Publications

It was proposed that acoustic perturbations generated by the QCD phase transition could create an inverse turbulent cascade [1]. An assumption was that propagation toward smaller momenta could reach the wavelength of a few km, the Universes size at the time. Such acoustic waves were proposed to be the source of gravity waves. The kilometer wavelength corresponds to year-long gravity wave today, which were recently discovered using pulsar correlations.This paper argues further that an acoustic turbulence must be an ensemble of shocks. This brings two consequences: First, shocks generate gravity waves much more efficiently than sound waves due to the intermittency of matter distribution. We reconsider gravity wave radiation, using universal emission theory for soft radiation, and argue that soft-momenta plateau should reach wavelengths of the order of shock mean free paths. Second, collisions of shock waves create local density excess, which may create primordial black holes. A good tool to settle this issue can be an evaluation of the trapped surfaces like it was done in studies of heavy ion collisions using AdS/CFT correspondence.

Applications of information theory span a broad range of disciplines today. This book presents a unified treatment of the subject for students and practitioners in the sciences. It teaches the tools universally used by physicists working on quantum computers and black holes, engineers designing self-driving cars, traders perfecting market strategies, chemists playing with molecules, biologists studying cells and living beings, linguists analyzing languages, and neuroscientists figuring out how the brain works. No matter what area of science you specialize in, The Physical Nature of Information unlocks the power of information theory to test the limits imposed by uncertainty.

Starting at the level of graduate students, this book offers a comprehensive introduction to a rapidly developing field of turbulence. It also presents the state of the art of the theory of wave turbulence in diverse media for researchers looking for a formalism to solve existing problems or for new research subjects and ideas. The book provides a general theory of developed wave turbulence in different media: plasmas, solids, atmosphere, oceans and space. The presentation starts with a simple and intuitive dimensional analysis and proceeds to a rigorous analytic theory with exact solutions for the stationary spectra of turbulence, stability theory of such spectra, description of nonstationary regimes and matching spectra with pumping and dissipation regions. The reader is provided with the necessary tools to study nonlinear waves and turbulence: Hamiltonian formalism, statistical description, derivation of kinetics equations and methods of finding their steady and non-steady solutions. In this second edition the book is brought up to date both in theoretical and experimental/observational aspects. In particular, the authors have updated and revised the description of nonstationary turbulence, turbulent entropy production, etc, and added discussions of master equation and several cases of strong turbulence and nonlocal cascades. The book now includes excercises, some with solutions.

We consider turbulence of waves interacting weakly via four-wave scattering (sea waves, plasma waves, spin waves, etc.). In the first order in the interaction, a closed kinetic equation has stationary solutions describing turbulent cascades. We show that the higher-order terms generally diverge both at small (IR) and large (UV) wave numbers for direct cascades. The analysis up to the third order identifies the most UV-divergent terms. To gain qualitative analytic control, we sum a subset of the most UV divergent terms, to all orders, giving a perturbation theory free from UV divergence, showing that turbulence becomes independent of the dissipation scale when it goes to zero. On the contrary, the IR divergence (present in the majority of cases) makes the effective coupling parametrically larger than the naive estimate and growing with the pumping scale (Formula presented) (similar to anomalous scaling in fluid turbulence). In such cases, the kinetic equation does not describe wave turbulence even of arbitrarily small level at a given (Formula presented) if (Formula presented) is large enough that is the cascade is sufficiently long. We show that the character of strong turbulence is determined by whether the effective four-wave interaction is enhanced or suppressed by collective effects. The enhancement possibly signals that strong turbulence is dominated by multiwave bound states (solitons, shocks, cusps), similar to confinement in quantum chromodynamics.

We suggest a new focus for turbulence studies-multimode correlations-which reveal the hitherto hidden nature of turbulent state. We apply this approach to shell models describing basic properties of turbulence. The family of such models allows one to study turbulence close to thermal equilibrium, which happens when the interaction time weakly depends on the mode number. As the number of modes increases, the one-mode statistics approaches Gaussian (like in weak turbulence), the occupation numbers grow, while the three-mode cumulant describing the energy flux stays constant. Yet we find that higher multimode cumulants grow with the order. We derive analytically and confirm numerically the scaling law of such growth. The sum of all squared dimensionless cumulants is equal to the relative entropy between the full multimode distribution and the Gaussian approximation of independent modes; we argue that the relative entropy could grow as the logarithm of the number of modes, similar to the entanglement entropy in critical phenomena. Therefore, the multimode correlations give the new way to characterize turbulence states and possibly divide them into universality classes.

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.

Aim: Bipolar disorder (BD) is a mood disorder with a high morbidity and death rate. Lithium (Li), a prominent mood stabilizer, is often used as a first-line treatment. However, clinical studies have shown that Li is fully effective in roughly 30% of BD patients. Our goal in this study was to use features derived from information theory to improve the prediction of the patient's response to Li as well as develop a diagnostic algorithm for the disorder. Methods: We have performed electrophysiological recordings in patient-derived dentate gyrus (DG) granule neurons (from a total of 9 subjects) for three groups: 3 control individuals, 3 BD patients who respond to Li treatment (LR), and 3 BD patients who do not respond to Li treatment (NR). The recordings were analyzed by the statistical tools of modern information theory. We used a Support Vector Machine (SVM) and Random forest (RF) classifiers with the basic electrophysiological features with additional information theory features. Results: Information theory features provided further knowledge about the distribution of the electrophysiological entities and the interactions between the different features, which improved classification schemes. These newly added features significantly improved our ability to distinguish the BD patients from the control individuals (an improvement from 60% to 74% accuracy) and LR from NR patients (an improvement from 81% to 99% accuracy). Conclusion: The addition of Information theory-derived features provides further knowledge about the distribution of the parameters and their interactions, thus significantly improving the ability to discriminate and predict the LRs from the NRs and the patients from the controls.

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.

Swirling vortices have been directly observed in a flow of electric current for the first time. Unlike conventional viscous fluids, collective fluid-like behaviour in this case is not caused by particle-particle collisions, but results from a previously unidentified mechanism involving single electrons scattering from material surfaces at small angles.Flows of electrons have now been shown to contain swirling whirlpools like those found in viscous fluids - and physicists have been figuring out what caused them.

This note is devoted to broken and emerging scale invariance of turbulence. Pumping breaks the symmetry: the statistics of every mode explicitly depend on the distance from the pumping. And yet the ratios of mode amplitudes, called Kolmogorov multipliers, are known to approach scale-invariant statistics away from the pumping. This emergent scale invariance deserves an explanation and a detailed study. We put forward the hypothesis that the invariance of multipliers is due to an extreme non-locality of their interactions (similar to the appearance of mean-field properties in the thermodynamic limit for systems with long-range interaction). We analyse this phenomenon in a family of models that connects two very different classes of systems: resonantly interacting waves and wave-free incompressible flows. The connection is algebraic and turns into an identity for properly discretized models. We show that this family provides a unique opportunity for an analytic (perturbative) study of emerging scale invariance in a system with strong interactions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

When two resonantly interacting modes are in contact with a thermostat, their statistics is exactly Gaussian and the modes are statistically independent despite strong interaction. Considering a noise-driven system, we show that when one mode is pumped and another dissipates, the statistics of such cascades is never close to Gaussian, no matter what is the relation between interaction and noise. One finds substantial phase correlation in the limit of strong interaction or weak noise. Surprisingly, the mutual information between modes increases and entropy decreases when interaction strength decreases. We use the model to elucidate the fundamental problem of far-from equilibrium physics: where the information, or entropy deficit, is encoded, and how singular measures form. For an instability-driven system, such as laser, even a small added noise leads to large fluctuations of the relative phase near the stability threshold, while far from the equilibrium the conversion into the second harmonic is weakly affected by noise.

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.

We consider the developed turbulence of capillary waves on shallow water. Analytic theory shows that an isotropic cascade spectrum is unstable with respect to small angular perturbations, in particular, to spontaneous breakdown of the reflection symmetry and generation of nonzero momentum. By computer modeling we show that indeed a random pumping, generating on average zero momentum, produces turbulence with a nonzero total momentum. A strongly anisotropic large-scale pumping produces turbulence whose degree of anisotropy decreases along a cascade. It tends to saturation in the inertial interval and then further decreases in the dissipation interval. Surprisingly, neither the direction of the total momentum nor the direction of the compensated spectrum anisotropy is locked by our square box preferred directions (side or diagonal) but fluctuate.

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.

