Publications

We suggest a new computer-assisted approach to the development of turbulence theory. It allows one to impose lower and upper bounds on correlation functions using sum-of-squares polynomials. We demonstrate it on the minimal cascade model of two resonantly interacting modes when one is pumped and the other dissipates. We show how to present correlation functions of interest as part of a sum-of-squares polynomial using the stationarity of the statistics. That allows us to find how the moments of the mode amplitudes depend on the degree of nonequilibrium (analog of the Reynolds number), which reveals some properties of marginal statistical distributions. By combining scaling dependence with the results of direct numerical simulations, we obtain the probability densities of both modes in a highly intermittent inverse cascade. As the Reynolds number tends to infinity, we show that the relative phase between modes tends to pi /2 and -pi/2 in the direct and inverse cascades, respectively, and derive bounds on the phase variance. Our approach combines computer-aided analytical proofs with a numerical algorithm applied to high-degree polynomials.

Vortices are the hallmarks of hydrodynamic flow. Strongly interacting electrons in ultrapure conductors can display signatures of hydrodynamic behaviour, including negative non-local resistance1,2,3,4, higher-than-ballistic conduction5,6,7, Poiseuille flow in narrow channels8,9,10 and violation of the WiedemannFranz law11. Here we provide a visualization of whirlpools in an electron fluid. By using a nanoscale scanning superconducting quantum interference device on a tip12, we image the current distribution in a circular chamber connected through a small aperture to a current-carrying strip in the high-purity type II Weyl semimetal WTe2. In this geometry, the Gurzhi momentum diffusion length and the size of the aperture determine the vortex stability phase diagram. We find that vortices are present for only small apertures, whereas the flow is laminar (non-vortical) for larger apertures. Near the vortical-to-laminar transition, we observe the single vortex in the chamber splitting into two vortices; this behaviour is expected only in the hydrodynamic regime and is not anticipated for ballistic transport. These findings suggest a new mechanism of hydrodynamic flow in thin pure crystals such that the spatial diffusion of electron momenta is enabled by small-angle scattering at the surfaces instead of the routinely invoked electronelectron scattering, which becomes extremely weak at low temperatures. This surface-induced para-hydrodynamics, which mimics many aspects of conventional hydrodynamics including vortices, opens new possibilities for exploring and using electron fluidics in high-mobility electron systems.

Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis involves both energy and entropy and elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with triplet interactions of neighboring modes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, three types of turbulence were found: single direct cascade, double cascade, and the first-ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, as quantified by the mutual information in pairs and the interaction information inside triplets.

Polymer molecules in a flow undergo a coil-stretch phase transition on an increase of the velocity gradients. Model-independent identification and characterization of the transition in a random flow has been lacking so far. Here we suggest to use the entropy of the extension statistics as a proper measure due to strong fluctuations around the transition. We measure experimentally the entropy as a function of the local Weisenberg number and show that it has a maximum, which identifies and quantifies the transition. We compare the new approach with the traditional one based on the theory using either linear Oldroyd-B or nonlinear finite extensible nonlinear elastic polymer models.

We discover dramatic impact of vortex formation in the transverse component of the Poynting vector of the fundamental core mode in solid core micro - structured optical fibers on the energy dissipation. The vortices can reduce losses of the mode by several orders of magnitude under proper selection of the fiber parameters at a given wavelength.

Viscous electron fluids have emerged recently as a new paradigm of strongly-correlated electron transport in solids. Here we report on a direct observation of the transition to this long-sought-for state of matter in a high-mobility electron system in graphene. Unexpectedly, the electron flow is found to be interaction-dominated but non-hydrodynamic (quasiballistic) in a wide temperature range, showing signatures of viscous flows only at relatively high temperatures. The transition between the two regimes is characterized by a sharp maximum of negative resistance, probed in proximity to the current injector. The resistance decreases as the system goes deeper into the hydrodynamic regime. In a perfect darkness-before daybreak manner, the interaction-dominated negative response is strongest at the transition to the quasiballistic regime. Our work provides the first demonstration of how the viscous fluid behavior emerges in an interacting electron system.

Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counter-rotating bladed disks reveal an apparent breaking of the law of similarity. Potentially, such breakdown could arise from several possible factors, including dependence on dimensionless numbers other than Re or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed that this unexpected result was due to background errors caused by minute misalignments which lead to a noisy and irreproducible torque signal at low rotation speeds and prevented correct background subtraction normally ascribed to frictional losses. An important lesson to be learnt is that multiple minute misalignments can nonlinearly couple to the torque signal and provide a dc offset that cannot be removed by averaging. That offset can cause the observed divergence of the friction coefficient C-f from its constant value observed in the turbulent regime. By significant modifications of the setup and conducting the experiment with one bladed disk and precisely aligned the disk, torque meter and motor shaft, we are able to conduct precise measurements close to the expected resolution at small torques at low rotation speeds and to confirm the similarity law in a wide range of Re, in particular, in low viscosity fluids.

This article introduces the Focus Issue on Two-Dimensional Turbulence appearing in Physics of Fluids (Volume 29, Issue 11, November 2017). Published by AIP Publishing.

Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons behave as a fluid. Electron viscous flows are governed by a nonlocal current-field relation which renders the spatial patterns of the current and electric field strikingly dissimilar. Notably, driven by the viscous friction force from adjacent layers, current can flow against the electric field, generating negative resistance, vorticity, and vortices. Moreover, different current flows can result in identical potential distributions. This sets a new situation where inferring the electron flow pattern from the measured potentials presents a nontrivial problem. Using the inherent relation between these patterns through complex analysis, here we propose a method for extracting the current flows from potential distributions measured in the presence of a magnetic field.

Quantum-critical strongly correlated electron systems are predicted to feature universal collision-dominated transport resembling that of viscous fluids. However, investigation of these phenomena has been hampered by the lack of known macroscopic signatures of electron viscosity. Here we identify vorticity as such a signature and link it with a readily verifiable striking macroscopic d.c. transport behaviour. Produced by the viscous flow, vorticity can drive electric current against an applied field, resulting in a negative nonlocal voltage. We argue that the latter may play the same role for the viscous regime as zero electrical resistance does for superconductivity. Besides offering a diagnostic that distinguishes viscous transport from ohmic currents, the sign-changing electrical response affords a robust tool for directly measuring the viscosity-to-resistivity ratio. A strongly interacting electron-hole plasma in high-mobility graphene affords a unique link between quantum-critical electron transport and the wealth of fluid mechanics phenomena.

Inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in half-space z>0 has been discovered by McKean [J. Math. Kyoto Univ.2, 227 (1963)] and then independently rediscovered by Cornellet al.[Phys. Rev. Lett.81, 1142 (1998)]. The essence of this phenomenon is that the particle arrives at the wall at z=0 with zero velocity after an infinite number of inelastic collisions if the restitution coefficient beta of particle velocity is smaller than the critical value betac=exp(-pi/radic3). We demonstrate that inelastic collapse takes place also in a wide class of models with spatially inhomogeneous random forcing and, what is more, that the critical value betacis universal. That class includes an important case of inertial particles in wall-bounded random flows. To establish how inelastic collapse influences the particle distribution, we derive the exact equilibrium probability density function rho(z,v) for the particle position and velocity. The equilibrium distribution exists only at β

We discuss general features of the operator product expansion and use it to infer multi-point manifestations of the energy cascade in turbulence. We calculate explicitly the possible form of the three-point velocity correlation function when one distance is smaller than two others. We elucidate manifestation of direct and inverse energy cascades in the three-point velocity correlation function.

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example - random fibre laser - we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

The ability to send a wave to fetch an object from a distance would find a broad range of applications. Quasi-standing Faraday waves on water create horizontal vortices(1,2), yet it is not known whether propagating waves can generate large-scale flows-small-amplitude irrotational waves only push particles in the direction of propagation(3-5). Here we show that when waves become three-dimensional as a result of the modulation instability, a floater can be forced to move towards the wave source. The mechanism for this is the generation of surface vortices by waves propagating away from vertically oscillating plungers. We introduce a new conceptual framework for understanding wave-driven flows, which enables us to engineer inward and outward surface jets, stationary vortices, and other complex flows. The results form a new basis for the remote manipulation of objects on fluid surfaces and for a better understanding of the motion of floaters in the ocean, the generation of wave-driven jets, and the formation of Lagrangian coherent structures.

