On the predictability of evolution of surface gravity and gravity–capillary waves

Abstract

The work is concerned with the fundamental problem of establishing limits of predictability of water wave evolution. The problem is addressed by simulating dynamics of waves on the free surface of heavy fluid using a new algorithm based on the integrodifferential Zakharov equation. Two classes of wave fields were considered: gravity and capillary waves. A gravity wave system in the generic case exhibits chaotic behaviour; any two initially close trajectories in the phase space diverge exponentially, until the distance between them becomes comparable to the size of the entire manifold. The divergence is fast and is found to have a universal character; the exponent is proportional to the square of the characteristic wave steepness, but otherwise shows surprisingly little variability. Due to the stochastization, a generic system of gravity waves loses all information on the initial conditions after a certain, relatively short, characteristic time τ ∗ . For wave slopes typical of natural basins conditions, τ ∗ is found to be of the order of 10 3 characteristic wave periods. Evolution of capillary waves also showed a tendency towards stochastization in most cases. However, in contrast to gravity waves, this phenomenon is not universal and strongly depends on initial conditions. Besides that, the divergence of trajectories proved to be an order of magnitude slower, in terms of the characteristic timescale.