How weak is the weak turbulence? Here, we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence. The mutual information between modes is stationary and small in thermal equilibrium, yet it is shown here to grow with time for weak turbulence in a finite box. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phase-space measure is far from Gaussian, as manifested by a large relative entropy. This is a rare piece of good news for turbulence modeling: the resolved scales carry significant information about the unresolved scales. The mutual information between large and small scales is the information capacity of turbulent cascade, setting the limit on the representation of subgrid scales in turbulence modeling.

In hydrodynamics, vortex generation upon the transition from smooth laminar flows to turbulence is generally accompanied by increased dissipation. However, vortices in the plane can provide transport barriers and decrease losses, as it happens in numerous geophysical, astrophysical flows and in tokamaks. Photon interactions with matter can affect light transport in ways resembling fluid dynamics. Here, we demonstrate significant impact of light vortex formation in micro-structured optical fibres on the energy dissipation. We show possibility of vortex formation in both solid core and hollow core fibres on the zero energy flow lines in the cladding. Through intensive numerical modelling using different independent approaches, we discovered a correlation between appearance of vortices and reduction of light leakage by three orders of magnitude, effectively improving wave guiding. This new effect potentially might have strong impact on numerous practical applications of micro-structured fibres. For instance, a strong light localization based on the same principle can also be achieved in the negative curvature hollow core fibres.

We show that rotating particles at the liquid-gas interface can be efficiently manipulated using the surface-wave analogue of optical lattices. Two orthogonal standing waves generate surface flows of counter-rotating half-wavelength unit cells, the liquid interface metamaterial, whose geometry is controlled by the wave phase shift. Here we demonstrate that by placing active magnetic spinners inside such metamaterials, one makes a powerful tool which allows manipulation and self-assembly of spinners, turning them into vehicles capable of transporting matter and information between autonomous metamaterial unit cells. We discuss forces acting on a spinner carried by a nonuniform flow and show how the forces confine spinners to orbit inside the same-sign vortex cells of the wave-driven flow. Reversing the spin, we move the spinner into an adjacent cell. By changing the spinning frequency or the wave amplitude, one can precisely control the spinner orbit. Multiple spinners within a unit cell self-organize into stable patterns, e.g., triangles or squares, orbiting around the center of the cell. Spinners having different frequencies can also be confined, such that the higher-frequency spinner occupies the inner orbit and the lower-frequency one circles on the outer orbit, while the orbital motions of both spinners are synchronized.

Electronic fluids bring into hydrodynamics a new setting: equipotential flow sources embedded inside the fluid. Here we show that the nonlocal relation between the current and electric field due to momentum-conserving interparticle collisions leads to a total or partial field expulsion from such flows. That results in freely flowing currents in the bulk and a boundary jump in the electric potential at current-injecting electrodes. We derive a new type of boundary conditions, appropriate for the case. We then analyze current distribution in free flows, discuss how the field expulsion depends upon the geometry of the electrode, and link the phenomenon to the breakdown of conformal invariance.

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.

An electric field that builds in the direction against current, known as negative nonlocal resistance, arises naturally in viscous flows and is thus often taken as a telltale of this regime. Here, we predict negative resistance for the ballistic regime, wherein the ee collision mean free path is greater than the length scale at which the system is being probed. Therefore, negative resistance alone does not provide strong evidence for the occurrence of the hydrodynamic regime; it must thus be demoted from the rank of irrefutable evidence to that of a mere forerunner. Furthermore, we find that negative response is log enhanced in the ballistic regime by the physics related to the seminal Dorfman-Cohen log divergence due to memory effects in the kinetics of dilute gases. The ballistic regime therefore offers a unique setting for exploring these interesting effects due to electron interactions.

Flows in fluid layers are ubiquitous in industry, geophysics, and astrophysics. Large-scale flows in thin layers can he considered two dimensional with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they arc driven. We argue that a wall-driven (Couctte) flow cannot sustain turbulence, no matter how small the viscosity and friction. Direct numerical simulations (DNSs) up to the Reynolds number Re = 10(6) confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, perturbations destroy the pressure-driven laminar (Poiseuille) flow. What appears instead is a traveling wave in the form of a jet slithering between wall vortices. For 5 x 10(3)

The multidisciplinary field of fluid mechanics is one of the most actively developing fields of physics, mathematics and engineering. This textbook, fully revised and enlarged for the second edition, presents the minimum of what every physicist, engineer and mathematician needs to know about hydrodynamics. It includes new illustrations throughout, using examples from everyday life, from hydraulic jumps in a kitchen sink to Kelvin-Helmholtz instabilities in clouds, and geophysical and astrophysical phenomena, providing readers with a better understanding of the world around them. Aimed at undergraduate and graduate students as well as researchers, the book assumes no prior knowledge of the subject and only a basic understanding of vector calculus and analysis. It contains forty-one original problems with very detailed solutions, progressing from dimensional estimates and intuitive arguments to detailed computations to help readers understand fluid mechanics.

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.

Strong interaction among charge carriers can make them move like viscous fluid. Here we explore alternating current (ac) effects in viscous electronics. In the Ohmic case, incompressible current distribution in a sample adjusts fast to a time-dependent voltage on the electrodes, while in the viscous case, momentum diffusion makes for retardation and for the possibility of propagating slow shear waves. We focus on specific geometries that showcase interesting aspects of such waves: current parallel to a one-dimensional defect and current applied across a long strip. We find that the phase velocity of the wave propagating along the strip respectively increases/decreases with the frequency for no-slip/no-stress boundary conditions. This is so because when the frequency or strip width goes to zero (alternatively, viscosity go to infinity), the wavelength of the current pattern tends to infinity in the no-stress case and to a finite value in a general case. We also show that for dc current across a strip with a no-stress boundary, there are only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.

Electron-electron (e-e) collisions can impact transport in a variety of surprising and sometimes counterintuitive ways(1-6). Despite strong interest, experiments on the subject proved challenging because of the simultaneous presence of different scattering mechanisms that suppress or obscure consequences of e-e scattering(7-11). Only recently, suffciently clean electron systems with transport dominated by e-e collisions have become available, showing behaviour characteristic of highly viscous fluids(12-14). Here we study electron transport through graphene constrictions and show that their conductance below 150 K increases with increasing temperature, in stark contrast to the metallic character of doped graphene(15). Notably, the measured conductance exceeds the maximum conductance possible for free electrons(16,17). This anomalous behaviour is attributed to collective movement of interacting electrons, which 'shields' individual carriers from momentum loss at sample boundaries(18,19). The measurements allow us to identify the conductance contribution arising due to electron viscosity and determine its temperature dependence. Besides fundamental interest, our work shows that viscous effects can facilitate high-mobility transport at elevated temperatures, a potentially useful behaviour for designing graphene-based devices.

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.

Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse cascade of kinetic energy up to a scale, where a typical velocity reaches the speed of sound; this creates shock waves, which provide for a compensating direct cascade. When the system size is less than, the steady state contains a system-size pair of long-living condensate vortices connected by a system of shocks. Thus turbulence in fluid layers processes energy via a loop: Most energy first goes to large scales via vortices and is then transported by waves to small-scale dissipation.

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.

An inverse cascade, energy transfer to progressively larger scales, is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean flow with zero total momentum was therefore believed to be unidirectional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole is expected. Using direct numerical simulations, we show that in fact neither is the box symmetry respected nor the largest scale realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can neither be deduced from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasilinear approximation.

Strongly interacting electrons can move in a neatly coordinated way, reminiscent of the movement of viscous fluids. Here, we show that in viscous flows, interactions facilitate transport, allowing conductance to exceed the fundamental Landauer's ballistic limit G_ball. The effect is particularly striking for the flow through a viscous point contact, a constriction exhibiting the quantum mechanical ballistic transport at T = 0 but governed by electron hydrodynamics at elevated temperatures. We develop a theory of the ballistic-to-viscous crossover using an approach based on quasi-hydrodynamic variables. Conductance is found to obey an additive relation G = G_ball + G_vis, where the viscous contribution G_vis dominates over G_ball in the hydrodynamic limit. The superballistic, low-dissipation transport is a generic feature of viscous electronics.

Interactions in electron systems can lead to viscous flows in which correlations allow electrons to avoid disorder scattering, reducing momentum loss and dissipation. We illustrate this behavior in a viscous pinball model, describing electrons moving in the presence of dilute point-like defects. Conductivity is found to obey an additive relation $\sigma=\sigma_0+\Delta\sigma$, with a non-interacting Drude contribution $\sigma_0$ and a contribution $\Delta\sigma>0$ describing conductivity enhancement due to interactions. The quantity $\Delta\sigma$ is enhanced by a logarithmically large factor originating from the Stokes paradox at the hydrodynamic lengthscales and, in addition, from an effect of repeated returns to the same scatterer due to backreflection in the carrier-carrier collisions occurring at the ballistic lengthscales. The interplay between these effects is essential at the ballistic-to-viscous crossover.