The turbulent energy flux through scales, remains constant and nonvanishing in the limit of zero viscosity, which results in the fundamental anomaly of time irreversibility. It was considered straightforward to deduce from this the Lagrangian velocity anomaly, du2/dt=-4 at t=0, where uâ is the velocity difference of a pair of particles, initially separated by a fixed distance. Here we demonstrate that this assumed first taking the limit tâ0 and then Îâ0, while a zero-friction anomaly requires taking viscosity to zero first. We find that the limits tâ0 and Îâ0 do not commute if particles deplete (accumulate) in shocks backward (forward) in time on the viscous time scale. We compute analytically the resultant Lagrangian anomaly for one-dimensional Burgers turbulence and find it completely altered: du2/dt has different values forward and backward in time. For incompressible flows, on the other hand, we show that the limits commute and the Lagrangian anomaly is still induced by the flux law, apparently due to a homogeneous distribution of fluid particles at all times.

The statistical properties of turbulence differ in an essential way from those of systems in or near thermal equilibrium because of the flux of energy between vastly different scales at which energy is supplied and at which it is dissipated. We elucidate this difference by studying experimentally and numerically the fluctuations of the energy of a small fluid particle moving in a turbulent fluid. We demonstrate how the fundamental property of detailed balance is broken, so that the probabilities of forward and backward transitions are not equal for turbulence. In physical terms, we found that in a large set of flow configurations, fluid elements decelerate faster than accelerate, a feature known all too well from driving in dense traffic. The statistical signature of rare "flight-crash" events, associated with fast particle deceleration, provides a way to quantify irreversibility in a turbulent flow. Namely, we find that the third moment of the power fluctuations along a trajectory, nondimensionalized by the energy flux, displays a remarkable power law as a function of the Reynolds number, both in two and in three spatial dimensions. This establishes a relation between the irreversibility of the system and the range of active scales. We speculate that the breakdown of the detailed balance characterized here is a general feature of other systems very far from equilibrium, displaying a wide range of spatial scales.

We consider developed turbulence in the Gross-Pitaevsky model, where a condensate appears due to an inverse cascade. Despite being fully turbulent, the system demonstrates nondecaying periodic oscillations around a steady state, when turbulence and condensate periodically exchange a small fraction of waves. We show that these collective oscillations are not of a predator-prey type, as was suggested earlier; rather, they are due to phase coherence and anomalous correlations imposed by the condensate.

We develop an analytic formalism and derive new exact relations that express the short-time dispersion of fluid particles via the single-time velocity correlation functions in homogeneous isotropic and incompressible turbulence. The formalism establishes a bridge between single-time Eulerian and long-time Lagrangian pictures of turbulent flows. In particular, we derive an exact formula for a short-term counterpart of the long-time Richardson law, and we identify a conservation law of turbulent dispersion which is true even in nonstationary turbulence.

We study numerically optical turbulence using the particular example of a recently created, ultra-long fibre laser. For normal fibre dispersion, we observed an intermediate state with an extremely narrow spectrum (condensate), which experiences instability and a sharp transition to a fluctuating regime with a wider spectrum. We demonstrate that the number of modes has an impact on the condensate's lifetime. The smaller the number of modes, the more resistant is the condensate to perturbations. Experimental results show a good agreement with numerical simulations.

A passive scalar field was studied under the action of pumping, diffusion and advection by a 2D smooth flow with Lagrangian chaos. We present theoretical arguments showing that the scalar statistics are not conformally invariant and formulate a new effective semi-analytic algorithm to model scalar turbulence. We then carry out massive numerics of scalar turbulence, focusing on nodal lines. The distribution of contours over sizes and perimeters is shown to depend neither on the flow realization nor on the resolution (diffusion) scale r_d for scales exceeding r_d. The scalar isolines are found to be fractal/smooth at scales larger/smaller than the pumping scale. We characterize the statistics of isoline bending by the driving function of the Löwner map. That function is found to behave like diffusion with diffusivity independent of the resolution yet, most surprisingly, dependent on the velocity realization and time (beyond the time on which the statistics of the scalar is stabilized).

