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TURBULENCE AS STATISTICS OF VORTEX CELLS

arXiv:hep-th/9306152v2 29 Jun 1993June ’93PUPT-1409TURBULENCE AS STATISTICS OF VORTEX CELLSA.A. MigdalPhysics Department, Princeton University,Jadwin Hall, Princeton, NJ 08544-1000.E-mail: migdal@acm.princeton.eduAbstractWe develop the formulation of turbulence in terms of the functional integralover the phase space configurations of the vortex cells.

The phase space consists ofClebsch coordinates at the surface of the vortex cells plus the Lagrange coordinatesof this surface plus the conformal metric. Using the Hamiltonian dynamics we findan invariant probability distribution which satisfies the Liouville equation.Theviolations of the time reversal invariance come from certain topological terms ineffective energy of our Gibbs-like distribution.

We study the topological aspects ofthe statistics and use the string theory methods to estimate intermittency.

Contents1Introduction22Loop functional43Vortex dynamics74Vortex statistics115Discussion186Acknowledgments20A Poisson brackets for the Euler equation20B Helmholtz vs Euler dynamics211

1IntroductionThe central problem of turbulence is to find the analog of the Gibbs distribution for theenergy cascade. The mathematical formulation of this problem is amazingly simple.

Ininertial range we could neglect the viscosity and forcing and study the Euler dynamics ofideal incompressible fluid∂t vα = vβωαβ −∂αh(1)Hereωαβ = ∂αvβ −∂βvα(2)is the vorticity field andh = p + 12v2α(3)is the enthalpy which is eliminated from the incompressibility condition∂αvα = 0(4)The key point is that the Euler dynamics can be regarded as a Hamiltonian flow infunctional phase space. This geometric view was first proposed by Arnold in 1966 (see hisfamous book [2]) and later developed by other mathematicians (see the references in theMoffatt’s lecture in the 1992 Santa Barbara conference proceedings[10]).

Here we derivethe Hamiltonian dynamics from scratch using the conventional physical terminology. Allthe necessary computations are presented in Appendix A.

Some of these results are new.The Hamiltonian here is just the fluid energyH =Zd3r12v2α(5)and the phase space corresponds to all the velocity fields subject to the incompressibilityconstraint. The Poisson brackets between the components of velocity field[vα(r1), vβ(r2)] =Zd3rTαµ (r1 −r) Tβν (r2 −r) ωµν(r)(6)whereTαβ(r) = δαβδ(r) + ∂α∂β14πr(7)is the projection operator.

The Euler equation can be written in a manifestly Hamiltonianform∂tvα =Zd3r′Tαβ (r −r′) ωβγ(r′)vγ(r′) = [vα, H](8)The Liouville theorem of the phase space volume conservation applies hereδδvα(r) [vα(r), H] = 0(9)(Dv) =Yrd3v(r) = const(10)2

The probability distributions P[v] compatible with this dynamics, must also be conserved,i.e. it must commute with the Hamiltonian[P[v], H] = 0(11)The Gibbs distributionP[v] = exp (−βH)(12)is the only known general solution of the Liouville equation (11).

The above mentionedcentral problem of turbulence is to find another one.This formulation of turbulence is significantly different from the popular formulationbased on the so called Wyld diagram technique [9]. There, the functional integral is inplace from the very beginning but the problem is to find its turbulent limit (zero viscosity).This functional integral involves time, so it describes the kinetic phenomena in additionto the steady state we are studying here.After trying for few years to do something with the Wyld approach I conclude thatthis is a dead end.

The best bet here would be the renormalization group, which magi-cally works in statistical physics. Those critical phenomena were close to Gaussian, whichallowed Wilson to develop the ǫ expansion by rearranging the ordinary perturbation ex-pansion.There is no such luck in turbulence.

The nonlinear effects are much stronger. Theobserved variety of vorticity structures with their long range interactions does not look likethe block spins of critical phenomena.

Moreover, there are notorious infrared divergencies,which make problematic the whole existence of the universal kinetics of turbulence. No!These old tricks are not going to work, we have to invent the new ones.If we are looking for something pure and simple this might be the steady state dis-tribution of vorticity structures.

Here the geometric methods may allow us to go muchfurther than the general methods of quantum field theory. Being regarded as a problemof fractal geometry rather than a nonlinear wave problem, turbulence may reveal somemathematical beauty to match the beauty of the Euler-Lagrange dynamics.This dream motivated the geometric approach to the vortex sheet dynamics in [6],where the attempt was made to simulate turbulence as the stochastic motion of thevortex sheet.

This project ran out of computer resources, as it happened before to manyother projects of that kind. However, the geometric tools developed in that work provedto be useful and we are going to use them here.Another false start: I tried to solve the Liouville equation variationally, with theGaussian Anzatz with anomalous dimensions for the velocity field.

[8] The numbers cameout too far from the experiment and it was hard to improve them. Similar attemptswith the Gaussian Clebsch variables [11] also failed to produce the correct numbers.

Itbecame clear to me that velocity field fluctuates too much to be used as a basic variable3

in turbulence.1We encounter the same problem in QCD where the gauge field strongly fluctuates,and its correlations do not decrease with distance. The problem is not yet solved there,but some useful tools were developed.

In particular, the Wilson loops (the averages ofthe ordered exponentials of the circulation of the gauge field) are known to be a betterfield variables. The Wilson loop is expected to be described by some kind of the stringtheory, though nobody managed to prove this.The dynamical equations for the Wilson loops as functionals of the shape of the loopwere derived, and studied for many years[4].

