Modulation instability and capillary wave turbulence

Formation of turbulence of capillary waves is studied in laboratory experiments. The spectra show multiple exponentially decreasing harmonics of the parametrically excited wave which nonlinearly broaden with the increase in forcing. Spectral broadening leads to the development of the spectral continuum which scales as ∝ f, in agreement with the weak turbulence theory (WTT). Modulation instability of capillary waves is shown to be responsible for the transition from discrete to broadband spectrum. The instability leads to spectral broadening of the harmonics, randomization of their phases, it isolates the wave field from the wall, eventually allows the transition from 4to 3-wave interactions as the dominant nonlinear process, thus creating the prerequisites assumed in WTT. Copyright c © EPLA, 2010 Broad frequency and wave number spectra characterize many turbulent systems. When energy is injected into a system in a limited range of wave numbers or frequencies, nonlinear interactions extend this range. In the flows described by the Navier-Stokes equations such spectral energy transfer at large Reynolds numbers is attributed to the energy cascades. The idea of the cascade also lies in the center of the weak wave turbulence theory (WTT) [1], where instead of strong nonlinearity of the Navier-Stokes equation, the cascade is driven by weakly interacting waves having almost random phases. Among the best known applications of the WTT to the surface waves is the theory of capillary wave turbulence of Zakharov and Filonenko [2]. The theory assumes that the dispersion relation of capillary waves ωk = (αk /ρ) (where α is the surface tension and ρ is the liquid density) allows simultaneous satisfaction of the matching rules for the wave triads in the frequency and in the wave number domains “for any possible choice of the wave vectors” k1, k2, k3 [3]: