T. G. Talipova

A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone

A nonlinear model is developed, based on the rotated-modified extended Korteweg-de Vries (reKdV) equation, of the evolution of an initially sinusoidal long wave in the coastal zone, representing an internal tide, into nonlinear waves including internal solitary waves. The coefficients of the basic equation are calculated using observed conditions for the north west shelf (NWS) of Australia. The roles of both quadratic and cubic nonlinearity, the Earths rotation, and frictional dissipation are discussed. The combined action of nonlinearity and rotation leads to a number of intersting features in the wave form including solitons of both polarities, “thick” solitons, and sharp waves with steep fronts. It is shown that rotation is important for modelling the evolution of the internal tide, even for the relatively low latitude on the NWS of 20°S. Rotation increases the phase speed of the long internal tide, reduces the number of internal solitary waves that form from a long wave, and changes the form of the waves. The effects of nonlinearity on the vertical modal structure of the internal waves are also discussed. Results of numerical simulations are compared with current and temperature observations of the internal wave field on the NWS which show many of the features produced by the generalized KdV model.

Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity

Abstract Solitary wave transformation in a zone with a sign-variable coefficient for the quadratic nonlinear term is studied in the framework of the variable-coefficient extended Korteweg–de Vries equation. This situation can be realised for internal waves in a stratified ocean, when the pycnocline lies midway between the sea bed and the sea surface. The effects of the cases when the coefficient of the cubic nonlinear term can be either positive or negative are investigated. For small-amplitude solitary waves, previous results (when the cubic nonlinear term is ignored) are confirmed; the initial solitary wave is destroyed, and solitary waves of the opposite polarity are generated after passage through the turning point (i.e. the location where the coefficient of the quadratic nonlinear term is zero). For large-amplitude solitary waves the cubic nonlinear term, and particularly its sign, influence significantly both the polarity and amplitude of the resulting solitary waves. If the cubic nonlinear term has a negative coefficient, the amplitude of the ‘large’ terminal wave is comparable with the initial value (it is less than 50% when the cubic nonlinear term is ignored). If the cubic nonlinear term has a positive coefficient, the result depends on the initial wave amplitude. Large-amplitude solitary waves pass through the transition zone keeping the solitary wave shape and its polarity. Moderate-amplitude solitary wave are destroyed and transformed to a strongly pulsating wave packet (breather).

Nonlinear wave focusing on water of finite depth

Abstract The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of the nonlinear Schrodinger equation and the Davey–Stewartson system, describing 2+1-dimensional surface wave groups on water of finite depth. In the 2+1-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The mechanism of dispersive focusing is compared with the wave enhancement due to the Benjamin–Feir instability, which is found to be extremely sensitive with respect to weak random perturbations.

Focusing of nonlinear wave groups in deep water

The freak wave phenomenon in the ocean is explained by the nonlinear dynamics of phase-modulated wave trains. It is shown that the preliminary quadratic phase modulation of wave packets leads to a significant amplification of the usual modulation (Benjamin-Feir) instability. Physically, the phase modulation of water waves may be due to a variable wind in storm areas. The well-known breather solutions of the cubic Schrödinger equation appear on the final stage of the nonlinear dynamics of wave packets when the phase modulation becomes more uniform.

Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.

Transformation of a soliton at a point of zero nonlinearity

The transformation of a soliton in a zone with a sign-changing nonlinearity has been investigated on the basis of the Korteweg-de Vries equation. It is shown that after passage through the critical zone a soliton with opposite polarity is formed in the wave field and, together with the previously known mechanism of secondary-soliton generation as a result of a pedestal formed at the adiabatic stage, there also exists another mechanism which is associated with the transformation of the wave in a zone of variable nonlinearity after the critical point. It is shown that both mechanisms make approximately the same contribution to the secondary-soliton energetics.

Propagation of solitary internal waves in two-layer ocean of variable depth

The propagation of internal solitons of moderate amplitude in a two-layer ocean of variable depth is studied in terms of the Gardner and Euler equations. An analytical solution is obtained with the use of asymptotic expansions on a small parameter (bottom slope). The theoretical results are compared with the numerical modeling results. The possibility of soliton shape preservation during pulse propagation is discussed. It is obtained that, as the initial amplitude increases, the pulse deviates from the soliton shape more rapidly.

Penetration of internal waves into the ocean’s thickness

The data of beam propagation of internal waves into the ocean are confirmed by several analytical solutions of the linear long wave theory. The obtained solutions are applied to calculate the mode of the internal waves.

Modulation instability of long internal waves with moderate amplitudes in a stratified horizontally inhomogeneous ocean

It has been shown that the known effect of the modulation instability of wavepackets can occur for long internal waves with a moderate amplitude in a stratified horizontally inhomogeneous ocean under certain conditions on the vertical structure of the density field and flows. The numerical calculations that have been performed for the transformation of wavepackets in some regions of the World Ocean indicate the possibility of the appearance of rogue waves in the bulk of the Ocean.

Modeling the dynamics of intense internal waves on the shelf

The transformation of the internal wave packet during its propagation over the shelf of Portugal was studied in the international experiment EU MAST II MORENA in 1994. This paper presents the results of modeling of the dynamics of this packet under hydrological conditions along the pathway of its propagation. The modeling was performed on the basis of the generalized Gardner-Ostrovskii equation, including inhomogeneous hydrological conditions, rotation of the Earth, and dissipation in the bottom boundary layer. We also discuss the results of the comparison of the observed and simulated forms and phases of individual waves in a packet at reference points.