Joint statistical distribution of surface elevation and water velocity for irregular waves propagating over a shoal

In Trulsen et al. (2020) we reported that when irregular waves propagate over a shoal the extreme wave statistics of surface elevation and water velocity can be dramatically different:  The surface elevation can have a local maximum of kurtosis some distance into the shallower part of the shoal, while it relaxes to normality after the shoal.  The velocity field can have a local maximum of kurtosis after the shoal, while it is close to normality over the shallower part of the shoal.  These two fields clearly do not coincide regarding the location of increased probability of extreme waves.

Here we consider the evolution of the irregular waves over the shoal as a multivariate stochastic process, with a view to reveal the evolution of the joint statistical distribution of surface elevation and water velocity.  Higher order multivariate moments, coskewness and cokurtosis, more commonly seen in mathematical finance theory, are employed to describe the joint extreme wave statistical distribution of the elevation and the velocity.

Trulsen, K., Raustøl, A., Jorde, S. & Rye, L. B. (2020) Extreme wave statistics of longcrested irregular waves over a shoal.  J. Fluid Mech. 882, R2.