TURBULENCE terval of scales in a cascade-like process. The cascade is a state of a nonlinear physical system that idea explains the basic macroscopic manifestation of has distribution over many degrees of freedom turbulence: the rate of of the dynamical in- strongly deviated from equilibrium. Turbulence is tegral of motion has a finite limit when the dissipation irregular both in time and in space. Turbulence can coefficient tends to zero. In other words, the mean rate be maintained by some external influence or it can of the viscous energy dissipation does not depend on decay on the way to relaxation to equilibrium. The at large Reynolds numbers. That means that term first appeared in fluid mechanics and was later symmetry of the inviscid equation (here, time-reversal generalized to include far-from-equilibrium states in invariance) is broken by the presence of the viscous solids and plasmas. term, even though the latter might have been expected If an obstacle of size L is placed in a fluid of viscosity to become negligible in the limit Re →∞. ν that is moving with velocity V , a turbulent wake The cascade idea fixes only the mean flux of the emerges for sufficiently large values of the Reynolds respective integral of motion, requiring it to be constant number across the inertial interval of scales. To describe an Re ≡ V L/ν. entire turbulence statistics, one has to solve problems on a case-by-case basis with most cases still unsolved. At large Re, flow perturbations produced at scale L experience, a viscous dissipation that is small Weak Turbulence compared with nonlinear effects. Nonlinearity then induces motions at smaller and smaller scales until From a theoretical point of view, the simplest case is viscous dissipation terminates the process at a scale the turbulence of weakly interacting . Examples much smaller than L, leading to a wide (so-called include waves on the water surface, waves in inertial) interval of scales where viscosity is negligible with and without a magnetic field, and spin waves in and nonlinearity plays a dominant role. magnetics. We assume spatial homogeneity and denote Examples of this phenomenon include waves excited by ak the amplitude of the wave with the wave vector on a fluid surface by or moving bodies and waves k. When the amplitude is small, it satisfies the linear in plasmas and solids that are excited by external equation electromagnetic fields. The state of such a system is called turbulent when the of the waves ∂a k =−iω a + f (t) − γ a . (1) excited greatly differs from the wavelength of the waves ∂t k k k k k that dissipate. Nonlinear interactions excite waves in the interval of (called the transparency Here, the law ωk describes wave propaga- window or inertial interval as in fluid turbulence) tion, γk is the decrement of linear damping, and fk between the injection and dissipation scales. describes pumping. For the linear system, ak is differ- The ensuing complicated and irregular dynamics ent from zero only in the regions of k-space where fk require a statistical description based on averaging is nonzero. To describe that involves over regions of space or intervals of time. Because wave numbers outside the pumping region, one must nonlinearity dominates in the inertial interval, it is account for the interactions among different waves. natural to ask to what extent the statistics are universal, Considering the wave system to be closed (no exter- in the sense of being independent of the details of nal pumping or dissipation) one can describe it as a excitation and dissipation. The answer to this question Hamiltonian system using wave amplitudes as normal is far from evident for non-equilibrium systems. A canonical variables (Zakharov et al., 1992). At small fundamental physical problem is to establish which amplitudes, the Hamiltonian can be written as an ex- statistical properties are universal in the inertial interval pansion over ak, where the second-order term describes of scales and which are features of different turbulent non-interacting waves and high-order terms determine systems. the interaction Constraints on dynamics are imposed by conserva-     tion laws, and therefore conserved quantities must play 2 ∗ ∗ H = ωk|ak| dk + V123a1a a + c.c. an essential role in turbulence. Although the conser- 2 3 4 vation laws are broken by pumping and dissipation, δ(k1 − k2 − k3) dk1 dk2 dk3 + O(a ). (2) these factors do not act in the inertial interval. Under incompressible turbulence, for example, the kinetic en- Here, V123 = V(k1, k2, k3) is the interaction vertex, ergy is pumped by external forcing and is dissipated and c.c. denotes complex conjugate. In this expansion, by viscosity. As suggested by in we presume every subsequent term smaller than the d 1921, kinetic energy flows throughout the inertial in- previous one, in particular, ξk =|Vkkkak|k /ωk  1.

