Correlations in a weakly interacting two-dimensional random flow.

We analytically examine fluctuations of vorticity excited by an external random force in two-dimensional fluid. We develop the perturbation theory enabling one to calculate nonlinear corrections to correlation functions of the flow fluctuations found in the linear approximation. We calculate the correction to the pair correlation function and the triple correlation function. It enables us to establish the criterion of validity of the perturbation theory for different ratios of viscosity and bottom friction. We find that the corrections to the second moment are anomalously weak in the cases of small bottom friction and small viscosity and relate the weakness to the energy and enstrophy balances. We demonstrate that at small bottom friction the triple correlation function is characterized by universal scaling behavior in some region of lengths. The developed perturbation method was verified and confirmed by direct numerical simulations.

Coherent vortex in two-dimensional turbulence: Interplay of viscosity and bottom friction.

We examine coherent vortices appearing as a result of the inverse cascade of two-dimensional turbulence in a finite box in the case of pumping with arbitrary correlation time in the quasilinear regime. We demonstrate that the existence of the vortices depends on the ratio between the values of the bottom friction coefficient α and the viscous damping of the flow fluctuations at the pumping scale νk_{f}^{2} (ν is the kinematic viscosity coefficient and k_{f} is the characteristic wave vector at the pumping scale). The coherent vortices appear if νk_{f}^{2}≫α and cannot exist if νk_{f}^{2}≪α. Therefore there is a border value α∼νk_{f}^{2} separating the regions. In numerical simulations, νk_{f}^{2}/α can be arbitrary, whereas in a laboratory experiment νk_{f}^{2}/α≲1 and the coherent vortices can be observed solely near the border value of νk_{f}^{2}/α.

Publisher Correction: Wave kinetics of random fibre lasers.

This corrects the article DOI: 10.1038/ncomms7214.

Structure of coherent vortices generated by the inverse cascade of two-dimensional turbulence in a finite box.

We discuss the structure and geometrical characteristics of coherent vortices appearing as a result of the inverse cascade in two-dimensional turbulence in a finite box. We demonstrate that the universal velocity profile, established by J. Laurie et al. [Phys. Rev. Lett. 113, 254503 (2014)], corresponds to the passive regime of flow fluctuations. We find the vortex core radius and the vortex size, and we argue that the amount of vortices generated in the box depends on the system parameters.

Wave kinetics of random fibre lasers.

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

Universal velocity profile for coherent vortices in two-dimensional turbulence.

Two-dimensional turbulence generated in a finite box produces large-scale coherent vortices coexisting with small-scale fluctuations. We present a rigorous theory explaining the eta=1/4 scaling in the V is proportional to r(-eta) law of the velocity spatial profile within a vortex, where r is the distance from the vortex center. This scaling, consistent with earlier numerical and laboratory measurements, is universal in its independence of details of the small-scale injection of turbulent fluctuations and details of the shape of the box.

Explicit solution of the optimal fluctuation problem for an elastic string in a random medium.

The free-energy distribution function of an elastic string in a quenched random potential, PL(F) , is investigated with the help of the optimal fluctuation approach. The form of the far-right tail of PL(F) is found by constructing the exact solution of the nonlinear saddle-point equations describing the asymptotic form of the optimal fluctuation. The solution of the problem is obtained for two different types of boundary conditions and for an arbitrary dimension of the imbedding space 1+d with d from the interval 0

Patch coalescence as a mechanism for eukaryotic directional sensing.

Eukaryotic cells possess a sensible chemical compass allowing them to orient toward sources of soluble chemicals. The extracellular chemical signal triggers separation of the cell membrane into two domains populated by different phospholipid molecules and oriented along the signal anisotropy. We propose a theory of this polarization process, which is articulated into subsequent stages of germ nucleation, patch coarsening, and merging into a single domain. We find that the polarization time, t{epsilon}, depends on the anisotropy degree through the power law t{epsilon} infinity epsilon{-2}, and that in a cell of radius R there should exist a threshold value epsilon{th} infinity R{-1} for the smallest detectable anisotropy.

Dynamics of energy condensation in two-dimensional turbulence.

We report a numerical study, supplemented by phenomenological explanations, of "energy condensation" in forced 2D turbulence in a biperiodic box. Condensation is a finite size effect which occurs after the standard inverse cascade reaches the size of the system. It leads to the emergence of a coherent vortex dipole. We show that the time growth of the dipole is self-similar, and it contains most of the injected energy, thus resulting in an energy spectrum which is markedly steeper than the standard k{-5/3} one. Once the coherent component is subtracted, however, the remaining fluctuations have a spectrum close to k{-1}. The fluctuations decay slowly as the coherent part grows.

Effects of surface tension on immiscible Rayleigh-Taylor turbulence.

We present phenomenology describing the internal structure of a turbulent zone, produced as the result of the push of a heavy fluid into a light one, for the case of immiscible fluids. One finds that the Kolmogorov cascade is realized within a range that grows with time, viz., scales between the mixing zone width, L proportional variant t(2), and the viscous scale, eta proportional variant t(-1/4). Surface-tension effects lead to formation of an emulsionlike state. Density fluctuations on scales larger than the typical drop size, l, are governed by the Obukhov-Corrsin cascade. If l >> eta, a wave energy cascade, related to capillary waves propagating along the surfaces of drops, is formed at scales below l, l proportional variant t(-2/5).

Shedding and interaction of solitons in weakly disordered optical fibers.

The propagation of the soliton pattern through optical fiber with weakly disordered dispersion coefficient is considered. Solitons perturbed by this disorder radiate and, as a consequence, decay. The average radiation profile is found. Emergence of a long-range intrachannel interaction between the solitons (mediated by this radiation) is reported. We show that soliton in a multisoliton pattern experiences a random jitter: intersoliton separation is zero mean Gaussian random field. Fluctuations of this separation are estimated by deltay approximately Dz(2)square root mu, where D measures the disorder strength, z is the propagation distance, and mu stands for the transmission rate (number of solitons per unit length of the fiber). Direct numerical simulations are used to validate theoretical predictions for single soliton decay and two-soliton interaction. Relevance of these results to fiber optics communication technology is discussed.

Probability of anomalously large bit-error rate in long haul optical transmission.

We consider a linear model of optical transmission through a fiber with birefringent disorder in the presence of amplifier noise. Both disorder and noise are assumed to be weak, i.e., the average bit-error rate (BER) is small. The probability distribution function (PDF) of rare violent events leading to the values of BER much larger than its typical value is estimated. We show that the PDF has a long algebraic-like tail.

Statistics of soliton-bearing systems with additive noise.

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