Nonlinear dynamics of flexural wave turbulence

TL;DR

This study measures the nonlinear timescale in flexural wave turbulence on a thin elastic plate, confirming the theoretical scale separation but revealing finite size effects and a crossover to discrete turbulence that alters the expected energy cascade.

Contribution

It provides the first direct measurement of the nonlinear timescale in flexural wave turbulence and investigates the effects of finite size and mode discreteness on the energy cascade.

Findings

Scale separation between dissipation, nonlinearity, and wave period is confirmed.

Finite size effects cause discrete modes that disrupt the theoretical energy cascade.

A crossover from continuous to discrete turbulence is observed when nonlinear timescale matches mode frequency separation.

Abstract

The Kolmogorov-Zakharov spectrum predicted by the Weak Turbulence Theory remains elusive for wave turbulence of flexural waves at the surface of an thin elastic plate. We report a direct measurement of the nonlinear timescale $T_{NL}$ related to energy transfer between waves. This time scale is extracted from the space-time measurement of the deformation of the plate by studying the temporal dynamics of wavelet coefficients of the turbulent field. The central hypothesis of the theory is the time scale separation between dissipative time scale, nonlinear time scale and the period of the wave ($T_d>>T_{NL}>>T$). We observe that this scale separation is valid in our system. The discrete modes due to the finite size effects are responsible for the disagreement between observations and theory. A crossover from continuous weak turbulence and discrete turbulence is observed when the nonlinear time scale is of the same order of magnitude as the frequency separation of the discrete modes. The Kolmogorov-Zakharov energy cascade is then strongly altered and is frozen before reaching the dissipative regime expected in the theory.