# Spiral chains in wavenumber space of two dimensional turbulence

## TL;DR

This paper introduces a novel spiral chain model in wave-number space to describe the self-similar, fractal nature of two-dimensional turbulence, capturing the dual cascade phenomenon through energy evolution.

## Contribution

It proposes a new spiral chain framework based on energy evolution that effectively models the dual cascade in 2D turbulence, overcoming limitations of previous Fourier-based models.

## Key findings

- Spiral chains can represent mono-fractal self-similar cascades.
- The proposed energy-based spiral chain model recovers the dual cascade behavior.
- Scaling factors relate to ratios like the plastic and supergolden ratios.

## Abstract

Abstract Self-similar, fractal nature of turbulence is discussed in the context of two dimensional turbulence, by considering the fractal structure of the wave-number domain using spirals. In loose analogy with phyllotaxis in plants, each step of the cascade can be represented by a rotation and a scaling of the interacting triad. Using a constant divergence angle and a constant scaling factor, one obtains a family of such fractals depending on the distance of interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the supergolden ratio or some small Pisot numbers. While spiral chains can represent mono-fractal models of self-similar cascade, which can span a very large range in wave-number domain with good angular coverage, it is also possible that spiral chains or chains of consecutive triads play an important role in the cascade. As numerical models, the spiral chain models based on decimated Fourier coefficients have problems such as the dual cascade being overwhelmed by statistical chain equipartition due to almost stochastic evolution of the complex phases. A generic spiral chain model based on evolution of energy is proposed, which is shown to recover the dual cascade behavior in two-dimensional turbulence.