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We consider an initial value problem for shell models that mimic turbulent velocity fluctuations over a geometric sequence of scales. Our goal is to study the convergence of solutions in the inviscid (more generally, vanishing regularization) limit and explain the universality of both the limiting solutions and the convergence process. We develop a renormalization group (RG) formalism representing this limit as dynamics in a space of flow maps. For the dyadic shell model, the RG dynamics has a fixed-point attractor, which determines universal limiting solutions. Deviations from the limiting solutions are also universal and given by a leading eigenmode (eigenvalue and eigenvector) of the linearized RG operator. Application to the Gledzer and Sabra shell models reveal the respective quasiperiodic and chaotic RG dynamics., Comment: 33 pages, 8 figures

Many fundamental problems in fluid dynamics are related to the effects of solid boundaries. In general, they install sharp gradients and contribute to the developement of small-scale structures, which are computationally expensive to resolve with numerical simulations. A way to access extremely fine scales with a reduced number of degrees of freedom is to consider the equations on logarithmic lattices in Fourier space. Here we introduce new toy models for flows with walls, by showing how to add boundaries to the logarithmic lattice framework. The resulting equations retain many important properties of the original systems, such as the conserved quantities, the symmetries and the boundary effects. We apply this technique to many flows, with emphasis on the inviscid limit of the Navier-Stokes equations. For this setup, simulations reach impressively large Reynolds numbers and disclose interesting insights about the original problem., Comment: 37 pages, 13 figures

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier-Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It ultimately identifies the scaling exponents as the eigenvalues of a Perron-Frobenius operator acting on invariant measures of the rescaled equations. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models.

We present a study of the intermittent properties of a shell model of turbulence with unprecedented statistics, about $\sim 10^7$ eddy turn over time, achieved thanks to an implementation on a large-scale parallel GPU factory. This allows us to quantify the inertial range anomalous scaling properties of the velocity fluctuations up to the 24th order moment. Through a careful assessment of the statistical and systematic uncertainties, we show that none of the phenomenological and theoretical models previously proposed in the literature to predict the anomalous power-law exponents in the inertial range is in agreement with our high-precision numerical measurements. We find that at asymptotically high order moments, the anomalous exponents tend towards a linear scaling, suggesting that extreme turbulent events are dominated by one leading singularity. We found that systematic corrections to scaling induced by the infrared and ultraviolet (viscous) cut-offs are the main limitations to precision for low-order moments, while high orders are mainly affected by the finite statistical samples. The unprecedentedly high fidelity numerical results reported in this work offer an ideal benchmark for the development of future theoretical models of intermittency in dynamical systems for either extreme events (high-order moments) or typical fluctuations (low-order moments). For the latter, we show that we achieve a precision in the determination of the inertial range scaling exponents of the order of one part over ten thousand (5th significant digit), which must be considered a record for out-of-equilibrium fluid-mechanics systems and models.

How predictable are turbulent flows? Here we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the non-uniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. These universal statistics are predictable, even though individual flow realizations are not. Any small-scale noise vanishing slowly enough with increasing Reynolds number can trigger spontaneous stochasticity and here we show that thermal noise alone, in the absence of any larger disturbances, would suffice. If confirmed for Navier-Stokes turbulence, our findings would imply that intrinsic stochasticity of turbulent fluid motions at all scales can be triggered even by unavoidable molecular noise, with implications for modeling in engineering, climate, astrophysics and cosmology.

In 1981, Frisch and Morf [1] postulated the existence of complex singularities in solutions of Navier-Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier-Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.

