A mathematical model for nonlinear gravity-capillary waves in a large temporal-spatial domain

Abstract

A mathematical model of Coupled Envelope Evolution Equations (CEEEs) has been derived by Li (2022) for the propagation of nonlinear surface gravity waves. Different from the High-Order Spectral Method (HOSM) (see, e.g., Dommermuth & Yue 1987; West 1987), this newly derived model proposes to use a new pair of canonical variables introduced in the work as the main unknown parameters in the approximate Hamiltonian dynamic equations for surface gravity waves. The canonical conjugates are the envelope of the velocity potential at the free water surface and the surface displacement. They are parameters which are slowly varying in both space and time, similar to the envelope used in a nonlinear Schrödinger equation (NLSE)-based model (see, Li 2021 among others). In the limiting cases of weakly nonlinear monochromatic and irregular waves, it is shown in Li (2022) that the CEEEs can recover the analytical results of the Stokes wave theory (Fenton 1985) and the semi-analytical framework by Li & Li (2021).  Similar to the HOSM, the new model is based on a perturbation expansion and can account for the physics up to arbitrary order in wave steepness. In contrast, it has a semi-analytical feature as it is analytical for the evolution of linear waves but requires additional numerical implementations when wave nonlinearity is accounted for. The new model is especially suitable for the extremely long-term evolution of surface waves in a very large domain in space, which is more so for waves with a narrower bandwidth. In this work, the CEEEs are explored for the nonlinear evolution of gravity-capillary waves on a finite water depth. The finite water waves in the neighborhood of kh ≈ 1.363 are investigated, where k and h denote the characteristic wavenumber and constant water depth, respectively. The roles of the nonlinear forcing of mean flows due to a moderately steep wave group in extremely large wave events are examined.Key words: waves/free-surface flow, gravity-capillary waves ReferencesDommermuth, D. G & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.Fenton, J.D. 1985 A fifth-order Stokes theory for steady waves. J. Waterway, Port, Coast. & Ocean Eng. 111 (2), 216–234.Li, Y. 2022 On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves. submitted to J. Fluid Mech (under review).Li, Y. & Li, X. 2021 Weakly nonlinear broadband and multidirectional surface waves on an arbitrary depth: A framework, Stokes drift, and particle trajectories. Phys. Fluids 33 (7), 076609.Li, Y. 2021 Three-dimensional surface gravity waves of a broad bandwidth on deep water. J. Fluid Mech. 926, 1–43.West, B. J., Brueckner, K A, Janda, R. S., Milder, D M & Milton, R. L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res.: Oceans 92 (C11), 11803–11824.