This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.

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 address theoretically the longstanding problem of particle dispersion in the lower atmosphere. The evolution of particle concentration under an absorbing boundary condition at the ground is described. We derive a close-form solution for the downwind surface density of deposited particles and find how the number of airborne particles decreases with time. The problem of the plume formation above the extended surface source is also solved analytically. At the end, we show how turbophoresis modifies the mean settling velocity of particles.

It was recently suggested that the direction of particle drift in inhomogeneous temperature or turbulence depends on the particle inertia: weakly inertial particles localize near minima of temperature or turbulence intensity (effects known as thermophoresis and turbophoresis), while strongly inertial particles fly away from minima in an unbounded space. The problem of a particle near minima of turbulence intensity is related to that of two particles in a random flow, so that the localization-delocalization transition in the former corresponds to the path-coalescence transition in the latter. The transition is signaled by the sign change of the Lyapunov exponent that characterizes the mean rate of particle approach to the minimum (a wall or another particle). Here we solve analytically this problem for inelastic collisions and derive the phase diagram for the transition in the inertia-inelasticity plane. An important feature of the diagram is the region of inelastic collapse: if the restitution coefficient β of particle velocity is smaller than the critical value β ₀ = exp(-π/√3), then the particle is localized for any inertia. We present direct numerical simulations which support the theory and in addition reveal the dependence of the transition of the flow correlation time, characterized by the Stokes number.

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu, Proc. Natl. Acad. Sci. USA 111, 7558 (2014)PNASA60027-842410.1073/pnas.1321682111], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling ([E(t)-E(0)]n)?t and (pn)?Ren-1 for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for (E(t)-E(0)). To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

We consider developed turbulence in the two-dimensional Gross-Pitaevskii model, which describes wide classes of phenomena from atomic and optical physics to condensed matter, fluids, and plasma. The well-known difficulty of the problem is that the hypothetical local spectra of both inverse and direct cascades in the weak-turbulence approximation carry fluxes that are either zero or have the wrong sign; Such spectra cannot be realized. We analytically derive the exact flux constancy laws (analogs of Kolmogorov's 4/5 law for incompressible fluid turbulence), expressed via the fourth-order moment and valid for any nonlinearity. We confirm the flux laws in direct numerical simulations. We show that a constant flux is realized by a nonlocal wave interaction in both the direct and inverse cascades. Wave spectra (second-order moments) are close to slightly (logarithmically) distorted thermal equilibrium in both cascades.

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.

A side remark in the old paper by Zeldovich and co-authors leads to the recent discovery of a universal conservation law of turbulent dispersion.

Under the pumping at intermediate scales and small-scale dissipation, an inverse cascade in a finite box tends to produce a coherent box-size flow called condensate. If that condensate is stable, it grows, as is the case for incompressible 2d turbulence and optical or Langmuir turbulence with repulsive (defocusing) nonlinearity. If the condensate is unstable, as is the case for optical and Langmuir turbulence with attractive (focusing) nonlinearity, it produces finite-time singularities (self-focusing or wave collapse) which stabilize turbulence at large scales. Here we discover a third possibility: inverse cascade in a compressible 2d turbulence (also turbulence in a thin fluid layer) produces box-size long-living vortices, which fluctuate and oscillate strongly and are accompanied by large-scale shock waves that dissipate the energy. Turbulence provides for a net energy flux into vortex but a zero net momentum flux. We analyze the spectra of kinetic energy and density in both intervals of direct and inverse cascade and show different physics that determines those cascades in compressible 2d turbulence.We present both basic analytic theory and extensive numerics.

We experimentally and theoretically describe formation of random fiber laser's optical spectrum. We propose a new concept of active cycled wave kinetics from which we derive first ever nonlinear kinetic theory describing laser spectrum.

We report on results from a set of numerical simulations of isothermal compressible turbulence forced at an intermediate scale with grid resolutions up to 8192². At a moderate pumping rate, the total energy of a typical frictionless system, subject to periodic boundary conditions, exhibits linear growth characteristic of an inverse cascade followedas in the incompressible caseby condensation. The energy capacity of the compressible condensate, however, is limited by shock dissipation. Effects of compressibility are discussed in both shaping the inverse energy cascade at moderate Mach numbers and mediating the energy condensate growth that involves the formation of strong shocks, connecting centers of large vortices and short-circuiting the energy cycle. In the saturated regime, the total energy exhibits strong irregular oscillations caused by metastability of the compressible condensate.

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate - a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

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 vortices1,2, yet it is not known whether propagating waves can generate large-scale flows-small-amplitude irrotational waves only push particles in the direction of propagation3-5. 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 newconceptual 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.

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as k^-5/3.

Small aerosols drift down a temperature or turbulence gradient since faster particles fly longer distances before equilibration. That fundamental phenomenon, called thermophoresis or turbophoresis, is widely encountered in nature and used in industry. It is universally believed that particles moving down the kinetic energy gradient must concentrate in minima (say, on walls in turbulence). Here, we show that this is incorrect: escaping minima is possible for inertial particles whose time of equilibration is longer than the time to reach the minimum. "The best way out is always through": particles escape by flying through minima or reflecting from walls. We solve the problem analytically and find the phase transition as a sign change of the mean velocity. That means separation: light particles concentrate in a minimum while heavy particles spread away from it (gravity can reverse the effect). That discovery changes our understanding of that fundamental phenomenon and may find numerous applications.

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.

In statistically homogeneous turbulent flows, pressure forces provide the main mechanism to redistribute kinetic energy among fluid elements, without net contribution to the overall energy budget. This holds true in both two-dimensional (2D) and three-dimensional (3D) flows, which show fundamentally different physics. As we demonstrate here, pressure forces act on fluid elements very differently in these two cases. We find in numerical simulations that in 3D pressure forces strongly accelerate the fastest fluid elements, and that in 2D this effect is absent. In 3D turbulence, our findings put forward a mechanism for a possibly singular buildup of energy, and thus may shed new light on the smoothness problem of the solution of the Navier-Stokes equation in 3D.

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 review recent progress in optical wave turbulence with a specific focus on the fast growing field of fiber lasers. Weak irregular nonlinear interactions between a large number of resonator modes are responsible for practically important characteristics of fiber lasers such as the spectral broadening of radiation. Wave turbulence is a fundamental nonlinear phenomenon which occurs in a variety of nonlinear wave-bearing physical systems. The experimental impediments and the computationally intensive nature of simulation of hydrodynamic or plasma wave turbulence often make it rather challenging to collect a significant number of statistical data. The study of turbulent wave behavior in optical devices offers quite a unique opportunity to collect an enormous amount of data on the statistical properties of wave turbulence using high-speed, high precision optical measurements during a relatively short period of time. We present recent theoretical, numerical and experimental results in optical wave turbulence in fiber lasers ranging from weak to strong turbulences for different signs of fiber dispersion. Furthermore, we report on our studies of spectral wave condensate in fiber lasers that make interdisciplinary links with a number of other research fields.

Using data from a large-scale three-dimensional simulation of supersonic isothermal turbulence, we have tested the validity of an exact flux relation derived analytically from the Navier-Stokes equation by Falkovich, Fouxon & Oz (J. Fluid Mech., vol. 644, 2010, p. 465). That relation, for compressible barotropic fluids, was derived assuming turbulence generated by a large-scale force. However, compressible turbulence in simulations is usually initialized and maintained by a large-scale acceleration, as in gravity-driven astrophysical flows. We present a new approximate flux relation for isothermal turbulence driven by a large-scale acceleration, and find it in reasonable agreement with the simulation results.

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.

In turbulence, ideas of energy cascade and energy flux, substantiated by the exact Kolmogorov relation, lead to the determination of scaling laws for the velocity spatial correlation function. Here we ask whether similar ideas can be applied to temporal correlations. We critically review the relevant theoretical and experimental results concerning the velocity statistics of a single fluid particle in the inertial range of statistically homogeneous, stationary and isotropic turbulence. We stress that the widely used relations for the second structure function, D₂(t) ≡ ([v(t) - v(0))]²∝ εt, relies on dimensional arguments only: no relation of D2(t) to the energy cascade is known, neither in two- nor in three-dimensional turbulence. State of the art experimental and numerical results demonstrate that at high Reynolds numbers, the derivative )d D(t) dt2t has a finite non-zero slope starting from t ≈ 2τν. The analysis of the acceleration spectrum Φ_A(ε) indicates a possible small correction with respect to the dimensional expectation Φ_A(ε;) ~ ε⁰ but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics.

