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

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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 us to correct the time evolution of surface waves, through a modified nonlinear 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 the 'improved choppy wave model' (ICWM) (with respect to Nouguier et al.'s J. Geophys. Res., vol. 114, 2009, p. C09012), 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- A nd 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 providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

Publication Title

Journal of Fluid Mechanics