Proof of Modulational Instability of Stokes Waves in Deep Water (original) (raw)

Modulational instability and wave amplification in finite water depth

Natural Hazards and Earth System Science, 2014

The modulational instability of a uniform wave train to side band perturbations is one of the most plausible mechanisms for the generation of rogue waves in deep water. In a condition of finite water depth, however, the interaction with the sea floor generates a wave-induced current that subtracts energy from the wave field and consequently attenuates the instability mechanism. As a result, a plane wave remains stable under the influence of collinear side bands for relative depths kh ≤ 1.36 (where k is the wavenumber of the plane wave and h is the water depth), but it can still destabilise due to oblique perturbations. Using direct numerical simulations of the Euler equations, it is here demonstrated that oblique side bands are capable of triggering modulational instability and eventually leading to the formation of rogue waves also for kh ≤ 1.36. Results, nonetheless, indicate that modulational instability cannot sustain a substantial wave growth for kh < 0.8.

Modulational instability in some shallow water wave models

2018

Modulational or Benjamin-Feir instability is a well known phenomenon of Stokes' periodic waves on the water surface. In this dissertation, we study this phenomenon for periodic traveling wave solutions of various shallow water wave models. We study the spectral stability or instability with respect to long wave length perturbations of small amplitude periodic traveling waves of shallow water wave models like Benjamin-Bona-Mahony and Camassa-Holm equations. We propose a bi-directional shallow water model which generalizes Whitham equation to contain the nonlinearities of nonlinear shallow water equations. The analysis yields a modulational instability index for each model which is solely determined by the wavenumber of underlying periodic traveling wave. For a fixed wavenumber, the sign of the index determines modulational instability. We also includes the effects of surface tension in full-dispersion shallow water models and study its effects on modulational instability.

Modulational instability and rogue waves in finite water depth

Natural Hazards and Earth System Sciences Discussions, 2013

The mechanism of side band perturbations to a uniform wave train is known to produce modulational instability and in deep water conditions it is accepted as a plausible cause for rogue wave formation. In a condition of finite water depth, however, the interaction with the sea floor generates a wave-induced current that subtracts energy from the wave field and consequently attenuates this instability mechanism. As a result, a plane wave remains stable under the influence of collinear side bands for relative water depths kh ≤ 1.36 (where k represents the wavenumber of the plane wave and h the water depth), but it can still destabilise due to oblique perturbations. Using direct numerical simulations of the Euler equations, it is here demonstrated that oblique side bands are capable of triggering modulational instability and eventually leading to the formation of rogue waves also for kh ≤ 1.36. Results, nonetheless, indicates that modulational instability cannot sustain a substantial wave growth for kh < 0.8.

Modulational instability in a full-dispersion shallow water model

Studies in Applied Mathematics

We propose a shallow water model which combines the dispersion relation of water waves and the Boussinesq equations, and which extends the Whitham equation to permit bidirectional propagation. We establish that its sufficiently small, periodic wave train is spectrally unstable to long wavelength perturbations, provided that the wave number is greater than a critical value, like the Benjamin-Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability away from the origin in the spectral plane to the leading order in the amplitude parameter. We discuss the effects of surface tension on the modulational instability. The results agree with those from formal asymptotic expansions and numerical computations for the physical problem.

Modulations of Deep Water Waves and Spectral Filtering

Studies in Applied Mathematics, 2003

Modulations of deep water waves are studied by a new formalism of spectral filtering. For single-mode dynamics, spectral filtering results in computable equations, which are counterpart to the nonlinear Schrödinger (NLS) equations. An essential feature of new equations is that bandwidth limitation is decoupled from small-amplitude assumption. The filtered equations have a substantially broader range of validity than the NLS equations, and may be viewed as intermediate between the NLS and Zakharov equations. The new single-mode equations reproduce exactly the conditions for nonlinear four-wave resonance ("figure 8" of Phillips [1]) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean [2].

Large transient waves generated through modulational instability in deep water

Journal of Hydrodynamics, Ser. B, 2010

The long-term evolution of nonlinear wave train in deep water with varied initial wave steepness is investigated experimentally in a super wave flume (300 m long, 5 m wide, 5.2 m deep). The initial wave train is the combination of one carrier wave and a pair of imposed sideband components. Increasing modulation of wave train is observed due to sideband instability until a critical value which either initiates wave breaking or reaches the maximum modulation. The observed maximum local wave steepness increases rapidly with the increase of the initial wave steepness, and levels off at initial wave steepness roughly equal to 0.15 despites that the data exhibits a little scattering. The normalized crest elevation at peak modulation increases rapidly with initial wave steepness and approached a maximum value almost equal to 3.5 which corresponds to initial wave steepness around 0.15 c c k a = . The results reveal that the large transient wave such as freak wave could be generated during the propagation of nonlinear wave trains in deep water through sideband instability.

The spectrum of finite depth water waves

European Journal of Mechanics - B/Fluids, 2014

In this contribution we study the spectrum of periodic traveling gravity waves on a two-dimensional fluid of finite depth. We extend the stable and highly accurate method of Transformed Field Expansions to the finite depth case in the presence of both simple and repeated eigenvalues, and then numerically simulate the changes in the spectrum as the wave amplitude is increased. We also calculate explicitly the first non-zero correction to the flat-water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disc of analyticity of the spectrum. In addition to computations of the spectrum, we also compute the radius of the disc of analyticity of the spectrum-the amplitude boundary beyond which neither the asymptotics nor the TFE method is applicable. We observe an instability which is analytically connected to the flat state for kh ∈ (0.855, 1).

On weakly nonlinear modulation of waves on deep water

Physics of Fluids, 2000

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.