Single-wavenumber Representation of Nonlinear Energy Spectrum in Elastic-Wave Turbulence of {F}\"oppl-von {K}\'arm\'an Equation: Energy Decomposition Analysis and Energy Budget

TL;DR

This paper introduces a novel single-wavenumber representation of the nonlinear energy spectrum in elastic-wave turbulence governed by the F"oppl-von K"arm"an equation, enabling detailed energy budget analysis and revealing energy transfer mechanisms.

Contribution

It provides the first analytical expressions for energy budgets in wave turbulence systems and demonstrates their application through numerical simulations of elastic-wave turbulence.

Findings

Kinetic and bending energies are comparable at large wavenumbers.

Stretching energy dominates at small wavenumbers, indicating strong nonlinearity.

A clear forward energy cascade is observed in the system.

Abstract

A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spectrum is found in elastic-wave turbulence governed by the F\"oppl-von K\'arm\'an (FvK) equation. The representation enables energy decomposition analysis in the wavenumber space, and analytical expressions of detailed energy budget in the nonlinear interactions are obtained for the first time in wave turbulence systems. We numerically solved the FvK equation and observed the following facts. Kinetic and bending energies are comparable with each other at large wavenumbers as the weak turbulence theory suggests. On the other hand, the stretching energy is larger than the bending energy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode $a_{\bm{k}}$ and its companion mode $a_{-\bm{k}}$ is observed at the small wavenumbers. Energy transfer shows that the energy is input into the wave field through stretching-energy transfer at the small wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the large wavenumbers. A total-energy flux consistent with the energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.