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

The work is concerned with the fundamental problem of establishing limits o f predictability of water wave evolution. The problem is addressed by simul ating dynamics of waves on the free surface of heavy fluid using a new algo rithm based on the integrodifferential Zakharov equation. Two classes of wa ve fields were considered: gravity and capillary waves. A gravity wave syst em in the generic case exhibits chaotic behaviour; any two initially close trajectories in the phase space diverge exponentially, until the distance b etween them becomes comparable to the size of the entire manifold. The dive rgence is fast and is found to have a universal character; the exponent is proportional to the square of the characteristic wave steepness, but otherw ise shows surprisingly little variability. Due to the stochastization, a ge neric system of gravity waves loses all information on the initial conditio ns after a certain, relatively short, characteristic time tau*. For wave sl opes typical of natural basins conditions, t* is found to be of the order o f 10(3) characteristic wave periods. Evolution of capillary waves also show ed a tendency towards stochastization in most cases. However, in contrast t o gravity waves, this phenomenon is not universal and strongly depends on i nitial conditions. Besides that, the divergence of trajectories proved to b e an order of magnitude slower, in terms of the characteristic timescale. ( C) 2001 Published by Elsevier Science B.V.