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).

Geometric statistics open the window into the most fundamental aspect of turbulence flows, their symmetries, both broken and emerging. On one hand, the study of the stochastic geometry of multi-point configurations reveals the statistical conservation laws which are responsible for the breakdown of scale invariance in direct turbulence cascades. On the other hand, the numerical and experimental studies of inverse cascade reveal that some families of isolines can be mapped to a Brownian walk (i.e. belong to the so-called SLE class) and are thus not only scale invariant but conformally invariant. That means that some aspects of turbulence statistics can be probably described by a conformal field theory. The talk is a review of broken and emerging symmetries in turbulence statistics.

We consider turbulence within the Gross-Pitaevsky model and look into the creation of a coherent condensate via an inverse cascade originating at small scales. The growth of the condensate leads to a spontaneous breakdown of statistical symmetries of overcondensate fluctuations: First, isotropy is broken, then a series of phase transitions marks the changing symmetry from twofold to threefold to fourfold. We describe respective anisotropic flux flows in the k space. At the highest level reached, we observe a short-range positional and long-range orientational order (as in a hexatic phase). In other words, the more one pumps the system, the more ordered the system becomes. The phase transitions happen when the system is pumped by an instability term and does not occur when pumped by a random force. We thus demonstrate nonuniversality of an inverse-cascade turbulence with respect to the nature of small-scale forcing.

The towering figure of Kolmogorov and his very productive school is what was perceived in the twentieth century as the Russian school of turbulence. However, important Russian contributions neither start nor end with that school.

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.

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, the Earth's atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection whereas horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales ^1,2 . The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale whereas 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions, thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow that secures an upscale energy transfer.

We study optical wave turbulence in Raman fibre lasers using particular examples of 13 km and 22 km long Fabry-Perot resonators. We demonstrate that the sign of the cavity dispersion has a critical impact on the spectral and temporal properties of generated radiation that are directly relevant to the fibre laser performance. For a normal dispersion, we observe in numerical modelling an intermediate state with an extremely narrow spectrum (condensate), which experiences instability and a sharp transition to a strongly fluctuating regime with a wider spectrum. The experimental results for the generated spectra demonstrate a good match with numerical simulations.

We study statistical properties of turbulent inverse cascades in a class of nonlinear models describing a scalar field transported by a two-dimensional incompressible flow. The class is characterized by a linear relation between the transported field and the velocity, and include several cases of physical interest, such as Navier-Stokes, surface quasi-geostrophic and Charney-Hasegawa-Mima equations. We find that some statistical properties of the inverse turbulent cascades in such systems are conformal invariant. In particular, the zero-isolines of the scalar field are statistically equivalent to conformal invariant curves within the resolution of our numerics. We show that the choice of the conformal class is determined by the properties of a transporting velocity rather than those of a transported field and discover a phase transition when the velocity turns from a large-scale field to a small-scale one.

We consider the steady-state statistics of turbulence in the inertial interval. The Kolmogorov flux relation (4/5-law) is shown to be a particular case of the general relation on the current-density correlation function. Using that, we derive an analogous flux relation for compressible turbulence and a new exact relation for incompressible turbulence.

We present experimental results on the properties of bounded turbulence in thin fluid layers. In contrast with the theory of two-dimensional (2D) turbulence, the effects of the bottom friction and of the spectral condensation of the turbulence energy are important in our experiment. Here we investigate how these two factors affect statistical moments of turbulent fluctuations. The inverse energy cascade in a bounded turbulent quasi-2D flow leads to the formation of a large coherent vortex (condensate) fed by turbulence. This vortex, depending on its strength, can substantially affect the turbulence statistics, even at small scales. Up to the intermediate strength of the condensate, the velocity moments similar to those in isotropic 2D turbulence are recovered by subtracting the coherent component from the velocity fields. A strong condensate leaves a footprint on the underlying turbulence; it generates stronger non-Gaussianity and reduces the efficiency of the inverse energy cascade. Remarkably, the energy flux in the cascade derived from the third-order structure function using the Kolmogorov flux relation gives physically meaningful values in a broad range of experimental parameters regardless of the condensate strength. This result has important implications for the analysis of the atmospheric wind data in upper troposphere and lower stratosphere.

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.

We study optical wave turbulence using as a particular example recently created ultralong-fiber laser. We show that the sign of the cavity dispersion has a critical impact on the spectral and temporal properties of generated radiation that are directly relevant to the fiber laser performance. For a normal dispersion, we observe an intermediate state with an extremely narrow spectrum (condensate), which experiences an instability and a sharp transition to a strongly fluctuating regime with a wide spectrum and increased probability of spontaneous generation of large-amplitude pulses.

A Reply to the Comment by Erik Lindborg.

The emphasis of this review is on fundamental properties, degree of universality and symmetries of the turbulent state. The central questions are which symmetries remain broken even when the symmetry-breaking factor reaches zero, and which symmetries, in contrast, emerge in the state of developed turbulence. We shall see that time reversibility is broken in all cases since turbulence is a far-from-equilibrium state accompanied by dissipation. As far as scale invariance is concerned, we argue that it is always broken in direct cascades (toward small scales) no matter how far one goes away from the pumping scale. In contrast, inverse cascades become scale invariant as they go toward large scales. Moreover, some properties of the inverse cascades seem to be conformal invariant and related to Schramm-Loewner evolution (a class of random curves that can be mapped to a 1D Brownian walk).

We present experimental results on turbulence generated in thin fluid layers in the presence of a large-scale coherent flow, or a spectral condensate. It is shown that the condensate modifies the third-order velocity moment in a much wider interval of scales than the second one. The modification may include the change of sign of the third moment in the inverse cascade. This observation may help resolve a controversy on the energy flux in mesoscale atmospheric turbulence (10-500 km): to recover a correct energy flux from the third velocity moment one needs first to subtract the coherent flow. We find that the condensate also increases the velocity flatness.

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.

This is a short course on developed turbulence, weak and strong. The main emphasis is on fundamental properties like universality and symmetries. Two main notions are explained: i) fluxes of dynamical integrals of motion, ii) statistical integrals of motion.

Measurements of atmospheric winds in the mesoscale range (10-500 km) reveal remarkably universal spectra with the $k^{-5/3}$ power law. Despite initial expectations of the inverse energy cascade, as in two-dimensional (2D) turbulence, measurements of the third velocity moment in atmosphere, suggested a direct energy cascade. Here we propose a possible solution to this controversy by accounting for the presence of a large-scale coherent flow, or a spectral condensate. We present new experimental laboratory data and show that the presence of a large-scale shear flow modifies the third-order velocity moment in spectrally condensed 2D turbulence, making it, in some conditions, similar to that observed in the atmosphere.

We describe a new effect of floaters clustering by surface waves. This clustering is a result of the surface tension force, which for small particles becomes comparable with their weight. Surface tension creates a difference between the masses of a particle and displaced liquid making the particle effectively inertial. Inertia, positive for hydrophobic or negative for hydrophilic particles, causes particle clustering in the nodes or antinodes of a standing wave and leads to chaotic mixing in random waves. Here we show experimentally that in a standing wave the clustering rate is proportional to the squared wave amplitude. In the case of random waves we demonstrate that inertia effects change statistics of floater distribution and particles concentrate on a multifractal set.

Methods of non-equilibrium statistical mechanics play an increasingly important role in modern turbulence research, yet the range of relevant tools and methods is so wide and developing so fast that until now there has not been a single book covering the subject. As an introduction to modern methods of statistical mechanics in turbulence, this volume rectifies that situation. The book comprises three harmonised lecture courses by world class experts in statistical physics and turbulence: John Cardy introduces Field Theory and Non-Equilibrium Statistical Mechanics; Gregory Falkovich discusses Turbulence Theory as part of Statistical Physics; and Krzysztof Gawedzki examines Soluble Models of Turbulent Transport. To encourage readers to deepen their understanding of the theoretical material, each chapter contains exercises with solutions. Essential reading for students and researchers in the field of theoretical turbulence, this volume will also interest any scientist or engineer who applies knowledge of turbulence and non-equilibrium physics to their work.

