Publications
2020

(2020) Physical Review Letters. 125, 10, 104501. Abstract
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 phasespace 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.

(2020) Scientific Reports. 10, p. 2507 Abstract
In hydrodynamics, vortex generation upon the transition from smooth laminar flows to turbulence is generally accompanied by increased dissipation. However, plane vortices can provide transport barriers and decrease losses, as it happens in numerous geophysical, astrophysical flows and in tokamaks. Photon interactions with matter can affectlight transportin waveguides in unexpected and somewhat counterintuitive ways resembling fluid dynamics. Here, we demonstrate dramatic impact of light vortex formation in microstructured optical fibres on the energy dissipation. We show possibility of vortices formationin both solid core and hollow core fibres on the zero amplitude lines in the cladding. We find that vortices reduce light leakage by three orders of magnitude, effectively improving wave guiding. A strong light localization based on the same principle can also be achieved in the negative curvature hollow core fibres.
2019

(2019) Proceedings of the National Academy of Sciences of the United States of America. 116, 51, p. 2542425429 Abstract
We show that rotating particles at the liquidgas interface can be efficiently manipulated using the surfacewave analogue of optical lattices. Two orthogonal standing waves generate surface flows of counterrotating halfwavelength 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 selfassembly 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 samesign vortex cells of the wavedriven 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 selforganize 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 higherfrequency spinner occupies the inner orbit and the lowerfrequency one circles on the outer orbit, while the orbital motions of both spinners are synchronized.

(2019) Physical Review Letters. 123, 2, 026801. Abstract
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 momentumconserving 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 currentinjecting 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.
2018

(2018) Physical Review Letters. 121, 16, 164501. Abstract
Flows in fluid layers are ubiquitous in industry, geophysics, and astrophysics. Largescale 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 walldriven (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 pressuredriven 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) <Re <3 x 10(4), the mean flow in most cases has remarkably simple structure: the jet is sinusoidal with a parabolic velocity profile, and vorticity is constant inside vortices, while the fluctuations are small. At higher Re, strong fluctuations appear, yet the mean traveling wave survives. Considering the momentum flux barrier in such a flow, we derive a new scaling law for the Re dependence of the friction factor and confirm it by DNS.

(2018) Nature Communications. 9, 4533. Abstract
Viscous electron fluids have emerged recently as a new paradigm of stronglycorrelated electron transport in solids. Here we report on a direct observation of the transition to this longsoughtfor state of matter in a highmobility electron system in graphene. Unexpectedly, the electron flow is found to be interactiondominated but nonhydrodynamic (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 darknessbefore daybreak manner, the interactiondominated 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.

(2018) Physical Review Letters. 121, 17, 176805. Abstract
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 DorfmanCohen 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.

(2018) Journal of Turbulence. 19, 8, p. 647663 Abstract
Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counterrotating 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 Cf 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.

(2018) Physical Review B. 97, 8, 085127. Abstract
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 timedependent 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 onedimensional 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 noslip/nostress 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 nostress case and to a finite value in a general case. We also show that for dc current across a strip with a nostress boundary, there are only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.
2017

(2017) Nature Physics. 13, p. 1182–1185 Abstract
Electronelectron (ee) collisions can impact transport in a variety of surprising and sometimes counterintuitive ways(16). Despite strong interest, experiments on the subject proved challenging because of the simultaneous presence of different scattering mechanisms that suppress or obscure consequences of ee scattering(711). Only recently, suffciently clean electron systems with transport dominated by ee collisions have become available, showing behaviour characteristic of highly viscous fluids(1214). 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 highmobility transport at elevated temperatures, a potentially useful behaviour for designing graphenebased devices.

(2017) Physics of Fluids. 29, 11, 110901. Abstract
This article introduces the Focus Issue on TwoDimensional Turbulence appearing in Physics of Fluids (Volume 29, Issue 11, November 2017). Published by AIP Publishing.

(2017) Physical Review Fluids. 2, 9, 92603. Abstract
Largescale turbulence in fluid layers and other quasitwodimensional 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 l, 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 l, the steady state contains a systemsize pair of longliving 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 smallscale dissipation.

(2017) Physical Review Letters. 119, 6, 066601. Abstract
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 currentfield 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.

(2017) Physical Review Fluids. 2, 3, 032602. Abstract
An inverse cascade, energy transfer to progressively larger scales, is a salient feature of twodimensional 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. Longtime averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can neither be deduced from the largestavailablescale argument, nor from the often employed maximumentropy principle or quasilinear approximation.

(2017) Proceedings of the National Academy of Sciences of the United States of America. 114, 12, p. 30683073 Abstract
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 Gball. 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 ballistictoviscous crossover using an approach based on quasihydrodynamic variables. Conductance is found to obey an additive relation G = G(ball) + G(vis), where the viscous contribution Gvis dominates over Gball in the hydrodynamic limit. The superballistic, lowdissipation transport is a generic feature of viscous electronics.
2016

Stokes Paradox, Back Reflections and InteractionEnhanced Conduction(2016) arXiv. Abstract
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 pointlike defects. Conductivity is found to obey an additive relation $\sigma=\sigma_0+\Delta\sigma$, with a noninteracting 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 carriercarrier collisions occurring at the ballistic lengthscales. The interplay between these effects is essential at the ballistictoviscous crossover.

(2016) Nature Physics. 12, p. 672676 Abstract
Quantumcritical strongly correlated electron systems are predicted to feature universal collisiondominated transport resembling that of viscous fluids(14). However, investigation of these phenomena has been hampered by the lack of known macroscopic signatures of electron viscosity(59). 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 signchanging electrical response affords a robust tool for directly measuring the viscositytoresistivity ratio. A strongly interacting electronhole plasma in highmobility graphene(1012) affords a unique link between quantumcritical electron transport and the wealth of fluid mechanics phenomena.

(2016) Proceedings Of The Royal Society AMathematical Physical And Engineering Sciences. 472, 2191, 20160287. Abstract
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 twodimensional 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 strongwhile 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.

(2016) Physical Review. E. 93, 5, 052206. Abstract
Inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in halfspace 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 wallbounded 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<βc and indicates that inelastic collapse does not necessarily imply nearwall localization.

