Stochastic asymmetry properties of 3D Gauss-Lagrange ocean waves with directional spreading

In the stochastic Lagrange model for ocean waves the vertical and horizontal location of
surface water particles are modeled as correlated Gaussian processes. In this article we investigate
the statistical properties of wave characteristics related to wave asymmetry in the 3D Lagrange
model. We present a modification of the original Lagrange model that can produce front-back
asymmetry both of the space waves, i.e. observation of the sea surface at a fixed time, and
of the time waves, observed at a fixed measuring station. The results, which are based on a
multivariate form of Rice’s formula for the expected number of level crossings, are given in
the form of the cumulative distribution functions for the slopes observed either by asynchronous
sampling in space, or at synchronous sampling at upcrossings and down-crossings, respectively,
of a specified fixed level. The theory is illustrated in a numerical section, showing how the
degree of wave asymmetry depends on the directional spectral spreading and on the mean wave
direction. It is seen that the asymmetry is more accentuated for high waves, a fact that may be
of importance in safety analysis of capsizing risk.