Developed turbulent motion of fluid still lacks an analytical description despite more than a century of active research. Nowadays phenomenological ideas are widely used in practical applications, such as small-scale closures for numerical simulations of turbulent flows. In the present work, we use a shell model of turbulence to construct a closure intended to have a solid theoretical background and to capture intrinsic probabilistic features of turbulence. Shell models of turbulence are dynamical deterministic systems used to model energy cascade and other key aspects of the Navier-Stokes such as intermittency. We rescale the variables of the Sabra model in a way which leads to hidden symmetries and universal distributions. We then use such fine distributions to write closures, i.e., missing expressions for some of the Sabra variables. Our closures rely on approximating probability density functions using a Gaussian Mixture Model, which makes them probabilistic by nature and allows us to write time-correlated closures. We also provide a framework where other Machine Learning tools can be employed with reduced black-box aspects., Comment: 23 pages, 5 figures

Intermittency is one of central obstacles for understanding small-scale dynamics in the fully developed hydrodynamic turbulence. The modern approach is largely based on the multifractal theory of Parisi and Frisch which is, however, phenomenological. It was shown recently that the intermittency can be related to the hidden scale invariance. The latter is a new statistical scaling symmetry unbroken in a rescaled (projected) formulation of equations of motion. In the present work, we consider a shell model of turbulence and describe how the hidden symmetry manifests itself through all scales, both in the inertial interval and in the transition to forcing and dissipation ranges. In the inertial interval, we derive anomalous scaling laws from the hidden symmetry. Then, we show how a complicated form of the dissipation range is controlled by intermittent rescaled Reynolds numbers within a large range of dissipation scales. This dissipative intermittency can be removed by using a special class of dissipation models. For such models, the hidden scale invariance is restored both in the inertial interval and the dissipation range. Overall, the presented approach deduces the multifractal theory and some of its basic conclusions from the hidden scaling symmetry of equations of motion., Comment: 42 pages, 15 figures

In this work we present a systematic numerical study of the post-blowup dynamics of singular solutions of the 1D focusing critical NLS equation in the framework of a nonlinear damped perturbation. The first part of this study shows that initially the post-blowup is described by the adiabatic approximation, in which the collapsing core approaches an universal profile and the solution width is governed by a system of ODEs (reduced system). After that, a non-adiabatic regime is observed soon after the maximum of the solution, in which our direct numerical simulations show a clear deviation from the dynamics based on the reduced system. Our study suggests that such non-adiabatic regime is caused by the increasing influx of mass into the collapsing core of the solution, which is not considered in the derivation of the reduced system. Also, adiabatic theoretical predictions related to the wave-maximum and wave-dissipation are compared with our numerical simulations. The second part of this work describes the non-adiabatic dynamics. Here, numerical simulations reveal a dominant quasi linear regime, caused by the rapid defocusing process. The collapsing core approaches the universal profile, after removing some oscillations resulting from the interference with the tail. Finally, our numerical study suggests that in the limit of vanishing dissipation, and in a free-space domain, the critical mass is radiated to infinity instantly at the collapse time.

We introduce a class of multi-scale systems with discrete time, motivated by the problem of inviscid limit in fluid dynamics in the presence of small-scale noise. These systems are infinite-dimensional and defined on a scale-invariant space-time lattice. We propose a qualitative theory describing the vanishing regularization (inviscid) limit as an attractor of the renormalization group operator acting in the space of flow maps or respective probability kernels. If the attractor is a nontrivial probability kernel, we say that the inviscid limit is spontaneously stochastic: it defines a stochastic (Markov) process solving deterministic equations with deterministic initial and boundary conditions. The results are illustrated with solvable models: symbolic systems leading to digital turbulence and systems of expanding interacting phases., Comment: 34 pages, 10 figures

We show that the intermittent dynamics observed in the inertial interval of Sabra shell model of turbulence can be rigorously related to the property of scaling self-similarity. In this connection, the space-time scaling symmetries (like in the K41 theory) are replaced by the new hidden scaling symmetry, which is an exact symmetry of inviscid dynamics represented in special rescaled coordinates and times. We derive the formulas expressing anomalous scaling exponents in terms of Perron-Frobenius eigenvalues of linear operators based on the self-similar statistics. Theoretical conclusions are verified by extensive numerical simulations., Comment: 25 pages, 7 figures

We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, like non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics., Comment: 20 pages, 5 figures

We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an ``effective field-theory'' valid below some high-wavenumber cutoff $\Lambda$, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million the strongest turbulent excitations observed in our simulation penetrate down to a length-scale of microns. However, for longer observation times or higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.

