An improved Lagrangian model for the time evolution of nonlinear surface waves

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time-evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second-order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows to correct the time-evolution of surface waves, through a modified non-linear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e., Stokes drift and mean vertical level. This results in a hybrid model, referred to as "Improved Choppy Wave Model" (ICWM) [with respect to Nouguier et al.'s (2009) 'Choppy Wave' Model for nonlinear gravity waves. J. Geo-phys. Res.: Oceans 114 (C9)], whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a High-Order Spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow-and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well-suited to provide accurate and efficient short-term ocean wave forecast (e.g., over a few peak periods). This aspect will be the object of future work.