Stochastic anomaly and large Reynolds number limit in hydrodynamic turbulence models

In this work we address the open problem of high Reynolds number limit in hydrodynamic turbulence, which we modify by considering a vanishing random (instead of deterministic) viscosity. In this formulation, a small-scale noise propagates to large scales in an inverse cascade, which can be described using qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the limit of the resulting probability distribution exists as $\mathrm{Re} \to \infty$, and the limiting flow at finite time remains stochastic even if forcing, initial and boundary conditions are deterministic. This conjecture is confirmed numerically for the Sabra model of turbulence, where the solution is deterministic before and random immediately after a blowup. Then, we derive a purely inviscid problem formulation with a stochastic boundary condition imposed in the inertial interval.Comment: 20 pages, 6 figure

Spontaneous stochasticity of velocity in turbulence models

We analyze the phenomenon of spontaneous stochasticity in fluid dynamics formulated as the nonuniqueness of solutions resulting from viscosity at infinitesimal scales acting through intermediate on large scales of the flow. We study the finite-time onset of spontaneous stochasticity in a real version of the GOY shell model of turbulence. This model allows high-accuracy numerical simulations for a wide range of scales (up to ten orders of magnitude) and demonstrates non-chaotic dynamics, but leads to an infinite number of solutions in the vanishing viscosity limit after the blowup time. Thus, the spontaneous stochasticity phenomenon is clearly distinguished from the chaotic behavior in turbulent flows. We provide the numerical and theoretical description of the system dynamics at all stages. This includes the asymptotic analysis before and after the blowup leading to universal (periodic and quasi-periodic) renormalized solutions, followed by nonunique stationary states at large times.Comment: 20 pages, 9 figure

Continuous representation for shell models of turbulence

In this work we construct and analyze continuous hydrodynamic models in one space dimension, which are induced by shell models of turbulence. After Fourier transformation, such continuous models split into an infinite number of uncoupled subsystems, which are all identical to the same shell model. The two shell models, which allow such a construction, are considered: the dyadic (Desnyansky--Novikov) model with the intershell ratio $\lambda = 2^{3/2}$ and the Sabra model of turbulence with $\lambda = \sqrt{2+\sqrt{5}} \approx 2.058$. The continuous models allow understanding various properties of shell model solutions and provide their interpretation in physical space. We show that the asymptotic solutions of the dyadic model with Kolmogorov scaling correspond to the shocks (discontinuities) for the induced continuous solutions in physical space, and the finite-time blowup together with its viscous regularization follow the scenario similar to the Burgers equation. For the Sabra model, we provide the physical space representation for blowup solutions and intermittent turbulent dynamics.Comment: 21 pages, 7 figure

Spontaneously stochastic solutions in one-dimensional inviscid systems

In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finite-time blowup, $t < t_b$, must be continued as a stochastic process after the blowup, $t > t_b$, representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for the logarithmic time $\tau = \log(t-t_b)$, which features a stable traveling wave solution for the inviscid Burgers equation, but a stochastic traveling wave for the Sabra model. The latter describes a universal onset of stochasticity immediately after the blowup

Universal structure of blow-up in 1D conservation laws

We discuss universality properties of blow-up of a classical (smooth) solutions of conservation laws in one space dimension. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of the form of a conservation law. This property is explained in terms of the renormalization group theory. A solitary wave appears in logarithmic coordinates of the Fourier space as a counterpart of this universality. Universality is demonstrated in two examples: Burgers equation and dynamics of ideal polytropic gas.Comment: 8 pages, 2 figure

Blowup as a driving mechanism of turbulence in shell models

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949 Onsager noticed that the scaling properties in inertial range imply non-differentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.Comment: 26 pages, 11 figure

Renormalization group formalism for incompressible Euler equations and the blowup problem

The paper discusses extensions of the renormalization group (RG) formalism for 3D incompressible Euler equations, which can be used for describing singularities developing in finite (blowup) or infinite time from smooth initial conditions of finite energy. In this theory, time evolution is substituted by the equivalent evolution for renormalized solutions governed by the RG equations. A fixed point attractor of the RG equations, if it exists, describes universal self-similar form of observable singularities. This universality provides a constructive criterion for interpreting results of numerical experiments. In this paper, renormalization schemes with multiple spatial scales are developed for the cases of power law and exponential scaling. The results are compared with the numerical simulations of a singularity in incompressible Euler equations obtained by Hou and Li (2006) and Grafke et al. (2008). The comparison supports the conjecture of a singularity developing exponentially in infinite time and described by a multiple-scale self-similar asymptotic solution predicted by the RG theory.Comment: 16 pages, 5 figure

Development of high vorticity structures in incompressible 3D Euler equations

We perform the systematic numerical study of high vorticity structures that develop in the 3D incompressible Euler equations from generic large-scale initial conditions. We observe that a multitude of high vorticity structures appear in the form of thin vorticity sheets (pancakes). Our analysis reveals the self-similarity of the pancakes evolution, which is governed by two different exponents $e^{-t/T_{\ell}}$ and $e^{t/T_{\omega}}$ describing compression in the transverse direction and the vorticity growth respectively, with the universal ratio $T_{\ell}/T_{\omega} \approx 2/3$. We relate development of these structures to the gradual formation of the Kolmogorov energy spectrum $E_{k}\propto\, k^{-5/3}$, which we observe in a fully inviscid system. With the spectral analysis we demonstrate that the energy transfer to small scales is performed through the pancake structures, which accumulate in the Kolmogorov interval of scales and evolve according to the scaling law $\omega_{\max} \propto \ell^{-2/3}$ for the local vorticity maximums $\omega_{\max}$ and the transverse pancake scales $\ell$.Comment: 31 pages, 18 figure

Light stops at exceptional points

Almost twenty years ago the light was slowed down to less than $10^{-7}$ of its vacuum speed in a cloud of ultracold atoms of sodium. Upon a sudden turn-off of the coupling laser, a slow light pulse can be imprinted on cold atoms such that it can be read out and converted into photon again. In this process, the light is stopped by absorbing it and storing its shape within the atomic ensemble. Alternatively, the light can be stopped at the band edge in photonic-crystal waveguides, where the group speed vanishes. Here we extend the phenomenon of stopped light to the new field of parity-time (PT) symmetric systems. We show that zero group speed in PT symmetric optical waveguides can be achieved if the system is prepared at an exceptional point, where two optical modes coalesce. This effect can be tuned for optical pulses in a wide range of frequencies and bandwidths, as we demonstrate in a system of coupled waveguides with gain and loss.Comment: 5 pages, 4 figure

The time-asymmetric quantum state exchange mechanism

We show here that due to non-adiabatic couplings in decaying systems applying the same time-dependent protocol in the forward and reverse direction to the same mixed initial state leads to different final pure states. In particular, in laser driven molecular systems applying a specifically chosen positively chirped laser pulse or an equivalent negatively chirped laser pulse yields entirely different final vibrational states. This phenomenon occurs when the laser frequency and intensity are slowly varied around an exceptional point (EP) in the laser intensity and frequency parameter space where the non-hermitian spectrum of the problem is degenerate. The protocol implies that a positively chirped laser pulse traces a counter-clockwise loop in time in the laser parameters' space whereas a negatively chirped pulse follows the same loop in the clockwise direction. According to this protocol one can choose the final pure state from any initial state. The obtained results imply the intrinsic non-adiabaticity of quantum transport around an EP, and offer a way to observe the EP experimentally in time-dependent quantum systems