Cantorian Fractal Patterns, Quantum-Like Chaos and Prime Numbers in Atmospheric Flows

TL;DR

This paper explores how atmospheric flows exhibit fractal, quantum-like chaos with long-range correlations, linking fractal structures, number theory, and prime number spectra to flow dynamics.

Contribution

It introduces a novel model connecting fractal atmospheric patterns with number theory, especially prime number distributions, to explain flow chaos.

Findings

Prime number spectrum matches eddy energy spectrum in atmospheric flows.

Fractal structures evolve through Fibonacci and Penrose tiling patterns.

Large eddy scales relate to prime number distributions.

Abstract

Atmospheric flows exhibit cantorian fractal space-time fluctuations signifying long-range spatiotemporal correlations. A recently developed cell dynamical system model shows that such non-local connections are intrinsic to quantum-like chaos governing flow dynamics. The dynamical evolution of fractal structures can be quantified in terms of ordered energy flow described by mathematical functions which occur in the field of number theory. The quantum-like chaos in atmospheric flows can be quantified in terms of the following mathematical functions / concepts: (1) The fractal structure of the flow pattern is resolved into an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure and is equivalent to a hierarchy of vortices. The incorporation of Fibonacci mathematical series, representative of ramified bifurcations, indicates ordered growth of fractal patterns. (2) The steady state emergence of progressively larger fractal structures incorporates unique primary perturbation domains of progressively increasing number equal to z/lnz where z, the length step growth stage is equal to the length scale ratio of large eddy to turbulent eddy. In number theory z/lnz gives the number of primes less than z. The model also predicts that z/lnz represents the normalised cumulative variance spectrum of the eddies and which follows statistical normal distribution. The important result of the study is that the prime number spectrum is the same as the eddy energy spectrum for quantum-like chaos in atmospheric flows.