1 TURBULENCE

Wave turbulence that satisfies that condition is called dominate over dissipation at k  kd . As has been noted, weak turbulence. Also, space dimensionality d can be wave turbulence appears when there is a wide (inertial) 1, 2, or 3. interval of scales where both pumping and damping A dynamic equation that accounts for pumping, are negligible, which requires kd  kf , the condition damping, wave propagation, and interaction thus has analogous to Re  1. the following form: (3) The presence of delta-function in Ik means that wave interaction conserves the quadratic ∂ak δH part of the energy E = ω n dk = E dk. For the =−i + f (t) − γ a . (3) k k k ∗ k k k cascade picture to be valid, the collision integral ∂t δak has to converge in the inertial interval which means that energy exchange is small between motions of It is likely that the statistics of the weak turbulence vastly different scales, a property called interaction at k  k is close to Gaussian for wide classes of f locality in k-space. Consider now a statistical steady pumping statistics (this has not been shown rigorously). state established under the action of pumping and It is definitely the case for a random force with the dissipation. Let us multiply (5) by ω and integrate it statistics close to Gaussian. We consider here and below k over either interior or exterior of the ball with radius k. a pumping by a Gaussian random force statistically   isotropic and homogeneous in space, and white in time. Taking kf k kd , one sees that the energy flux Thus, through any spherical surface ( is a solid angle),   k  ∗  = k + k −  = d−1 (3) fk(t)fk (t ) F(k)δ( )δ(t t ), (4) Pk k dk dωkIk , 0 where angular brackets imply spatial averages, and is constant in the inertial interval and is equal to the F(k) is assumed nonzero only around some kf .For energy production/dissipation rate: waves to be well defined, we assume γk  ωk. Because the dynamic equation (3) contains a   quadratic nonlinearity, the statistical description in Pk = ε = ωkFk dk = γkEk dk. (6) terms of moments encounters the closure problem: the time derivative of the second moment is expressed via the third one, the time derivative of the third Let us assume now that the medium (characterized moment is expressed via the fourth one, and so on. by ωk and V123) can be considered isotropic at Fortunately, weak turbulence in the inertial interval is the scales in the inertial interval. In addition, for expected to have the statistics close to Gaussian so scales much larger or much smaller than a typical one can express the fourth moment as the product of scale in the medium (like the Debye radius in two second ones. As a result, one gets a closed kinetic plasma or the depth of the water), the Hamiltonian equation for the single-time pair correlation function = α  coefficients are usually scale invariant: ω(k) ck and akak =nkδ(k + k ) (Zakharov et al., 1992): | k k k |2 = 2 2m k k V( , 1, 2) V0 k χ( 1/k, 2/k) with χ 1. ∂n Remember that we presumed statistically isotropic k = F − γ n + I (3) ,  k k k k force. In this case, the pair correlation function that ∂t  (3) describes a steady cascade is also isotropic and scale I = Uk12 − U1k2 − U2k1 dk1 dk2 , k   invariant: 2 U123 = π n2n3 − n1(n2 + n3) |V123| − − − δ(k1 − k2 − k3)δ(ω1 − ω2 − ω3). 1/2 1 m d nk ε V0 k . (7) (5)

(3) This is called the kinetic equation for waves. One can show that (7) reduces Ik to zero (see (3) The collision integral Ik describes three-wave Zakharov et al. 1992). interactions: the first term in the integral corresponds If the dispersion relation ω(k) does not allow for to a decay of a given wave while the second and third the condition ω(k1)+ω(k2) = ω(|k1 + k2|) terms correspond to a confluence with other waves. then the three-wave collision integral is zero One can estimate from (5) the inverse time of nonlin- and one has to account for four-wave scatter- ear interaction at a given k as |V (k,k,k)|2n(k)kd /ω(k). ing which is always resonant, that is what- We define kd as the wave number where this inverse ever ω(k) one can always find four wave vec- time is comparable to γ(k)and assume nonlinearity to tors that satisfy ω(k1) + ω(k2) = ω(k3) + ω(k4) and

