Yury Stepanyants

Internal solitons in the ocean and their effect on underwater sound

Nonlinear internal waves in the ocean are discussed (a) from the standpoint of soliton theory and (b) from the viewpoint of experimental measurements. First, theoretical models for internal solitary waves in the ocean are briefly described. Various nonlinear analytical solutions are treated, commencing with the well-known Boussinesq and Korteweg-de Vries equations. Then certain generalizations are considered, including effects of cubic nonlinearity, Earths rotation, cylindrical divergence, dissipation, shear flows, and others. Recent theoretical models for strongly nonlinear internal waves are outlined. Second, examples of experimental evidence for the existence of solitons in the upper ocean are presented; the data include radar and optical images and in situ measurements of wave forms, propagation speeds, and dispersion characteristics. Third, and finally, action of internal solitons on sound wave propagation is discussed. This review paper is intended for researchers from diverse backgrounds, including acousticians, who may not be familiar in detail with soliton theory. Thus, it includes an outline of the basics of soliton theory. At the same time, recent theoretical and observational results are described which can also make this review useful for mainstream oceanographers and theoreticians.

Internal solitons in the ocean

Internal waves (IW) are among the important factors affecting sound propagation in the ocean. A special role may be played by solitary IWs because of their spatial localization and high magnitudes. Here, nonlinear IWs are discussed (a) from the standpoint of soliton theory and (b) from the viewpoint of experimental measurements. First, basic theoretical models for solitary IWs in the ocean are described, and various analytical solutions are treated, commencing with the well‐known Korteweg–de Vries equation and its important generalizations including effects of rotation, cylindrical divergence, eddy viscosity, shear flows and instabilities, and turbulence. Experimental evidence for the existence of solitons in the upper ocean is presented both for shallow and deep sea regions. The data include radar and optical images and in situ measurements of waveforms, propagation speeds, and dispersion characteristics. It is suggested that internal solitons in the ocean are ubiquitous and are generated primarily by ti...

Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations

We analyze a heuristic numerical method suggested by V. I. Petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations. The method is used to construct numerically the solitary wave solutions, such as solitons, lumps, and vortices, in a space of one and higher dimensions. Assuming that the stationary solution exists, we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest. The theory is illustrated with examples of physical interest such as generalized Korteweg--de Vries, Benjamin--Ono, Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and Klein--Gordon equations.

Internal solitons in laboratory experiments: Comparison with theoretical models

Nonlinear internal solitary waves observed in laboratory experiments are discussed from the standpoint of their relation to different soliton theories, from the classical integrable models such as the Korteweg-de Vries, Gardner, Benjamin-Ono, and Joseph-Kubota-Ko-Dobbs equations and their modifications, through the nonintegrable models describing higher-order nonlinear effects, viscosity, rotation, and cylindrical spreading, to the strongly nonlinear models. First, these theoretical models are briefly described and, then, laboratory data and their comparison with the theory are presented.

Propagation of waves in shear flows

Present-day results in the theory of oscillatory and wave phenomena in hydrodynamic flows are presented. A unified approach is used for waves of different physical origins. A characteristic feature of this approach is that hydrodynamic phenomena are considered in terms of physics, which complements significantly the traditional formal mathematical approach. Some physical concepts such as wave energy and momentum in a moving fluid are analyzed taking into account induced mean flows. The physical mechanisms that are responsible for hydrodynamic instability of shear flows are considered within the concept of negative energy waves. The phenomenon of over-reflection for waves of different types is analysed. A number of well-known theorems of hydrodynamic theory of stability are interpreted in terms of the interaction of waves having different energy signs. Great attention is drawn to the plasma-hydrodynamic analogy which is a powerful tool for physical analyses of general mechanisms of wave amplification and absorption in flows. A lot of hydrodynamical, acoustical and geophysical phenomena may be classified on the basis of this analogy. Various wave-flow interaction problems are considered, for instance, wave generation in whistlers, wave scattering and amplification by vortices, methods of wave remote sounding, some nonlinear dynamical phenomena, etc. The book is intended for researchers specialized in wave theory, aero-acoustics, geophysical and astrophysical fluid dynamics, and related fields.

Stationary bathtub vortices and a critical regime of liquid discharge

A modified Lundgren model is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in the horizontally unbounded domain. The liquid is assumed to be undisturbed at infinity and its depth is taken to be constant. Three different drainage regimes are studied: (i) subcritical, where whirlpool dents are less than the fluid depth; (ii) critical, where the whirlpool tips touch the outlet orifice; and (iii) supercritical, where surface vortices entrain air into the intake pipe. Particular attention is paid to critical vortices; the condition for their existence is determined and analysed. The influence of surface tension on subcritical whirlpools is investigated. Comparison of results with known experimental data is discussed.

Modelling internal solitary waves on the Australian North West Shelf

The transformation of the non-linear internal tide and the consequent development of internal solitary waves on the Australian North West Shelf is studied numerically in the framework of the generalised rotation-modified Korteweg–de Vries equation. This model contains both non-linearity (quadratic and cubic), the Coriolis effect, depth variation and horizontal variability of the density stratification. The simulation results demonstrate that a wide variety of non-linear wave shapes can be explained by the synergetic action of non-linearity and the variability of the hydrology along the wave path.

Decay of cylindrical and spherical solitons in rotating media

Decay laws of solitary waves due to geometrical divergence in rotating media are analysed within the framework of Ostrovskys equation. Dependences of solitons amplitudes on distance both for KdV and for Ostrovskys solitons are obtained and compared on the basis of approximate analytical approach and by means of numerical modelling.

Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave.

Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions

A brief review of soliton dynamics constituting one-dimensional periodic chains is presented. It is shown that depending on the governing equation, solitons may have either exponential or oscillatory-exponential decaying tails. Under certain conditions, solitons interaction can be considered within the framework of Newtonian equations describing the dynamics of classical particles. Collective behaviour of such particles forming a one-dimensional chain may be simple or complex and even chaotic. Specific features of soliton motions are presented for some popular models of nonlinear waves (Korteweg-de Vries, Toda, Benjamin-Ono, Kadomtsev-Petviashvili, and others).