(2016) BoundaryLayer Meteorology. 159, p. 2340 Abstract
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 closeform 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.

(2016) Journal of Physics A: Mathematical and Theoretical. 49, 3, 035102. Abstract
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 localizationdelocalization transition in the former corresponds to the pathcoalescence 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 inertiainelasticity plane. An important feature of the diagram is the region of inelastic collapse: if the restitution coefficient beta of particle velocity is smaller than the critical value beta(0) = exp(pi/root 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.
2015

(2015) Journal of Physics A: Mathematical and Theoretical. 48, 18, 18FT02. Abstract
We discuss general features of the operator product expansion and use it to infer multipoint manifestations of the energy cascade in turbulence. We calculate explicitly the possible form of the threepoint velocity correlation function when one distance is smaller than two others. We elucidate manifestation of direct and inverse energy cascades in the threepoint velocity correlation function.

(2015) Physical Review E. 91, 4, 43022. Abstract
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 et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)], 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)] proportional to t and [p(n)] proportional to Ren1 for the power as a function of the Reynolds number. Both observations can be explained using the "flightcrash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the prefactor for [E(t)  E(0)]. To complete the singleparticle 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 longtime 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 "flightcrash" events related to shock collisions.

(2015) Physical Review E. 91, 4, 41201. Abstract
We consider developed turbulence in the twodimensional GrossPitaevskii model, which describes wide classes of phenomena from atomic and optical physics to condensed matter, fluids, and plasma. The wellknown difficulty of the problem is that the hypothetical local spectra of both inverse and direct cascades in the weakturbulence 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 fourthorder 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 (secondorder moments) are close to slightly (logarithmically) distorted thermal equilibrium in both cascades.

(2015) Nature Communications. 6, 6214. Abstract
Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak nonlinear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with nonlinear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by nonuniform doublescale 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 examplerandom fibre laserwe show that a model describing such a system is close to integrable nonlinear Schrodinger equation and needs a new formalism of wave kinetics, developed here. We derive a nonlinear 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.

(2015) Abstract
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.

(2015) Journal of Plasma Physics. 81, 2, 395810201. Abstract
A side remark in the old paper by Zeldovich and coauthors leads to the recent discovery of a universal conservation law of turbulent dispersion.
2014

(2014) Physical Review Letters. 113, 25, Abstract
An inverse turbulent cascade in a restricted twodimensional periodic domain creates a condensatea pair of coherent systemsize 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 smallscale dissipation, smallscale 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.

(2014) Physical Review X. 4, 4, Abstract
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 twodimensional (2D) and threedimensional (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 NavierStokes equation in 3D.

(2014) Nature Physics. 10, 9, p. 658663 Abstract
The ability to send a wave to fetch an object from a distance would find a broad range of applications. Quasistanding Faraday waves on water create horizontal vortices(1,2), yet it is not known whether propagating waves can generate largescale flowssmallamplitude irrotational waves only push particles in the direction of propagation(35). Here we show that when waves become threedimensional as a result of the modulation instability, a floater can be forced to move towards the wave source. The mechanism for this is the generation of surface vortices by waves propagating away from vertically oscillating plungers. We introduce a new conceptual framework for understanding wavedriven 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 wavedriven jets, and the formation of Lagrangian coherent structures.

(2014) Journal of Statistical Physics. 156, 1, p. 1054 Abstract
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 smallscale 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 longtime longdistance scaling limit, that lives in a degenerate conical geometry, and find that the energycontaining 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 noncascade 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 timesaturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the timesaturated spectrum behaves as k(5/3).

(2014) Physical Review Letters. 113, 2, 24501. Abstract
The turbulent energy flux through scales, (epsilon) over bar, 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, <du(2) / dt > = 4 (epsilon) over bar at t = 0, where (u) over right arrow 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 t > 0 and then v > 0, while a zerofriction anomaly requires taking viscosity to zero first. We find that the limits t > 0 and v > 0 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 onedimensional Burgers turbulence and find it completely altered: <du(2) / 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.

(2014) Physical Review Letters. 112, 23, 234502. Abstract
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.

(2014) Proceedings of the National Academy of Sciences of the United States of America. 111, 21, p. 75587563 Abstract
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 "flightcrash" events, associated with fast particle deceleration, provides away 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.
2013

(2013) Nature Photonics. 7, 10, p. 783786 Abstract
Studying the transition from a linearly stable coherent laminar state to a highly disordered state of turbulence is conceptually and technically challenging, and of great interest because all pipe and channel flows are of that type(1,2). In optics, understanding how a system loses coherence, as spatial size or the strength of excitation increases, is a fundamental problem of practical importance(35). Here, we report our studies of a fibre laser that operates in both laminar and turbulent regimes. We show that the laminar phase is analogous to a onedimensional coherent condensate and the onset of turbulence is due to the loss of spatial coherence. Our investigations suggest that the laminarturbulent transition in the laser is due to condensate destruction by clustering dark and grey solitons. This finding could prove valuable for the design of coherent optical devices as well as systems operating far from thermodynamic equilibrium.

(2013) Physical Review E. 87, 6, Abstract
We consider developed turbulence in the GrossPitaevsky 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 predatorprey type, as was suggested earlier; rather, they are due to phase coherence and anomalous correlations imposed by the condensate.


(2013) Physical Review Letters. 110, 21, Abstract
We develop an analytic formalism and derive new exact relations that express the shorttime dispersion of fluid particles via the singletime velocity correlation functions in homogeneous isotropic and incompressible turbulence. The formalism establishes a bridge between singletime Eulerian and longtime Lagrangian pictures of turbulent flows. In particular, we derive an exact formula for a shortterm counterpart of the longtime Richardson law, and we identify a conservation law of turbulent dispersion which is true even in nonstationary turbulence.

Optical Wave Turbulence(2013) Advances In Wave Turbulence. p. 113163 (World Scientific Series on Nonlinear Science Series A). Abstract
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 wavebearing 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 highspeed, 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.
2012

(2012) Journal of Fluid Mechanics. 713, p. 482490 Abstract
Using data from a largescale threedimensional simulation of supersonic isothermal turbulence, we have tested the validity of an exact flux relation derived analytically from the NavierStokes 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 largescale force. However, compressible turbulence in simulations is usually initialized and maintained by a largescale acceleration, as in gravitydriven astrophysical flows. We present a new approximate flux relation for isothermal turbulence driven by a largescale acceleration, and find it in reasonable agreement with the simulation results.

(2012) Proceedings Of The Royal Society AMathematical Physical And Engineering Sciences. 468, 2145, p. 24962508 Abstract
We study numerically optical turbulence using the particular example of a recently created, ultralong 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.

(2012) Physics of Fluids. 24, 5, Abstract
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) equivalent to proportional to epsilon t, relies on dimensional arguments only: no relation of D2(t) to the energy cascade is known, neither in two nor in threedimensional turbulence. State of the art experimental and numerical results demonstrate that at high Reynolds numbers, the derivative dD(2)(t)/dt has a finite nonzero slope starting from t approximate to 2 tau(eta). The analysis of the acceleration spectrum Phi(A)(omega) indicates a possible small correction with respect to the dimensional expectation Phi(A)(omega) similar to omega(0) but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4711397]

(2012) Journal of Statistical Physics. 147, 2, p. 424435 Abstract
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 semianalytic 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 Lowner 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).

Stochastic geometry of turbulence(2012) APS Meeting Abstracts. Abstract
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 multipoint 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 socalled 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.

(2012) Physical Review E. 85, 1, Abstract
We consider turbulence within the GrossPitaevsky 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 shortrange positional and longrange 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 inversecascade turbulence with respect to the nature of smallscale forcing.
2011

(2011) A Voyage Through Turbulence. Sreenivasan K. R., Moffatt K., Davidson P. A. & Kaneda Y.(eds.). p. 209237 Abstract
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.

(2011) Cambridge: . Vol. 9781107005754. Abstract
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 Kelvin–Helmholtz instabilities in clouds, the book provides readers with a better understanding of the world around them. It teaches the art of fluidmechanical 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.

(2011) Physical Review E. 83, 4, 045301. Abstract
For the direct cascade of steady twodimensional (2D) NavierStokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When pi is the vorticity coarsegrained over a scale R, the probability density function (PDF), P(pi), has a universal asymptotic behavior lnP similar to pi/pi(rms) at pi >> pi(rms) = [H ln(L/R)](1/3), where H is the enstrophy flux and L is the pumping length. Therefore, the PDF has exponential tails and is selfsimilar, that is, it can be presented as a function of a single argument, pi/pi(rms), in distinction from other known direct cascades.

(2011) Nature Physics. 7, 4, p. 321324 Abstract
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 threedimensional (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 largescale 2D flow or from a smallscale 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 largescale vortex and smallscale 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 boxsize vortex provides for a selforganized planar flow that secures an upscale energy transfer.
2010

Conformal Invariance in Inverse Turbulent Cascades(2010) arXiv. 1012.3868. Abstract
We study statistical properties of turbulent inverse cascades in a class of nonlinear models describing a scalar field transported by a twodimensional 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 NavierStokes, surface quasigeostrophic and CharneyHasegawaMima equations. We find that some statistical properties of the inverse turbulent cascades in such systems are conformal invariant. In particular, the zeroisolines 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 largescale field to a smallscale one.

(2010) Journal of Fluid Mechanics. 644, p. 465472 Abstract
We consider the steadystate statistics of turbulence in the inertial interval. The Kolmogorov flux relation (4/5law) is shown to be a particular case of the general relation on the currentdensity correlation function. Using that, we derive an analogous flux relation for compressible turbulence and a new exact relation for incompressible turbulence.
2009

(2009) Physics of Fluids. 21, 12, Abstract
We present experimental results on the properties of bounded turbulence in thin fluid layers. In contrast with the theory of twodimensional (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 quasi2D 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 nonGaussianity and reduces the efficiency of the inverse energy cascade. Remarkably, the energy flux in the cascade derived from the thirdorder 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.

(2009) Journal of Fluid Mechanics. 638, p. 14 Abstract
A finiteamplitude 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 smallamplitude surface and internal waves. Now, this is extended by Buhler & HolmesCerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 526) to waves in a rotating shallowwater system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finiteamplitude 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.

(2009) Physical Review A. 80, 3, Abstract
We study optical wave turbulence using as a particular example recently created ultralongfiber 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 largeamplitude pulses.


(2009) JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL. 42, 12, Abstract
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 symmetrybreaking 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 farfromequilibrium 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 SchrammLoewner evolution (a class of random curves that can be mapped to a 1D Brownian walk).
2008

(2008) Physical Review Letters. 101, 19, Abstract
We present experimental results on turbulence generated in thin fluid layers in the presence of a largescale coherent flow, or a spectral condensate. It is shown that the condensate modifies the thirdorder 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 (10500 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.


(2008) Journal Of Statistical MechanicsTheory And Experiment. Abstract
We discuss fluctuation relations in simple cases of nonequilibrium Langevin dynamics. In particular, we show that, close to nonequilibrium steady states with nonvanishing probability currents, some of these relations reduce to a modified version of the fluctuationdissipation theorem. The latter may be interpreted as the equilibriumlike relation in the reference frame moving with the mean local velocity determined by the probability current.

(2008) New Journal of Physics. 10, Abstract
We present a meanfield 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 largescale 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 raininitiation 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 overseeding even a part of a cloud by small hygroscopic nuclei, one can substantially delay the onset and increase the amount of precipitation.

(2008) New Journal of Physics. 10, Abstract
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 submicron to kilometres. Some of these interactions are subtle and others are more straightforward. Large and smallscale 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.

(2008) Abstract
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.

What drives mesoscale atmospheric turbulence?(2008) arXiv. 0805.0390. Abstract
Measurements of atmospheric winds in the mesoscale range (10500 km) reveal remarkably universal spectra with the $k^{5/3}$ power law. Despite initial expectations of the inverse energy cascade, as in twodimensional (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 largescale coherent flow, or a spectral condensate. We present new experimental laboratory data and show that the presence of a largescale shear flow modifies the thirdorder velocity moment in spectrally condensed 2D turbulence, making it, in some conditions, similar to that observed in the atmosphere.


(2008) Cambridge: . Vol. 355. (London Mathematical Society lecture note series). Abstract
Methods of nonequilibrium 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 NonEquilibrium 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 nonequilibrium physics to their work.


(2008) IUTAM SYMPOSIUM ON HAMILTONIAN DYNAMICS, VORTEX STRUCTURES, TURBULENCE. Borisov AV., Mamaev IS., Sokolovskiy MA. & Kozlov VV.(eds.). Vol. 6. p. 257267 (IUTAM Bookseries). Abstract
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.
2007

(2007) Journal Of The Atmospheric Sciences. 64, 12, p. 44974505 Abstract
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 samesize water droplets is described and evaluated. It is shown that already for turbulence with the dissipation rate 10(3) cm(2) s(3), the sling effect gives a contribution to the collision rate of 15mu m 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.

(2007) Physical Review Letters. 99, 16, Abstract
The first direct experimental evidence of the suppression of quasitwodimensional 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 forcingscale vortices reduces the energy input, leading to a reduction in the turbulence level.

(2007) Abstract
A review of recent experimental studies of turbulence suppression by mean flows in quasitwodimensional fluids is presented. Largescale 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 KolmogorovKraichnan k −5/3 power law spectrum in the energy range.

(2007) Physical Review E. 76, 2, Abstract
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 aggregationdisorder 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 longcorrelated flows are discussed, as well as the fluidtracer limit.

(2007) Physical Review E. 76, 2, Abstract
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 finitecorrelated noise. In one dimension, we are able to find analytically the longtime 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 Cramer 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 twodimensional 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 alreadyknown deltacorrelated case.

(2007) Physical Review E. 75, 6, Abstract
The growth rate of smallscale 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.

(2007) European Physical JournalSpecial Topics. 145, p. 211216 Abstract
The statistics of the nodal lines of scalar fields in twodimensional (2d) turbulence is found to be conformal invariant and equivalent to that of cluster boundaries in critical phenomena. That allows for a rich variety of exact analytic results,. rst time in turbulence studies. In particular, the statistics of zerovorticity lines in NavierStokes turbulence is found to be equivalent to that of critical percolation. The statistics of the zerotemperature lines in surface quasigeostrophic (SQG) turbulence is found to be equivalent to that of the isolines of a Gaussian (free) field.

(2007) European Physical JournalSpecial Topics. 145, p. 125136 Abstract
We argue theoretically and demonstrate experimentally that in a standing wave floating particles drift towards the nodes or antinodes depending on their hydrophilic or hydrophobic properties. We explain this effect as the breakdown of Archimedes' law by a surface 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 smallamplitude wave and show that the drift appears as a second order effect in wave amplitude. We con. rm 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 a multifractal set with caustics.

(2007) Russian Mathematical Surveys. 62, 3, p. 497510 Abstract
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 twodimensional turbulence belong to the class of conformally invariant curves called SLE (ScrammLowner 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.

(2007) Physical Review Letters. 98, 2, Abstract
We offer a new example of conformal invariance (local scale invariance) far from equilibriumthe inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a onedimensional Brownian walk (called SchrammLoewner evolution or SLE kappa). The diffusivity is close to kappa = 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 twodimensional NavierStokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.

(2007) Journal of Turbulence. 8, 16, p. 118 Abstract
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.
2006

(2006) Physical Review Letters. 97, 24, Abstract
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.



(2006) Journal of Low Temperature Physics. 145, 4Jan, p. 297310 Abstract
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.

(2006) Journal of Applied Meteorology and Climatology. 45, 4, p. 591599 Abstract
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.

(2006) Physics Today. 59, 4, p. 4349 Abstract
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.

(2006) Nature Physics. 2, 2, p. 124128 Abstract
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 nonuniform 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 nonequilibrium 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.

Explosive growth of inhomogeneities in the distribution of droplets in a turbulent air(2006) arXiv. arXiv:nlin. Abstract
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.

(2006) Mathematical and Physical Theory of Turbulence. Shivamoggi B. & Cannon J.(eds.). New York: . Vol. 250. p. 746 (Lecture notes in pure and applied mathematics). Abstract
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 smallscale inhomogeneities of the transported fields, which are then killed by molecular diffusion, and cases when largescale structures of the fields appear. Our goal is to describe temporal and spatial statistical properties of transported fields.

Introduction to developed turbulence(2006) Lecture Notes On Turbulence And Coherent Structures In Fluids, Plasmas And Nonlinear Media. 4, p. 120 Abstract
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.
2005

Clustering of floaters by waves(2005) arXiv. nlin/05110. Abstract
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 multifractal set.

(2005) Physical Review E. 71, 6, Abstract
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 shortcorrelated 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.


(2005) Physical Review Letters. 94, 21, Abstract
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 shortcorrelated velocity we identify the new type of zero modes, which break scale invariance and determine an anomalously slow decay of correlations at large scales.

(2005) Europhysics Letters. 70, 6, p. 775781 Abstract
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.
2004

Intermittent Distribution of Heavy Inertial Particles in Turbulent Flows(2004) Abstract
The phenomenon of preferential concentration of inertial particles is studied by following lagrangian trajectories. Elementary properties of the coarsegrained distribution of heavy particles in simple turbulent flows are investigated by direct numerical simulations. In the small Stokes number case, we compute the coarsegrained 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.

(2004) Physica DNonlinear Phenomena. 195, 2Jan, p. 128 Abstract
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 saddlepoint approximation in the path integral. We first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear Schrodinger equation. We then consider model modifications due to inline (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 errorcausing 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. (C) 2004 Elsevier B.V. All rights reserved.

(2004) Physics of Fluids. 16, 7, p. L47L50 Abstract
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 coarsegrained over the scale r behaves as an approximate power law: similar tor(alpha). We study the dependencies of the exponent alpha on the Reynolds number, of the Stokes number, and on the settling velocity. We find numerically that the theoretical lower bound previously suggested [Falkovich , 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. (C) 2004 American Institute of Physics.