Turbulent flows frequently accompany physical, chemical and biological processes, such as mixing, two-phase flow, combustion and even foraging by bacteria and plankton larvae, all of which are in principle subject to thermal fluctuations already on scales of several microns. Nevertheless the large separation between the millimeter scale at which turbulent fluctuations begin to be strongly damped and the mean free path of the fluid has been generally assumed to imply that thermal fluctuations are irrelevant to the turbulent dissipation range. Here we use statistical mechanical estimates to show that thermal fluctuations are not negligible compared to turbulent eddies in the dissipation range. Simulation of the Sabra shell model shows that intermittent bursts of turbulence lead to a fluctuating length scale below which thermal fluctuations are important: over three decades of length, from sub-millimeter scales down to the mean free path, thermal fluctuations coexist with hydrodynamics. Our results imply that thermal fluctuations cannot be neglected when modeling turbulent phenomena in the far dissipation range.

We expose a hidden scaling symmetry of the Navier-Stokes equations in the limit of vanishing viscosity, which stems from dynamical space-time rescaling around suitably defined Lagrangian scaling centers. At a dynamical level, the hidden symmetry projects solutions which differ up to Galilean invariance and global temporal scaling onto the same representative flow. At a statistical level, this projection repairs the scale invariance, which is broken by intermittency in the original formulation. Following previous work by the first author, we here postulate and substantiate with numerics that hidden symmetry statistically holds in the inertial interval of fully developed turbulence. We show that this symmetry accounts for the scale-invariance of a certain class of observables, in particular, the Kolmogorov multipliers., Comment: 18 pages, 4 figures

We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence., Comment: 12 pages, 2 figures

We develop a multicomponent lattice Boltzmann (LB) model for the 2D Rayleigh--Taylor turbulence with a Shan-Chen pseudopotential implemented on GPUs. In the immiscible case this method is able to accurately overcome the inherent numerical complexity caused by the complicated structure of the interface that appears in the fully developed turbulent regime. Accuracy of the LB model is tested both for early and late stages of instability. For the developed turbulent motion we analyze the balance between different terms describing variations of the kinetic and potential energies. Then, we analyze the role of interface in the energy balance, and also the effects of the vorticity induced by the interface in the energy dissipation. Statistical properties are compared for miscible and immiscible flows. Our results can also be considered as a first validation step to extend the application of LB model to 3D immiscible Rayleigh-Taylor turbulence., Comment: 14 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2009.00054

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the Quasi--Lagrangian description, making the developed theory applicable to the Euler and Navier-Stokes systems., Comment: 56 pages

We studied turbulence induced by the Rayleigh-Taylor (RT) instability for 2D immiscible two-component flows by using a multicomponent lattice Boltzmann method with a Shan-Chen pseudopotential implemented on GPUs. We compare our results with the extension to the 2D case of the phenomenological theory for immiscible 3D RT studied by Chertkov and collaborators ({\it Physical Review E 71, 055301, 2005}). Furthermore, we compared the growth of the mixing layer, typical velocity, average density profiles and enstrophy with the equivalent case but for miscible two-component fluid. Both in the miscible and immiscible cases, the expected quadratic growth of the mixing layer and the linear growth of the typical velocity are observed with close long-time asymptotic prefactors but different initial transients. In the immiscible case, the enstrophy shows a tendency to grow like $\propto t^{3/2}$, with the highest values of vorticity concentrated close to the interface. In addition, we investigate the evolution of the typical drop size and the behavior of the total length of the interface in the emulsion-like state, showing the existence of a power law behavior compatible with our phenomenological predictions. Our results can also be considered as a first validation step to extend the application of lattice Boltzmann tool to study the 3D immiscible case., Comment: 25 pages, 12 figures

