Figure 4. Inthe one-cascade cases, both skewness and the form of thespectrum are practically independent of the mode num-ber, as seen from Figures 2,5.The fact that the deviations from Gaussianity grow along our inverse cascade, in distinction from all the in-verse cascades known before, calls for reflection. We usedto think about the anomalous scaling and intermittencyin spatial terms: Direct cascades proceed inside the forcecorrelation radius, which imposes non-locality, while ininverse cascades one effectively averages over many small-scale fluctuations, which bring scale invariance [19, 20].The emphasis on the spatial features was reinforced bythe success of the Kraichnan’s model of passive tracerturbulence, where it has been shown that the spatial(rather than temporal) structure of the velocity field isresponsible for an anomalous scaling and intermittencyof the tracer. There is no space in our case, so appar-ently it is all about time. Indeed, as we have seen, all ourcascades propagate from slow to fast modes, which leadsto the build-up of non-Gaussianity and correlations. Asa result, the entropy of every mode decreases and theinter-mode information grows along the cascade. Thisdiminishes the overall entropy compared to the entropyof the same number of modes in thermal equilibrium withthe same total energy.Despite qualitative similarity, there is a quantitativedifferences between our direct and inverse cascades. Fig-ures 5,6 show that the one-mode statistics and its mo-ments faster deviate from Gaussian as one proceeds alongthe inverse cascade than the direct one. And yet onecan see from Figures 6,7 that the one-mode entropy isessentially the same in both cascades, as well as the mu-tual information between two neighboring modes and thethree-mode mutual information. The mutual informa-tion between non-neighboring modes I is about twicesmaller, as seen in Figure 8. This difference can proba-bly be related to the dynamics, which in our system isthe coalescence of two neighboring modes into the nextone and the inverse process of decay of one into two. Inthe dynamical equation (16), only one (first) term is re-sponsible for the direct process (and the direct cascade),while two terms are responsible for the inverse process(and the inverse cascade).An important distinction between double-cascade andsingle-cascade turbulence in our system is the dependenceon the system size. The degree of non-Gaussianity of thecomplex amplitudes is fixed in the dissipation regions ofthe double cascade, so that in the thermodynamic limitthe statistics is Gaussian in the inertial intervals. On thecontrary, the statistics of the amplitudes is fixed at theforcing scale for a single cascade, and it deviates more andmore from Gaussianity as one goes along the cascade.We end this section by a short remark on the produc-tion balance of the total entropy S = −(cid:104) ln ρ ( a , . . . , a N ) (cid:105) .Here ρ ( a , . . . , a N ) is the full N -mode PDF. Since waveinteraction does not change the total entropy, then theentropy absorption by the dissipation must be equal tothe entropy production by the pumping [18, 29]: P (cid:90) (cid:89) i da i da ∗ i ρ (cid:12)(cid:12)(cid:12) ∂ρ∂a p (cid:12)(cid:12)(cid:12) = 2 (cid:88) k γ k , . (23)0For a single-cascade cases ( V i = 1 and V i = F i ), the en-ergy balance P F p = 2 γF d n d means that the left handside of (23) must be much larger than the Gaussian es-timate P/n p [18]. It may seem to contradict our numer-ical finding that the pumping-connected mode a p hasits one-mode statistics close to Gaussian. Of course,there are nonzero triple correlation and the mutual in-formation with two neighboring modes in the directionof the cascade. Yet since ξ (cid:39)
1, then the triple moment J p (cid:39) n / p both in direct and inverse cascades, so that thecontribution to the left hand side of (23) is comparablewith P/n p . We conclude then that even the pumping-connected mode must have strong correlations with manyother modes. Since the triple correlation function of non-adjacent modes are zero, such correlations must be en-coded in higher cumulants. That deserves further study. VI. KOLMOGOROV MULTIPLIERS ANDSELF-SIMILARITY
Unbounded decrease of entropy along a single cascadeprompts one to ask whether the total entropy of turbu-lence is extensive (that is proportional to the number ofmodes) or grows slower than linear with the number ofmodes, so there could be some “area law of turbulence”(like for the entropy of black holes). This question canbe answered with the help of the so-called Kolmogorovmultipliers, σ i = ln | a i /a i − | [30]. Figure 9 shows thatin our cascades the multipliers have universal statisticsindependent of i , similar to shell models [31–34]. Oneconsequence of the scale invariance of the statistics of themultipliers is that the entropy of the system is extensive,that is proportional to the number of modes. Of course,the entropy depends on the representation. From the in-formation theory viewpoint, the Kolmogorov multipliersrealize representation by (almost) independent compo-nent, that is allow for maximal entropy. In other words,computing or measuring turbulence in terms of multi-pliers gives maximal information per measurement (theabsolute maximum is achieved by using the flat distribu-tion, that is the variable u ( σ ) defined by du = P ( σ ) dσ ).The amplitudes are expressed via the multipliers: X k = ln x k = ln | a k |√ n k = ln x p + p + k (cid:88) i = p +1 σ i + 12 log n p n k . The first term is due to the pumping-connected mode,which correlates weakly with σ i in the inertial interval.As shown below, the correlation between multipliers de-cays fast with the distance between them. That suggeststhat the statistics of the amplitude logarithm at large k must have asymptotically a large-deviation form:ln P ( X k ) = − kH ( X k /k ) . (24)Indeed, the three upper curves in the top row of Figure 5collapse in these variables, as shown in the bottom row of Figure 9. The self-similar distribution of the logarithm ofamplitude, (24), is a dramatic simplification in compari-son with the general multi-fractal form (20). Technically,it means that g ( x k /F hk ) = g ( e X k − kh ln φ ) is such a sharpfunction that the integral in (20) is determined by thesingle X k -dependent value, h ( X k ) = X k /k ln φ . We thenidentify f = − H/ ln φ .The self-similarity of the amplitude distribution (plusthe independence of the phase distribution on the modenumber) is great news, since it allows one to predictthe statistics of long cascades (at higher Reynolds num-ber) from the study of shorter ones. In our case, Fig-ure 9 shows that 28-th mode already has the form closeto asymptotic. Self-similarity and finite correlation ra-dius of the Kolmogorov multipliers has been also estab-lished experimentally for Navier-Stokes turbulence [35].To avoid misunderstanding, let us stress that the self-similarity is found for the probability distribution of thelogarithm of the amplitude, which does not contradictthe anomalous scaling of the amplitude moments withthe exponents ζ q determined by the Legendre transformof f or H .If the multipliers were statistically independent, onewould compute ln P ( X ) = − kH ( X/k ) or ζ q proceed-ing from P ( σ ) by a standard large-deviation formalism: H ( y ) = min z [ zy − G ( z )], where G ( z ) = ln (cid:82) dσe zσ P ( σ ).Such derivation would express (cid:104)| a k | q (cid:105) via (cid:104) e qσ k (cid:105) , which isimpossible since the former moments exist for all q , whilethe latter do not because of the exponential tails of P ( σ ),see also [35, 36].Therefore, to describe properly the scaling of the am-plitudes one needs to study correlations between mul-tipliers. Physically, it is quite natural that the law ofthe distribution change along the cascade must be en-coded in correlations between the steps of the cascade.Indeed, we find that the neighboring multipliers are de-pendent, albeit weakly, as expressed in their mutual in-formation (traditionally used pair correlation function[32, 33, 35] is not a proper measure of correlation for non-Gaussian statistics). We find that for the inverse cascade, I ( σ i , σ i +1 ) (cid:39) . II ( σ i , σ i +1 , σ i +2 ) (cid:39) − .
1. For thedirect cascade, I ( σ i , σ i +1 ) (cid:39) . II (( σ i , σ i +1 , σ i +2 ) (cid:39)− .
08. No discernible I ( σ i , σ i + k ) were found for k > σ i and σ i +2 are practically uncorrelated, there issome small synergy in a triplet.To appreciate these numbers, let us present for com-parison the statistics of the Kolmogorov multipliers inthermal equilibrium. Normalized for zero mean and unitvariance, we have P ( σ ) = (cid:90) (cid:90) ∞ dxdy e − x − y δ (cid:0) σ −
12 ln xy (cid:1) = 12 cosh σ ,P ( σ i , σ i +1 ) = 8 e σ i +2 σ i +1 (cid:2) e σ i (cid:0) e σ i +1 (cid:1)(cid:3) . (25)That gives I ( σ i , σ i +1 ) b = ln 2 − / ≈ . -6 -4 -2 -8 -4 0 4 8 a = 0 p r obab ili t y s i p - i = 42306 -6 -4 -2 -8 -4 0 4 8 a = 1 p r obab ili t y s i i - p = 63042 -0.4-0.20 0 0.05 0.1 0.15 a = 0 l n ( P ( X )) / k X / k p - i = 423426 -0.4-0.20 0 0.05 0.1 0.15 a = 1 l n ( P ( X )) / k X / k i - p = 463830
FIG. 9. Top: probability distributions of the Kolmogorovmultipliers σ i = ln | a i /a i − | for different positions in the in-verse (left) and direct (right) turbulent cascades. Solid linescorrespond to the thermal equilibrium P ( σ ) = 1 / ( σ − ¯ σ ), where ¯ σ = − (1 /
3) ln φ for the inverse cascade and ¯ σ = − (2 /
3) ln φ for the direct one. Bottom: probability distribu-tions of X = ln | a k | collapse to the large-deviation form faraway from the pumping, that is for large k = | i − p | . tics of a single multiplier. The joint PDFs P ( σ i , σ i +1 )are shown in Figure 10 for thermal equilibrium and fortwo cascades. Again, the Gaussian statistics representsturbulence remarkable well. The differences between thethree cases are most pronounced around the peak at theorigin, while the distant contours are hardly distinguish-able. In plain words, the probabilities of strong fluctua-tions of the multipliers are the same in thermal equilib-rium as in turbulence cascades. This is remarkably differ-ent from the statistics of the complex amplitudes, whichdemonstrate most difference between the three cases forstrong fluctuations and for high moments. There seemsto be a certain duality between fluctuations of the ampli-tudes and multipliers: strong fluctuations of the multi-pliers correspond to weakly correlated amplitudes, whilestrong fluctuations of the amplitudes may require theirstrong correlations and thus correspond to multipliersclose to their mean values. Whether this duality can beexploited for an analytic treatment remains to be seen.The information about the anomalous scaling exponentsof the amplitudes in turbulence must be encoded in thecorrelations between multipliers. Note that the mutualinformation I ( σ i , σ i +1 ) for both cascades ( I = 0 .
