A brief overview of works on transformation of surface and internal gravity waves over a bottom step is presented. The generalization of Lamb formulae for the transformation coefficients derived in the longwave approximation is discussed for waves of arbitrary length in the fluid of a finite length. The rigorous approach to calculate the transformation coefficients in the linear approximation is described both for the surface and internal waves in two-layer fluid. The problems associated with the application of the rigorous approach are noticed. The various approximate approaches are considered, as well as their compliance with the rigorous theory and numerical and experimental results. Within the framework of the rigorous approach the transformation coefficients of travelling waves and the excitation coefficients of evanescent modes are calculated. It is shown that wavelength of a quasi-monochromatic wavetrain changes after transformation on a bottom step proportionally to the phase speed, whereas the length of the envelope changes proportionally to the group speed. A comparison of theoretical results with numerical data and laboratory experiments is presented. 

The Korteweg—de Vries equation is a standard model for the description of long nonlinear internal waves in the ocean. When the effect of weak transverse variations and the Earth's background rotation are taken into account, this is replaced by the rotation-modified Kadomtsev—Petviashvili equation. In this short note we revisit the asymptotic derivation of this equation, incorporating a background shear flow as well as the background stratification. 

The internal gravity waves fields in a stratified fluid layer of variable depth are considered. Assuming a linear slope bottom, the exact solutions are obtained using the Kantorovich—Lebedev transform which describe an individual wave mode and a full wave field of internal gravity waves. Individual wave mode is expressed in terms of the hypergeometric function, the full wave field is described by semi-logarithmic function. WKBJ-asymptotics are constructed both for individual wave mode and for a full wave field. For the parameters of the stratified medium, which is characteristic for a real ocean (Bay of Biscay), the results of numerical calculations of the wave fields of asymptotic formulas are presented. A comparison with the results of numerical simulation of the complete system of hydrodynamic equations describing the evolution of nonlinear wave disturbances over a non-uniform ocean bottom, and it is shown that the amplitude-phase structure of the wave field is described by obtained in the paper asymptotic formulas. Available measurements data also show that the wave pattern with a strong beam structure can be observed in a real ocean, especially in the study of the evolution of a package of internal gravity waves over a non-uniform ocean bottom. In particular, analytical, numerical and measurements data show that the width of the wave ray decreases when approaching the shore. Typical ray pattern of the internal gravity waves in a stratified fluid layer of variable depth have been obtained without using of WKBJtechnics.

The propagation of finite amplitude internal waves over an uneven bottom is considered. One of the specific features of the large amplitude internal waves is the ability of the waves to carry fluid in the «trapped core» for a long distance. The velocity of particles in the «trapped core» is very close and, even, exceeds the wave speed. Such waves are detected in different parts of seas and oceans as internal waves of depression and elevation as well as short intrusions at interfaces. Laboratory experiments on the generation, interaction and decay of solitary waves in a two-layer fluid are discussed. Analytical and numerical solutions describing the evolution of internal waves in a shelf zone are constructed by the three-layer shallow water model. Laboratory investigations of the different types of internal waves (bottom, subsurface and interlayer waves) are demonstrated, that the model can be effectively applied to the numerical solution of unsteady wave motions, and the travelling waves, which can be found from the model in rather simple form, give the realistic form and governing parameters of internal waves in laboratory and field observations. The basic features of the large amplitude solitary waves and nonlinear wave trains evolution over a shelf can be represented by the model. 

The dynamics and energetics of a frontal collision of internal solitary waves of high amplitude propagating in a two-layer stratified fluid are studied numerically. The computations are carried out within the framework of the Navier—Stokes equations in the Boussinesq approximation. It was shown that the frontal collision of internal solitary waves of moderate amplitude leads to a small phase shift and to the generation of dispersive wavetrain trailing behind transmitted solitary wave. The phase shift grows with increasing amplitudes of the interacting waves and approaches the limiting value at large amplitudes of the waves. The deviation of the maximum wave height during collision from the twice the amplitude does not grow with increasing amplitude in the case of interaction of wave of large amplitude in contrast to the moderate amplitude waves. It was shown that the interaction of waves of large amplitude leads to the shear instability and the formation of Kelvin—Helmholtz vortices in the interface layer, however, subsequently waves again become stable. 