(2004) Physical Review Letters. 92, 24, Abstract
We consider the flow being a superposition of random waves and describe the evolution of the spectrum of the passive scalar in the leading (fourth) order with respect to the wave amplitudes. We find that wave turbulence can produce an exponential growth of the passive scalar fluctuations when either both solenoidal and potential components are present in the flow or there are potential waves with the same frequencies but different wave numbers.

(2004) New Journal of Physics. 6, Abstract
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 twopoint correlation function for any time and any distance from equilibrium. The longtime limit gives the sum of the Lyapunov exponents expressed via the formula of GreenKubo 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.
2003

Statistics of turbulenceinduced fluctuations of particle concentration(2003) Sedimentation And Sediment Transport, Proceedings. p. 155158 Abstract
Keywords: HOMOGENEOUS ISOTROPIC TURBULENCE; SETTLING VELOCITY; HEAVYPARTICLES; CLOUDS; FLOW; INTERMITTENCY; ACCELERATION; INERTIA
2002

(2002) Nature. 419, 6903, p. 151154 Abstract
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 similarsized droplets exceeding radii of a few micrometres, and therefore rain prediction requires a quantitative description of droplet collision in turbulence(15). Turbulent vortices act as small centrifuges that spin heavy droplets out, creating concentration inhomogeneities (614) 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.

(2002) Physical Review E. 65, 5, Abstract
The logarithmic renormalization predicted by Kraichnan (1971) for the direct cascade of enstrophy in the inertial range of twodimensional turbulence has been observed in a numerical simulation. A moderate resolution allows for a very long time integration that provides very good statistics. Deviations from Gaussianity in the vorticity probability distribution are observed.

(2002) Optics Letters. 27, 1, p. 1315 Abstract
We consider two solitons propagating under a filtercontrol scheme and describe the timing jitter that is caused by spontaneousemission noise and enhanced by attraction between solitons. We find the biterror rate as a function of system parameters (filtering and noise level), timing, initial distance, and the phase difference between solitons. (C) 2002 Optical Society of America.
2001

(2001) Physical Review E. 64, 6, Abstract
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 nonGaussian tail of the PDF to be exponential. The propagation distance that corresponds to a given biterror 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.

(2001) Reviews of Modern Physics. 73, 4, p. 913975 Abstract
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 scaleinvariance 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.

(2001) Abstract
We present analytical method to acculate nonGaussian intensity and timing jitters statistics in optical communication systems. Using developed method we have calculated probability density function for optical regenerated soliton systems with inline synchronous intensity modulation and filtering.

(2001) Physical Review Letters. 86, 13, p. 27902793 Abstract
We consider inertial particles suspended in an incompressible turbulent flow. Because of particles' inertia their flow is compressible, which leads to fluctuations of concentration significant for heavy particles. We show that the statistics of these fluctuations is independent of details of the velocity statistics, which allows us to predict that the particles cluster on the viscous scale of turbulence and describe the probability distribution of concentration fluctuations. We discuss the possible role of the clustering in the physics of atmospheric aerosols, in particular, in cloud formation.

(2001) Physical Review E. 63, 2, Abstract
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 nonGaussian) which are beyond perturbation theory. Our method is a development of the instanton formalism (method of optimal fluctuation) based on a saddlepoint approximation in the path integral. We first solve a fundamental problem of soliton statistics governed by a noisy nonlinear Schrodinger equation. We then apply our method to optical soliton transmission systems using signal control elements (filters and amplitude and phase modulators).

(2001) New trends in turbulence. Turbulence: nouveaux aspects. Yaglom A. M., David F. & Lesieur M..(eds.). Berlin: . p. 505554 (Les Houches  Ecole d'Ete de Physique Theorique). Abstract
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 manyparticle configurations. Section 2 describes the statistics of the passive scalar that can be inferred from the particle analysis. In Section 3, we discuss the Navier—Stokes 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

Growth of magnetic fluctuations in a turbulent flow(2001) Intermittency In Turbulent Flows. p. 105117 Abstract
Keywords: PASSIVE SCALAR; VELOCITYFIELD; DYNAMO

Feedback of a smallscale magnetic dynamo(2001) Physical Review E. 63, 1, Abstract
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 smallscale magnetic fluctuations in a largescale 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 smallscale magnetic dynamo.
2000

(2000) Journal of Turbulence. 1, 12, N12. Abstract
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]
1999

Clustering of inertial particles in turbulent flows(1999) arXiv. chaodyn/9. Abstract
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.

(1999) Physical Review Letters. 83, 20, p. 40654068 Abstract
Kinematic dynamo theory is presented here for turbulent conductive fluids. We describe how inhomogeneous magnetic fluctuations are generated below the viscous scale of turbulence where the spatial smoothness of the velocity permits a systematic analysis of the Lagrangian path dynamics. We find analytically the moments and multipoint correlation functions of the magnetic field at small yet finite magnetic diffusivity. We show that the fil:ld is concentrated in long narrow strips and describe anomalous scalings and angular singularities of the multipoint correlation functions which are manifestations of the field's intermittency. The growth rate of the magnetic field in a typical realization is found to be half the difference of two Lyapunov exponents of the same sign.

(1999) Physics of Fluids. 11, 8, p. 22692279 Abstract
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. (C) 1999 American Institute of Physics. [S10706631(99)030081].
1998

(1998) Physical Review Letters. 80, 10, p. 21212124 Abstract
The probability density function (PDF) of passive scalar dissipation P(epsilon) is found analytically in the limit of large Peclet and Prandtl numbers (BatchelorKraichnan regime) in two dimensions. The tail of PDF at epsilon >> [epsilon] is shown to be stretched exponent ln P(epsilon) proportional to epsilon(1/3); at epsilon

(1998) Physical Review E. 57, 2, p. R1231R1234 Abstract
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 onedimensional compressible random how. [S1063651X(98)506022].