The QCD loop equations proved to be veryhard to solve, because of the singularities at self intersections.In my recent paper[12] I derived similar equations for the averages of the exponentialsof velocity circulation in (forced) Navier-Stokes equation. These equations have no singu-larities at self intersections, in addition they are linear, unlike the loop equations of QCD.This raised the hopes to find exact solution in terms of the string functional integrals.The theory developed below started as the solution the (Euler limit of the) loopequations.

However, later I found how to derive it from the Liouville equation, whithoutunjustified assumptions of the loop calculus. This is how I am presenting this theory here.2Loop functionalIt is generally believed that vorticity is more appropriate than velocity for the descriptionof turbulence.

Vorticity is invariant with respect to Galilean transformations which shiftthe space independent part of velocityvα(r) ⇒vα(r −ut) + uα; ωαβ(r) ⇒ωαβ(r −ut)(13)The correlation functions of velocity field involve the infrared divergencies coming fromthis part. Say, in the two- point correlation function this would be the energy density⟨vα(r)vα(r′)⟩= 2HV−12⟨(vα(r) −vα(r′))2⟩(14)which formally diverges asHV ∝Z ∞1/L dkk−53 ∼L23(15)according to Kolmogorov scaling [1].The infrared divergencies are absent in vorticity correlations.The correspondingFourier integral is ultraviolet divergent due to extra factor of k2, but this is healthy.

The1This does not mean that the simple Gaussian models in velocity or Clebsch variables could not explainobserved turbulence in finite systems. We are talking about idealized problem of isotropic homogeneousturbulence with infinite Reynolds number.4

physical observables involve the vorticity correlations at split points, where the integralsconverge.The complete set of such observables is generated by velocity circulations for variousloops in the fluidΓC[v] =IC drαvα(r)(16)The constant part of velocity drops here after the integration over the closed loop, thusthe circulation is Galilean invariant. The loop gets translatedC ⇒C −ut(17)but all the equal time correlations of the circulation stay invariant in virtue of translationsymmetry.One can express the circulation in terms of vorticity via the Stokes theoremΓC[v] =Xµ<νZS drµ ∧drν ωµν(r)(18)where S is an arbitrary surface bounded by C. In particular, for infinitesimal loop wewould get the local vorticity.

The moments of the circulation⟨(ΓC[v])n⟩=Z[Dv]P[v] (ΓC[v])n(19)all converge in the ultraviolet as well as in the infrared domain. The infrared convergenceis guaranteed since these are surface integrals of vorticity, and the ultraviolet one isguaranteed since these are line integrals of velocity.This nice property suggests to study the distribution of the velocity circulationPC[Γ] =Z[Dv]P[v]δ (Γ −ΓC[v])(20)It is more convenient to study the Fourier transformΨC(γ) =Z ∞−∞dΓ exp (ı γΓ) PC(Γ) =Z[Dv]P[v] exp (ı γΓC[v])(21)We expect these functionals to exist in the turbulent limit unlike the distribution P[v].The dynamical equation for these functionals (the loop equation) was derived in myprevious work[12].

The time derivative of the circulation reads∂t ΓC[v] =IC drαωαβ(r)vβ(r)(22)All the nonlocal terms reduced to the closed loop integrals of the total derivatives and van-ished. Being averaged with appropriate measure, the remaining terms in time derivativemust vanish according to the Liouville equation⟨∂t ΓC[v] exp (ı gΓC[v])⟩= 0(23)5

Note that this is not the Kelvin theorem of conservation of the circulation.Thecirculation is conserved in a Lagrange sense, at purely kinematical levelddtΓC[v] = ∂t ΓC[v] +Idθ δΓC[v]δCβ(θ)vβ (C(θ)) = 0(24)In other words, the Euler derivative (22) is exactly compensated by the shift of everypoint at the loop by local velocity. Now, according to the Liouville equation, each ofthese equal terms must vanish in average.The formal derivation goes as followsZ[Dv]P[v] exp (ı γΓC[v]) [ı γΓC[v], H](25)=Z[Dv]P[v] [exp (ı γΓC[v]) , H]= −Z[Dv] [P[v], H]exp (ı γΓC[v]) = 0and the physics is obvious: the average of any time derivative with time independentweight must vanish.Let us interpret in these terms the solution for ΨC(γ) found in the previous paper [12]ΨC(γ) = f(Σαβ); Σαβ =IC drαrβ(26)In virtue of linearity of the Liouville equation it suffices to check the Fourier transformexp (ı γRαβΣαβ)(27)In terms of the velocity field this corresponds to global rotationvα = Rαβrβ; ωαβ = 2Rαβ(28)The corresponding Poisson brackets reads[ΓC[v], H] =IC drαωαβvβ = 4ΣαγRαβRβγ = 0(29)The sum over tensor indexes vanished by symmetry.This solution is always present in fluid mechanics due to the conservation of the angularmomentum.

Unfortunately, it has nothing to do with turbulence, contrary to the hopesexpressed in my previous work.Let us also note, that the Gibbs solution does not apply here. One could formallycompute the loop functional for the Gibbs distribution, but the result is singular2ΨC(γ) = exp −γ22βIC drαIC dr′αδ3(r −r′)!

(30)2This is yet another advantage of the loop functional: the Gibbs solution for the loop functional doesnot exist, which forces us to look for alternative invariant distributions.6

Should we cut offthe wavevectors at k ∼1r0 this would becomeΨC(γ) ≈exp −γ2βr0IC |dr|! (31)This is so called perimeter law, characteristic to the local vector fields.