2 TURBULENCE k1 + k2 = k3 + k4. The collision integral that de- (acoustic) dispersion where ω(k)= ck for arbitrarily scribes scattering, small amplitudes. Indeed, there is no dispersion of wave    velocity for acoustic waves, so waves moving in the (4) = | |2 + Ik Tk123 n2n3(n1 nk) same direction interact strongly and produce shock 2  waves when viscosity is small. Formally, there is a − + k + k − k − k n1nk(n2 n3) δ( 1 2 3) singularity due to the coinciding arguments of delta- ×δ(ωk + ω1 − ω2 − ω2) dk1 dk2 dk3, functions in (5) (and in the higher terms of perturbation (8) expansion for ∂nk/∂t), which is thus invalid at however small amplitudes. Still, some features of the statistics conserves the energy and also the wave action N = of acoustic turbulence can be understood even without nk dk (which can also be called the number of waves). a closed description. Pumping generally provides for an input of both E and Consider a one-dimensional case which pertains, for N. If there are two inertial intervals (at k  kf and instance, to sound propagating in long pipes. Because k  kf ), then there should be two cascades. Indeed, if weak shocks are stable with respect to transversal ω(k)grows with k then absorbing finite amount of E at perturbations (Landau & Lifshitz, 1987), quasi-one- kd →∞ corresponds to an absorption of an infinitely dimensional perturbations may propagate in two and small N. It is thus clear that the flux of N has to go in three dimensions as well. In a reference frame that the opposite direction, that is to snall wave numbers. moves with the sound velocity, weakly compressible A so-called inverse cascade with a constant flux of N 1-d flows (u  c) are described by the Burgers’ can thus be realized at k  kf . A sink at small k can equation (Landau & Lifshitz, 1987) be provided by wall in the container or by long waves leaving the turbulent region in open spaces (as + − = in storms). ut uux νuxx 0. (9) (3) The collision integral Ik involves products of two 1/2 nk so that flux constancy requires Ek ∝ ε while The Burgers’ equation has a propagating shock-wave 1/3 = { + [ − ]} − 1 for the four-wave case, one has Ek ∝ ε . In many solution u 2v 1 exp v(x vt)/ν with the en- 2 cases (when there is complete self-similarity), that ergy dissipation rate ν ux dx independent of ν. knowledge is sufficient to obtain the scaling of Ek from The shock width ν/v is a dissipative scale, and we a dimensional reasoning without actually calculating consider acoustic turbulence produced by a pump- V and T . For example, short waves in deep water ing correlated on much larger scales (i.e., pump- are characterized by the surface tension σ and density ing a pipe from one end by much less 3 than cv/ν). After some time, the system will de- ρ so the dispersion relation must be ωk ∼ σk /ρ, which allows for the three-wave resonance and thus velop shocks at random positions. Here we con- E ∼ ε1/2(ρσ )1/4k − 7/4. For long waves in deep water, sider the single-time statistics of the Galilean in- k = − the surface-restoring force is dominated by gravity so variant velocity difference δu(x, t) u(x, t) u(0,t). The moments of δu are called structure functions that the gravitational acceleration√ g replaces σ as a S (x, t) =[u(x, t) − u(0,t)]n . Quadratic nonlinear- defining parameter and ωk ∼ gk. Such a dispersion n law does not allow for three-wave resonance so that ity allows the time derivative of the second moment to the dominant interaction is four-wave scattering which be expressed via the third one: permits two cascades. The direct − ∼ 1/3 2/3 1/2 5/2 2 corresponds to Ek ε ρ g k . The inverse ∂S2 ∂S3 ∂ S2 cascade carries the flux of N which we denote Q,ithas =− − 4ε + ν . (10) ∂t 3∂x ∂x2 the dimensionality [Q]=[ε]/[ωk] and corresponds to 1/3 2/3 2/3 − 7/3 Ek ∼ Q ρ g k . =  2 Because the statistics of weak turbulence is near Here ε ν ux is the mean energy dissipation rate. Gaussian, it is completely determined by the pair Equation (10) describes both a free decay (then correlation function, which is in turn determined by the ε depends on t) and the case of a permanently respective flux. We thus conclude that weak turbulence acting pumping which generates turbulence statistically is universal in the inertial interval. steady at scales less than the pumping length. 3 In the first case, ∂S2/∂t S2u/L  ε u /L (where L is a typical distance between shocks) Strong Wave Turbulence while in the second case, ∂S2/∂t = 0 so that One cannot treat wave turbulence as a set of weakly S3 = 12εx + ν∂S2/∂x. Consider now the limit ν → 0 interacting waves when the wave amplitudes are large at fixed x (and t for decaying turbulence). Shock (ξk ≥ 1) and also in the particular case of linear dissipation provides for a finite limit of ε at ν → 0,

