Abstract
A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.
Highlights
An important task in the study of surface gravity waves is the development of the theoretical description of flow fields
This paper aims to make attempts to fill in gaps in the aforementioned two aspects, through the development of a new framework that would allow for the study of three-dimensional waves of a broad bandwidth and would take into account the structures of flow fields below the surface without compromising much the computational efficiency of an nonlinear Schrödinger equation (NLSE)-based model
This section focuses on three aspects in order to explore and validate the theory presented, including the second-order solutions presented in § 3, sideband instability of Stokes waves and the roles of wave directionality
Summary
An important task in the study of surface gravity waves is the development of the theoretical description of flow fields. Theoretical models of surface gravity waves are 926 A34-1. Theoretical findings have contributed to providing possible explanations of the formation mechanism of extremely large waves that appear suddenly with much larger amplitude than their surroundings, known as ‘rogue’ or ‘freak’ waves. A few examples of the possible mechanisms proposed are modulational or Benjamin–Feir instability in deep water (Onorato et al 2009), refraction by ambient currents or bathymetry (Janssen & Herbers 2009; Onorato, Proment & Toffoli 2011) and effects on weakly nonlinear waves of a depth change in shallow or intermediate water depth (Trulsen et al 2020; Li et al 2021a,b)
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