Predictability of surface gravity waves

This work concerns the fundamental problem of establishing the limits of predictability for the evolution of surface gravity waves. For a gravity wave system, in the absence of strong irreversible process as wave breaking or turbulent mixing, the unpredictability of evolution is linked to inaccuracies in the initial data conditions and the question can be addressed by numerical simulation. In this thesis the problem is faced by simulating dynamics of gravity waves on the surface of sea water, considering the fluid as incompressible, inviscid and irrotational, using the fourth-order Runge-Kutta algorithm on the four-wave discretized Zakharov equation. A gravity wave system can exhibits chaotic behaviour: after a characteristic time Tp, as known as ¿time of predictability¿, any two initially infinitesimally close trajectories in the phase system space diverge exponentially, until the distance between them becomes comparable to the size of the entire manifold. This entails that the system loses all information on the initial conditions, and it can be studied only statistically. In order to study the limits of the deterministic approach and estimate the time of predictability of the system, in this thesis we evaluate the rate of divergence of phase trajectories through the numerical calculation of the first Lyapunov exponent (LE). This is done by using an efficient numerical technique, the algorithm of Benettin, observing the behaviour of the LE varying the number of interacting modes and the initial amplitudes in the Zakharov equation.