3 TURBULENCE then wave turbulence, being a set of weakly interacting plane waves, can be studied uniformly for different S3 =−12εx. (11) systems (Zakharov et al., 1992). On the contrary, when nonlinearity is comparable to or exceeds dispersion, different structures appear in different This formula is a direct analog of (6). Indeed, the systems. Identifying structures and the role they play Fourier transform of (10) describes the energy density in determining different statistical characteristics of E =|u |2 /2: (∂ − νk2)E =−∂P /∂k where the k k t k k strong wave turbulence remains to be investigated for k-space flux most cases. Broadly, one distinguishes conservative   k ∞    structures (like and vortices) from dissipative Pk = dk dxS3(x)k sin(k x)/24. structures which usually appear as a result of finite-time 0 −∞ singularity of the nondissipative equations (like shocks, It is thus the flux constancy that fixes S3(x) which is light self-focusing, or wave collapse). For example, universal (determined solely by ε) and depends neither nonlinear wave packets are described by the nonlinear on the initial statistics for decay nor on the pumping Schrödinger equation, for steady turbulence. On the contrary, other structure functions S (x) are not given by (εx)n/3. Indeed, 2 n it +  + T ||  = 0. (12) the scaling of the structure functions can be readily understood for any dilute set of shocks (that is, when | |2 shocks do not cluster in space) which seems to be the Weak wave turbulence is determined by T and is the case both for smooth initial, conditions and large-scale same both for T<0 (wave repulsion) and T>0(wave attraction). At high levels of nonlinearity, different pumping in Burgers turbulence. In this case, Sn(x) ∼ | |n +  | | signs of T correspond to dramatically different physics: Cn x Cn x , where the first term comes from the regular (smooth) parts of the velocity while the second At T<0, one has a stable condensate, solitons and comes from O(x) probability to have a shock in the vortices, while at T>0, instabilities dominate and wave collapse is possible at d = 2, 3. No analytic theory interval x. The scaling exponents, ξn = dlnSn/dlnx, is yet available for strong turbulence described by (12). thus behave as follows: ξn = n for n ≤ 1 and ξn = 1 for n>1. That means that the probability density function Because the parameter of nonlinearity ξ(k)generally (PDF) of the velocity difference in the inertial interval depends on k then there may exist a weakly turbulent ∼ P(δu,x) is not scale-invariant, that is, the function cascade until some k∗ where ξ(k∗) 1, and strong of the rescaled velocity difference δu/xa cannot be turbulence beyond this, wave number; thus weak and made scale-independent for any a. As one goes to strong turbulence can coexist in the same system. smaller scales, the lower-order moments decrease faster Presuming that some mechanism (for instance, wave than the higher-order ones, that means that the smaller breaking) prevents the appearance of wave amplitudes  the scale the more probable are large fluctuations. In that correspond to ξk 1, one may hypothetize that other words, the level of fluctuations increases with the some cases of strong turbulence correspond to the resolution. When the scaling exponents ξ do not lie on balance between dispersion and nonlinearity local in n = a straight line, this is called an anomalous scaling since k-space so that ξ(k) constant throughout its domain it is related again to the symmetry (scale invariance) of in k-space. That would correspond to the spectrum ∼ 3 − d | |2 the PDF broken by pumping and not restored even when Ek ωkk / Vkkk which is ultimately universal, x/L → 0. As an alternative to the description in terms that is independent even of the flux (only the boundary of structures (shocks), one can relate the anomalous k∗ depends on the flux). For gravity waves, this = − 3 scaling in Burgers turbulence to the additional integrals gives Ek ρgk , the same spectrum one obtains 2n presuming the wave profile to have cusps (another type of motion. Indeed, the integrals En = u dx/2 are all conserved by the inviscid Burgers’ equation. Any shock of dissipative structure leading to whitecaps in stormy —see Phillips, 1977). It is unclear if such flux- dissipates the finite amount of En at the limit ν → 0so ˙ independent spectra are realized. that similar to (11) one denotes  En =εn and obtains S2n + 1 =−4(2n + 1)εnx/(2n − 1) for integer n. ∝| | ∝ − 2 Note that S2(x) x corresponds to E(k) k , Incompressible Turbulence which is natural since every shock gives uk ∝ 1/k at k  v/ν, that is, the energy spectrum is determined Incompressible fluid flow is described by the NavierÐ by the type of structures (shocks) rather than by Stokes equation energy flux constancy. Similar ideas were suggested 2 for other types of strong wave turbulence assuming ∂t v(r,t)+ v(r,t)· ∇v(r,t)− ν∇ v(r,t) them to be dominated by different structures. Weak =−∇p(r,t), div v = 0.

