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Open AccessArticle

Turbulence as a Network of Fourier Modes

1
Laboratoire de Physique des Plasmas, CNRS, Ecole Polytechnique, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91120 Palaiseau, France
2
Southwestern Institute of Physics, Chengdu 610041, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 530; https://doi.org/10.3390/math8040530
Received: 3 March 2020 / Revised: 20 March 2020 / Accepted: 22 March 2020 / Published: 3 April 2020
(This article belongs to the Special Issue Turbulence Modeling)
Turbulence is the duality of chaotic dynamics and hierarchical organization of a field over a large range of scales due to advective nonlinearities. Quadratic nonlinearities (e.g., advection) in real space, translates into triadic interactions in Fourier space. Those interactions can be computed using fast Fourier transforms, or other methods of computing convolution integrals. However, more generally, they can be interpreted as a network of interacting nodes, where each interaction is between a node and a pair. In this formulation, each node interacts with a list of pairs that satisfy the triadic interaction condition with that node, and the convolution becomes a sum over this list. A regular wavenumber space mesh can be written in the form of such a network. Reducing the resolution of a regular mesh and combining the nearby nodes in order to obtain the reduced network corresponding to the low resolution mesh, we can deduce the reduction rules for such a network. This perspective allows us to develop network models as approximations of various types of turbulent dynamics. Various examples, such as shell models, nested polyhedra models, or predator–prey models, are briefly discussed. A prescription for setting up a small world variants of these models are given. View Full-Text
Keywords: turbulence; navier-stokes; networks turbulence; navier-stokes; networks
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Gürcan, Ö.D.; Li, Y.; Morel, P. Turbulence as a Network of Fourier Modes. Mathematics 2020, 8, 530.

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