Experimental Quantization of Exact Wave Turbulence I: Spatial Quantization

Abstract

In previous articles, the exact solutions for deterministic chaos, stochastic chaos, and wave turbulence have been developed in terms of exponential oscillons and pulsons, which are governed by the nonstationary three-dimensional Navier-Stokes equations. We have later considered theoretical quantization of the deterministic chaos in invariant structures and experimental quantization in spatial and temporal eigenfunctions with the help of inhomogeneous Fourier expansions. The study of exact wave turbulence was also continued with the theoretical quantization of stochastic chaos and wave turbulence. The current paper proceeds with experimental quantization of the stochastic chaos and the wave turbulence in spatial x -eigenfunctions. The method of inhomogeneous Fourier expansions in the deterministic eigenfunctions has been extended to deterministic-random, random-deterministic, random, external, and internal eigenfunctions. The previous results on theoretical quantization in invariant structures have been confirmed, analyzed, and visualized in this work using experimental quantization in the novel eigenfunctions. Arguments of exact solutions for quantized oscillons and pulsons are given by 1-, 2-, 3-, 4-, 5-, 6-, 8-, 12-, 15-, 16, and 32-tuples of the spatial eigenfunctions. The exact solutions are grouped into the vector, deterministic-random, external oscillons, the vector, random-deterministic, external oscillons, the vector, deterministic-random, internal oscillons, the vector, turbulent, external oscillons, the vector, turbulent, diagonal oscillons, the vector, turbulent, internal, oscillons, and the vector, turbulent pulsons. We compute independent random parameters with the help of the random model of oscillatory cn-noise. Computation is performed by experimental and theoretical programming in Maple. The obtained results demonstrate a strong dependence of the quantized oscillons and pulsons on the Reynolds number.