4 TURBULENCE

We are again interested in the structure functions n Sn(r,t)=[(v(r,t)− v(0,t))· r/r] and treat first the three-dimensional case. Similar to (10), one considers distance r smaller than the force correlation scale for a steady case and smaller than the size of the turbulent region for a decay case. For such r, one can derive the KarmanÐHowarth relation between S2 and S3 (see Landau & Lifshitz, 1987):

∂S 1 ∂ 4ε 2ν ∂ ∂S 2 =− (r4S ) + + r4 2 . (13) ∂t 3r4 ∂r 3 3 r4 ∂r ∂r

Here ε = ν  (∇v)2 is the mean energy dissipation rate. Neglecting the time derivative (which is zero in a steady state and small compared to ε for decaying turbulence), one can multiply (13) by r4 and integrate: S3(r) =−4εr/5 + 6ν dS2(r)/dr. Andrei Kolmogorov in 1941 considered the limit ν → 0 for fixed r and assumed nonzero limit for ε, which gives the 4 so-called 5 law (see Landau & Lifshitz, 1987; Frisch, n/3 ζ 1995): n σ ξ n n =−4 1 S3 5 εr. (14)

This relation is a direct analog of (6) and (11). It also means that the kinetic energy has a constant flux in Exponents of structure functions 13 n the inertial interval of scales (the viscous scale η is 2 Figure 1. The scaling, exponents of the structure functions ξn defined by νS2(η) εη ). Law (14) implies that the for Burgers, ζn for NavierÐStokes, and σn for the passive scalar. third-order moment is universal, that is, it does not The dotted straight line is the Kolmogorov hypothesis n/3. depend on the details of the turbulence production but is determined solely by the mean energy dissipation rate. The rest of the structure functions have not yet been derived. Kolmogorov (and also Werner Heisenberg, Karl von Weizsacker, and Lars Onsager) presumed in particular, that detailed computer simulation of water 4 7 the pair correlation function to be determined only or oil pipe flows (Re ∼ 10 − 10 ) or turbulent clouds 2/3 ∼ 6 − 9 by ε and r which would give S2(r) ∼ (εr) and the (Re 10 10 ) is out of question for the foreseeable 2/3 −5/3 energy spectrum Ek ∼ ε k . Experiments suggest future. To calculate correctly at least the large-scale that ζn = dlnSn/dlnr lie on a smooth concave curve part of the flow, it is desirable to have some theoretical sketched in Figure 1. While ζ2 is close to 2/3, it has model to parametrize the small-scale motions, the main to be a bit larger because experiments show that the obstacle being our lack of qualitative understanding 1 = and quantitative description of how turbulence statistics slope at zero dζn/dn is larger than 3 while, ζ(3) 1 in agreement with (14). As in Burgers turbulence, the changes as one goes downscale. PDF of velocity differences in the inertial interval is Large-scale motions in a shallow fluid can be not scale-invariant in 3-d incompressible turbulence. approximately considered two dimensional. When the No one has yet found an explicit relation between the velocities of such motions are much smaller than the anomalous scaling for 3-d NavierÐStokes turbulence velocities of the surface waves and the velocity of and either structures or additional integrals of sound, such flows can be considered incompressible. motion. Their description is important for understanding While not exact, the Kolomogorov approximation atmospheric and oceanic turbulence at the scales larger 2/3 S2(η) (εη) can be used to estimate the viscous than atmosphere height and depth. scale: η LRe − 3/4. The number of degrees of freedom Vorticity ω = curl v is a scalar in a two-dimensional involved in 3-d incompressible turbulence can thus be flow. It is advected by the velocity field and dissipated roughly estimated as N ∼ (L/η)3 ∼ Re9/4. That means, by viscosity. Taking the curl of the NavierÐStokes

5 TURBULENCE equation, one gets If the source ϕ produces fluctuations of θ on some scale L then the inhomogeneous velocity field stretches, ∂ ω + (v · ∇)ω = ν∇2ω. (15) contracts, and folds the field θ producing progressively t smaller and smaller scales. If the rms velocity gradient is  then molecular diffusion is√ substantial at scales Two-dimensional incompressible inviscid flow just less than the diffusion scale rd = κ/. The ratio transports vorticity from place to place and thus conserves spatial averages of any function of vorticity. Pe = L/rd In particular, we now have the second quadratic inviscid invariant (in addition to energy) which is is called the Peclet number. It is an analog of the called enstrophy: ω2 dr. Since the spectral density Reynolds number for passive scalar turbulence. When 2 Pe  1, there is a long inertial interval where the flux of the energy is |vk| /2 while that of the enstrophy is 2 constancy relation derived by A.M. Yaglom in 1949 |k × vk| , Robert Kraichnan suggested in 1967 that the direct cascade (towards large k) is that of enstrophy holds: while the inverse cascade is that of energy. Again, for the inverse energy cascade, there is no consistent theory (v1 · ∇1 + v2 · ∇2)θ1θ2 =2P, (18) except for the flux relation that can be derived similar to (14): where P = κ  (∇θ)2 and subscripts denote the spatial points. In considering the passive scalar problem, the S3(r) = 4εr/3. (16) velocity statistics is presumed to be given. Still, the correlation function (18) mixes v and θ and does The inverse cascade is observed in the atmosphere (at not generally allow one to make a statement on any scales of 30Ð500 km) and in laboratory experiments. correlation function of θ. The proper way to describe the Experimental data suggest that there is no anomalous correlation functions of the scalar at scales much larger n/3 2/3 than the diffusion scale is to employ the Lagrangian scaling; thus Sn ∝ r . In particular, S2 ∝ r which corresponds to E ∝ k − 5/3. It is ironic that probably description, that is, to follow fluid trajectories. Indeed, k if we neglect diffusion, then Equation (17) can be the most widely known statement on turbulence, the 5 3 solved along the characteristics R(t) which are called spectrum suggested by Kolmogorov for the 3-d case, is Lagrangian trajectories and satisfy dR/dt = v(R,t). not correct in this case (even though the true scaling Presuming zero initial conditions at t →−∞, we write is close), while it is probably exact in Kraichnan’s  inverse 2-d cascade. Qualitatively, it is likely that the   t      absence of anomalous scaling in the inverse cascade is θ R(t), t = ϕ R(t ), t dt . (19) associated with the growth√ of the typical turnover time −∞ (estimated, say, as r/ S2) with the scale. As the inverse cascade proceeds, the fluctuations have enough time to In that way, the correlation functions of the scalar get smoothed out as opposed to the direct cascade in Fn =θ(r1,t)...θ(rn,t) can be obtained by in- three dimensions, where the turnover time decreases in tegrating the correlation functions of the pumping the direction of the cascade. along the trajectories that satisfy the final conditions Before discussing the direct (enstrophy) cascade, we Ri(t) = ri. describe a similar yet somewhat simpler problem of Consider first, the case of pumping which is Gaus- passive scalar turbulence, which allows one to introduce sian, statistically homogeneous, and isotropic in space the necessary notions of Lagrangian description of the and white in time:  ϕ(r1,t1)ϕ(r2,t2) =(|r1 − r2|) fluid flow. Consider a scalar quantity θ(r,t) that is δ(t1 − t2) where the function  is constant at r  L subject to molecular diffusion and advection by the and goes to zero at r  L. The pumping provides for fluid flow but has no back influence on the velocity symmetry θ →−θ which makes only even correlation (i.e., passive): functions F2n nonzero. The pair correlation function is  2 t   ∂t θ + (v · ∇)θ = κ∇ θ + ϕ. (17)   F2(r, t) =  R12(t ) dt . (20) −∞ Here κ is molecular diffusivity. In the same 2-d flow, ω    and θ behave in the same way, but vorticity is related Here R12(t ) =|R1(t ) − R2(t )| is the distance be- to velocity while the passive scalar is not. Examples tween two trajectories and R12(t) = r. The function  of passive scalar are smoke in air, salinity in water, essentially restricts the integration to the time interval  and temperature when one neglects thermal convection. when the distance R12(t ) ≤ L. Simply speaking, the

6 TURBULENCE stationary pair correlation function of a tracer is (0) it cannot be perfectly smooth to provide for a (which is twice the injection rate of θ 2) times the aver- nonzero vorticity dissipation in the inviscid limit, age time T2(r, L) that two fluid particles spend within but the possible singularitites are indeed shown to the correlation scale of the pumping. The larger r, the be no stronger than logarithmic). That makes the less time it takes for the particles to separate from r to L vorticity cascade similar to the Batchelor regime and the smaller is F2(r). Of course, T12(r, L) depends of passive scalar cascade with a notable change on the properties of the velocity field. A general the- in that the rate of stretching λ acting on a ory is available only when the velocity field is spatially given scale is not a constant but is logarithmically smooth at the scale of scalar pumping L. This so-called growing when the scale decreases. Since λ scales Batchelor regime happens, in particular, when the scalar as vorticity, the law of renormalization can be cascade occurs at the scales less than the viscous scale established from dimensional reasoning and one gets of fluid turbulence. This requires the Schmidt number ω(r, t)ω(0,t) ∼[D ln(L/r)]2/3 which corresponds 2/3 − 3 − 1/3 ν/κ (called the Prandtl number when θ is temperature) to the energy spectrum Ek ∝ D k ln (kL). to be large, which is the case for very viscous liquids. Higher-order correlation functions of vorticity are In this case, one can approximate the velocity differ- also logarithmic, for instance,  ωn(r,t)ωn(0,t) ∼ 2n/3 ence v(R1,t)− v(R2,t) ≈ˆσ(t)R12(t) with the La- [D ln(L/r)] . Note that both passive scalar in grangian strain matrix σij (t) =∇j vi. In this regime, the the Batchelor regime and vorticity cascade in two distance obeys the linear differential equation dimensions are universal, that is, determined by the single flux (P and D, respectively) despite ˙ R12(t) =ˆσ(t)R12(t). (21) the existence of higher-order conserved quantities. Experimental data and numeric simulations support The theory of such equations is well developed and these conclusions. related to what is called Lagrangian chaos, as fluid trajectories separate exponentially as is typical for Zero Modes and Anomalous Scaling systems with dynamical chaos (see, e.g. Falkovich et al., 2001): At t much larger than the correlation time Let us now return to the Lagrangian description and discuss it when velocity is not spatially smooth, for of the random process σ(t)ˆ , all moments of R12 grow example, that of the energy cascades in the inertial exponentially with time and  ln[R12(t)R12(0)] =λt, where λ is called a senior Lyapunov exponent of the interval. One can assume that it is Lagrangian statistics flow (note that for the description of the scalar we need that are determined by the energy flux when the the flow taken backwards in time which is different from distances between fluid trajectories are in the inertial that taken forward because turbulence is irreversible). interval. That assumption leads, in particular, to the Dimensionally, λ = f (Re) where the limit of the Richardson law for the asymptotic growth of the function f at Re →∞ is unknown. We thus obtain interparticle distance:

= −1 = −1  2 ∼ 3 F2(r) (0)λ ln(L/r) 2Pλ ln(L/r). (22) R12(t) εt , (24)

In a similar way, one shows that for n  ln(L/r), all which was first established from atmospheric observa- Fn are expressed via F2 and the structure functions tions (in 1926) and later confirmed experimentally for 2n n S2n =[θ(r,t)− θ(0,t)] ∝ ln (r/rd ) for n  energy cascades both in 3-d and in 2-d. There is no ln(r/rd ). This can be generalized for an arbitrary statis- consistent theoretical derivation of (24) and it is un- tics of pumping as long as it is finite-correlated in time clear whether it is exact (likely to be in 2-d) or just (Falkovich et al., 2001). approximate (possible in 3-d). The semi-heuristic ar- One can use the analogy between passive scalar gument usually presented in textbooks is based on the ˙ 1/3 and vorticity in two dimensions as has been shown mean-field estimate: R12 = δv(R12,t) ∼ (εR12) by Falkovich and Lebedev in 1994 following the line 2/3 − 2/3 ∼ which upon integration gives: R12 (t) R12 (0) suggested by Kraichnan in 1967. For the enstrophy ε1/3t. For the passive scalar it gives, by virtue of (20), cascade, one derives the flux relation analogous to (18): − 1/3 2/3 2/3 F2(r) ∼ (0)ε [L − r ] which was suggested by S. Corrsin and A.M. Oboukhov. The structure func-  v · ∇ + v · ∇ = − 1/3 2/3 ( 1 1 2 2)ω1ω2 2D, (23) tion is then S2(r) ∼ (0)ε r . Experiments mea- suring the scaling exponents σn = dlnSn(r)/dlnr gen- 2 where D =ν(∇ω) . The flux relation along with erally give σ2 close to 2/3 but higher exponents devi- ω = curl v suggests the scaling δv(r)∝ r, that is, ating from the straight lineane even stronger than the velocity being close to spatially smooth (of course, exponents of the velocity in 3-d. Moreover, the scalar