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 effect of turbulence on the collision rate between droplets in clouds is investigated. Because of their inertia, water droplets can be shot out of curved streamlines of the turbulent airflow. The contribution of such a "sling effect" in the collision rate of the same-size water droplets is described and evaluated. It is shown that already for turbulence with the dissipation rate 10³ cm² sL^-3, the sling effect gives a contribution to the collision rate of 15-μm droplets comparable to that due to the local velocity gradient. That may explain why the formulas based on the local velocity gradient consistently underestimate the turbulent collision rate, even with the account of preferential concentration.

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.

A review of recent experimental studies of turbulence suppression by mean flows in quasi-two-dimensional fluids is presented. Large-scale mean flows develop during spectral condensation of 2D turbulence as a result of the inverse energy cascade in spatially bounded flow. The spectral energy which is accumulated at the largest scale supports the mean flow which in turn affects turbulence. We show that such a flow can reduce the energy flux in the inverse energy cascade range via shearing and sweeping of the turbulent eddies. The former mechanism is more efficient at larger scales, while the latter acts on the smaller scales. Similar suppression of turbulence has been found in the presence of externally imposed flows. Turbulent (inverse energy) cascade is reduced in the presence of imposed flow, but still supports Kolmogorov-Kraichnan k −5/3 power law spectrum in the energy range.

The statistics of the nodal lines of scalar fields intwo-dimensional (2d) turbulence is found to be conformal invariantand equivalent to that of cluster boundaries in criticalphenomena. That allows for a rich variety of exact analyticresults, first time in turbulence studies. In particular, thestatistics of zero-vorticity lines in Navier-Stokes turbulence isfound to be equivalent to that of critical percolation. Thestatistics of the zero-temperature lines in surfacequasi-geostrophic (SQG) turbulence is found to be equivalent tothat of the isolines of a Gaussian (free) field.

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.

We present a model for the Lagrangian dynamics of inertial particles in a compressible flow, where fluid velocity gradients are modeled by a telegraph noise. The model allows for an analytic investigation of the role of time correlation of the flow in the aggregation-disorder transition of inertial particles. The dependence on the Stokes number St and the Kubo number Ku of the Lyapunov exponent of particle trajectories reveals the presence of a region in parameter space (St, Ku), where the leading Lyapunov exponent changes sign, thus signaling the transition. The asymptotics of short- and long-correlated flows are discussed, as well as the fluid-tracer limit.

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.

We study how the spatial distribution of inertial particles evolves with time in a random flow. We describe an explosive appearance of caustics and show how they influence an exponential growth of clusters due to smooth parts of the flow, leading in particular to an exponential growth of the average distance between particles. We demonstrate how caustics restrict applicability of Lagrangian description to inertial particles.

This short survey is written by a physicist. It contains neither theorems nor precise definitions. Its main content is a description of the results of numerical solution of the equations of fluid mechanics in the regime of developed turbulence. Due to limitations of computers, the results are not very precise. Despite being neither exact nor rigorous, the findings may nevertheless be of interest for mathematicians. The main result is that the isolines of some scalar fields (vorticity, temperature) in two-dimensional turbulence belong to the class of conformally invariant curves called SLE (Scramm-Löwner evolution) curves. First, this enables one to predict and find a plethora of quantitative relations going far beyond what was known previously about turbulence. Second, it suggests relations between phenomena that seemed unrelated, like the Euler equation and critical percolation. Third, it shows that one is able to get exact analytic results in statistical hydrodynamics. In short, physicists have found something unexpected and hope that mathematicians can help to explain it.

We study experimentally and theoretically how waves affect the distribution of particles floating on a liquid surface. According to the Archimedes' law the weight of floating particle is equal to the weight of displaced liquid. This law is not quite precise for small floating objects. An additional force generating by surface tension pulls a hydrophilic particle deeper into the water so that the mass of the displaced liquid exceeds the particle mass. This mass mismatch makes the floating particle effectively inertial that may lead to the clustering of particles when they move by the surface waves. Here we report the results of our recent work showing that particles gather in the nodes or antinodes of a standing surface wave depending on the sign of capillarity effect. Experimentally measured rate of particles gathering is shown to be proportional to the square of the wave amplitude which agrees with the theoretical model.

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.

A mean field model is presented that describes droplet growth resulting from condensation and collisions and droplet loss resulting from fallout. The model allows for an effective numerical simulation. The numerical scheme that is conservative in water mass and keeps accurate count of the number of droplets is applied, and the way in which the rain initiation time depends on different parameters is studied. In particular, it is shown that the rain initiation time depends nonmonotonically (has a minimum) on the number of cloud condensation nuclei. Also presented is a simple model that allows one to estimate the rain initiation time for turbulent clouds with an inhomogeneous concentration of cloud condensation nuclei. It is argued that by overseeding even a part of a cloud by small hygroscopic nuclei one can substantially delay the onset of precipitation.

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.

We study how the spatial distribution of inertial particles evolves with time in a random flow. We describe an explosive appearance of caustics and show how they influence an exponential growth of clusters due to smooth parts of the flow, leading in particular to an exponential growth of the average distance between particles.

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.

This is a short course on developed turbulence, weak and strong. The main emphasis is on fundamental properties like universality and symmetries. Two main notions are explained: i) fluxes of dynamical integrals of motion, ii) statistical integrals of motion.

We study experimentally how waves affect the distribution of particles that float on a liquid surface. We show that clustering of small particles in a standing wave is a nonlinear effect with the clustering time decreasing as the square of the wave amplitude. In a set of random waves, we show that small floaters concentrate on a multifractal set with caustics.

We study experimentally how waves affect distribution of particles that float on a water surface. We show that clustering of small particles in a standing wave is a nonlinear effect with the clustering time decreasing as the square of the wave amplitude. In a set of random waves, we show that small floaters concentrate on a multi-fractal set.

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.

The problem of droplet growth by condensation in a turbulent flow of nearly saturated vapour is addressed theoretically and numerically. We show how the presence of an underlying turbulent velocity field induces a correlation between droplet trajectories and supersaturation. This leads both to the enhancement of the droplet growth rate and to a fast spreading of the droplet size distribution.

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.

The phenomenon of preferential concentration of inertial particles is studied by following lagrangian trajectories. Elementary properties of the coarse-grained distribution of heavy particles in simple turbulent flows are investigated by direct numerical simulations. In the small Stokes number case, we compute the coarse-grained particle distribution, and we demonstrate that the second moment behaves as an approximate power law. The dependence of the exponent as a function of the Reynolds and of the Stokes number is studied in the small Stokes number limit. Our results show a strong dependence of the level of fluctuation of the particle distribution as a function of the Reynolds number.

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 retardation of weakly inertial particles depends on the acceleration of the ambient fluid, so the particle concentration n is determined by the divergence of Lagrangian acceleration which we study by direct numerical simulations. We demonstrate that the second moment of the concentration coarse-grained over the scale r behaves as an approximate power law: (n̄_r²) ∼ r^α. We study the dependencies of the exponent α on the Reynolds number, of the Stokes number, and on the settling velocity. We find numerically that the theoretical lower bound previously suggested [Falkovich et al., Nature 419, 151 (2002)] correctly estimates the order of magnitude (within a factor 2 to 4) as well as the dependencies on the Reynolds, Stokes, and Froude numbers. The discrepancy grows with the Reynolds number and the Froude number. We analyze the possible physical mechanism responsible for that behavior.

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.

In this paper we consider systems deviated from equilibrium by some external factors and discuss the internal entropy production and entropy extraction by the environment. For a system moving away from equilibrium, we express the entropy extraction via a two-point correlation function for any time and any distance from equilibrium. The long-time limit gives the sum of the Lyapunov exponents expressed via the formula of Green-Kubo type. We discuss what is known about the entropy production from deviations away from the equilibrium and back to equilibrium and for turbulent states. In particular, we show that the entropy production is due to the degrees of freedom participating in a direct cascade but not in an inverse cascade.

To describe transport by a random flow, one needs to apply the statistical methods to the motion of fluid particles, that is, to the Lagrangian dynamics. We first present the propagators describing evolving probability distributions of different configurations of fluid particles. We then use those propagators to describe decay and steady states of a passive scalar field transported by random flows.

Keywords: HOMOGENEOUS ISOTROPIC TURBULENCE; SETTLING VELOCITY; HEAVY-PARTICLES; CLOUDS; FLOW; INTERMITTENCY; ACCELERATION; INERTIA

Vapour condensation in cloud cores produces small droplets that are close to one another in size. Droplets are believed to grow to raindrop size by coalescence due to collision^1,2. Air turbulence is thought to be the main cause for collisions of similar-sized droplets exceeding radii of a few micrometres, and therefore rain prediction requires a quantitative description of droplet collision in turbulence^1-5. Turbulent vortices act as small centrifuges that spin heavy droplets out, creating concentration inhomogeneities^6-14 and jets of droplets, both of which increase the mean collision rate. Here we derive a formula for the collision rate of small heavy particles in a turbulent flow, using a recently developed formalism for tracing random trajectories^15,16. We describe an enhancement of inertial effects by turbulence intermittency and an interplay between turbulence and gravity that determines the collision rate. We present a new mechanism, the 'sling effect', for collisions due to jets of droplets that become detached from the air flow. We conclude that air turbulence can substantially accelerate the appearance of large droplets that trigger rain.