Open problems in fluid dynamics, such as the existence of finite-time singularities (blowup), explanation of intermittency in developed turbulence, etc., are related to multi-scale structure and symmetries of underlying equations of motion. Significantly simplified equations of motion, called toy-models, are traditionally employed in the analysis of such complex systems. In such models, equations are modified preserving just a part of the structure believed to be important. Here we propose a different approach for constructing simplified models, in which instead of simplifying equations one introduces a simplified configuration space: velocity fields are defined on multi-dimensional logarithmic lattices with proper algebraic operations and calculus. Then, the equations of motion retain their exact original form and, therefore, naturally maintain most scaling properties, symmetries and invariants of the original systems. Classification of such models reveals a fascinating relation with renowned mathematical constants such as the golden mean and the plastic number. Using both rigorous and numerical analysis, we describe various properties of solutions in these models, from the basic concepts of existence and uniqueness to the blowup development and turbulent dynamics. In particular, we observe strong robustness of the chaotic blowup scenario in the three-dimensional incompressible Euler equations, as well as the Fourier mode statistics of developed turbulence resembling the full three-dimensional Navier-Stokes system., Comment: 39 pages, 6 figures

It is known that scale invariance is broken in the developed hydrodynamic turbulence due to intermittency, substantiating complexity of turbulent flows. Here we challenge the concept of broken scale invariance by establishing a hidden self-similarity in intermittent turbulence. Using a simplified (shell) model, we derive a nonlinear spatiotemporal scaling symmetry of inviscid equations, which are reformulated in terms of intrinsic times introduced at different scales of motion. Numerical analysis persuasively confirms that this symmetry is restored in a statistical sense within the inertial interval. At the end, we discuss implications of this result for the Navier-Stokes system., Comment: 5 pages, 2 figures

The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually predicted that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect, which is radically different from chaos, have remained out of reach for detailed physical observations. Here, we substantiate this scenario by showing that it is inherent to the elementary Kelvin--Helmholtz hydrodynamical instability of an initially singular shear layer. We moreover provide evidence that the resulting macroscopic flow displays universal statistical properties that are triggered by, but independent of specific physical properties at micro-scales. This spontaneous stochasticity is interpreted as an Eulerian counterpart to Richardson's relative dispersion of Lagrangian particles, giving substance to the intrinsic nature of randomness in turbulence.

The efficient transfer of excitations between different levels of a quantum system is a task with many applications. Among the various protocols to carry out such a state transfer in driven systems, rapid adiabatic passage (RAP) is one of the most widely used. Here we show both theoretically and experimentally that adding a suitable amount of loss to the driven Hamiltonian turns a RAP protocol into a scheme for encircling an exceptional point including the chiral state transfer associated with it. Our work thus discloses an intimate connection between a whole body of literature on RAP and recent studies on the dynamics in the vicinity of an exceptional point, which we expect to serve as a bridge between the disjoint communities working on these two scenarios., Comment: 5 pages, 4 figures in main text; 14 pages supplementary material

We study a class of ordinary differential equations with a non-Lipschitz point singularity, which admit non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on the parameter $\nu$: the regularized dynamics is globally defined for each $\nu > 0$, and the original singular system is recovered in the limit of vanishing $\nu$. We prove that this limit yields a unique statistical solution independent of regularization, when the deterministic system possesses certain chaotic properties. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution., Comment: 28 pages, 7 figures, 1 supplementary video

We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$, which we term `blowup'. However, infinitely many solutions may exist for longer times. To study continuation past blowup, we introduce physically motivated regularizations: they consist of smoothing the vector field in a $\nu$--ball around the origin and then removing the regularization in the limit $\nu\to 0$. We show that this limit can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization procedure. This procedure maps the pre-blowup dynamics, $t < t_b$, to the solution ending at infinitely large renormalized time. In particular, the asymptotic behavior as $t \nearrow t_b$ is described by an attractor. The post-blowup dynamics, $t > t_b$, is mapped to a different renormalized solution starting infinitely far in the past. Consequently, it is associated with another attractor. The $\nu$-regularization establishes a relation between these two different "lives" of the renormalized system. We prove that, in some generic situations, this procedure selects a unique global solution (or a family of solutions), which does not depend on the details of the regularization. We provide concrete examples and argue that these situations are qualitatively similar to post-blowup scenarios observed in infinite-dimensional models of turbulence., Comment: 27 pages, 8 figures