Clearly this isnot the case in turbulence, as velocity field is highly nonlocal. Also, the odd correlationsof velocity, such as the triple correlation, which are present due to the time irreversibility,would, in general make the loop functional complex.3Vortex dynamicsLet us study the dynamics of the vortex structures from the Hamiltonian point of view.We shall assume that vorticity is not spread all over the space but rather occupies somefraction of it.

It is concentrated in some number of cells Di of various topology movingin their own velocity field.Clearly, this picture is an idealization. In the real fluid, with finite viscosity, therewill always be some background vorticity between cells.

In this case, the cells could bedefined as the domains with vorticity above this background. The reason we are doingthis is obvious: we would like to work with the Euler equation with its symmetries.We shall use two types of tensor and vector indexes.

The Euler (fixed space) tensorswill be denoted by Greek subscripts such as rα.The Lagrange tensors (moving withfluid) will be denoted by latin subscripts such as ρa. The field Xα(ρ) describes the instantposition of the point with initial coordinates ρa.

The transformation matrix from theLagrange to Euler frame is given by ∂aXα(ρ).The contribution of each cell D to the net velocity field can be written as followsvα(r) = −eαβγ∂βZD d3ρΩγ(ρ)4π|r −X(ρ)|(32)whereΩγ(ρ) = Ωa(ρ)∂aXγ(ρ)(33)is the vorticity vector in the Euler frame. The vorticity vector Ωa(ρ) in the Lagrangeframe is conserved∂t Ωa(ρ) = 0(34)The physical vorticity tensor ωαβ = ∂αvβ−∂βvα inside the cell can be readily computedfrom velocity integral.

The gradients produce the δ function so that we getωαβ(X(ρ)) = ∂(X1, X2, X3)∂(ρ1, ρ2, ρ3)!−1eαβγΩγ(ρ)(35)7

or in Euler frame, inverting ρ = X−1(r)ωαβ(r) = 16 eλµν eabc∂λρa∂µρb∂νρc eαβγ Ωf(ρ)∂fXγ = eabc∂αρa(r)∂βρb(r) Ωc(ρ(r))(36)The inverse matrix ∂αρa = (∂aXα)−1 relates the Euler indexes to the Lagrange ones, asit should. These relations between the Euler and Lagrange vorticity are equivalent to theconservation of the vorticity 2-formΩ=Xα<βωαβ(X)dXα ∧dXβ =Xa

We show that this is equivalent to the Hamiltoniandynamics with the (degenerate) Poisson brackets[Xα(ρ), Xβ(ρ′)] = −δ3(ρ −ρ′)eαβγΩγ(ρ)Ω2µ(ρ)(39)The Hamiltonian is given by the same fluid energy, with the velocity understood as func-tional of X(.) The degenaracy of the Poisson brackets reflects the fact that there are onlytwo independent degrees of freedom at each point.

This leads to the gauge symmetrywhich we discuss below.The Hamiltonian variation readsδHδXα(ρ) = eαβγvβ (X(ρ)) Ωγ(ρ)(40)This variation is orthogonal to velocity, which provides the energy conservation. It is alsoorthogonal to vorticity vector which leads to the gauge invariance.

The gauge transfor-mationsδXα(ρ) = f(ρ)Ωα(ρ)(41)leave the Hamiltonian invariant. These transformations reparametrize the coordinatesδρa = f(ρ)Ωa(ρ)(42)The vorticity density transforms as followsδΩa = −Ωa∂b(fΩb) + fΩb∂bΩa + Ωb∂b(fΩa) = 2fΩb∂bΩa(43)8

The first term comes from the volume transformation, the second one - from the argumenttransformation and the third one - from the vector index transformation of Ωso thatd3ρ Ωa∂a = inv(44)The identity∂aΩa = 0(45)was taken into account. We observe that these gauge transformations leave invariant thewhole velocity field, not just the Hamiltonian.The vorticity 2-form simplifies in the Clebsch variablesΩa = eabc∂bφ1(ρ)∂cφ2(ρ)(46)Ω= dφ1 ∧dφ2 = inv(47)The Clebsch variables provide the bridge between the Lagrange and the Euler dynamics.By construction they are conserved, as they parametrize the conserved vorticity.

TheEuler Clebsch fields Φi(r) can be introduced by solving the equation r = X(ρ)Φi(r) = φiX−1(r)(48)However, unlike the vorticity field, the Clebsch variables cannot be defined globally inthe whole space. The inverse map ρ = X−1(r) is defined separately for each cell, thereforeone cannot write vα = Φ1∂αΦ2 + ∂αΦ3 everywhere in space.

Rather one should add thecontributions from all cells to the Poisson integral, as we did before.This explains the notorious helicity paradox. The conserved helicity integralH =Zd3ρ vc(ρ) Ωc(ρ)(49)whereva(ρ) = φ1∂aφ2 + ∂aφ3; ∂2φ3 = −∂a (φ1∂aφ2)(50)is the initial velocity field (to be distinguished from the physical velocity field vα(r) whichcannot be paramatrized globally by the Clebsch variables).The helicity integral for the finite cell could be finite.

