The design of structures at sea requires knowledge on how large and steep waves can be. Although extreme waves are very rare, their consequences in terms of structural loads, such as wave impact or ringing, are critical. However, modelling the physical properties of steep waves along with their probability of occurrence in given sea states has remained a challenge. On the one hand, standard linear and weakly nonlinear wave theories are computationally efficient, but since they assume that the steepness parameter is small, they are unable to capture extreme waves. On the other hand, recent simulation methods based on CFD or fully nonlinear potential solvers are able to capture the physics of steep waves up to the onset on breaking, but their large computational cost makes it difficult to investigate rare events. Between these two extremes, the High-Order Spectral (HOS) method, which solves surface equations, is both efficient and able to capture highly nonlinear effects. It may then represent a good compromise for long simulations of steep waves. Unfortunately, it is based on a perturbation expansion where the small parameter is the wave steepness, and consequently, simulations tend to become unstable when steep wave events occur.

In this work, we investigate the properties of irregular waves simulated with a modified HOS method, in which the sea surface is described with a Lagrangian representation, i.e. by computing the position and the velocity potential of individual surface particles. By doing so, nonlinear properties of the surface elevation are simply captured by the modulation of the horizontal and vertical particle motion. The same steep wave is then described more linearly with a Lagrangian representation, which reduces the instabilities of the HOS method. The paper focuses on bi-chromatic waves and irregular waves simulated from a JONSWAP spectrum. We compare simulations performed with the standard HOS and the modified Lagrangian methods for various HOS-orders.

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