Experimental Quantization of Exact Wave Turbulence II: Temporal Quantization

Abstract

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 already treated theoretical quantization of the deterministic chaos in invariant structures and experimental quantization in spatial and temporal eigenfunctions using the inhomogeneous Fourier expansions. Theoretical quantization of the stochastic chaos and the wave turbulence has been considered together with experimental quantization of the stochastic chaos and the wave turbulence in spatial x -eigenfunctions. In the present paper, experimental quantization of the stochastic chaos and the wave turbulence in temporal eigenfunctions proceeds experimental quantization of the stochastic chaos and the wave turbulence in the spatial x -eigenfunctions. The method of inhomogeneous Fourier expansions in the spatial x -eigenfunctions has been extended to deterministic-random, random-deterministic, random, external, internal, and temporal eigenfunctions. Exact solutions for quantized oscillons and pulsons depend on 1-, 2-, 3-, 4-, 5-, 6-, 8-, 9-, 12-, 13-, 16-, and 32-tuples of the temporal eigenfunctions. Similar to spatial quantization, 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 are computed with the help of the random model of oscillatory cn-noise. Computation is performed using experimental and theoretical programming in Maple. The obtained results show a strong dependence of the quantized oscillons and pulsons on the Reynolds number. Contrary to spatial quantization, where oscillons and pulsons are displayed as multi-mode waves, the quantized oscillons and pulsons in the case of temporal quantization are visualized as fringed waves, which qualitatively correlate with experimental data.