This lecture course is intended to bring home to a reader two main lessons: power of the Lagrangian approach to fluid turbulence and importance of statistical integrals of motion for systems far from equilibrium. We present the description of turbulence from a Lagrangian viewpoint that is based on the motion of fluid particles. Section 1 describes the statistics of one, two and many-particle configurations. Section 2 describes the statistics of the passive scalar that can be inferred from the particle analysis. In Section 3, we discuss the NavierStokes equation from a Lagrangian viewpoint. Conclusion lists the most important open problems. We restrict ourselves to incompressible flows. Lagrangian description of compressible flows (as well as other subjects like magnetic dynamo, Lagrangian numerics etc.) can be found in the extended version

The logarithmic renormalization predicted by Kraichnan (1971) for the direct cascade of enstrophy in the incrtial 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 arc observed.

The role of interaction in causing errors in optical soliton transmission was studied. Two solitons propagating under a filter-control scheme was considered and the timing jitter caused by spontaneous-emission noise and enhanced by attraction between solitons was described. Results showed the bit-error rate as a function of system parameters, timing, initial distance and the phase difference between solitons.

We examine statistics of two interacting optical solitons and describe timing jitter caused by spontaneous emission noise and enhanced by pulse interaction. Dynamics of phase difference is shown to be of crucial importance in determining the probability distribution function (PDF) of the distance between solitons. We find analytically the non-Gaussian tail of the PDF to be exponential. The propagation distance that corresponds to a given bit-error rate is described as a function of system parameters (filtering and noise level), initial distance, and initial phase difference between solitons. We find the interval of parameters where a larger propagation distance can be achieved for higher density of information.

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.

A statistical theory based on a Lagrangian description of turbulence was developed to account for the fluctuations of suspended particles in turbulent flows. By considering a small spherical particle, it was shown that the flow of inertial particles is compressible. In addition, the initial growth of concentration fluctuations from a uniform state and its saturation due to finite-size effects were illustrated.

We present analytical method to acculate non-Gaussian intensity and timing jitters statistics in optical communication systems. Using developed method we have calculated probability density function for optical regenerated soliton systems with in-line synchronous intensity modulation and filtering.

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.

We present analytical method to calculate non-Gaussian intensity and timing jitters statistics in optical communication systems. Using developed method we have calculated probability density function for optical regenerated soliton systems with in-line synchronous intensity modulation and filtering.

We present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise. Even though the noise is weak, we are interested in probabilities of large fluctuations (generally non-Gaussian) which are beyond perturbation theory. Our method is a development of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve a fundamental problem of soliton statistics governed by a noisy nonlinear Schrödinger equation. We then apply our method to optical soliton transmission systems using signal control elements (filters and amplitude and phase modulators).

Keywords: PASSIVE SCALAR; VELOCITY-FIELD; DYNAMO

I cannot resist quoting the beginning of the Introduction by Chaté, Villermaux and Chomaz verbatim: Mixing is a subject which suffers from the Bourgeois Gentilhomme complex. Like Monsieur Jourdain in Moliere's play (1670), scientists, engineers, and indeed all of us, often do mixing without even knowing it (just think about yourself trying to prepare mayonnaise...). [first paragraph]

We consider inertial particles suspended in an incompressible turbulent flow. Due to inertia of particles, their velocity field acquires small compressible component. Its presence leads to a new qualitative effect --- possibility of clustering. We show that this effect is significant for heavy particles, leading to strong fluctuations of the concentration.

We consider statistics of the passive scalar on distances much larger than the pumping scale. Such statistics is determined by statistics of Lagrangian contraction, that is by probabilities of initially distant fluid particles coming close. At the Batchelor limit of spatially smooth velocity, the breakdown of scale invariance is established for scalar statistics.

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 field is concentrated in long narrow strips and describe anomalous scalings and angular singularities of the multipoint correlation functions which are manifestations of the fields 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.

Probability density function (PDF) of passive scalar dissipation (formula presented) is found analytically in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. The tail of PDF at (formula presented) is shown to be stretched exponent (formula presented). At (formula presented).

I briefly describe here emerging methods designed for the description of probability density function (PDF) of the gradients of turbulent fields. The formalism is applied to Burgers turbulence and to the scalar advected by both compressible and incompressible turbulence.

We describe two complementary formalisms designed for the description of the probability density function (PDF) of the gradients of turbulent fields. The first approach, we call it adiabatic, describes the PDF at the values much less than dispersion. The second, instanton, approach gives the tails of the PDF at the values of the gradient much larger than dispersion. Together, both approaches give a satisfactory description of gradient PDFs, as illustrated here by an example of a passive scalar advected by a one-dimensional compressible random flow.

We show that the tails of the single-point velocity probability distribution function (PDF) are generally non-Gaussian in developed turbulence. By using instanton formalism for the randomly forced Navier-Stokes equation, we establish the relation between the PDF tails of the velocity and those of the external forcing. In particular, we show that a Gaussian random force having correlation scale L and correlation time T produces velocity PDF tails ln P(v)∝-v⁴ at v ≫ v_rms, L/τ. For a short-correlated forcing when ∝ ≪L/v_rms there is an intermediate asymptotics P(v)∝-v₃ at L/τ ≫v≫v_rms.

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).

We demonstrate that if the exponent γ that measures nonsmoothness of the velocity field is small then the isotropic zero modes of the scalar's triple correlation function have the scaling exponents proportional to √γ. Therefore, zero modes are subleading with respect to the forced solution that has normal scaling with the exponent γ.

We consider the tails of probability density function (PDF) for the velocity that satisfies Burgers equation driven by a Gaussian 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. For the PDFs of velocity and its derivatives u^(k) = ∂^k_xu, the general formula is found: 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 < In (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.

We consider a passive scalar convected by a multiscale random velocity field with short yet finite temporal correlations. Taking Kraichnans limit of a white Gaussian velocity as a zero approximation we develop the perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus nonuniversal as was predicted by Shraiman and Siggia on a phenomenological ground.

Keywords: Engineering, Mechanical; Mechanics

Two-dimensional turbulence of the waves with linear dispersion law is analyzed numerically at small Mach numbers and large Reynolds numbers. It is shown that the energy-flux relation is close to E∝[Formula Presented] as for a one-dimensional system. The analysis of the wave distribution in k space shows that the anisotropic large-scale pumping produces turbulence as a set of narrow jets that do not smear as the cascade proceeds towards high wave numbers. The energy spectrum along the direction of a jet is close to E([Formula Presented][Formula Presented] due to shock waves, while the spectrum per unit interval of wave numbers is E(k)∝[Formula Presented][Formula Presented] contrary to all previous predictions. Probability density functions of the velocity and velocity differences are found and compared with recent theoretical predictions.

For Kraichnans problem of passive scalar advection by a velocity field delta correlated in time, the limit of large space dimensionality d ≫ 1 is considered.Scaling exponents of the scalar field are analytically found to be ζ 2n = n ζ 2-2(2-ζ 2)n (n -1)/d, while those of the dissipation field are μ n = -2(2-ζ 2)n (n -1)/d for orders n ≪d.The refined similarity hypothesis ζ 2n = n ζ 2 + μ n is thus established by a straightforward calculation for the case considered.

We describe the method for finding the non-Gaussian tails of the probability distribution function (PDF) for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configurationthe instanton. As an example, we examine the correlation functions of the passive scalar u advected by a large-scale velocity field δ correlated in time. We find the instanton determining the tails of the generating functional, and show that it is different from the instanton that determines the probability distribution function of high powers of u. We discuss the simplest instantons for the Navier-Stokes equation.