It can be written in invariantterms of the vorticity formsHD =ZD d3ρeabc∂aφ1∂bφ2∂cφ3 =ZD dφ1 ∧dφ2 ∧dφ3 =Z∂D φ3 dφ1 ∧dφ2(51)from which representation it is clear that it is conservedHD =Z∂D Φ3 dΦ1 ∧dΦ2(52)Our gauge transformation leave the Clebsch field invariantδφi(ρ) = f(ρ)Ωa(ρ)∂aφi(ρ) = 0(53)9

The velocity integral in Clebsch variables reduces to the 3-formvα(r) = −eαβγ∂βZDdXγ ∧dφ1 ∧dφ24π|r −X|(54)This gauge invariance is less than the full diffeomorphism group which involves arbi-trary function for each component of ρ. This is a subtle point.

The field Ωa(ρ) has nodynamics, it is conserved. However, the initial values of Ωa can be defined only modulodiffeomorphisms, as the physical observables are parametric invariant.

So, we could as wellaverage over reparametrizations of these initial values, which would make the parametricinvariance complete.Another subtlety. The equations of motion, which literally follow from above Poissonbrackets describe the motion in the direction orthogonal to the gauge transformations,namely∂t Xα(ρ) = δαβ −Ωα(ρ)Ωβ(ρ)Ω2µ(ρ)!vβ (X(ρ)) = vα(X(ρ)) + f(ρ)Ωα(ρ)(55)The difference is unobservable, due to gauge invariance.We could have defined theHelmholtz equation this way from the very beginning.

The conventional Helmholtz dy-namics represents so called generalized Hamiltonian dynamics, which cannot be describedin terms of the Poisson brackets. The formula (40) cannot be solved for the velocity fieldbecause the matrixΩαβ(ρ) = eαβγΩγ(ρ)(56)cannot be inverted (there is the zero mode Ωβ(ρ)).

The physical meaning is the gauge in-variance which allows us to perform the gauge transformations in addition to the Lagrangemotion of the fluid element.Formally, the inversion of the Ω−matrix can be performed in a subspace which isorthogonal to the zero mode. The inverse matrix Ωβγ in this subspace satisfies the equationΩαβΩβγ = δαγ −ΩαΩγΩ2µ(57)which has the unique solutionΩβγ = −eαβγΩαΩ2µ(58)This solution leads to our Poisson brackets.Now we see how the correct number of degrees of freedom is restored.

The Hamilto-nian vortex dynamics locally has only two degrees of freedom, those orthogonal to thegauge transformations. We could have obtained the same Poisson brackets by canonicaltransformations in the vorticity form from the Clebsch variables to the X−variables.

Thiscanonical transformation describes the surface in the X−space. The vorticity form at thissurface can be treated as a degenerate form in the 3-dimensional space.10

One may readily check the conservation of the volume element of the cell∂t∂(X1, X2, X3)∂(ρ1, ρ2, ρ3)= ∂(X1, X2, X3)∂(ρ1, ρ2, ρ3) ∂αvα(X) = 0(59)Using the formulas for the variations of the velocity field derived in Appendix B onemay prove the stronger statement of the phase volume element conservation (Liouvilletheorem)δvα (X(ρ))δXα(ρ)= 0(60)(DX) =Yρd3X(ρ) = const(61)4Vortex statisticsLet us recall the foundation of statistical mechanics. The Gibbs distribution can be for-mally derived from the Liouville equation plus extra requirement of multiplicativity.

Ingeneral, any additive conserved functional E could serve as energy in the Gibbs distribu-tion.The physical mechanism is the energy exchange between the small subsystem understudy and the rest of the system (thermostat). The conditional probability for a subsystemis obtained from the microcanonical distribution dΓ′dΓδ(E′+E−E0) for the whole systemby integrating out the configurations dΓ′ of the thermostat.The corresponding phase space volume eS(E′) =R dΓ′δ(E′ +E −E0) of the thermostatdepends upon its energy E′ = E0 −E where the contribution −E from the subsystemrepresents the small correction.

Expanding S(E0 −E) = S(E0) −βE we arrive at theGibbs distribution.In case of the vortex statistics we may try the same line of arguments. The importantaddition to the general Gibbs statistics is the parametric invariance.

Reparametrizations,or gauge transformations, are part of the dynamics, as have seen above.3 So, the Gibbsdistribution should be both gauge invariant and conserved.The net volume of the set of vortex cells V =Pi V (Di) whereV (D) =ZD d3ρ∂(X1, X2, X3)∂(ρ1, ρ2, ρ3)=ZD dX1 ∧dX2 ∧dX3(62)is the simplest term in effective energy of the Gibbs distribution. It is gauge invariant,additive and positive definite.It is bounded from above by the volume of the system, sothat there could be no infrared divergencies.43This makes so hard the numerical simulation of the Lagrange motion.

The significant part of notoriousinstability of the Lagrange dynamics is the reparametrization of the volume inside the vortex cell.4The Hamiltonian does not exist in the turbulent flow because the energy spectrum diverges at smallwavevectors. The net Hamiltonian grows faster than the volume of the system which is unacceptable forthe Gibbs distribution.11

The mechanism leading to the thermal equilibrium is quite transparent here. Thevolume of a little cell surrounded by the large amount of other cells, would fluctuate dueto exchange with neighbors.5These are the viscous effects, in the same way as the energyexchange mechanisms in the ideal gas were the effects of interaction.

The relaxation timeis inversely proportional to the strength of interaction (viscosity in our case).We must take these effects into account in kinetics, but the resulting statistical distri-bution involves only the energy of the ideal system. This was the most impressive partof the achievement of Gibbs and Boltzman.

They found the shortcut from mechanics tostatistics, avoiding the kinetics. All the interactions are hidden in the temperature andchemical potentials.Mechanically, the cells avoid each other as well as themselves and preserve their topol-ogy but the implicit viscous interactions would lead to fluctuations.

