Numerical Simulation of Rogue Waves in Coastal Waters

Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchive

Damping of large-amplitude solitary waves

Soliton damping in weakly dissipative media has been studied for several decades, usually using the asymptotic theory of the slowly-varying solitary wave solution of the Korteweg-de Vries equation. Damping then occurs according to the momentum balance equation, and a shelf is generated behind the soliton. However, for many cases in nonlinear wave dynamics, such as for internal solitary waves in the ocean and atmosphere, the nonlinearity is not so weak as is implied by the Korteweg-de Vries equation. In the next order of the perturbation theory, a higher-order equation can be obtained, which in general includes cubic nonlinearity, fifth-order linear dispersion, and nonlinear dispersion. For certain environmental conditions when the quadratic nonlinear term is small, the cubic nonlinear term becomes the major term and it should be taken into account together with the quadratic nonlinear term. The corresponding equation is known as the extended Korteweg-de Vries equation, or the Gardner equation. Our purpose here is to consider this equation supplemented with a damping term. If this term is small, the damping of a solitary wave can be studied with the use of asymptotic methods developed for perturbed solitons. Our aim is to obtain some new nontrivial results for the damping of large-amplitude solitary waves

Fission of a weakly nonlinear interfacial solitary wave at a step

The transformation of a weakly nonlinear interfacial solitary wave in an ideal two-layer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a weak transformation for propagation into deeper water. Far from the step, the wave dynamics is described by the Korteweg-de Vries equation which is fully integrable. In the vicinity of the step, the reflected and transmitted waves have soliton-like shapes, but their parameters do not satisfy the steady-state soliton solutions. Using the inverse scattering technique it is shown that the reflected wave evolves into a single soliton and dispersing radiation if the wave propagates from deep to shallow water, and only a dispersing radiation if the wave propagates from shallow to deep water. The dynamics of the transmitted wave is more complicated. In particular, if the coefficient of the nonlinear quadratic term in the Korteweg-de Vries equation is not changed in sign in the region after the step, the transmitted wave evolves into a group of solitons and radiation, a process similar to soliton fission for surface gravity waves at a step. But if the coefficient of the nonlinear term changes sign, the soliton destroys completely and transforms into radiation. The effects of cubic nonlinearity are studied in the framework of the extended Korteweg-de Vries (Gardner) equation which is also integrable. The higher-order nonlinear effects influence the amplitudes of the generated solitons if the amplitude of the transformed wave is comparable with the thickness of lower layer, but otherwise the process of soliton fission is qualitatively the same as in the framework of the Korteweg-de Vries equation

Modeling internal solitary waves in the coastal ocean

In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography can generate large-amplitude, horizontally propagating internal solitary waves. These waves often occur in regions where the waveguide properties vary in the direction of propagation. We consider the modeling of these waves by nonlinear evolution equations of the Korteweg-de Vries type with variable coefficients, and we describe how these models to describe the shoaling of internal solitary waves over the continental shelf and slope. The theories are compared with various numerical simulations

Soliton dynamics in a strong periodic field: the Korteweg-de Vries framework

Nonlinear long wave propagation in a medium with periodic parameters is considered in the framework of a variable-coefficient Korteweg-de Vries equation. The characteristic period of the variable medium is varied from slow to rapid, and its amplitude is also varied. For the case of a piecewise constant coefficient with a large scale for each constant piece, explicit results for the damping of a soliton damping are obtained. These theoretical results are confirmed by numerical simulations of the variable-coefficient Korteweg-de Vries equation for the same piecewise constant coefficient, as well as for a sinusoidally-varying coefficient. The resonance curve for soliton damping is predicted, and the maximum damping is for a soliton whose characteristic timescale is of the same order as the coefficient inhomogeneity scale. If the variation of the nonlinear coefficient is very large, and includes the a critical point where the nonlinear coefficient equals to zero, the soliton breaks and is quickly damped

Wave group dynamics in weakly nonlinear long-wave models

Wave group dynamics is studied in the framework of the extended Korteweg-de Vries equation. The nonlinear Schrodinger equation is derived for weakly nonlinear wave packets, and the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended Korteweg-de Vries equation, and for a high carrier frequency. At the boundary of this parameter space, a modified nonlinear Schrodinger equation is derived, and its steady-state solutions, including an algebraic soliton, are found. The exact breather solution of the extended Korteweg-de Vries equation is analyzed. It is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear Schrodinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behavior of the solutions of the extended Korteweg-de Vries equation, compared with those of the nonlinear Schrodinger equation

Short-lived large-amplitude pulses in the nonlinear long-wave model described by the modified Korteweg–de Vries equation

The appearance and disappearance of short-lived large-amplitude pulses in a nonlinear long wave model is studied in the framework of the modified Korteweg-de Vries equation. The major mechanism of such wave generation is modulational instability leading to the generation and interaction of the breathers. The properties of breathers are studied both within the modified Korteweg -de Vries equation, and also within the nonlinear Schrödinger equation derived by an asymptotic reduction from the modified Korteweg -de Vries for weakly nonlinear wave packets, The associated spectral problems (AKNS or Zakharov-Shabat) of the inverse-scattering transform technique also utilized. Wave formation due to this modulational instability is investigated for localized and for periodic disturbances. Nonlinear-dispersive focusing is identified as a possible mechanism for the formation of anomalously large pulses

Two-soliton interaction as an elementary act of soliton turbulence in integrable systems

Two-soliton interactions play a definitive role in the formation of the structure of soliton turbulence in integrable systems. To quantify the contribution of these interactions to the dynamical and statistical characteristics of the nonlinear wave field of soliton turbulence we study properties of the spatial moments of the two-soliton solution of the Korteweg-de Vries (KdV) equation. While the first two moments are integrals of the KdV evolution, the 3rd and 4th moments undergo significant variations in the dominant interaction region, which could have strong effect on the values of the skewness and kurtosis in soliton turbulence