The multidisciplinary field of fluid mechanics is one of the most actively developing fieldsof physics, mathematics and engineering. In this book, the fundamental ideas of fluid mechanics are presented from a physics perspective. Using examples taken from everyday life, from hydraulic jumps ina kitchen sink to KelvinHelmholtz instabilities in clouds, the book provides readers with a better understanding of the world around them. It teaches the art of fluid-mechanical estimates and shows how the ideas and methods developed to study the mechanics of fluids are used to analyze other systems with many degrees of freedom in statistical physics and field theory. Aimed at undergraduate and graduate students, the book assumes no prior knowledge of the subject and only a basic understanding of vector calculus and analysis. It contains 32 exercises of varying difficulties, from simple estimates to elaborate calculations, with detailed solutions to help readers understand fluid mechanics.

For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When is the vorticity coarse-grained over a scale R, the probability density function (PDF), P, has a universal asymptotic behavior lnP~-_rms at _rms=[Hln(L/R)]1^/3, where H is the enstrophy flux and L is the pumping length. Therefore, the PDF has exponential tails and is self-similar, that is, it can be presented as a function of a single argument, _rms, in distinction from other known direct cascades.

A finite-amplitude propagating wave induces a drift in fluids. Understanding how drifts produced by many waves disperse pollutants has broad implications for geophysics and engineering. Previously, the effective diffusivity was calculated for a random set of small-amplitude surface and internal waves. Now, this is extended by Bhler & Holmes-Cerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 526) to waves in a rotating shallow-water system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finite-amplitude waves affect particle velocity in subtle ways. An expression describing the particle diffusivity as a function of scale is derived, showing that the diffusivity can be substantially reduced by rotation.

A Reply to the Comment by Erik Lindborg.

We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that, close to non-equilibrium steady states with non-vanishing probability currents, some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.

We present a mean-field model of cloud evolution that describes droplet growth due to condensation and collisions and droplet loss due to fallout. The model accounts for the effects of cloud turbulence both in a large-scale turbulent mixing and in a microphysical enhancement of condensation and collisions. The model allows for an effective numerical simulation by a scheme that is conservative in water mass and keeps accurate count of the number of droplets. We first study the homogeneous situation and determine how the rain-initiation time depends on the concentration of cloud condensation nuclei (CCN) and turbulence level. We then consider clouds with an inhomogeneous concentration of CCN and evaluate how the rain initiation time and the effective optical depth vary in space and time. We argue that over-seeding even a part of a cloud by small hygroscopic nuclei, one can substantially delay the onset and increase the amount of precipitation.

Cloud physics has for a long time been an important segment of atmospheric science. It is common knowledge that clouds are crucial for our understanding of weather and climate. Clouds are also interesting by themselves (not to mention that they are beautiful). Complexity is hidden behind the common picture of these beautiful and interesting objects. The typical school textbook definition that a cloud is 'a set of droplets or particles suspended in the atmosphere' is not adequate. Clouds are complicated phenomena in which dynamics, turbulence, microphysics, thermodynamics and radiative transfer interact on a wide range of scales, from sub-micron to kilometres. Some of these interactions are subtle and others are more straightforward. Large and small-scale motions lead to activation of cloud condensation nuclei, condensational growth and collisions; small changes in composition and concentration of atmospheric aerosol lead to significant differences in radiative properties of the clouds and influence rainfall formation. It is justified to look at a cloud as a composite, nonlinear system which involves many interactions and feedback. This system is actively linked into a web of atmospheric, oceanic and even cosmic interactions.

The first direct experimental evidence of the suppression of quasi-two-dimensional turbulence by mean flows is presented. The flow either is induced externally or appears in the process of spectral condensation due to an inverse cascade in bounded turbulence. The observed suppression of large scales is consistent with an expected reduction in the correlation time of turbulent eddies due to shearing. At high flow velocities, sweeping of the forcing-scale vortices reduces the energy input, leading to a reduction in the turbulence level.

We argue theoretically and demonstrate experimentally that in a standing wave floating particles drift towards the nodes oranti-nodes depending on their hydrophilic or hydrophobic properties. We explain this effect as the breakdown of Archimedes' law by asurface tension, which creates a difference between the masses of the floater and displaced liquid, making the particle effectively inertial. We describe analytically the motion of a small floating particle in a small-amplitude wave and show that the drift appears as a second order effect in wave amplitude. We confirm experimentally that indeed the clustering rate is proportional to the square of the wave amplitude. In the case of surface random waves we show experimentally that the inertial effects significantly change the statistics of floater distribution on a liquid surface. The analysis of particle concentration moments and probability distribution functions shows that particle concentrate on amulti-fractal set with caustics.