The nonlinear Schrödinger equation with repulsion (also called the Gross-Pitaevsky equation) is solved numerically with damping at small scales and pumping at intermediate scales and without any large-scale damping. Inverse cascade creating a wave condensate is studied. At moderate pumping, it is shown that the evolution comprises three stages: (i) short period (few nonlinear times) of setting the distribution of fluctuations with the flux of waves towards large scales, (ii) long intermediate period of self-saturated condensation with the rate of condensate growth being inversely proportional to the condensate amplitude, the number of waves growing as √t, the total energy linearly increasing with time and the level of over-condensate fluctuations going down as 1/√t, and (iii) final stage with a constant level of over-condensate fluctuations and with the condensate linearly growing with time. Most of the waves are in the condensate. The flatness initially increases and then goes down as the over-condensate fluctuations are suppressed. At the final stage, the second structure function 〈[Formula Presented]-[Formula Presented][Formula Presented]〉∝ln[Formula Presented] while the fourth and sixth functions are close to their Gaussian values. Spontaneous symmetry breaking is observed: turbulence is much more anisotropic at large scales than at pumping scales. Another scenario may take place for a very strong pumping: the condensate contains 2530 % of the total number of waves, the harmonics with small wave numbers grow as well.

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.

Three-dimensional turbulence of incompressible fluid is described by using Clebsch canonical variables. This reveals the families of new local integrals of motion so that there are additional cascade spectra besides the energy cascade. A weakly anisotropic spectrum of developed turbulence is shown to be as universal as isotropic Kolmogorov spectrum. The correlation functions of three-dimensional incompressible turbulence approach their isotropic values in the inertial interval so that the share taken by the anisotropic parts of velocity correlators decrease with the wavenumber as k^{- 2 3}, which satisfactorily fits the experimental data. The complementarity of the turbulence description in Clebsch and velocity variables is demonstrated.

Evolution of decay turbulence of capillary waves in deep water is considered in the framework of the isotropic kinetic equation. It is shown that the evolution comprises of two stages. During the first stage an arbitrary localized large-scale wave distribution explosively evolves into a small-scale Kolmogorov spectrum. The second stage starts at the moment the Kolmogorov spectrum reaches dissipative scales. For systems with non-linear damping, the characteristic time of this stage is much longer (up to thousand times) than the first stage. The energy distribution is close to the Kolmogorov spectrum and decaying follows a self-similar law.

Evolution of decay turbulence of capillary waves on a deep water is considered in the framework of the isotropic kinetic equation. It is shown that the evolution comprises two stages. During the first stage an arbitrary localized large-scale wave distribution explosively evolves into a small-scale Kolmogorov spectrum. The second stage starts at the moment the Kolmogorov spectrum reaches dissipative scales. The characteristic time of this stage is much longer (up to thousand times) than that of the first one. The energy distribution is close to the Kolmogorov spectrum and decays by a self-similar law.

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.

Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. This corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach Gaussian statistics with increase in the Péclet number Pe (the pumping-to-diffusion scale ratio). The first n

Wave turbulence for systems with only direct (small-scale) turbulent cascades is analyzed at scales much larger than the scale of the pumping. At such scales, the turbulence spectrum is shown to turn into an equilibrium Rayleigh-Jeans distribution with the temperature determined by the pumping scale and energy dissipation rate (the turbulent flux). The behavior of the damping of the waves changes drastically at a scale determined by the mean free path of turbulent waves. Two particular examples of acoustic and capillary-wave turbulence are considered. We also carried out numerics which confirm the theoretical predictions.

Weakly anisotropic steady are found for both inverse and direct cascades in two-dimensional turbulence of an incompressible fluid. The degree of anisotropy is shown to increase for both spectra: as (kL) ^{-2 3} upscales and as (kΔ)² downscales from the pump. A weakly anisotropic intermediate-scale pumping may thus produce a substantially anistropic turbulence in the inertial intervals of scales.

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.

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 ω in the inertial interval: ωn(r1)ωn(r2)ln2n/3(L/r1 -r2), with L being the scale of the external pump.

It is shown that if the pair correlation function of any tracer in incompressible turbulent flow is scale invariant with the exponent ζ2, then the exponent of two-point function of 2nth order does not equal nζ2. In this case, the probability distribution should depend, generally speaking, on an infinite number of parameters (fluxes of the integrals). Three examples are considered: two-dimensional vorticity cascade, action cascade in Clebsch variables, and entropy cascade in inhomogeneously heated fluid.

The equations describing the interaction of long inertio-gravity (IG) waves with the Rossby waves are derived. Due to remarkable cancellations, the interaction is shown to be anomalously weak. As a result, an inverse cascade of turbulence produces wave condensate of large amplitude so that wave breaking with front creation can occur.

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.

It is shown how viscosity increases turbulence level in the inertial interval by suppressing turbulent transfer.

Nonlinear random classical waves driven far off equilibrium by the steady input of energy can support propagating collective modes analogous to zero sound in Fermi liquids. The conditions for the existence of these collisionless and dispersionless modes are presented. Applications to a variety of systems as well as experiments to test the theory are suggested. In particular, this article predicts that for gravity waves on the surface of a liquid both longitudinal and transverse collective modes are possible in the collisionless regime.

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

The picture of cascade turbulence suggested by Richardson, Kolmogorov and Obukhov is based on the concept of interaction locality [15]. That means that those modes (vortices or waves) effectively interact which are of comparable scales only. The question naturally arise: whether a locality property should be satisfied on the steady Kolmogorov-like spectrum only or on the slightly differing distributions as well? Proceeding from continuity-like speculations, one might suppose that in general case interaction locality for Kolmogorov distribution leads to that for close ones. Such a supposition is, however, incorrect since Kolmogorov spectrum usually possesses higher degree of symmetry (for example, being isotropic) than arbitrary yet close distributions. A stationary locality does not mean thus an evolutionary locality as it was stated in [4,6].

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 critical analysis of the conformal approach to the theory of 2D turbulence is delivered. It is shown, in particular, that conformal minimal models cannot give a general turbulent solution, which should provide for constant fluxes of all vorticity integrals of motion.

A new type of instability of Kolmogorov-like wave turbulence spectra is found. Such an instability is due to an interaction nonlocal in k-space and it strongly modifies the angular structure of the turbulence spectrum. However, the spectrum dependence on the modulus k is still a Kolmogorov-like one corresponding to energy transfer local in k-space. The specific case of capillary waves on shallow water is considered in detail. It is shown that the energy transfer is local while that of momentum is nonlocal in k-space.

Since the human organism is itself an open system, we are naturally curious about the behavior of other open systems with fluxes of matter, energy or information. Of the possible open systems, it is those endowed with many degrees of freedom and strongly deviating from equilibrium that are most challenging. A simple but very significant example of such a system is given by developed turbulence in a continuous medium, where we can discern astonishing features of universality. This two-volume monograph deals with the theory of turbulence viewed as a general physical phenomenon. In addition to vortex hydrodynamic turbulence, it considers various cases of wave turbulence in plasmas, magnets, atmosphere, ocean and space. A sound basis for discussion is provided by the concept of cascade turbulence with relay energy transfer over different scales and modes. We shall show how the initial cascade hypothesis turns into an elegant theory yielding the Kolmogorov spectra of turbulence as exact solutions. We shall describe the further development of the theory discussing stability prob­ lems and modes of Kolmogorov spectra formation, as well as their matching with sources and sinks. This volume is dedicated to developed wave turbulence in different media.

The stationary spectrum of hydrodynamic convective turbulence is shown to be defined by influxes of two independent motion integrals: entropy and mechanical energy. A careful analysis of the conservation laws is performed. It is shown that in the inertial range of scales kinetic energy converts into potential energy due to presence of temperature fluctuations independently of the type of long-scale stratification (stable or unstable one). Under a purely entropic excitation (for example, by horizontal temperature gradient) the spectrum with constant entropy flux, F_vv ∼ k ^{-21 5}, fills the whole of the inertial interval and crossover to the Kolmogorov-Obukhov spectrum with constant energy flux, F_vv ∼ k ^{-11 3}, is absent. An estimate for crossover scale is obtained for a mixed method of excitation with both nonzero energy pumping and nonzero entropy extraction caused by an environment. A simple but consistent differential model is suggested for the description of the fluxes of energy and entropy in k-space. Two-flux universal spectraof the velocity and temperature fluctuations are obtained.

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.

It is shown that an inverse cascade of the turbulence of inertio-gravity waves produces a long-scale wave condensate. A new nonlinear equation is derived for long waves on rotating shallow water. It is proven that steady localized solutions are absent and that the condensate (a uniform inertial oscillation) is stable with respect to small perturbations. Wave self-interaction could not thus stop an inverse cascade of mesoscale geophysical turbulence. The implication of the existence of a condensate for the problem of tidal dissipation and retardation of the Earth's rotation is discussed.

