Golden Ratio Phenomenon of Random Data Obeying von Karman Spectrum

Abstract

von Karman originally deduced his spectrum of wind speed fluctuation based on the Stokes-Navier equation. Taking into account, the practical issues of measurement and/or computation errors, we suggest that the spectrum can be described from the point of view of the golden ratio. We call it the golden ratio phenomenon of the von Karman spectrum. To depict that phenomenon, we derive the von Karman spectrum based on fractional differential equations, which bridges the golden ratio to the von Karman spectrum and consequently provides a new outlook of random data following the von Karman spectrum in turbulence. In addition, we express the fractal dimension, which is a measure of local self-similarity, using the golden ratio, of random data governed by the von Karman spectrum.