(1998) Physica A. 249, 4Jan, p. 3646 Abstract
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)] proportional to exp at different values of the ratio between the correlation and turnover rimes. We find the stretching correlation time which determines the dependence of [R1R2] on the difference t(1)t(2). The calculation of the nest cumulant of R2 shows that statistics of R2 is nearly Gaussian at small times (as long as d much greater than 1) and becomes lognormal at large times when larged approach fails for highorder 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 much less than tau. We establish the dispersion of the In(R2) in the lognormal statistics. (C) 1998 Elsevier Science B.V. All rights reserved.

Intermittent dissipation in developed turbulence(1998) Advances In Turbulence Vii. 46, p. 207210 Abstract
Keywords: Engineering, Mechanical; Mechanics
1997

(1997) Physical Review Letters. 79, 21, p. 41594161 Abstract
We show that the tails of the singlepoint velocity probability distribution function (PDF) are generally nonGaussian in developed turbulence. By using instanton formalism for the randomly forced NavierStokes 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 tau produces velocity PDF tails In P(v) proportional to  v(4) at v much greater than v(rms), L/tau. For a shortcorrelated forcing when tau much less than L/v(rms) there is an intermediate asymptotics In P(v) proportional to  v(3) at L/tau much greater than v much greater than v(rms).

(1997) International Journal of Modern Physics B. 11, 2627, p. 32233245 Abstract
We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a largescale force. The saddlepoint approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special fieldforce configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient partial derivative(x)u and find out that they correspond to the PDF with ln[P(partial derivative(x)u)] proportional to (partial derivative(x)u/Re)(3/2) where Re is the Reynolds number. That stretched exponential form is valid for negative partial derivative(x)u with the modulus much larger than its rootmeansquare (rms) value. The respective tail of PDF for negative velocity differences w is steeper than Gaussian, ln P(w) similar to (w/u(rms))(3), as well as singlepoint velocity PDF lnP(u) similar to (\u\/u(rms))(3). For high velocity derivatives u(k) = partial derivative(x)(k)u, the general formula is found: lnP(\u((k))\) proportional to  (\u((k))\/Rek)(3/(k+1)).

Threepoint correlation function of a scalar mixed by an almost smooth random velocity field(1997) Physical Review E. 55, 5, p. R4881R4884 Abstract
We demonstrate that if the exponent gamma 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 root gamma. Therefore, zero modes are subleading with respect to the forced solution that has normal scaling with the exponent gamma.

(1997) Physical Review Letters. 78, 8, p. 14521455 Abstract
We consider the tails of probability density function (PDF) for the velocity that satisfies Burgers equation driven by a Gaussian largescale force. The saddlepoint approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special fieldforce configuration (instanton) that realizes the extremum of probability. For the PDFs of velocity and its derivatives u((k)) = partial derivative(x)(k)u, the general formula is found: In P(\u((k))\) proportional to (\u((k))\/Rek)(3/(k+1)).
1996

(1996) Physical Review E. 54, 5, p. 50955099 Abstract
The nonlinear Schrodinger equation with repulsion (also called the GrossPitaevsky equation) is solved numerically with damping at small scales and pumping at intermediate scales and without any largescale 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 selfsaturated condensation with the rate of condensate growth being inversely proportional to the condensate amplitude, the number of waves growing as root t, the total energy linearly increasing with time and the level of overcondensate fluctuations going down as 1/root t, and (iii) final stage with a constant level of overcondensate 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 overcondensate fluctuations are suppressed. At the final stage, the second structure function [\psi(1)psi(2)\(2)]proportional to lnr(12) 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.

(1996) Physical Review E. 54, 5, p. 48964907 Abstract
We describe the method for finding the nonGaussian 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 NavierStokes equation, etc. The existence of such ails 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 fieldforce configurationthe instanton. As an example, we examine the correlation functions of the passive scalar u advected by a largescale velocity field delta 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 NavierStokes equation.

(1996) Physical Review E. 54, 4, p. 44314434 Abstract
Twodimensional turbulence of the waves with linear dispersion law is analyzed numerically at small Mach numbers and large Reynolds numbers, It is shown that the energyflux relation is close to E proportional to p(2/3) as for a onedimensional system. The analysis of the wave distribution in k space shows that the anisotropic largescale 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(k(parallel to))proportional to k(parallel to)(2) due to shock waves, while the spectrum per unit interval of wave numbers is E(k)proportional to P(2/3)k(1) contrary to all previous predictions. Probability density functions of the velocity and velocity differences are found and compared with recent theoretical predictions.

(1996) International Journal of Modern Physics B. 10, 1819, p. 22732309 Abstract
The steady statistics of a passive scalar advected by a random twodimensional 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 <ln (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 nonGaussian tails of the probability distributions at finite Pe are found to be exponential.

(1996) Physical Review Letters. 76, 20, p. 37073710 Abstract
We consider a passive scalar convected by a multiscale random velocity field with short yet finite temporal correlations. Taking Kraichnan's limit of a white Gaussian velocity as a zero approximation we develop the perturbation theory with respect to a small correlation time and small nonGaussianity of the velocity. We derive the renormalization (due to temporal correlations and nonGaussianity) 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 nonGaussianity. The scalar exponents are thus nonuniversal as was predicted by Shraiman and Siggia on a phenomenological ground.

(1996) Physical Review Letters. 76, 15, p. 27062709 Abstract
For Kraichnan's 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 zeta(2n) = (n) zeta(2)  2(2  zeta(2))n(n  1)/d, while those of the dissipation field are mu(n) = 2(2  zeta(2))n(n  1)/d for orders n


Anomalous scaling exponents of a passive scalar advected by turbulence(1996) Advances In Turbulences Vi. 36, p. 577580 Abstract
Keywords: Engineering, Mechanical; Mechanics
1995

(1995) Physical Review E. 52, 5, p. 49244941 Abstract
For a shortcorrelated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the fourpoint 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 fourpoint 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 lid perturbation theory. Anomalous dimensions are found analytically both for the scalar held and for its derivatives, in particular, for the dissipation held.

(1995) Chaos Solitons & Fractals. 5, 10, p. 18551869 Abstract
Threedimensional 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 threedimensional 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.

(1995) Physical Review E. 52, 4, p. 45374540 Abstract
Wave turbulence for systems with only direct (smallscale) 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 RayleighJeans 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 capillarywave turbulence are considered. We also carried out numerics which confirm the theoretical predictions.