We study the statistics of the relative separation between two fluid particles in a spatially smooth and temporally random flow. The Lagrangian strain is modeled by a telegraph noise, which is a stationary random Markov process that can only take two values with known transition probabilities. The simplicity of the model enables us to write closed equations for the interparticle distance in the presence of a finite-correlated noise. In one dimension, we are able to find analytically the long-time growth rates of the distance moments and the senior Lyapunov exponent, which consistently turns out to be negative. We also find the exact expression for the Cramér function and show that it satisfies the fluctuation relation (for the probability of positive and negative entropy production) despite the time irreversibility of the strain statistics. For the two-dimensional incompressible isotropic case, we obtain the Lyapunov exponent (positive) and the asymptotic growth rates of the moments in two opposite limits of fast and slow strain. The quasideterministic limit (of slow strain) turns out to be singular, while a perfect agreement is found with the already-known δ -correlated case.

The growth rate of small-scale density inhomogeneities (the entropy production rate) is given by the sum of the Lyapunov exponents in a random flow. We derive an analytic formula for the rate in a flow of weakly interacting waves and show that in most cases it is zero up to the fourth order in the wave amplitude. We then derive an analytic formula for the rate in a flow of waves and currents. Estimates of the rate and the fractal dimension of the density distribution show that the interplay between waves and currents is a realistic mechanism for providing patchiness of the pollutant distribution on the ocean surface.

We offer a new example of conformal invariance (local scale invariance) far from equilibrium-the inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a one-dimensional Brownian walk (called Schramm-Loewner evolution or SLEκ). The diffusivity is close to κ=4, that is, isotemperature curves belong to the same universality class as domain walls in the O(2) spin model. Several statistics of temperature clusters and isolines are shown to agree with the theoretical expectations for such a spin system at criticality. We also show that the direct cascade in two-dimensional Navier-Stokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.

This review is an abridged and updated version of Falkovich et al.[1]. The subject is the combined effect of molecular diffusion and random flow on scalar and vector fields transported by a fluid. We want to understand first when there is mixing and when, on the contrary, inhomogeneities are created and enhanced. We want to distinguish between cases when flow creates small-scale inhomogeneities of the transported fields, which are then killed by molecular diffusion, and cases when large-scale structures of the fields appear. Our goal is to describe temporal and spatial statistical properties of transported fields.

Turbulent flows, with their irregular behavior, confound any simple attempts to understand them. But physicists have succeeded in identifying some universal properties of turbulence and relating them to broken symmetries.

The simplicity of fundamental physical laws manifests itself in fundamental symmetries. Although systems with an infinite number of strongly interacting degrees of freedom (in particle physics and critical phenomena) are hard to describe, they often demonstrate symmetries, in particular scale invariance. In two dimensions (2D) locality often extends scale invariance to a wider class of conformal transformations that allow non-uniform rescaling. Conformal invariance enables a thorough classification of universality classes of critical phenomena in 2D. Is there conformal invariance in 2D turbulence, a paradigmatic example of a strongly interacting non-equilibrium system? Here, we show numerically that some features of a 2D inverse turbulent cascade show conformal invariance. We observe that the statistics of vorticity clusters are remarkably close to that of critical percolation, one of the simplest universality classes of critical phenomena. These results represent a key step in the unification of 2D physics within the framework of conformal symmetry.

We analyze the multipoint correlation functions of a tracer in an incompressible flow at scales far exceeding the scale L at which fluctuations are generated (quasiequilibrium domain) and compare them with the correlation functions at scales smaller than L (turbulence domain). We demonstrate that scale invariance can be broken in the equilibrium domain and trace this breakdown to the statistical integrals of motion (zero modes) as has been done before for turbulence. Employing the Kraichnan model of short-correlated velocity we identify the new type of zero modes, which break scale invariance and determine an anomalously slow decay of correlations at large scales.

We consider a passive pollutant advected by the flow due to linear random waves with finite attenuation. We derive the equation that governs the evolution of the pair correlation function of pollutant concentration and show that it coincides with the equation for the case of a short-correlated velocity. Due to a finite wave attenuation, nontrivial evolution (particularly, the growth of inhomogeneities) appears already in the second order in wave amplitudes. We show that random potential waves lead to the growth of concentration inhomogeneities. We identify two stationary solutions for the spectral density of concentration, equipartition, and flux state. Which one is established depends on the relation between mean square velocity gradients due to potential and solenoidal parts of the flow, respectively. We also analyze transient regimes and show how periodic component in the concentration distribution appears and disappears.