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem., Comment: 7 pages, 5 figures, 1 supplemental video

We perform direct numerical simulations of three dimensional Rayleigh-Taylor turbulence with a nonuniform singular initial temperature background. In such conditions, the mixing layer evolves under the driving of a varying effective Atwood number; the long-time growth is still self-similar, but not anymore proportional to $t^2$ and depends on the singularity exponent $c$ of the initial profile $\Delta T \propto z^c$. We show that the universality is recovered when looking at the efficiency, defined as the ratio of the variation rates of the kinetic energy over the heat flux. A closure model is proposed that is able to reproduce analytically the time evolution of the mean temperature profiles, in excellent agreement with the numerical results. Finally, we reinterpret our findings in the light of spontaneous stochasticity where the growth of the mixing layer is mapped into the propagation of a wave of turbulent fluctuations on a rough background.

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 figures

Considering two-dimensional potential ideal flow with free surface and finite depth, we study the dynamics of small-amplitude and short-wavelength wavetrains propagating on the background of a steepening nonlinear wave. This can be seen as a model for small ripples developing on slopes of breaking waves in the surf zone. Using the concept of wave action as an adiabatic invariant, we derive an explicit asymptotic expression for the change of ripple steepness. Through this expression, nonlinear effects are described using the intrinsic frequency and intrinsic gravity along Lagrangian (material) trajectories on a free surface. We show that strong compression near the tip on the wave leads to an explosive (super-exponential) ripple instability. This instability may play important role for understanding fragmentation and whitecapping at the surface of breaking waves. Analytical results are confirmed by numerical simulations using a potential theory model., Comment: 16 pages, 11 figures

We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reproducing all known properties of the large-scale dynamics including anomalous scaling. We found small but systematic discrepancies only for a range of scales close to the sub-grid threshold, which do not tend to disappear by increasing the order of the approximation. We speculate that the lack of convergence might be due to a structural instability, at least for the evolution of very fast degrees of freedom at small scales. Connections with similar problems for Large Eddy Simulations of the three-dimensional Navier-Stokes equations are also discussed. Postprint version of the article published on Phys. Rev. E 95, 043108 (2017) DOI: 10.1103/PhysRevE.00.003100, Comment: 27 pages

We study a nonlinear wave for a system of balance laws in one space dimension, which describes combustion for two-phase (gas and liquid) flow in porous medium. The problem is formulated for a general $N$-component liquid for modeling the strong multicomponent effects reported recently for an application to light oil recovery by air injection. Despite the immense complexity of the model, the problem allows analytic solution. The clue to this solution is a special form of a folding singularity, which occurs at an internal point of the wave profile. Analysis of this singularity provides a missing determining relation for wave parameters. This result is not only interesting for the application under consideration, but also motivates a deeper mathematical study of such singularities for general systems of balance laws., Comment: 23 pages, 5 figures

In this work we suggest that a turbulent phase of the Rayleigh-Taylor instability can be explained as a universal stochastic wave traveling with constant speed in a properly renormalized system. This wave, originating from ordinary deterministic chaos in a renormalized time, has two constant limiting states at both sides. These states are related to the initial discontinuity at large scales and to stationary turbulence at small scales. The theoretical analysis is confirmed with extensive numerical simulations made for a new shell model, which features basic properties of the phenomenological theory for the Rayleigh-Taylor instability., Comment: 22 pages

Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations in helical eigenstates. We find that depending on the helical structure of each model, instantons are chaotic or regular. Some instantonic solutions tend to recover mirror symmetry for scales small enough. Models that have anomalous scaling develop regular non chaotic instantons. Conversely, models that have non anomalous scaling in the stationary regime are those that have chaotic instantons. The direction of the energy carried by each single instanton tends to coincide with the direction of the energy cascade in the stationary regime. Finally, we find that whenever the small-scale stationary statistics is intermittent, the instanton is less steep than the dimensional Kolmogorov scaling, independently of whether or not it is chaotic. Our findings further support the idea that instantons might be crucial to describe some aspects of the multi-scale anomalous statistics of shell models.