Even if we start fromone spherical cell it would inevitably touch itself in course of the time evolution. At thevicinity of the touching point the viscous effects show up, which break the topologicalconservation laws of the Euler dynamics.

The result could be a handle, or the splittinginto two cells. After long evolution we would end up with the ensemble of cells Di withvarious number hi of handles.The related subject is the vorticity vector field Ωa(ρ) inside the cells.

In the Euler dy-namics it is conserved, but the viscosity-generated interaction would lead to fluctuations.The invariant measure is(DΩ) =Yρd3Ω(ρ)δ[∂aΩa(ρ)](63)In terms of the Clebsch variables (46) the measure is simply(Dφ) =Yρd2φ(ρ)(64)as these are canonical hamiltonian variables. As discussed above, there are no globalClebsch variables in Euler sense.

These Clebsch variables are defined separately in eachcell and the net velocity field is a sum of contributions from all cells rather than the singleClebsch-parametrized expression Φ1∂αΦ2 + ∂αΦ3.What could be an effective energy for the vorticity? The helicity integral is excludedas a pseudoscalar, besides, it is nonlocal, like the Hamiltonian.

We insist on parametricinvariance and locality in a sense that the cell splitting and joining do not change thisenergy.The Clebsch variables are defined modulo additive constants, they could bemultivalued in complex topology, therefore we have to use the vorticity field itself. Thegeneric Ω−invariant in d dimensions isZD ddρqdet Ωab(65)5The volume, as well as any other local functional of the cell does not change at splitting/joining, there-fore these processes go with significant probability.

For the nonlocal functionals, such as the hamiltonian,there are long range interactions, which makes the exchange process less probable.12

In even dimensions d = 2k this invariant reduces to the PfaffianZD d2kρqdet Ωab = 1k!ZD Ω∧Ω. .

. ∧Ω=ZDΩkk!

(66)In two dimensions it would be simply the net vorticity of the cellZD d2ρΩ12(67)However, in odd dimensions it vanishes so that there is no Ω−invariant. We see that thereis a significant difference in the vortex statistics in even and odd dimensions.Another interesting comment.

For odd k = d2 the Ω−invariant is an odd functionalof Ωwhich breaks time reversal invariance. The vorticity stays invariant under spacereflection but changes sign at time reversal.

This simple local mechanism of the timeirreversibility is present only at d = 4k + 2, k = 0, 1, . .

.. In three dimensions we live in itis absent.Let us turn to the boundary terms.

The boundary of the cell S = ∂D is described bycertain parametric equationS : ρa = Ra (ξ1, ξ2)(68)Clearly, in our case this is a self- avoiding surface. Here we could add the following localsurface terms to the energyEφ =XiZ∂Did2ξa√g + b√ggij∂jφk∂jφk(69)wheregij(ξ) = ∂iRa(ξ)∂jRa(ξ)(70)is the induced metric.There is also a topological term for each non-contractible loop L of ∂DEΘ = ΘZL φ1dφ2(71)This is simply the velocity circulation around such loop.

Assuming continuity (i.e. van-ishing) of vorticity at the boundary of the cell, this circulation can be also written as theintegral in the external space.

Then is it obvious that this integral does not depend uponthe shape of the loop L, it is given by the invariant vorticity flux through the cross sectionΣ of the corresponding handleEΘ = ΘZΣ Ω; L = ∂Σ(72)This term breaks the time reversal! This is the only possible source of the irreversibilityin this theory in three dimensions.13

The following observation leads to crucial simplifications. The only term which de-pends upon the Lagrange field Xα(ρ) is the volume term, which in fact is the functionalof the bounding surface X(∂D)V =ZD dX1 ∧dX2 ∧dX3 =Z∂D X3 dX1 ∧dX2(73)The rest of the terms in the effective energy also depending only upon the boundary,this lowers the dimension of our effective field theory.

We are dealing with the theoryof self-avoiding random surfaces rather that the 3-d Lagrange dynamics.With somemodifications the methods of the string theory can be applied to this problem.The invariant distance in the X(ξ) functional space is||δX||2 =Z∂D d2ξ√g (δXα(ξ))2(74)where g as usual stands for the determinant of the metric tensor. One may check thatthe corresponding volume element(DX) =Yξ∈∂DδX(75)is conserved in the Euler-Helmholtz dynamics as well as the complete volume element.The key point in this extension of the Liouville theorem is the observation that the matrixtrace of the functional derivative vanishesδvα(X(ρ))δXα(ρ′)= 0(76)for arbitrary ρ, ρ′, including the boundary points.

The metric tensor gij does not introduceany complications, as it is X independent.This metric is the motion invariant in the vortex dynamics. In the statistics, accordingto the general philosophy these invariants become variables.

We see that the field Ra(ξ)enter only via the induced metric, which allows us to introduce the latter as a collectivefield variable.One has to introduce the functional space of all metric tensors with the Polyakovdistance [3]||δg||2 =ZS d2ξ√g (δgijδgkl)Agijgkl + Bgikgjl(77)The parametric invariance can be most conveniently fixed by the conformal gaugegij(ξ) = ˆgij(ξ)eαϕ(ξ)(78)where ˆg is some background metric parametrizing the surface with given topology. Unlikethe internal metric, the background metric does not fluctuate.