We find the probability distribution of the fluctuating parameters of a soliton propagating through a medium with additive noise. Our method is a modification of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear Schrödinger equation. We then consider model modifications due to in-line (filtering, amplitude and phase modulation) control. It is examined how control elements change the error probability in optical soliton transmission. Even though a weak noise is considered, we are interested here in probabilities of error-causing large fluctuations which are beyond perturbation theory. We describe in detail a new phenomenon of soliton collapse that occurs under the combined action of noise, filtering and amplitude modulation.

The growth of density inhomogeneities was investigated in a flow of wave turbulence. The conditions for a nonzero sum of the Lyapunov exponents was found which provides for an exponential growth of density inhomogeneities. The leading order contribution came from the pair of waves having coinciding frequencies but producing different stokes drifts. It was observed that nonzero Lyapunov exponents appear in two and three dimensions both for potential and solenoidal waves.

The logarithmic renormalization predicted by Kraichnan (1971) for the direct cascade of enstrophy in the inertial range of two-dimensional turbulence has been observed in a numerical simulation. A moderate resolution allows for a very long time integration that provides very good statistics. Deviations from Gaussianity in the vorticity probability distribution are observed.

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

We develop a WKB approach to the rapid distortion theory for magnetohydrodynamic turbulence with large magnetic Prandtl number. Within this theory, we study the growth of small-scale magnetic fluctuations in a large-scale velocity field being initially a pure strain. We show that the magnetic Lorentz force excites a secondary flow in the form of counterrotating vortices on the periphery of the magnetic spot. Those vortices slow down stretching of the magnetic spot and thus provide a negative feedback for a small-scale magnetic dynamo.

Kinematic dynamo theory is presented here for turbulent conductive fluids. We describe how inhomogeneous magnetic fluctuations are generated below the viscous scale of turbulence where the spatial smoothness of the velocity permits a systematic analysis of the Lagrangian path dynamics. We find analytically the moments and multipoint correlation functions of the magnetic field at small yet finite magnetic diffusivity. We show that the fil:ld is concentrated in long narrow strips and describe anomalous scalings and angular singularities of the multipoint correlation functions which are manifestations of the field's intermittency. The growth rate of the magnetic field in a typical realization is found to be half the difference of two Lyapunov exponents of the same sign.

We study the statistics of relative distances R(t) between fluid particles in a spatially smooth random flow with arbitrary temporal correlations. Using the space dimensionality d as a large parameter we develop an effective description of Lagrangian dispersion. We describe the exponential growth of relative distances 〈R²(t)〉 ∝ exp2λ̄t at different values of the ratio between the correlation and turnover times. We find the stretching correlation time which determines the dependence of 〈R₁R₂〉 on the difference t₁ - t₂. The calculation of the next cumulant of R² shows that statistics of R² is nearly Gaussian at small times (as long as d ≫ 1) and becomes log-normal at large times when large-d approach fails for high-order moments. The crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t≪τ. We establish the dispersion of the ln(R²) in the log-normal statistics.

We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient ∂_xu and find out that they correspond to the PDF with In[P(∂_xu)] ∝ - (-∂_xu/ℜe)^3/2 where ℜe is the Reynolds number. That stretched exponential form is valid for negative ∂_xu with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences w is steeper than Gaussian, In P (w) ∼ - (w/u_rms)³, as well as single-point velocity PDF In P (u) ∼ - (|u|/u_rms)³. For high velocity derivatives u (k) = ∂^k_xu, the general formula is found: In P (|u^(k)|) ∝ -(|u^(k)|/ℜe^k)^3/(k+1).

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n

Keywords: Engineering, Mechanical; Mechanics

Amplitude equations are derived that describe the interaction between high-frequency inertio-gravity waves and low-frequency Rossby waves on rotating shallow water. A cyclone is shown to cause a local maximum to appear in the density of inertio-gravity waves. A packet of inertio-gravity waves is shown to produce a cyclone-anticyclone pair. The interaction between inertio-gravity and Rossby waves could therefore be an additional mechanism which sustains persistent atmospheric anomalies like the blocking phenomenon. However, we have found that neither a bound state nor a collapsing cavern may show up in the course of evolution which implies that the interaction with high-frequency waves does not, by itself, explain the blocking.

