Abstract
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.
Highlights
Many fundamental concepts in nature are defined through dualities
Turbulence exists in the real world, where the ideal has to adjust itself to the constraints of the real, somewhat similar to real world fractals, manifesting fractal behavior only over a finite range of scales
While here we do not intend to go into details of how one can propose a model for the transfer rate, a statistical closure such as the eddy damped quasi-normal Markovian approximation (EDQNM) for the network nodes could be use for such purpose [23]
Summary
Many fundamental concepts in nature are defined through dualities. A duality is an opposition/complementarity, represented by the canonical example of the wave-particle duality in modern physics, and commonly described through the principle of “yin and yang” in ancient Eastern tradition or through dialectics in modern Western thought. The archetypical example of turbulence is the Navier–Stokes equation, which describes the self-advection of a divergence-free velocity field that is dissipated at small scales due to kinematic viscosity ν as: When we mix this system (via external forcing not shown in the above equation) with a spectrally localized large-scale forcing and have sufficiently small dissipation that is localized naturally at very small scales because of its form, it is well known to display universal behavior over a range of scales in between the scales of energy injection and those of dissipation. This paper is an initial attempt at providing a bridge between the turbulence and network theories from the perspective of wave-number space evolution of turbulence It poses many new questions without really trying to answer them, with a focus on laying out the foundation, and demonstrating the ideas with a few simple examples such as shell models.
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