Physical systems with loss or gain feature resonant modes that are decaying or growing exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, a so-called "exceptional point" occurs, around which many fascinating phenomena have recently been reported to arise. Particularly intriguing behavior is predicted to appear when encircling an exceptional point sufficiently slowly, like a state-flip or the accumulation of a geometric phase. Experiments dedicated to this issue could already successfully explore the topological structure of exceptional points, but a full dynamical encircling and the breakdown of adiabaticity inevitably associated with it remained out of reach of any measurement so far. Here we demonstrate that a dynamical encircling of an exceptional point can be mapped onto the problem of scattering through a two-mode waveguide, which allows us for the first time to access the elusive effects occurring in this context. Specifically, we present experimental results from a waveguide structure that steers incoming waves around an exceptional point during the transmission process. In this way mode transitions are induced that make this device perfectly suited as a robust and asymmetric switch between different waveguide modes. Our work opens up new and exciting avenues to explore exceptional point physics at the crossroads between fundamental research and practical applications.

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.

Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)] we introduce a modified version of helical shell models for turbulence with non-local triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties.

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 figures

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 figures

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 figures

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.

We analyze the blowup (finite-time singularity) in inviscid shell models of convective turbulence. We show that the blowup exists and its internal structure undergoes a series of bifurcations under a change of shell model parameter. Various blowup structures are observed and explained, which vary from self-similar to periodic, quasi-periodic and chaotic regimes. Though the blowup takes sophisticated forms, its asymptotic small-scale structure is independent of initial conditions, i.e., universal. Finally, we discuss implications of the obtained results for the open problems of blowup in inviscid flows and for the theory of turbulence., Comment: 22 pages, 10 figures

In atomic physics, adiabatic evolution is often used to achieve a robust and efficient population transfer. Many adiabatic schemes have also been implemented in optical waveguide structures. Recently there has been increasing interests in the influence of decay and absorption, and their engineering applications. Here it is shown that even a small decay can significantly influence the dynamical behaviour of a system, above and beyond a mere change of the overall norm. In particular, a small decay can lead to a breakdown of adiabatic transfer schemes, even when both the spectrum and the eigenfunctions are only sightly modified. This is demonstrated for the generalization of a STIRAP scheme that has recently been implemented in optical waveguide structures. Here the question how an additional absorption in either the initial or the target waveguide influences the transfer property of the scheme is addressed. It is found that the scheme breaks down for small values of the absorption at a relatively sharp threshold, which can be estimated by simple analytical arguments., Comment: 8 pages, 7 figures, revised and extended

We show that multiscaling properties of developed turbulence in shell models, which lead to anomalous scaling exponents in the inertial range, are determined exclusively by instanton dynamics. Instantons represent correlated extreme events localized in space-time, whose structure is described by self-similar statistics with a single universal scaling exponent. We show that anomalous scaling exponents appear due to the process of instanton creation. A simplified model of instanton creation is suggested, which adequately describes this anomaly., Comment: 4 pages, 5 figures

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 figures

We consider self-similar solutions describing intermittent bursts in shell models of turbulence, and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows., Comment: 20 pages, 3 figures

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 figures

The paper studies the structure of high-order adiabatic approximation of a wave function for slowly changing Hamiltonians. A constructive technique for explicit separation of fast and slow components of the wave function is developed. The fast components are determined by dynamic phases, while the slow components are given by asymptotic series evaluated by means of an explicit recurrent procedure. It is shown that the slow components represent quasiadiabatic states, which play conceptually the same role as energy levels in stationary systems or Floquet states in time-periodic systems. As an application, we derive high-order asymptotic expressions for quasienergies of periodic Hamiltonians. As examples, a two-state (spin-1/2) system in periodically changing magnetic filed, and a particle in moving square potential well are studied., Comment: This paper has been withdrawn by the author due to new references information