Weakly nonlinear broadband and multi-directional surface waves on an arbitrary depth: A framework, Stokes drift, and particle trajectories

Abstract

Surface gravity waves in coastal waters are broadband and multi-directional, whose quadratic properties are of considerable engineering and scientific interest. Based on a Stokes expansion and an envelope-type framework, a new semi-analytical approach is proposed in this paper for the description of weakly nonlinear broadband and multi-directional surface waves. This approach proposes solving for the second-order wave fields through the separation of harmonics, by using a Fast Fourier transform and a time integration method. Different from some other methods, e.g., the High-Order Spectral method, the approach introduces a spectral shift for the superharmonic waves, leading to computationally efficient and accurate spectral predictions. The approach has been validated through comparisons with the results based on Dalzell [“A note on finite depth second-order wave–wave interactions,” Appl. Ocean Res. 21, 105–111 (1999)]. An envelope-type framework for the fast prediction of particle trajectories and Stokes drifts up to the second order in wave steepness is also derived in this paper, based on the semi-analytical approach. This paper shows that the results based on a narrowband assumption lead to underestimates of Stokes drift velocities driven by broadband unidirectional focused wave groups. The cases, examined for particle trajectories below broadband unidirectional focused wave groups, show that a larger bandwidth and water depth can enhance the differences in the net mean horizontal displacement of particles at water surface relative to these at seabed.