The distinctive features of particle transport during the propagation of long nonlinear localized wave packets (breathers) in quasi three-layer fluid are investigated in the framework of weakly nonlinear theory. To describe the displacement at the maximum of vertical baroclinic mode the Gardner equation is used. To determine the fields of displacements, vertical and horizontal velocities, which are used for finding of Lagrangian particle trajectories, three variants of the wave fields’ vertical structure are used: linear mode, weakly nonlinear approximation (with taking into account the first nonlinear amendment to linear mode) and weakly nonlinear weakly dispersive approximation (with taking into account both the first nonlinear amendment and the first dispersive amendment to linear mode). Since the velocity fields induced by nonlinear wave packets change immediately, the processes of particle transport are investigated for different initial configurations of breathers. The comparison of the form of particle trajectories for different horizons and different breathers’ configurations is made. It is shown that the use of the weakly nonlinear model is sufficient to determine the trajectories of fluid particles. Taking into account the first dispersive amendment to the modal function almost does not affect the quality and quantity of particles’ displacements. A significant difference between solutions of the problem of fluid particles’ trajectories for two types of nonlinear wave motions in a stratified fluid — solitons and breathers — is revealed. 

The numerical modeling of dynamics of the internal bore is done for the area in the Pechora Sea where the internal bore had been observed. The bore record obtained in 1998 is used as initial condition for numerical model based on the Gardner equation and analysis of bore evolution is done. The influences of hydrology variations and wave dissipation in the bottom boundary layer as well as the horizontal diffusion on the forecasted wave shape have been studied. It is shown that although internal wave characteristics are responsive to these factors nevertheless the abrupt overfall of isotherms in the thermocline (internal bore) is saved in one-two kilometers from the point of observation, and after that it is transformed into solibore (shock wave with ondulations). The process of internal bore generation from the train of nonlinear internal waves is studied as the result of the dispersive focusing. 

A series of numerical experiments is carried out for simulation of the stationary circulation and semidiurnal surface and internal tides in the region of the Kara Gates Strait. They are caused by either stationary contrasts of the free surface level at the open boundary of the domain under study, or by tidal elevations at the above boundary, or by means of both concurrently. It is shown that the predicted amplitudes of internal tidal waves on the bottom uplift in the strait amount to about 10—16 m under mean (over a syzygy-quadrature cycle) conditions. It is remarkable that the maximum internal tidal waves’ amplitudes are detected where internal tidal waves propagate against the stationary flow. Modeling results also show the slight western Litke current, a tendency to the appearance of the degenerate amphidrome with its center at the Vaigach Island and changes in tidal phases from 0 to 30° and from 330 to 0° in the adjacent parts of the Batrents and Kara Seas, respectively. 

We compare moored measurements of currents and temperature in the Gibraltar and Bab-el-Mandeb straits. In both straits the vertical displacements caused by semidiurnal internal waves are extremely high. Both straits are characterized by permanent strong two-layer currents of opposing directions. The hydrological regimes of the Gibraltar and Bab-el-Mandeb straits are very similar. Each strait connects an inland sea with the ocean. Strong evaporation exists in both seas and the runoff of fresh water by rivers is small in both seas. Evaporation is compensated by the surface flow of water from the ocean. Due to evaporation in the seas the saltier and dense water descends to the deep layers of the seas and then flows to the ocean as a bottom current of the opposite direction to the surface flow. A barotropic tidal wave is superimposed on this current system and generates a tidal internal wave during the flow of the currents over the sills in the straits. Internal tides near the sills are characterized by the amplitudes of vertical displacements that reach 160—200 m. Internal tides propagate to both sides of the sills rapidly losing their energy. Internal tide is generated over the background of two-layer shear current, which intensifies the amplitude of internal waves. The amplitudes of internal tides propagating opposite to the mean flow intensify due to the decrease in the wave length.