Direct observation of vortices in an electron fluid.

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

Emerging scale invariance in a model of turbulence of vortices and waves.

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

Localization-delocalization transitions in turbophoresis of inertial particles.

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.

Surface tension: floater clustering in a standing wave.

How do waves affect the distribution of small particles that float on water? Here we show that drifting small particles concentrate in either the nodes or antinodes of a standing wave, depending on whether they are hydrophilic or hydrophobic, as a result of a surface-tension effect that violates Archimedes' law of buoyancy. This clustering on waves may find practical application in particle separation and provides insight into the patchy distribution on water of, for example, plastic litter or oil slicks.

Second-harmonic generation as a minimal model of turbulence.

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

Entropic characterization of the coil-stretch transition of polymers in random flows.

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

Publisher Correction: Wave kinetics of random fibre lasers.

This corrects the article DOI: 10.1038/ncomms7214.

Inertial particles driven by a telegraph noise.

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

Inelastic collapse and near-wall localization of randomly accelerated particles.

Inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in half-space z>0 has been discovered by McKean [J. Math. Kyoto Univ. 2, 227 (1963)] and then independently rediscovered by Cornell et al. [Phys. Rev. Lett. 81, 1142 (1998)PRLTAO0031-900710.1103/PhysRevLett.81.1142]. 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 ß of particle velocity is smaller than the critical value ß_{c}=exp(-π/sqrt[3]). 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 ß_{c} is universal. That class includes an important case of inertial particles in wall-bounded random flows. To establish how inelastic collapse influences the particle distribution, we derive the exact equilibrium probability density function ρ(z,v) for the particle position and velocity. The equilibrium distribution exists only at ß<ß_{c} and indicates that inelastic collapse does not necessarily imply near-wall localization.

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.

Feedback of a small-scale magnetic dynamo.

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

Statistics of interacting optical solitons.

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

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).

Turbulence-condensate interaction in two dimensions.

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