We have the freedom tochoose any parametrization of the background metric.14

The effective energy which emerges after all computations of the functional jacobiansassociated with the gauge fixing reads6Eϕ = 14πZd2ξqˆg12ˆgij∂iϕ∂jϕ −Q ˆRϕ + µeαϕ(79)where ˆR is the scalar curvature in the background metric.The parameters Q, α should be found from the self-consistency requirements. In caseof the ordinary string theory in d dimensional space the requirement of cancellation ofconformal anomalies yields [5]α =√1 −d −√25 −d2√3; Q =s25 −d3(80)In three dimensions α is a complex number, which is fatal for the string theory.

Fortu-nately this formula does not apply to turbulence, because the dynamics of the X field iscompletely different here. Later we speculate about the values of these parameters.To summarize, the total phase space volume element of our string theorydΓ =Ycells(DX)(Dφ)(Dϕ)(81)and the total effective energyE =XcellsV + Eφ + Eϕ +XloopsEΘ(82)The grand partition functionZ =XN,hiexp −µN −λNXi=1hi!

ZdΓ exp (−βE)(83)The interaction between cells comes from the excluded volume effect.The vorticity correlations are generated by the loop functional[12]ΨC(γ) = ⟨exp (ı ΓC[v])⟩(84)whereΓC[v] =IC drαvα(r)(85)is the velocity circulation. Our solution for the loop functional readsΨC(γ) = Z−1 XN,hiexp −µN −λNXi=1hi!

Z ′dΓ exp (−βE + ı γΓC[v])(86)6 We cut some angles here. This effective measure was obtained [5] with some extra locality assump-tions in addition to the honest calculations.

These assumptions were never rigorously proven, but theyare known to work. All the results which were obtained with this measure coincide with those obtainedby the mathematically justified method of dynamical triangulations and matrix models.15

whereR ′ implies that the cells also avoid the loop C.There are now various topological sectors.In the trivial sector, the loop can becontracted to a point without crossing the cells.Clearly, circulation vanishes in thissector. In the nontrivial sectors, there is one or more handles encircled by the loop, sothat the circulation is finite.

Here is an example of such topologyCD2D1The circulation can be reduced to the Stokes integral of the vorticity 2-form over thesurface SC encircled by the loop C. Only the parts si = SC ∩Di passing through thecells contribute. In each such part the vorticity 2-form can be transformed to the Clebschcoordinates which reduces it to the sum of the topological termsΓC =XiZsiΩ=XiZsidφ1 ∧dφ2 =XiZ∂siφ1dφ2(87)In presence of the same terms in the effective energy the loop functional would be com-plex, as the positive and negative circulations would be weighted differently.This ismanifestation of the time irreversibility.The interesting thing is that the interaction between the X field and the rest offield variables originates in the topological restrictions.

The circulation reduces to thenet flux from the handles encircled by the loop in physical space (this is a restrictionon the X field). Also, the requirement that the loop is avoided by cells imposes anotherrestriction on the X field.

There are no explicit interaction terms in the energy. The wholedependence of the loop functional on the shape of the loop C comes from the excludedvolume effect.

The implications of this remarkable property are yet to be understood.Let us now check the Liouville equation. The time derivative of the circulation reads(see(22))∂t ΓC[v] =IC drαωαβ(r)vβ(r)(88)But the vorticity vanishes at the loop, because the cells avoid it!

Therefore⟨∂t ΓC[v] exp (ı γΓC[v])⟩= 0(89)which is the Liouville equation.16

For the readers of the previous paper[12] let us briefly discuss the loop equation. Theessence of the loop equation is the representation of the vorticity as the area derivativeacting on the loop functionalˆωαβ(r) = −ıγδδσαβ(r)(90)The formal definition of the area derivative in terms of the ordinary functional derivativeswas discussed before [4, 12].

The geometric meaning is simple: add the little loop tothe original one and find the term linear in the area δσαβ enclosed by this little loop.The parametric invariant functionals like this one can always be regularized so that thevariation would start from δσαβ. The area derivatives of the length and the minimal areainside the loop were derived in [4].The velocity operator is related to vorticity by the Poisson integralˆvα(r) = −∂βZd3R ˆωαβ(R)4π|r −R|(91)The geometric meaning is as follows.

The vorticity operator adds the little loop δCR,R atthe point R offthe original loop C. By adding the couple of straight line integralsW(r, R) = exp ı γZLr,Rdrαvα(r)!= W −1(R, r)(92)we reduce this to one loop of the singular shapeCr˜C = {Cr,r, Lr,R, δCR,R, LR,r}(93)ΓC + ΓδCR,R = Γ ˜C(94)which can be obtained from original one as certain variation[12]. Then the loop equationis simplyIC drαˆωαβ(r)ˆvβ(r)ΨC(γ) = 0(95)It was shown in [12] that these two operators commute as they should.In our case this equation is satisfied in a trivial way.

Regardless the subtleties in thedefinition of ˆω, ˆv, as long as these operators act on the vortex sheet circulation as theyshould, they insert the vorticity and velocity at the loop. The rest of the argument is thesame as in the Liouville equation.Note that this trick would not solve the Hopf equation for the velocity generatingfunctionalH[J] = ⟨expZd3rJα(r)vα(r)⟩(96)17

The Hopf equation would require⟨Zd3r′Tαβ (r −r′) ωβγ(r′)vγ(r′) expZd3rJα(r)vα(r)⟩= 0(97)which we cannot satisfy since the volume integralR d3r′ would inevitably overlap withcells where vorticity is present.Apparently we found invariant probability distribution for vorticity but not for thevelocity. The velocity distribution may not exist in the infinite system, due to the infrareddivergencies.