7 TURBULENCE exponents σn are anomalous even when advecting ve- a zero mode Z4 of the operator L4: L4Z4 = 0. Such locity has a normal scaling like in the 2-d energy cas- zero modes necessarily appear (to satisfy the bound- cade. ary conditions at r L) for all n>1 and the scal- To better understand the Lagrangian dynamics (and ing exponents of Z2n are generally different from nγ passive scalar statistics) in a spatially nonsmooth that is anomalous. In calculating the scalar structure velocity, Kraichnan suggested considering the model functions, all terms cancel out except a single zero of a velocity field having the simplest statistical and mode (called irreducible because it involves all dis- temporal properties, namely Gaussian velocity which tances between 2n points). Calculation of Zn and their is white in time: scaling exponents σn were carried out analytically at γ  1, 2 − γ  1 and d  1, and numerically for all γ  i r j = − r = v ( ,t)v (0, 0) δ(t) D0δij dij ( ) , and d 2, 3 (Falkovich et al., 2001). 2−γ ij i j −2 That gives σ lying on a convex curve (as in Figure 1) dij = D1 r (d + 1 − γ)δ + (γ − 2)r r r . n which saturates to a constant at large n. Such saturation (25) (confirmed by experiments) is a signature that most singular structures in a scalar field are shocks (as in Here the exponent γ ∈[0, 2] is a measure of the Burgers’ turbulence), the value σ at n →∞ is the velocity nonsmoothness with γ = 0 corresponding to a n fractal codimension of fronts in space. Interestingly, the smooth velocity and γ = 2 corresponding to a velocity Kraichnan model enables one to establish the relation very rough in space (distributional). RichardsonÐ between the anomalous scaling and conservation laws Kolmogorov scaling of the energy cascade corresponds of a new type. Thus, the combinations of distances to γ = 2/3. Lagrangian flow is a Markov random between points that constitute zero modes are the process for the Kraichnan ensemble (25). Every fluid statistical integrals of Lagrangian evolution. To give particle undergoes a Brownian random walk with the a simple example, in a Brownian walk, the mean so-called diffusivity D . The PDF for two particles 0 distance between every two particles grows with time, to be separated by r after time t satisfies the diffusion  R2 (t) =R2 (0) + κt, while  R2 − R2 and equation (see, e.g., Falkovich et al., 2001) lm lm lm pq  2(d + 2)R2 R2 − d(R4 + R4 ) (and an infinity = lm pq lm pq ∂t P(r,t) L2P(r,t), of similarly built harmonic polynomials) are conserved. i j 1−d d+1−γ L2 = dij (r)∇ ∇ = D1(d − 1)r ∂r r ∂r , Note that the integrals are not dynamical, they are (26) conserved only in average. In a turbulent flow, the form of such conserved quantities is more complicated − 2 − γ with the scale-dependent diffusivity D1(d 1)r . but the essence is the same: the increase of averaged The asymptotic solution of (26) is lognormal for the distances between fluid particles is compensated by Batchelor case while for γ>0 the decrease in shape fluctuations. The existence   of statistical conserved quantities breaks the scale − P(r,t)= rd 1td/γ exp −const rγ /t . (27) invariance of scalar statistics in the inertial interval and explains why scalar turbulence knows more about pumping than just the value of the flux. Note that both = For γ 2/3, it reproduces, in particular, the Richardson symmetries, one broken by pumping (scale invariance) law. Multiparticle probability distributions also satisfy and another by damping (time reversibility) are not diffusion equations in the Kraichnan model as well as restored even when r/L→ 0 and rd /r → 0. all the correlation functions of θ. Multiplying equation For the vector field (like velocity or magnetic field (17) by θ2 ...θ2n and averaging over the Gaussian in magnetohydrodynamics), the Lagrangian statistical statistics of v and ϕ, one derives  integrals of motion may involve both the coordinate of the fluid particle and the vector it carries. Such ∂t F2n = L2nF2n + F2n−2(rlm), integrals of motion were built explicitly and related l,m  to the anomalous scaling for the passively advected = r ∇i∇j L2n dij ( lm) l m. (28) magnetic field in the Kraichnan ensemble of velocities (Falkovich et al., 2001). Doing that for velocity that This equation enables one, in principle, to derive induc- satisfies the NavierÐStokes equation remains a task for tively all steady-state F2n starting from F2. The equa- the future. tion ∂t F2(r, t) = L2F2(r, t) + (r) has a steady so- γ GREGORY FALKOVICH lution F2(r) = 2[(0)/γ d(d − 1)D1][dL /(d − γ)− rγ ], which has the CorrsinÐOboukhov form for See also Chaos vs. turbulence; Development of sin- γ = 2/3. Further, F4 contains the so-called forced solu- gularities; Intermittency; Kolmogorov cascade; La- tion having the normal scaling 2γ but also, remarkably, grangian chaos; Magnetohydrodynamics; Mixing;

8 TURBULENCE

Navier–Stokes equation; Nonlinear Schrödinger Frisch, U. 1995. Turbulence: The Legacy of A.N. Kolmogorov, equations; Water waves; Wave packets, linear and Cambridge and New York: Cambridge University Press nonlinear Landau, L. & Lifshitz, E. 1987. Fluid Mechanics, 2nd edition, Oxford and New York: Pergamon Press Phillips, O. 1977. The Dynamics of the Upper Ocean, 2nd Further Reading edition, Cambridge and New York: Cambridge University Press Falkovich, G., Gaw¸edzki, K. & Vergassola, M. 2001. Particles Zakharov, V., L’vov, V. & Falkovich, G. 1992. Kolmogorov and fields in fluid turbulence, Reviews of Modern Physics, 73: Spectra of Turbulence, Berlin and New York: Springer 913Ð975

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