(1995) PHYSICA D. 87, 4Jan, p. 285289 Abstract
Amplitude equations are derived that describe the interaction between highfrequency inertiogravity waves and lowfrequency Rossby waves on rotating shallow water A cyclone is shown to cause a local maximum to appear in the density of inertiogravity waves. A packet of inertiogravity waves is shown to produce a cycloneanticyclone pair. The interaction between inertiogravity 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 highfrequency waves does not, by itself, explain the blocking.

(1995) International Journal of NonLinear Mechanics. 30, 4, p. 609616 Abstract
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 largescale wave distribution explosively evolves into a smallscale Kolmogorov spectrum. The second stage starts at the moment the Kolmogorov spectrum reaches dissipative scales. For systems with nonlinear 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 selfsimilar law.


(1995) Europhysics Letters. 29, 1, p. 16 Abstract
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 largescale wave distribution explosively evolves into a smallscale 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 selfsimilar law.
1994

(1994) PHYSICA D. 78, 2Jan, p. 1129 Abstract
Weakly anisotropic steady spectra are found for both inverse and direct cascades in twodimensional turbulence of an incompressible fluid. The degree of anisotropy is shown to increase for both spectra: as (kL)2/3 upscales and as (k LAMBDA)2 downscales from the pump. A weakly anisotropic intermediatescale pumping may thus produce a substantially anisotropic turbulence in the inertial intervals of scales.


(1994) Physics of Fluids. 6, 4, p. 14111414 Abstract
It is shown how viscosity increases turbulence level in the inertial interval by suppressing turbulent transfer.

NONLOCAL VORTICITY CASCADE IN 2 DIMENSIONS(1994) Physical Review E. 49, 3, p. R1800R1804 Abstract
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)] isproportionalto ln2n/3(L/\r1  r2\), with L being the scale of the external pump.

(1994) Physical Review E. 49, 3, p. 24682471 Abstract
It is shown that if the pair correlation function of any tracer in incompressible turbulent flow is scale invariant with the exponent zeta2, then the exponent of twopoint function of 2nth order does not equal nzeta2. In this case, the probability distribution should depend, generally speaking, on an infinite number of parameters (fluxes of the integrals). Three examples are considered: twodimensional vorticity cascade, action cascade in Clebsch variables, and entropy cascade in inhomogeneously heated fluid.

Nonlinear interaction between long inertiogravity and rossby waves(1994) Nonlinear Processes in Geophysics. 1, 3Feb, p. 168171 Abstract
The equations describing the interaction of long inertiogravity (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.
1993

(1993) Physical Review Letters. 71, 21, p. 34543457 Abstract
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.

(1993) Physical Review B. 48, 13, p. 98559857 Abstract
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.

(1993) Нелинейные волны  физика и астрофизика. GaponovGrekhov A. V. & Rabinovich M. I.(eds.). Moscow: . p. 280285 (Нелинейные волны). Abstract
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 параметра—потока энергии в кпространстве [1—3]. В соответствии с этой гипотезой были получены изотропные однонараметрические спектры (называемые обычно колмогоровскими) как для вихревой гидродинамической [1—4], 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.

REVISED UNIVERSALITY CONCEPT IN THE THEORY OF TURBULENCE(1993) Nonlinear Waves And Weak Turbulence With Applications In Oceanography And Condensed Matter Physics. 11, p. 1944 Abstract
Keywords: Mathematics, Applied; Mechanics; Physics, Fluids & Plasmas

(1993) Nonlinear Processes in Physics. Fokas A. S., Newell A. C., Kaup D. & Zakharov V. E.(eds.). Berlin: . p. 271274 (Springer Series in Nonlinear Dynamics). Abstract
The picture of cascade turbulence suggested by Richardson, Kolmogorov and Obukhov is based on the concept of interaction locality [1–5]. 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 Kolmogorovlike spectrum only or on the slightly differing distributions as well? Proceeding from continuitylike 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].

SINGULARITIES OF THE VELOCITYFIELD AND INTERACTION LOCALITY OF TURBULENCE(1993) Singularities In Fluids, Plasmas, And Optics. 404, p. 7591 Abstract
Keywords: Mathematics, Applied; Mechanics; Physics, Fluids & Plasmas
1992

(1992) Physical Review Letters. 69, 22, p. 31733176 Abstract
It is shown that an inverse cascade of the turbulence of inertiogravity waves produces a longscale 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 selfinteraction thus could not stop an inverse cascade of mesoscaie 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.

(1992) Physical Review A. 46, 8, p. 47624772 Abstract
The problem of interaction locality in k space is studied in a diagrammatic perturbation approach for the NavierStokes equation in quasiLagrangian variables. Analyzing the whole diagram series we have found an exact relation between the asymptotic behavior of the triplecorrelation function of velocities that governs the energy transfer over scales and the doublecorrelation function giving the energy distribution. Namely, at r > k) are shown to decrease by the same law with the distance in k space, such as (k1/k)2zeta2 and (k/k2)2zeta2. 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 [deltav (r) isproportionalto r(h)] which provides the locality of interaction in the k space. It is shown that the condition of infrared locality of interaction (with larger k1eddies) could give only the upper restriction for the exponent. The upper limit thus found (h(max) = 1) coincides with the boundary exponent of singularity of energy dissipation. As far as an interaction locality in the ultraviolet limit (k2 >> k) is concerned, we prove that any reasonable dimension function D(h) provides locality whatever small h is considered.

(1992) Physics Letters A. 168, 2, p. 127132 Abstract
A new type of instability of Kolmogorovlike wave turbulence spectra is found. Such an instability is due to an interaction nonlocal in kspace and it strongly modifies the angular structure of the turbulence spectrum. However, the spectrum dependence on the modulus k is still a Kolmogorovlike one corresponding to energy transfer local in kspace. 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 kspace.

(1992) Berlin: . (Springer Series in Nonlinear Dynamics). Abstract
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 twovolume 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.

