Turbulence in Noninteger Dimensions by Fractal Fourier Decimation

Frisch, U.; Pomyalov, Anna; Procaccia, Itamar; Ray, Samriddhi Sankar

doi:10.1103/physrevlett.108.074501

Cited by 66 publications

(86 citation statements)

References 17 publications

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“…For example, for approaches based on Galerkin projections, we cite the idea of keeping modes in Fourier space corresponding to wavenumbers logarithmically equispaced (Grossmann et al 1996), or on a Fractal set (Frisch et al 2012). Similarly, Laval, Dubrulle & Nazarenko (2001) proposed multiscale triad pruning, in order to distinguish the role played by local and non-local Fourier interactions in the growth of intermittency.…”

Section: Discussionmentioning

confidence: 99%

Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

2013

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We investigate the transfer properties of energy and helicity fluctuations in fully developed homogeneous and isotropic turbulence by changing the nature of the nonlinear Navier-Stokes terms. We perform a surgery of all possible interactions, by keeping only those triads that have sign-definite helicity content. In order to do this, we apply an exact decomposition of the velocity field in a helical Fourier basis, as first proposed by Constantin & Majda (Commun. Math. Phys, vol. 115, 1988, p. 435) and exploited in great detail by Waleffe (Phys. Fluids A, vol. 4, 1992, p. 350), and we evolve the Navier-Stokes dynamics keeping only those velocity components carrying a well-defined (positive or negative) helicity. The resulting dynamics preserves translational and rotational symmetries but not mirror invariance. We give clear evidence that this three-dimensional homogeneous and isotropic chiral turbulence is characterized by a stationary inverse energy cascade with a spectrum E back (k) ∼ k −5/3 and by a direct helicity cascade with a spectrum E forw (k) ∼ k −7/3 . Our results are important to highlight the dynamics and statistics of those subsets of all possible Navier-Stokes interactions responsible for reversal events in the energy-flux properties, and demonstrate that the presence of an inverse energy cascade is not necessarily connected to a two-dimensionalization of the flow. We further comment on the possible relevance of such findings to flows of geophysical interest under rotations and in thin layers. Finally we propose other innovative numerical experiments that can be achieved by using a similar decimation of degrees of freedom.

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Section: Discussionmentioning

confidence: 99%

Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

2013

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“…[18] we define the fractal Fourier decimation operator P D acting on a generic field u(x, t) = k e ikxû k (t) as:…”

Section: The Burgers Equation On a Fractally Decimated Fourier Setmentioning

confidence: 99%

“…In a recent work [18], the idea of fractal decimation was introduced, with the aim of studying the evolution of the Navier-Stokes equations on an effective dimension D out of an integer ddimensional embedding manifold. This is done by introducing a quenched mode-reduction in Fourier space such that in a sphere of radius k the number of modes that are involved in the dynamics scale as k D (where D < d is the effective fractal dimension of the system) for large k [18,19]. This approach allows us to decimate the number of triad interactions in Fourier space as a function of the wavenumbers involved as well as to consider the problem in non-integer, fractal dimensions D. In Ref.…”

Section: Introductionmentioning

confidence: 99%

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Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

et al. 2016

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We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension D.We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing D. We find that a very small percentage of mode-reduction (D 1) is enough to destroy most of the characteristics of the original non-decimated equation. In particular, we observe a suppression of intermittent fluctuations for D < 1 and a quasi-singular transition from the fully intermittent (D = 1) to the non-intermittent case for D 1. Our results indicate that the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the result of highly entangled correlations amongst all Fourier modes.

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“…Recent work on turbulence on fractal decimated Fourier space [32,33] show the role and the importance of equipartition in reduced Fourier space and its connection to cascades. In general if a binning is used, one that contains equal number of modes per bin gives the same equipartition solution as the original system [27].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

et al. 2016

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A detailed systematic derivation of a logarithmically discretized model for two dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with non-local interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seems to dominate over the dual cascade, which is a common problem of two dimensional shell models.

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Turbulence in Noninteger Dimensions by Fractal Fourier Decimation

Cited by 66 publications

References 17 publications

Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

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