Modulational instability and statistical properties of irregular waves in finite water depth

Abstract

Modulational instability is one of the possible basic mechanisms for the generation of rogue waves in deep water. Under the condition of finite depth, this instability is reduced and disappears as k1h<1.363 (k1 is the wavenumber of the carrier wave and h is the water depth). In this study, we addressed this phenomenon through three approaches, theoretical analysis of the Zakharov’s equation and the nonlinear Schrödinger equation and direct numerical simulations of the Euler equation. Their results on the modulational instability over arbitrary depths, particularly the growth rates of sidebands, are analysed and compared. In addition, the consequences of the suppression of modulational instability for realistic ocean waves are illustrated by direct simulations of long-crested, irregular waves at finite water depth. The results show that many phenomena related to the third-order nonlinearity (i.e. the downshift of the spectral peak frequency, the enhancement of kurtosis of surface elevations and rouge wave occurrence probability) are suppressed with decreasing depth k1h and eventually cease to exist as k1h<1.363. Considering the dependence of wave evolution on water depth, two Benjamin–Feir indices for arbitrary depths derived from Zakharov’s equation and nonlinear Schrödinger equation are used as indicators to estimate the modulational instability effect on nonlinear wave fields. Our results confirm that both parameters provide reasonable predictions of the modulational instability and probability of rogue wave occurrence.