In practice this would mean that the velocity distribution would depend ofthe details of the large scale energy pumping, but vorticity distribution would be universal.The correlation functions of vorticity field can be obtained from the multiloop func-tionals by contracting loops to points. In case when the loops encircle one or more handlesthere would be nonvanishing correlation function.

Here is an example of the topology withthe two point correlation which also has nontrivial helicity because of the knotted handle.C1C25DiscussionSo far we do not have much to tell to the engineers. Even if this statistics of the vor-tex structures will be confirmed by further study, we shall face the formidable task ofcomputing the correlation functions.Let us speculate what could come out of thesecomputations.The qualitative picture of intermittent distribution of vorticity will be the same as inthe multifractal models[10].

In fact these models inspired our study to some extent. Ourbasic idea is that the part V of the volume occupied by vorticity fluctuate.

In numericaland real experiments[10] the high vorticity structures were clearly seen. The cells take18

the shape of long sausages rather than spheres, which does not contradict our generalphilosophy but still lacks an explanation.Let us try to estimate the intermittency effects in the vortex cells statistics. Let uscontract the loop to a point r around some handle.

What we get in the limit can beexpressed in terms of the vertex operator of the string theoryΓC ∝ZS d2ξ0√gZS d2η0√gδ3(X(ξ) −r)δ3(X(η) −r′); r′ →r(98)The points ξ0 and η0 are mapped to the same point r′ →r in physical space: this isthe handle strangled by the loop.The important detail here is the factor √g = eαϕcorresponding to the metric tensor at the surface.The properties of such metric tensors were studied in the string theory[3, 5]. Themoments of ΓC would behave as⟨ΓnC⟩∝r−∆(n)0; ∆(n) = 12nα(nα + Q)(99)where r0 is the short-distance cutoff.The parameters α, Q are to be found from theselfconsistency conditions.In case of the turbulence theory we know that the third moment has no anomalousdimension, due to the Kolmogorov’s 45 law.

This implies thatQ = −3α(100)after which we exactly reproduce the anomalous dimensions of so called Kolmogorov-Obukhov intermittency model.They assumed in 1961 the log-normal distribution of the energy dissipation rate asa modification of the Kolmogorov scaling. Various multifractal models generalizing thisdistribution, and the corresponding experimental and numerical data are discussed inproceedings of the 1991 conference in Santa Barbara [10].The physics of the fractal dimensions in our theory is almost the same as in theKolmogorov-Obukhov model.

The local energy dissipation rate at the edge can be esti-mated as Γ3C ∝e3αϕ. This quantity is, indeed, an exponential of the gaussian fluctuatingvariable.

The variance is proportional to log r since it is the two dimensional field withthe logarithmic propagator (how could they have guessed that! ).Strictly speaking, the dimensions of powers of ΓC were never measured.

People con-sidered the moments of velocity differences instead. In principle, the potential part of vwhich is present in these moments may change the trajectory, so we must be cautious.The point is, the potential part has completely different origin in our theory.

It comesfrom the nonlocal effects, involving all the scales, including the energy pumping scales. Inshort, this part is infrared divergent.

The vorticity part which we compute, is ultravioletdivergent, it it determined by the small scale fluctuations of the vortex sheet, representedby the Liouville field.19

In absence of direct evidence we may try to stretch the rules and estimate our inter-mittency exponents from the moments of velocity. I would expect this to be an upperestimate, as intermittency tends to decrease with the removal of the large scale effects.In my opinion, there is still no direct evidence for the anomalous dimensions in vorticitycorrelators.

It would be most desirable to fill this gap in real or numerical experiments.Let us stress once again, that above speculations do not pretend to be a theory ofturbulence. Still they may give us an idea how to build one.In statistical mechanics the Gibbs distribution was the beginning, not the end of thetheory.

If this approach is correct, which remains to be seen, then all the work is alsoahead of us in the string theory of turbulence. The string theory methods should be fittedfor this unusual case, and, perhaps, some scaling and area laws could be established.

Iappeal to my friends in the string community. Look at the turbulence, this is a beautifulexample of the fractal geometry with extra advantage of being guaranteed to exist!6AcknowledgmentsI am grateful to Albert Schwarz for stimulating discussions at initial stage of this work andto Herman Verlinde and members of the Santa Barbara workshop in Quantum Gravity andString Theory for discussion of the intermediate results.

Final results were discussed withVadim Borue, Andrew Majda and Mark Wexler who made various useful comments.Thisresearch was sponsored in part by the Air Force Office of Scientific Research (AFSC) undercontract F49620-91-C-0059. The United States Government is authorized to reproduceand distribute reprints for governmental purposes notwithstanding any copyright notationhereon.

This research was supported in part by the National Science Foundation underGrant No. PHY89-04035 in ITP of Santa Barbara.APoisson brackets for the Euler equationOne may derive the Poisson brackets for the Euler equation by comparing the right sideof this equation with[vα(r), H] =Zd3r′ [vα(r), vβ(r′)]δHδvβ(r′) =Zd3r′ [vα(r), vβ(r′)] vβ(r′)(101)Comparing this with the Euler equation we find[vα(r), vβ(r′)] = Tαν(r −r′)ωνβ(r′) + .

. .