(1992) PHYSICA D. 57, 2Jan, p. 8595 Abstract
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 longscale stratification (stable or unstable one). Under a purely entropic excitation (for example, by horizontal temperature gradient) the spectrum with constant entropy flux, F(vv) approximately k21/5, fills the whole of the inertial interval and crossover to the KolmogorovObukhov spectrum with constant energy flux, F(vv) approximately k11/3, 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 kspace. Twoflux universal spectra of the velocity and temperature fluctuations are obtained.

(1992) Europhysics Letters. 19, 4, p. 279284 Abstract
A steady Kolmogorovlike spectrum of turbulence is found as an exact solution of the kinetic equation for inertialgravity waves. The spectrum obtained satisfactorily fits the results of atmospheric observations for mesoscale motions (from hundred to thousand kilometers).

(1992) Physics Of Fluids BPlasma Physics. 4, 3, p. 594598 Abstract
Weak developed turbulence in the framework of both scalar and vector nonlinear Schrodinger equations is considered. It corresponds to waves with a quadratic dispersion law omega(k) = omega(o) + betak2 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 Kolmogorovlike cascade spectra with the fluxes of energy and wave action.
1991

(1991) Journal of Nonlinear Science. 1, 4, p. 457480 Abstract
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 kspace comes into a universal, Kolmogorovtype spectrum in a finite time. Before and after the formation of the Kolmogorov spectrum, two different selfsimilar regimes of evolution occur: the first one is responsible for explosively forming the universal spectrum and the second one determines energy dissipation.
1990

(1990) Soviet Physics JETPUSSR. 71, 6, p. 10851090 Abstract
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, gravitationalcapillary waves in shallow water and threedimensional sound with positive dispersion. [(Russian original  ZhETF, Vol. 98, No. 6, p. 1931, December 1990, Vliianie dissipatsii na strukturu statsionarnogo spektra volnovoi turbulentnosti)]

ON THE IMPOSSIBILITY OF WEAKLY DAMPED 2ND SOUND IN TURBULENT MEDIA(1990) Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki. 97, 6, p. 18471851 Abstract

(1990) NONLINEAR EVOLUTION OF SPATIOTEMPORAL STRUCTURES IN DISSIPATIVE CONTINUOUS SYSTEMS. BUSSE FH. & KRAMER L.(eds.). p. 541544 (NATO ADVANCED SCIENCE INSTITUTES SERIES, SERIES B, PHYSICS). Abstract

(1990) Nonlinear Waves 3. GaponovGrekhov A. V., Engelbrecht J. & Rabinovich M. I.(eds.). Berlin: . Vol. 3. p. 256261 (Research Reports in Physics). Abstract
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 twoflux spectra (bearing the energy and momentum fluxes) are obtained analytically. The universality hypothesis for the developed wave turbulence is formulated.
1989

UNIVERSAL DOUBLEFLOW SPECTRA OF WEAK SOUND TURBULENCE(1989) Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki. 96, 6, p. 20332037 Abstract

(1989) Journal of Applied Mechanics and Technical Physics. 30, 1, p. 95100 Abstract
Underlying the presentday approach to the problem of the onset of turbulence is the hypothesis that the phenomena determining the development of instability are finitedimensional. 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 infinitedimensional 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 onetoone 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 laminarturbulent transition in circular Couette flow are analyzed in regard to dimension. Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 99–104, January–February, 1989.
1988

The effect of weak anisotropy of the source on the Kolmogorov spectrum of acoustic turbulence(1988) Soviet Physics  Doklady. 33, 7, p. 488490 Abstract

(1988) Soviet Physics JETPUSSR. 68, 1, p. 13931397 Abstract
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 kspace is found. [Russian original  ZhETF, Vol. 94, No. 1(7), p. 172, July 1988]
1987

(1987) PHYSICA D. 27, 3, p. 399411 Abstract
The stability problem of Kolmogorov spectra of a weak turbulence is analytically solved for the first time in the framework of a threewave kinetic equation. The spectrum of isotropic perturbations of a stationary notinequilibrium 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 kspace. 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.

(1987) Soviet Physics JETPUSSR. 66, 1, p. 97100 Abstract
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 steadystate 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]
1986

(1986) Soviet Physics JETPUSSR. 63, 6, p. 12701272 Abstract
It is shown that the singular character of the distribution of parametrically excited magnons in kspace (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]

(1986) Soviet Physics JETPUSSR. 63, 5, p. 10451053 Abstract
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]
1985

(1985) Soviet Physics JETPUSSR. 62, 1, p. 146152 Abstract
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 selffocusing is proved. Classes of initial data which give rise to a selffocusing sound wave are identified. (Russian original  ZhETF, Vol. 89, No. 1, p. 258, July 1985 )
1984

(1984) Soviet Physics JETPUSSR. 60, 1, p. 118122 Abstract
The paper reports an experimental investigation of the thresholds for spinwave 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 spinwave 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 )

(1984) Physica D: Nonlinear Phenomena. 10, 3, p. 379386 Abstract
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 onedimensional waves are unstable, and for negative dispersion such waves are stable.

Destruction of Stationary Solutions and Collapse in the Nonlinear String Equation(1984) Nonlinear and Turbulent Processes in Physics. Sagdeev R. Z.(eds.). Chur: . Vol. 2. p. 10691072 Abstract
1983

(1983) Physics Letters A. 99, 67, p. 271274 Abstract
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.
1982

INTERACTION BETWEEN PARAMETRICALLY EXCITED SPINWAVES AND THERMAL SPINWAVES(1982) Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki. 82, 5, p. 15621577 Abstract
1981

(1981) Physics Letters A. 86, 4, p. 203204 Abstract
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 selfsimilar solution in the Burgers model.

(1981) Soviet Physics JETPUSSR. 53, 2, p. 299300 Abstract
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, 592596 (February 1981)]

О НЕЛИНЕЙНОЙ ТЕОРИИ ДИЭЛЕКТРИЧЕСКОЙ РЕЛАКСАЦИИ В КРИСТАЛЛАХ(1981) Fizika Tverdogo Tela. 23, 1, p. 324326 Abstract
1979

(1979) JETP Letters. 30, 6, p. 303305 Abstract
It is shown that a weak shock wave is unstable relative to transverse modulations.