Shell Model Of Turbulence Research Articles

Irreversibility and response functions in the turbulent energy cascade

Abstract The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed “direct energy cascade”. Here, we aim at characterizing the non-equilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We shall contrast the behavior of these two observables in both non-equilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we shall show that equilibrium and non-equilibrium physics coexist in the same system, namely at scales larger and smaller, respectively, of the forcing scale.

Extreme statistics and extreme events in dynamical models of turbulence.

We present a study of the intermittent properties of a shell model of turbulence with statistics of ∼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 are in agreement with our high-precision numerical measurements. We find that at asymptotically high-order moments, the anomalous exponents tend toward 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) cutoffs are the main limitations to precision for low-order moments, while high orders are mainly affected by the finite statistical samples.. The 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 (fifth significant digit), which may be considered a record for out-of-equilibrium fluid-mechanics systems and models.

Data-based approach for time-correlated closures of turbulence models.

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 paper, 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.

Renormalization of the shell model of turbulence.

Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ renormalization group (RG) to the shell model of turbulence and show that the RG equationis satisfied by |u_{n}|^{2}=K_{Ko}ε^{2/3}k_{n}^{-2/3}, where Ko is the Kolmogorov constant and ν_{n}=ν_{*}sqrt[K_{Ko}]ε^{1/3}k_{n}^{-4/3}, where k_{n}andu_{n} are the wave number and velocity of shell n; ν_{*}andK_{Ko} are RG and Kolmogorov's constants; and ε is the energy dissipation rate. We find that ν_{*}≈0.5 and K_{Ko}≈1.7, consistent with earlier RG works on the Navier-Stokes equation. We verify the theoretical predictions using numerical simulations.

Turbulence with Magnetic Helicity That Is Absent on Average

Magnetic helicity plays a tremendously important role when it is different from zero on average. Most notably, it leads to the phenomenon of an inverse cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as well as some less common examples of magnetic evolution under the Hall effect and ambipolar diffusion, as well as cases in which the magnetic field evolution is constrained by the presence of an asymmetry in the number density of chiral fermions, whose spin is systematically either aligned or anti-aligned with its momentum. In all those cases, there is a new conserved quantity: the Hosking integral. We present quantitative scaling results for the magnetic integral scale as well as the magnetic energy density and its spectrum. We also compare with cases were a magnetic version of the Saffman integral is initially finite. Rotation in MHD turbulence tends to suppress nonlinearity and thereby also inverse cascading. Finally, the role of the Hosking and magnetic Saffman integrals in shell models of turbulence is examined.

Hidden scale invariance of turbulence in a shell model: From forcing to dissipation scales

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.

Nonconservative Cascades in a Shell Model of Turbulence

Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.

Large deviation principle for a class of stochastic hydrodynamical type systems driven by multiplicative Lévy noises

This article is devoted to the large deviation principle for a wide class of stochastic hydrodynamical systems driven by multiplicative Lévy noise. The model covers many equations arising form fluid dynamics such as 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also shell models of turbulence. The main difficulty in proving the large deviation principle for the system is overcame by using the weak convergence method introduced by Budhiraja, Dupuis and Maroulas (Ann. Probab. 36: 1390–1420, 2008 and Annales De L Institut Henri Poincare. 47: 725–747, 2011).

Numerical proof of shell model turbulence closure

The development of turbulence closure models, parametrizing the influence of small non-resolved scales on the dynamics of large resolved ones, is an outstanding theoretical challenge with vast applicative relevance. We present a closure, based on deep recurrent neural networks, that quantitatively reproduces, within statistical errors, Eulerian and Lagrangian structure functions and the intermittent statistics of the energy cascade, including those of subgrid fluxes. To achieve high-order statistical accuracy, and thus a stringent statistical test, we employ shell models of turbulence. Our results encourage the development of similar approaches for 3D Navier-Stokes turbulence.

Interior and H∞ feedback stabilization for sabra shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations that exhibit at most resemblance with turbulent flows. One of the mathematically well‐studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model, namely, feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus, only finitely many modes of the shell model would suffice to stabilize the system. Later, we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max‐min control problem. We first prove the stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic Riccati equation. Using the same Riccati operator, we establish asymptotic stability of the nonlinear system.

Shell model intermittency is the hidden self-similarity

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.

Well-posedness of stochastic 2D hydrodynamics type systems with multiplicative Lévy noises

We establish the existence and uniqueness of solutions to an abstract nonlinear equation driven by a multiplicative noise of L\'evy type, which covers many hydrodynamical models including 2D Navier-Stokes equations, 2D MHD equations, the 2D Magnetic Bernard problem, and several Shell models of turbulence. In the existing literature on this topic, besides the classical Lipschitz and one sided linear growth conditions, other assumptions, which might be untypical, are also required on the coefficients of the stochastic perturbations. This paper is to get rid of these untypical assumptions. Our assumption on the coefficients of stochastic perturbations is new even for the Wiener cases, and in some sense, is shown to be quite sharp. A new cutting-off argument and energy estimation procedure play an important role in establishing the existence and uniqueness under this assumption.

An efficient continuous data assimilation algorithm for the Sabra shell model of turbulence.

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas of research. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous-in-time data assimilation scheme is developed, which exploits closed analytic formulas for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It is shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It is also shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model with partial observations.

Solvable Intermittent Shell Model of Turbulence

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.

Hidden scale invariance of intermittent turbulence in a shell model

A study shows that the shell model of turbulence has a hidden scaling symmetry, which involves nonlinear transformations of shell speeds and time. This symmetry is restored statistically in the developed turbulence despite all original scaling symmetries being broken by the intermittency.

Generation of equations for shell models of turbulence in the Maple system

The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.

Asymptotic log-Harnack inequality and applications for stochastic 2D hydrodynamical-type systems with degenerate noise

In this paper, an asymptotic log-Harnack inequality and some consequent properties are established via the asymptotic coupling method for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise. The main results are applicable to the stochastic 2D Navier–Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 2D Boussinesq equations, stochastic 2D magnetic Benard problem, stochastic 3D Leray- $$\alpha $$ model and also stochastic shell models of turbulence in the degenerate noise case.

Construction of complex shell models of MHD turbulence in computer algebra systems

The paper describes the developed by authors technique for construct-ing complex shell models of turbulence. The compilation of the equa-tions of this model and its exactly solution are implemented using by computer algebra system. The technique allows one to vary the sizes of nonlocality of nonlinear interaction in the space of scales, expressions for shell analogues of conservation laws, and the nature of stationary solutions with different power distribution.

Application of computer algebra for the construction of shell models of turbulence

The technique for automatic constructing of shell models of turbulence has been developed. The compilation of a model equations and its exactly solution is implemented using computer algebra (symbolic calculation) systems. The technique allows one to vary the scaling nonlocality of nonlinear interaction, form of expressions for conservation laws in models, and the form of stationary solutions with power distributions to scales.

Equivalence of nonequilibrium ensembles in turbulence models.

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of nonequilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.