Chaotic behavior of Eulerian magnetohydrodynamic turbulence

Ho, Richard D. J. G.; Berera, Arjun; Clark, Daniel

doi:10.1063/1.5092367

Cited by 6 publications

(10 citation statements)

References 55 publications

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“…Similar to the hydrodynamic case, whilst the λ are well approximated by a Gaussian distribution, the value for Re is not. As is known, analysis of the chaotic properties of MHD simulations presents noisier results than the NSE case [17]. As well, we find that characteristic quantities and their fluctuations are less correlated.…”

Section: Mhdsupporting

confidence: 66%

“…The advantage of the FTLE method is that it produces a sample of Lyapunov exponents that allows us to perform a statistical analysis of the measurements. Other methods exist in literature, for instance, the direct method consists in introducing a small perturbation once the system reaches a steady state and then letting the two fields evolve until they become uncorrelated [14,17]. This method reveals interesting features because it displays the long time evolution of the spectrum of the field difference |δu| (in which a transient and a saturation stage are found [14]).…”

Section: A Finite Time Lyapunov Exponents Methodsmentioning

confidence: 99%

“…In recent times, high performance computing has made possible to study chaotic properties of 3D flows using direct numerical simulations (DNS) [14][15][16][17][18][19][20]. This consists in evolving the Navier-Stokes equations (NSE) using no modelling.…”

Section: Introductionmentioning

confidence: 99%

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Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

Armua

Berera

2020

Phys. Rev. Fluids

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In the context of the analysis of the chaotic properties of homogeneous and isotropic turbulence, direct numerical simulations are used to study the fluctuations of the finite time Lyapunov exponent (FTLE) and its relation to Reynolds number, lattice size and the choice of the steptime used to compute the Lyapunov exponents. The results show that using the FTLE method produces Lyapunov exponents that are remarkably stable under the variation of the steptime and lattice size. Furthermore, it reaches such stability faster than other characteristic quantities such as energy and dissipation rate. These results remain even if the steptime is made arbitrarily small. A discrepancy is also resolved between previous measurements of the dependence on the Reynolds number of the Lyapunov exponent. The signal produced by different variables in the steady state is analyzed and the self decorrelation time is used to determine the run time needed in the simulations to obtain proper statistics for each variable. Finally, a brief analysis on MHD flows is also presented, which shows that the Lyapunov exponent is still a robust measure in the simulations, although the Lyapunov exponent scaling with Reynolds number is significantly different from that of magnetically neutral hydrodynamic fluids. * richard.ho@ed.ac.uk † andres.armua@ed.ac.uk ‡ ab@ph.ed.ac.uk 2

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Section: Mhdsupporting

confidence: 66%

Section: A Finite Time Lyapunov Exponents Methodsmentioning

confidence: 99%

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Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

Armua

Berera

2020

Phys. Rev. Fluids

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“…This approach differs from the more standard statistical approach [5] in the sense that, instead of considering averaged properties of flows, we consider the properties of individual trajectories in a suitably defined state space of the system. Through such methods, a diverse range of problems in fluid dynamics have been studied including in weather and atmospheric predictability [6][7][8], as well as for the solar wind and other magneto-hydrodynamic systems [9][10][11][12].…”

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confidence: 99%

Information production in homogeneous isotropic turbulence

Berera

Clark

2019

Phys. Rev. E

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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 finite around λ ≈ 0 contrary to a possible divergence suggested by Ruelle. The relevance of the Kolmogorov-Sinai entropy and Lyapunov spectra in comparing complex physical systems is discussed. PACS numbers: 47.27.Gs, 47.27.ek One of the most striking features of turbulent fluid flows is their seemingly random and unpredictable nature. However, since such fluids are described by the Navier-Stokes equations, which are entirely deterministic in nature, their motion cannot be truly random. In reality, turbulent flows exhibit what is commonly referred to as deterministic chaos [1,2]. The time evolution of such systems is characterized by an extreme sensitivity to initial conditions which has profound consequences for their predictability.Studying turbulence through the lens of chaos theory and the related, but wider encompassing, area of dynamical systems theory has its roots in the seminal work of Ruelle and Takens [3] alongside that of Lorenz [4]. This approach differs from the more standard statistical approach [5] in the sense that, instead of considering averaged properties of flows, we consider the properties of individual trajectories in a suitably defined state space of the system. Through such methods, a diverse range of problems in fluid dynamics have been studied including in weather and atmospheric predictability [6][7][8], as well as for the solar wind and other magneto-hydrodynamic systems [9][10][11][12].A central theme for a large proportion of the literature investigating the chaotic properties of homogeneous isotropic turbulence (HIT) is the concept of the Lyapunov exponents. Put briefly, these exponents describe the rate of exponential stretching and contracting in the state space and are thus intimately related to the aforementioned sensitivity to initial conditions. For a given system, there exist as many Lyapunov exponents as degrees of freedom, which for real world Eulerian fluid turbulence is presumably infinite. To date the majority of such work, at least in the case of direct numerical simulation (DNS), has been concerned with the calculation of * Electronic address: ab@ph.ed.ac.uk † Electronic address: daniel-clark@ed.ac.uk only the largest Lyapunov exponent [13,14]. It is, however, possible to compute multiple exponents, leading to a partial Lyapunov spectrum, to obtain a more in-depth understanding of the chaotic properties of the system. Moreover, if all positive exponents are calculated, the Kolmogorov-...

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“…These authors used statistical closures such as the test field model (TFM) and EDQNM to study this problem (Leith 1971;Leith & Kraichnan 1972). In the past few decades, the rapid increase in computational power has allowed DNS to be used in the study of the sensitivity of turbulence to initial conditions (Boffetta et al 1997;Boffetta & Musacchio 2001;Mukherjee, Schalkwijk & Jonker 2016;Boffetta & Musacchio 2017;Mohan, Fitzsimmons & Moser 2017;Berera & Ho 2018;Ho, Berera & Clark 2019;Ho, Armua & Berera 2020;Nastac et al 2017;Yoshimatsu & Ariki 2019;Li et al 2020;Clark et al 2020Clark et al , 2021a.…”

Section: Chaos and Predictability In Turbulencementioning

confidence: 99%

Critical transition to a non-chaotic regime in isotropic turbulence

Clark

Armua

et al. 2021

J. Fluid Mech.

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We study the properties of homogeneous and isotropic turbulence in higher spatial dimensions through the lens of chaos and predictability using numerical simulations. We employ both direct numerical simulations and numerical calculations of the eddy damped quasi-normal Markovian closure approximation. Our closure results show a remarkable transition to a non-chaotic regime above the critical dimension, $d_c$ , which is found to be approximately 5.88. We relate these results to the properties of the energy cascade as a function of spatial dimension in the context of the idea of a critical dimension for turbulence where Kolmogorov's 1941 theory becomes exact.

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Chaotic behavior of Eulerian magnetohydrodynamic turbulence

Cited by 6 publications

References 55 publications

Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

Information production in homogeneous isotropic turbulence

Critical transition to a non-chaotic regime in isotropic turbulence

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