Spiral chain models of two-dimensional turbulence

Reduced models, mirroring self-similar, fractal nature of two-dimensional turbulence, are proposed, using logarithmic spiral chains, which provide a natural generalization of shell models to two dimensions. In a turbulent cascade, where each step can be represented by a rotation and a scaling of the interacting triad, the use of a spiral chain whose nodes can be obtained by scaling and rotating an original wave vector provides an interesting perspective. A family of such spiral chain models depending on the distance of interactions can be obtained by imposing a logarithmic spiral grid with a constant divergence angle and a constant scaling factor and imposing the condition of exact triadic interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the super-golden ratio, or some small Pisot numbers. While spiral chains can represent monofractal models of a self-similar cascade, which can span a large range of wave numbers and have 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 the usual problems of shell models of two-dimensional turbulence such as the dual cascade being overwhelmed by statistical chain equipartition due to an 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.