In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for sufficiently small viscosity term $ν$, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to $\log_λν^{-1}$ for all values of the governing parameter $ε$, except for $ε=1$. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, we show different scenarios of the transition to chaos for different parameters regime and for specific forcing. In the ``three-dimensional'' regime of parameters this scenario changes when the parameter $ε$ becomes sufficiently close to 0 or to 1. We also show that in the ``two-dimensional'' regime of parameters for a certain non-zero forcing term the long-time dynamics of the model becomes trivial for any value of the viscosity.
Complex nonlinear turbulent dynamical systems are ubiquitous in many areas. 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 formulae 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 a noise inflation. In such a way, many complic…
The aim of the present work is to derive rigorous estimates for turbulent MHD flow quantities such as the size and anisotropy of the dissipative scales, as well as the transition between 2D and 3D state. To this end, we calculate an upper bound for the attractor dimension of the motion equations, which indicates the number of modes present in the fully developed flow. This method has already been used successfully to derive such estimates for 2D and 3D hydrodynamic turbulence as a function of the $\mathcal L_\infty$ norm of the dissipation, as in \cite{doering95}. We tackle here the problem of…
Steady Low $R_m$ MHD turbulence is investigated here through estimates of upper bounds for attractor dimension. A flow between two parallel walls with an imposed perpendicular magnetic field is considered. The flow is defined by its maximum velocity and the intensity of the magnetic field. Given the corresponding Reynolds and Hartmann numbers, we derive an upper bound for the dimension of the attractor and find out which modes must be chosen to achieve this bound. Thier, are compared to heuristics estimates for the size of the smallest scales and anisotropy of the turbulence. We identify bound…
Competition between two main attractors of the distributed chaos, one associated with translational symmetry (homogeneity) and another associated with rotational symmetry (isotropy), has been studied in freely decaying turbulence. It is shown that, unlike the case of statistically stationary homogeneous isotropic turbulence, the attractor associated with rotational symmetry (and controlled by Loitsyanskii integral) can dominate turbulent local dynamics in an intermediate stage of the decay, because the attractor associated with translational symmetry (and controlled by Birkhoff-Saffman integra…
In this paper, we derive estimates for size of the small scales and the attractor dimension in low $Rm$ magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any $n$-dimensional volume of the phase space by the evolution operator associated to the Navier-Stokes equations. As explained by Constantin et al. (J. Fluid Mech., 1985), The value of $n$ for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the …
Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J.Fluid Mech. 893, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number $Re=3000$, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, $Wi$ and increasin…
Data-driven low-dimensional model for the onset of turbulence in plane Couette flow
Graduate School of Engineering Science, the University of Osaka（大阪大学工学研究科）；Fluids Laboratory, Department of Mechanical Engineering, University of the Philippines Diliman（菲律宾大学迪利曼分校机械工程系流体实验室）
Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as 'elastic turbulence' when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics requires an assumption that an attractor of the Oldroyd-B model exists : numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combin…
We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e., ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshots to snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse-graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO i…
The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties
University of Cambridge（剑桥大学）；Université Côte d’Azur, CNRS, Institut de Physique de Nice（蔚蓝海岸大学、法国国家科学研究中心、尼斯物理研究所）；Imperial College London（伦敦帝国理工学院）；Indian Institute of Science（印度科学学院）
We study the Reynolds number scaling of the Kolmogorov-Sinai entropy and attractor dimension for three dimensional homogeneous isotropic turbulence through the use of direct numerical simulation. To do so, we obtain Lyapunov spectra for a range of different Reynolds numbers by following the divergence of a large number of orthogonal fluid trajectories. We find that the attractor dimension grows with the Reynolds number as Re$^{2.35}$ with this exponent being larger than predicted by either dimensional arguments or intermittency models. The distribution of Lyapunov exponents is found to be fini…
One of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose internal wave attractors in the large amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochro-matic input to multi-scale internal wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how beyond this regime, we have a clear transition to a regime of small…
Dynamical slowdown, bottlenecks, and multiscaling in Voigt-regularised turbulence
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata（印度科学教育与研究学院加尔各答分校物理科学系）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Department of Mathematics, Texas A & M University（德克萨斯农工大学数学系）；Department of Computer Science and Applied Mathematics, Weizmann Institute of Science（魏茨曼科学研究所计算机科学与应用数学系）
Turbulent flows governed by the Navier-Stokes equations (NSE) generate an out-of-equilibrium time irreversible energy cascade from large to small scales. In the NSE, the energy transfer is due to the nonlinear terms that are formally symmetric under time reversal. As for the dissipative term: first it explicitly breaks time reversibility; second it produces a small-scale sink for the energy transfer that remains effective even in the limit of vanishing viscosity. As a result, it is not clear how to disentangle the time irreversibility originating from the non-equilibrium energy cascade from th…
Forcing regimes in the two-dimensional Navier-Stokes equations
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院国际理论科学中心）；Department of Mathematics, Imperial College London（帝国理工学院数学系）；Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉布雷亚斯-雅克·达迪厄实验室）