For a -function-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality d. It is shown that anomalous scaling appears in the first order of 1/d perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for its derivatives, in particular, for the dissipation field.

The whole set of simultaneous correlation functions describing steady vorticity cascade is obtained from the Euler equation by a straightforward procedure. Nonlocality of the cascade provides for a large logarithmic parameter that enables one to obtain a universal set of the correlation functions of the vorticity omega in the inertial interval: [omega(n)(r1)omega(n)(r2)] is-proportional-to ln2n/3(L/\r1 - r2\), with L being the scale of the external pump.

We consider the correlation functions of vorticity in the region of the direct cascade in a steady two-dimensional turbulence. The nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates, and the main problem is to take those logarithms into account. Our procedure starts directly with the Euler equation rewritten in the comoving reference frame. We express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain. It enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval. We find the indices characterizing the logarithmic behavior of different correlation functions. For example, the two-point simultaneous functions are as follows: n (r1)n(r2)[P2ln(L/r1-r2)]2n3, where L is the pumping scale. We demonstrate that the form of those correlation functions is universal, i.e., independent of the pumping. The only pumping-related value which enters the expressions is the enstrophy production rate P2. The contributions related to pumping rates Pn of the higher-order integrals of motion are demonstrated to be small in comparison with the ones induced by P2. We establish also the time dependence of the correlation functions, the correlation time in the comoving reference frame is the same for the vorticity and strain and is scale dependent: ln2/3(L/r). We reformulate our procedure in the diagrammatic language to reinforce the conclusions.

Обычно в утверждение об универсальности спектра развитой турбулентности вкладывают следующий смысл: в интервале масштабов, промежуточных между источником и стоком, турбулентность изотропна и распределение энергии по масштабам зависит от единственного внешнего параметрапотока энергии в к-пространстве [13]. В соответствии с этой гипотезой были получены изотропные однонараметрические спектры (называемые обычно колмогоровскими) как для вихревой гидродинамической [14], так и для волновой турбулентности в гидродинамике, физике плазмы, акустике [5]. Следует указать, однако, что во всех случаях взаимодействие и волн, и вихрей, помимо энергии, сохраняет суммарный импульс. Любой же реальный источник турбулентности является анизотропным и несимметричным, что приводит к появлению ненулевого импульса системы. Переносящие малый поток импульса К стационарные поправки 5пъ к несущим поток энергии Р слаботурбулентным колмогоровским решениям пк были построены в работе [6]. Для волн со степенным законом дисперсии сок эти так называемые дрейфовые поправки имеют простой вид Зик/ид^(Кк) сол/(Ле2)«" со $0*(шк/к)^ ка~ гсо $ 0к,(1) обусловленный тем, что у=(Кк) сок/(Рк2) является единственным безразмерным параметром, который можно составить из рассматриваемых величин.

Keywords: Mathematics, Applied; Mechanics; Physics, Fluids & Plasmas

A steady Kolmogorov-like spectrum of turbulence is found as an exact solution of the kinetic equation for inertial-gravity waves. The spectrum obtained satisfactorily fits the results of atmospheric observations for mesoscale motions (from hundred to thousand kilometers).

Weak developed turbulence in the framework of both scalar and vector nonlinear Schrödinger equations is considered. It corresponds to waves with a quadratic dispersion law ω_k = ω_o + βk² and with a zero scaling exponent of the interaction coefficient. The consideration thus embraces the turbulence of envelopes (usually referred to as optical turbulence) as well as Langmuir turbulence in nonisothermal plasma and other examples. Steady spectra of turbulence are shown to be close to Kolmogorov-like cascade spectra with the fluxes of energy and wave action.

Weak turbulence of waves with a weakly damped dispersion law is discussed. In the stationary case, small anisotropic additions to an isotropic spectrum of the Kolmogorov type are found. It is shown that a small anisotropic source located in a region of small k leads to an essentially anisotropic spectrum in a region of large k. Strongly anisotropic stationary spectra are found. [Russian original - Zh. Eksp. Teor. Fiz. 80, 592-596 (February 1981)]