23 and I = 0 .
30) is not that much higher than in thermal equilib-rium ( I = 0 .
19 bits). Physicists tend to be much excitedabout any broken symmetry; it is refreshing to noticethat relatively small information is needed to encode the − − − σ i − − − σ i + FIG. 10. Joint probability distributions of two neighboringKolmogorov multipliers shifted to zero means. The contoursare at log ( P ) = − . , − , − , − , − , −
5. Inverse cascade( α = 0) is red, direct cascade ( α = 1) is blue, black is theequilibrium distribution (25). broken scale invariance in turbulence. How to decodethis information from the joint statistics of multipliersremains the task for the future VII. DISCUSSION
Maybe the most surprising finding is the existence ofan inverse-only cascade and its anomalous scaling. In allcases known before, an inverse cascade appears only as aplan B for an extra invariant that cannot ride along thedirect cascade with other invariant(s). In a truly weakturbulence, when the whole statistics is close to Gaus-sian, an inverse-only cascade is indeed impossible, sinceit would require an environment that provides ratherthan extracts entropy, which contradicts the second lawof thermodynamics [18, 29]. Here we have shown that aninverse-only cascade is possible in a strong turbulence.As far as an anomalous scaling is concerned, we relateit to the change of the interaction time along the cas-cade. All the inverse cascades known before run fromfast to slow modes and have a normal scaling. In ourcase, as in all shell models, cascades always proceed fromslow to fast modes. Apparently, this is the reason thatnon-Gaussianity increases along all our cascades, and ananomalous scaling takes place in both single inverse andsingle direct cascades. Indeed, proceeding from fast toslow modes (in inverse cascades known before) involvesan effective averaging over fast degrees of freedom, which2diminishes intermittency. On the contrary, our cascadesbuild up intermittency as they proceed.Another unexpected conclusion follows from the en-tropy production balance in a steady turbulent state:even though the marginal statistics of the pumping-connected mode (averaged over all other modes) can beclose to Gaussian, the correlations of that mode withother modes cannot be weak.Most of the present work was devoted to disentanglingof the information encoded in strong turbulence. It waspredicted that in weak turbulence most of the informa-tion is encoded in the three-mode statistics [18], and herewe confirm this prediction. Yet in strong turbulence, wefind that as much information is encoded in one-mode asin two-mode statistics, while three-mode statistics doesnot add much. This is potentially great news for tur-bulence studies. An important lesson is that measur-ing or computing mode amplitudes (or velocity structure functions) brings diminishing returns, that is less andless information, as one goes deep into the cascade. Themaximal information is encoded in the statistics of theKolmogorov multipliers. Most of that information is en-coded in the statistics of a single multiplier; less than10% is encoded in the correlation of neighbors. How todecode it is the task for the future.We wish to thank Yotam Shapira for helpful discus-sions. The work was supported by the Scientific Excel-lence Center and Ariane de Rothschild Women DoctoralProgram at WIS, grant 662962 of the Simons foundation,grant 075-15-2019-1893 by the Russian Ministry of Sci-ence, grant 873028 of the EU Horizon 2020 programme,and grants of ISF, BSF and Minerva. NV was in partsupported by NSF grant number DMS-1814619. Thiswork used the Extreme Science and Engineering Discov-ery Environment (XSEDE), which is supported by NSFgrant number ACI-1548562, allocation DMS-140028. [1] V. Arnold , Ann. Inst. Fourier, , 319-361 (1966).[2] A. M. Obukhov, Dokl. Akad. Nauk SSSR, 184:2, (1969).[3] R. Peierls, Ann. Phys. (N.Y.) 3, 1055 (1929).[4] V. Zakharov, V. Lvov and G. Falkovich, KolmogorovSpectra of Turbulence (Springer 1991).[5] A. Newell and B. Rumpf, Annu. Rev. Fluid Mech. 43, 59(2011).[6] S. Nazarenko,
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