Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface Waves

Abstract

A two dimensional model is introduced to study pattern formation, secondary instabilities and the transition to spatiotemporal chaos (weak turbulence) in parametric surface waves. The stability of a periodic standing wave state above onset is studied against Eckhaus, zig-zag and transverse amplitude modulations (TAM) as a function of the control parameter $\varepsilon$ and the detuning. A mechanism leading to a finite threshold for the TAM instability is identified. Numerical solutions of the model are in agreement with the stability diagram, and also reveal the existence of a transition to spatiotemporal chaotic states at a finite $\varepsilon$. Power spectra of temporal fluctuations in the chaotic state are broadband, decaying as a power law of the frequency $\omega^{-z}$ with $z \approx 4.0$.