Spiral chain models of two-dimensional turbulence

Shell models on recurrent sequences: Fibonacci, Padovan, and other series

A class of shell models is proposed where the shell variables are defined on a recurrent sequence of integer wave numbers such as the Fibonacci or the Padovan series or their variations, including a sequence made of square roots of Fibonacci numbers rounded to the nearest integer. Considering the simplest model, which involves only local interactions, the interaction coefficients can be generalized in such a way that the inviscid invariants, such as energy and helicity, can be conserved even though there is no exact self-similarity. It is shown that these models basically have identical features with standard shell models and produce the same power-law spectra, similar spectral fluxes, and analogous deviation from self-similar scaling of the structure functions, implying comparable levels of turbulent intermittency. Such a formulation potentially opens up the possibility of using shell models, or their generalizations along with discretized regular grids such as those found in direct numerical simulations, as either diagnostic tools or subgrid models. It also allows us to develop models where the wave-number shells can be interpreted as sparsely decimated sets of wave numbers over an initially regular grid. In addition to conventional shell models with local interactions that result in forward cascade, a particular a helical shell model with long-range interactions is considered on a similarly recurrent sequence of wave numbers, corresponding to the Fibonacci series, and found to result in the usual inverse cascade.

Turbulence as a Network of Fourier Modes

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.