The problem of interaction locality in k space is studied in a diagrammatic perturbation approach for the Navier-Stokes equation in quasi-Lagrangian variables. Analyzing the whole diagram series we have found an exact relation between the asymptotic behavior of the triple-correlation function of velocities that governs the energy transfer over scales and the double-correlation function giving the energy distribution. Namely, at rR, we obtain S3(r,R,R-r)S2(R)(r/R) [S3(r)/S2(r)]R2ζ-1r22-ζ, where ζ2 is the static exponent of double-velocity moment. This relation between two different physical quantities (in principle, measurable independently) is accessible to an experimental check. Also, this relation allows us to describe an energy exchange between distant scales in k space: For any steady spectrum carrying constant energy flux, the interactions of the given k-eddies with large (k1k) and small eddies (k2k) are shown to decrease by the same law with the distance in k space, such as (k1/k)22-ζ and (k/k2)22-ζ. It means a balance of interactions for such a spectrum. Considering, in particular, the multifractal picture of developed turbulence, we analyze the range of exponents h of the velocity field [δv(r)rh] which provides the locality of interaction in the k space. It is shown that the condition of infrared locality of interaction (with larger k1-eddies) could give only the upper restriction for the exponent. The upper limit thus found (hmax=1) coincides with the boundary exponent of singularity of energy dissipation. As far as an interaction locality in the ultraviolet limit (k2k) is concerned, we prove that any reasonable dimension function D(h) provides locality whatever small h is considered.

Nonstationary regimes of the wave turbulence evolution are considered in the framework of isotropic kinetic equation. It is predicted analytically and confirmed by numerical experiment that there is a class of wave systems in which any initial distribution of the turbulence energy in k-space comes into a universal, Kolmogorovtype spectrum in a finite time. Before and after the formation of the Kolmogorov spectrum, two different self-similar regimes of evolution occur: the first one is responsible for explosively forming the universal spectrum and the second one determines energy dissipation.

The development of wave turbulence is considered in which the wave excitation and attenuation regions are separated by an extended inertia interval. The conditions which the function describing the dissipation must satisfy in order for a nonequilibrium stationary distribution to exist are found. The effect of dissipation on the structure of the stationary turbulence spectrum is described for both the inertial interval (in which the effect is small) and the region of strong dissipation. The general theory is verified in numerical experiments for three physical systems: capillary waves in deep water, gravitational-capillary waves in shallow water and three-dimensional sound with positive dispersion. [(Russian original - ZhETF, Vol. 98, No. 6, p. 1931, December 1990, Vliianie dissipatsii na strukturu statsionarnogo spektra volnovoi turbulentnosti)]

The structural instability of the isotropic Kolmogorov spectrum of a weak wave turbulence is discussed. The physical criterion for the existence of the drift Kolmogorov distribution is formulated. In the case of weak acoustic turbulence, universal two-flux spectra (bearing the energy and momentum fluxes) are obtained analytically. The universality hypothesis for the developed wave turbulence is formulated.

Underlying the present-day approach to the problem of the onset of turbulence is the hypothesis that the phenomena determining the development of instability are finite-dimensional. Experimental data on the laminar turbulent transition [i, 2] support this standpoint, although rigorous mathematical formulation of this assertion, the theorem of the central manifold [3, 4], has been proven only for bifurcations of loss of stability by a stationary point. Intuitive arguments at present [5] suggest the probable existence of a finite set of excited degree of freedom, which determine the dynamics of the system over long times for more complex regimes of motion than the limit cycle.In view of this, it is of considerable interest to determine the number of independent variables, which uniquely describe the potentially infinite-dimensional motion of a dissipative continuous medium, when the number of degrees of freedom, really set in motion, is not known beforehand. The necessary number of such variables is assigned by a one-to-one mapping of the phase space of asymptotic motion in Euclidean space, whose dimension will be called the dimension of embedding.In this communication we propose a direct method of determining the dimension of embedding directly from experimental data, on the basis of an examination of the functional relation between the variables. Along with the embedding dimension we consider the scaling dimensions [6, 7] and the possibility of measuring them experimentally. The experimental data obtained in a study of the laminar-turbulent transition in circular Couette flow are analyzed in regard to dimension. Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 99104, JanuaryFebruary, 1989.

For media with a decreasing dispersion law it is shown that the matching of the Kolmogorov weak turbulence spectrum, realized at large k, to a spectrally narrow source situated at small values of k is realized in terms of an intermediate solution. This solution has the form of a chain of peaks of decreasing amplitude on a background which decreases more slowly. The dependence of the energy flux carried off by the turbulence spectrum on the position of the source in k-space is found. [Russian original - ZhETF, Vol. 94, No. 1(7), p. 172, July 1988]

The stability problem of Kolmogorov spectra of a weak turbulence is analytically solved for the first time in the framework of a three-wave kinetic equation. The spectrum of isotropic perturbations of a stationary not-in-equilibrium distribution is found for the capillary waves on a shallow water surface. It is shown, in the isotropic case, that the Kolmogorov solution is stable with respect to excitations local in k-space. The perturbations drift to the damping region without growth of the magnitude. The structural instability of the isotropic spectrum is found by computer simulation: a small pumping anisotropy causes the spectrum to be essentially anisotropic within the inertial range.

The problem of the stability of the Kolmogorov spectra of weak turbulence is analytically solved for the first time. The spectrum of the isotropic perturbations of the steady-state distribution of the capillary waves on the surface ofshallow water is found. It is shown that the Kolmogorov solution is stable against excitations of packets localized in k space: the packets are carried into the runoff region without increasing in size. [Russian original - ZhETF, Vol. 93, No. 1, p. 172, July 1987]

It is shown that the singular character of the distribution of parametrically excited magnons in k-space (points, lines, surfaces) leads to singularities in the spectra of other quasiparticles. For antiferromagnets the positions and types of the singularities arising in the spectra of themagnons of the antiferromagnetic branch and of the phonons upon parametric excitation of magnons of the ferromagnetic branch are found. [Russian original - ZhETF, Vol. 90, No. 6, p. 2167, June 1986]

A study is made of the quasiparticle interaction processes that limit the amplitude of parametrically excited magnons in an antiferromagnet with a decay spectrum. It is shown that even a small excess above the parametric instability threshold gives rise to a sequence of kinetic instabilities which create several groups of nonequilibrium quasiparticles. A detailed analysis is made of the development of a hierarchy of kinetic instabilities in iron borate under experimental conditions [B. Ya. Kotyuzhanskii and L. A. Prozorova, Sov. Phys. JETP 54, 1013 (1981 ); 56,903 (1982); 59, 384 (1984); B. Ya. Kotyuzhanskiy, L. A. Prozorova, and L. E. Svistov, Sov. Phys. JETP 59, 644 ( 1984) 1. Calculations are reported of the dependences of the number and of the spectral width of the distribution of parametrically excited magnons on the pump power and on the intensity of the static magnetic field. [Russian original - ZhETF, Vol. 90, No. 5, p. 1781, May 1986]

Truncated equations describing the evolution of sound waves in antiferromagnets are derived. The crystallographic anisotropy is taken into account. These equations are used to study the stability of plane solitons. A theorem on self-focusing is proved. Classes of initial data which give rise to a self-focusing sound wave are identified. (Russian original - ZhETF, Vol. 89, No. 1, p. 258, July 1985 )

The paper reports an experimental investigation of the thresholds for spin-wave excitation under the action on an yttrium iron garnet sample of two pumps, one of which is powerful enough to excite a secondary (kinetic) instability of the spin waves. A diagram of the spin-wave instabilities occurring in various parts of the spectrum, depending on the powers of the pumps, is constructed and explained. (Russian original - ZhETF, Vol. 87, No. 1, p. 205, July 1984 )

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.

We consider the evolution, as a result of three-wave processes, of a narrow wave packet with a number of waves which is much larger than the number of thermal waves which interact with it. It is shown that the system of kinetic equations which describe the simultaneous evolution of a nonlinear wave packet and the thermal waves which interact with it, has a Hamiltonian structure which is not violated by the introduction of linear sources or damping decrements of the waves. We prove that the steady state, not coincident with equilibrium, is stable. We describe the consequences of the quasidynamical character of the kinetics: anomalously large times to reach the steady state, and the possibility of oscillations of the occupation numbers.

The stability of stationary waves is investigated in the framework of the nonlinear string equation. Waves of sufficiently large amplitude reveal instability resulting in collapse.

A new method of investigation of the stability of a solution of a nonlinear equation is suggested, which is based on the isospectral transformation and is applied to the problem of the stability of a self-similar solution in the Burgers model.

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)]

It is shown that a weak shock wave is unstable relative to transverse modulations.