Effects of discretization of the spectrum in water-wave turbulence

Abstract

The effects of discretization of the spectrum on the evolution of weak turbulence of surface gravity waves are investigated by direct numerical simulations. Discretization of the spectrum is implemented by imposing periodic boundary conditions on the horizontal plane. Initial wave fields are constructed so that they have the JONSWAP spectrum, and the nonlinear energy transfer among component waves is estimated by following their evolution deterministically according to the primitive governing equations. It is found that the discretized system can produce the same nonlinear energy transfer as that predicted theoretically for a continuous spectrum, even when the distribution of component waves on the wavenumber plane is so sparse that the corresponding area of the water surface on the horizontal plane is just 4 λ p×4 λ p, with λ p being the wavelength corresponding to the peak of the spectrum, provided that the ensemble averaging is taken over a sufficient number of realizations. The regime of “frozen” turbulence, which is known to appear in the weak turbulence of capillary waves with a discretized spectrum, is not observed in the case of gravity waves. This difference in the effects of discretization of the spectrum on the weak turbulence of capillary waves and that of gravity waves is discussed in relation to the number of trios or quartets of component waves which satisfy the conditions for quasi-resonant nonlinear interactions.