Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation

Abstract

The nonlinear stage of the modulational (Benjamin -Feir) instability of unidirectional deep water surface gravity waves is simulated numerically by the firth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower order envelope models (the 3-order nonlinear Schrödinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations. IntroductionThe concept of a wave envelope is very efficient in the situations of slow wave modulations, i.e., narrow spectra. It reduces the problem of description of individual waves to a simpler problem of description of the wave modulation. The nonlinear Schrödinger equation (NLS) for water waves was first derived by Benney & Newell (1967) and then by Zakharov (1968) and Hasimoto & Ono (1972) as the leading-order approximation for weakly nonlinear waves, with dominating four-wave interactions. The corresponding nonlinear term is of the order O(ε 3 ), where the small parameter ε << 1 is related to the small wave steepness, ε = kH/2 (k is the wavenumber and H is the wave height). In this theory the dispersion is assumed to be of the same order of magnitude as the nonlinearity, Δk/k = O(ε) (Δk -is the characteristic width of the wavenumber spectrum). However, realistic sea waves are not narrow-banded, and the NLS equation turns out to be a rather rough model for the wave description; its practical applications are very limited due to this reason. Kristian Dysthe derived an improved theory № 5. 5176.2017/8.9)

On the Deterministic Prediction of Water Waves

“…However, in most of the cases, the MNLSE is applied assuming infinite water depth and is typically referred as Dysthe equation. Recently, Slunyaev and Pelinovsky [39] extended the Dysthe equation to the next order (O( 5 )) referring to higher-order Dysthe equation.…”

Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

“…A higher-order approximation was proposed by Dysthe [9] for deep water, using the perturbative method of multiple scales. It was later extended to other settings such as finite depth [1], gravity-capillary waves [14], exact linear dispersion [22], waves in the presence of dissipation [13], even to higher order [20]. The Dysthe equation and its variants have been widely used in the water wave community due to their efficiency at describing realistic waves, in particular waves with moderately large steepness.…”