Adiabatic decay of internal solitons in a rotating ocean (original) (raw)
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Chaos (Woodbury, N.Y.), 2018
The adiabatic decay of different types of internal wave solitons caused by the Earth's rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions: (u + c u + α u u + α u u + β u) = γ u. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth's rotation described by the parameter γ. The characteristic decay time is derived for different types of solitons for positive and negative coefficients of cubic nonlinearity α (both signs of that parameter may occur in the oceans). The coefficient of quadratic nonlinearity α determines only a polarity of solitary wave when α < 0 or the asymmetry of solitary waves of opposite polarity when α > 0. It is found that the adiabatic theory describes well the decay of solitons having bell-shaped profiles. In contrast to that, l...
Internal Solitons in the Oceans
2006
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, Earth's 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 waveforms, 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.
Izvestiya, Atmospheric and Oceanic Physics, 2020
This review presents theoretical, numerical, and experimental results of a study of the structure and dynamics of weakly nonlinear internal waves in a rotating ocean accumulated over the past 40 years since the time when the approximate equation, called the Ostrovsky equation, was derived in 1978. The relationship of this equation with other well-known wave equations, the integrability of the Ostrovsky equation, and the condition for the existence of stationary solitary waves and envelope solitary waves are discussed. The adiabatic dynamics of Korteweg-de Vries solitons in the presence of fluid rotation is described. The mutual influence of the ocean inhomogeneity and rotation effect on the dynamics of solitary waves is considered. The universality of the Ostrovsky equation as applied to waves in other media (solids, plasma, quark-gluon plasma, and optics) is noted.
Internal solitons in the ocean
The Journal of the Acoustical Society of America, 1995
Chair's IntroductionS:25 Invited Papers 8:30 laAO1. Internal waves and tides in shallow water. Albert J. Plueddemann and James F. Lynch (Woods Hole Oceanogr. Inst.,
Interaction of solitons with long waves in a rotating fluid
Physica D: Nonlinear Phenomena, 2016
h i g h l i g h t s • Interaction of a KdV soliton with a long wave is studied in a rotating ocean. • Long background waves are sinusoidal wave and periodic sequence of parabolic arcs. • The model dynamical system is derived and studied analytically and numerically. • Solitons riding on long wave can propagate on long distances in a rotating ocean.
Dynamics of Internal Envelope Solitons in a Rotating Fluid of a Variable Depth
Fluids, 2019
We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrödinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrödinger equation. When such an envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase in soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow non-stationary solitary wave. The dependences of the soliton parameters on the distance when it m...
arXiv (Cornell University), 2019
Interaction of a solitary wave with a long background wave is studied within the framework of rotation modified Benjamin-Ono equation describing internal waves in a deep fluid. With the help of asymptotic method, we find stationary and nonstationary solutions for a Benjamin-Ono soliton trapped within a long sinusoidal wave. We show that the radiation losses experienced by a soliton and caused by the Earth rotation can be compensated by the energy pumping from the background wave. However, the back influence of the soliton on the background wave eventually leads to the destruction of the coherent structure and energy dispersion in a quasi-random wave field.
Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow
Mathematics
The results of a theoretical study on Benjamin–Ono (BO) soliton evolution are presented in a simple model of a two-layer ocean with a shear flow and viscosity. The upper layer is assumed to move with a constant speed relative to the lower layer with a tangential discontinuity in the flow profile. It is shown that in the long-wave approximation, such a model can be appropriate. If the flow is supercritical, i.e., its speed (U) exceeds the speed of long linear waves (c1), then BO solitons experience “explosive-type” enhancement due to viscosity, such that their amplitudes increase to infinity in a finite time. In the subcritical regime, when U
Some properties of deep water solitons
1976
Envelope solitons for surface waves in deep water are studied using the coupled equation for the Fourier amplitudes of the surface displacement. Comp3.rison is. made with some wave-tank experiments .of Feir. A linear stability analysis is made for an imposed transverse ripple.. A slowly growing instability is found at wavelengths comparable to, or longer than, the length of the soliton. A slowly developing instability is found also for a soliton propagating through a train of waves of wavelength appreciably smaller than that of the soliton. A soliton propagating through a train of waves with wavelength much larger than that of the soliton exhibits gross distortion due to the orbital fluid velocity of the wavetrain. This distortion is to some extent reversible, as the soliton tends to "recover" when •the wavetrain is damped to zero amplitude. Some comments are given concerning the statistics of a wave field containing solitons.
On radiating solitons in a model of the internal wave–shear flow resonance
Journal of Fluid Mechanics, 2006
The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including 'delocalized solitons' with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term 'soliton' as synonymous with 'solitary wave' and do not imply any integrability of the system). These 'solitons' are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the 'delocalized' ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.