(102)where dots stand for the ∂β terms which drop in the above integral. These terms shouldbe restored in such a way that the Poisson brackets would become skew symmetric plus20

they must be divergenceless in r′ as well as in r∂∂rα[vα(r), vβ(r′)] =∂∂r′β[vα(r), vβ(r′)] = 0(103)The unique solution is given by the formula in the text.It is also worth noting that one could derive the Poisson brackets for velocity fieldusing Clebsch variables, which are known to be an ordinary (p, q) pair. The velocity fieldis represented as an integralvα(r) =Zd3r′Tαβ(r −r′)Φ1(r′)∂βΦ2(r′)(104)and the corresponding vorticity is localωαβ(r) = eij∂αΦi∂βΦj(105)The Euler equations are equivalent to the following equations˙Φi + vβ∂βΦi = 0(106)which simply state that the Clebsch fields are passively advected by the flow.Theseequations have an explicit Hamiltonian form˙Φi = −eij δHδΦj(107)where it is implied that the Hamiltonian is the same, with velocity expressed in Clebschfields.Now the Poisson brackets for the velocity fields can be computed in a standard way[vα(r1), vβ(r2)] =Zd3reij δvα(r1)δΦi(r)δvβ(r2)δΦj(r)(108)which again yields the formula in the text.

This derivation is not as general as the previousone, since there are some flows which cannot be globally described in Clebsch variables.BHelmholtz vs Euler dynamicsLet us study the variation of the vortex cell velocity field. Our first objective would beto show that the time variation according to the Helmholtz equationδXα(ρ) = dt vα(X(ρ))(109)reproduces the Euler equationδvα(r) = dt (vβ(r)ωαβ(r) −∂αh(r))(110)21

Simple calculation of the variation of the initial definition with integration by partsyieldsδvα(r) = (eαβν∂γ −eαβγ∂ν) ∂βZD d3ρδXν(ρ)Ωγ(ρ)4π|r −X(ρ)|(111)Now we use the identityeαβν∂γ −eαβγ∂ν = eβγν∂α −eαγν∂β(112)(the difference between the left and the right sides represents the completely skew sym-metric tensor of the fourth rank in three dimensions, which must vanish). This gives usthe following two terms in velocity variationδvα(r) = −∂αδh(r) + eαγνZd3ρ δXν(ρ)Ωγ(ρ)δ3(r −X(ρ))(113)whereδh(r) = eγβν∂βZd3ρ δXν(ρ)Ωγ(ρ)4π|r −X(ρ)|(114)The first term is purely potential, it can be reconstructed from the second term bysolving ∂αδvα(r) = 0.

Now we see that the Helmholtz variation reproduces the Eulerequation with the enthalpyh(r) = eγβν∂βZd3ρ vν(X(ρ))Ωγ(ρ)4π|r −X(ρ)|(115)At the same time we see that the Helmholtz variation is defined modulo gauge trans-formationδXν(ρ) ⇒δXν(ρ) + f(ρ)Ων(ρ)(116)which leaves the velocity variation invariant.Note that the functional derivativeδvα(r)δXν(ρ) = (eαβν∂γ −eαβγ∂ν) ∂βΩγ(ρ)4π|r −X(ρ)|(117)is a traceless tensor in α, ν . This is sufficient for the volume conservation∂t (DX) ∝Zd3ρδvα(X(ρ))δXα(ρ)= 0(118)which is the Liouville theorem for the vortex dynamics.We found the following Poisson bracket for the X field[Xα(ρ), Xβ(ρ′)] = −δ3(ρ −ρ′)eαβγΩγ(ρ)Ω2µ(ρ)(119)The equations of motion corresponding to these Poisson brackets read∂t Xα(ρ) = [Xα(ρ), H] =Zd3ρ′ [Xα(ρ), Xβ(ρ′)]δHδXβ(ρ′)(120)22

Let us compare these equations with the usual Helmholtz dynamics. The variation ofthe Hamiltonian readsδH =Zd3rvα(r)δvα(r)(121)Substituting here the velocity variation (113) we could drop the ∂αδh term as it vanishesafter integration by parts.

As a result we findδHδXα(ρ) = eαβγvβ (X(ρ)) Ωγ(ρ)(122)Finally, in the equation of motion we have∂t Xα(ρ) = −Zd3ρ′δ3(ρ −ρ′)eαβγΩγ(ρ)Ω2µ(ρ)eβµνvµ (X(ρ)) Ων(ρ)(123)= δαβ −Ωα(ρ)Ωβ(ρ)Ω2µ(ρ)!vβ (X(ρ))So, the Poisson brackets correspond to the motion in transverse direction to the gaugetransformations. Or, to put it in different terms, we could modify the Helmholtz dynamicsby adding the time dependent gauge transformations δρ(t) to the time shift of the vortexsheet∂t Xα(ρ) = vα (X(ρ)) + ∂aXα(ρ)∂t δρa(124)where∂t δρa = −Ωa(ρ)Ωβ(ρ)vβ (X(ρ))Ω2µ(ρ)(125)Should we insist on the unmodified Helmholtz dynamics, we would have to admit thatthis cannot be achieved by any Poisson brackets.

This is so called generalized Hamiltoniandynamics, where the formula (122) for the variation of the Hamiltonian cannot be uniquelysolved for the velocity.The terms corresponding to the gauge transformation remainunspecified. In our opinion, this difference is immaterial, as the Helmholtz dynamics isindistinguishable from the one with the Poisson brackets.23

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[7] Anselmet et.al. Phys.

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Kluwer Academic Publishers, 1992. [11] A.A. Migdal, M. Wexler, Gaussian Clebsch variables and the Kolmogorov scaling law,1992, unpublished .

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