Train Of Solitary Waves Research Articles

Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su–Gardner (or one-dimensional Green–Naghdi) system. Our analysis is based on the properties of the characteristics of the associated Whitham modulation system which describes an intermediate “undular bore” stage of the evolution. The resulting formula represents a “non-integrable” analogue of the well-known semi-classical distribution for the Korteweg–de Vries equation, which is usually obtained through the inverse scattering transform. Our analytical results are shown to agree with the results of direct numerical simulations of the Su–Gardner system. Our analysis can be generalised to other weakly dispersive, fully nonlinear systems which are not necessarily completely integrable.

Three-dimensional localized coherent structures of surface turbulence. I. Scenarios of two-dimensional–three-dimensional transition

The evolution of naturally excited disturbances on a thin liquid film falling down an inclined planar substrate undergoes several transitions between different wave regimes starting from two-dimensional (2D, one spatial dimension and height) solitary pulses at small Reynolds numbers to the “surface turbulence” stage for sufficiently large Reynolds numbers where the surface is randomly covered by localized three-dimensional (3D, two spatial dimensions and height) coherent structures. In this study, we analyze for the first time the instability of 2D pulses to 3D disturbances and the transitions of 2D pulses to fully developed 3D waves. The main instability mechanism responsible for the transition to 3D localized patterns is the Rayleigh instability of well-separated (isolated) 2D solitary waves. On the other hand, the physical mechanism for the 3D instability of 2D periodic waves and wave trains of 2D solitary waves sufficiently close to each other is not related to the Rayleigh instability but is due to wave-wave interaction and mass exchange between neighboring waves. These instabilities are characteristic of small inclination angles but under special conditions can be realized also for the vertical flow.

Solitary wave trains in granular chains: experiments, theory and simulations

The features of solitary waves observed in horizontal monodisperse chain of barely touching beads not only depend on geometrical and material properties of the beads but also on the initial perturbation provided at the edge of the chain. An impact of a large striker on a monodisperse chain, and similarly a sharp decrease of bead radius in a stepped chain, generates a solitary wave train containing many single solitary waves ordered by decreasing amplitudes. We find, by simple analytical arguments, that the unloading of compression force at the chain edge has a nearly exponential decrease. The characteristic time is mainly a function involving the grains’ masses and the striker mass. Numerical calculations and experiments corroborate these findings.

Tsunami generation and propagation from the Mjølnir asteroid impact

Abstract— In the late Jurassic period, about 142 million years ago, an asteroid hit the shallow paleo‐Barents Sea, north of present‐day Norway. The geological structure resulting from the impact is today known as the Mjølnir crater. The present work attempts to model the generation and the propagation of the tsunami from the Mjølnir impact. A multi‐material hydrocode SOVA is used to model the impact and the early stages of tsunami generation, while models based on shallow‐water theories are used to study the subsequent wave propagation in the paleo‐Barents Sea. We apply several wave models of varying computational complexity. This includes both three‐dimensional and radially symmetric weakly dispersive and nonlinear Boussinesq equations, as well as equations based on nonlinear ray theory. These tsunami models require a reconstruction of the bathymetry of the paleo‐Barents Sea.The Mjølnir tsunami is characteristic of large bolides impacting in shallow sea; in this case the asteroid was about 1.6 km in diameter and the water depth was around 400 m. Contrary to earthquake‐ and slide‐generated tsunamis, this tsunami featured crucial dispersive and nonlinear effects: a few minutes after the impact, the ocean surface was formed into an undular bore, which developed further into a train of solitary waves. Our simulations indicate wave amplitudes above 200 m, and during shoaling the waves break far from the coastlines in rather deep water. The tsunami induced strong bottom currents, in the range of 30–90 km/h, which presumably caused a strong reworking of bottom sediments with dramatic consequences for the marine environment.

Solitary and shock waves in discrete strongly nonlinear double power-law materials

A laminar metamaterial supporting strongly nonlinear solitary and shock waves with impact energy mitigating capabilities is presented. It consists of steel plates with intermittent polymer toroidal rings acting as strongly nonlinear springs with large allowable strain. The force-displacement relationship of a compressed o-ring is described by the addition of two power-law relationships resulting in a solitary wave speed and width depending on the amplitude. This double nonlinearity allows splitting of an initial impulse into three separate strongly nonlinear solitary wave trains. Solitary and shock waves are observed experimentally and analyzed numerically in an assembly with Teflon o-rings.

Winter PRIMER Ocean-Acoustic Solitary Wave Modeling Studies

In this paper, we present results from a joint oceanographic-acoustic study of solitary waves and their effects during the 1997 winter PRIMER4 experiment on the shelfbreak south of Cape Cod, MA. The study addresses the acoustic effects induced by solitary waves and associated oceanographic phenomena. Solitary wave generation and propagation simulations are produced by the Lamb model [J. Geophys. Res., vol. 99, pp. 848-864, 1994]. The model is nonhydrostatic and is formulated in 2.5 dimensions using terrain following coordinates. Acoustic field calculations are performed with a parabolic equation acoustic model along the path of solitary wave train propagation. The oceanographic model is initialized from density profiles derived from conductivity-temperature-depth (CTD) casts using analytical functions. The model is forced with a prescribed semidiurnal tidal velocity. An ocean background current is introduced. Simulations based on parameters derived from measurements show the following: 1) internal solitary waves of elevation propagate onto the shelfbreak region; 2) opposing ocean currents enhance the formation of solitary waves at the shelfbreak; 3) deepening of the winter mixed layer results in less penetration of the solitary waves on to the shelf; 4) density structure, mixed-layer depth, tidal forcing, and ocean currents control the formation of solitary waves of elevation at the shelfbreak; 5) energy conversion, from semidiurnal barotropic to semidiurnal barcoclinic tides and to internal solitary waves, occurs; 6) amplitudes and periods of modeled solitary waves are in the range of thermistor chain measurements; and 7) lower mixed-layer densities increase the phase speed of simulated solitary waves. Acoustic field calculations are coupled to the propagation of the solitary wave packets through the sound-speed changes that are derived from the oceanographic simulations. Acoustic model predictions show signal intensity fluctuations similar to the anomalous loses in acoustic energy observed in the Yellow Sea data taken by Zhou [J. Acoust. Soc. Amer., vol. 90, pp. 2042-2054, 1991]. In some cases, the presence of solitary waves on the shelf enhances the propagation of acoustic energy onto the shelf. This was observed for acoustic simulations where the acoustic source was located beyond the shelfbreak and at a depth greater than the shelf depth.

Nonlinear deformation waves in solids and dispersion

In contemporary technology, high-speed loading, high frequencies and high-amplitude excitations are widely used. For proper analysis, the microstructure of solids must then be taken into account. Together with nonlinear effects, the dispersive effects due to the microstructure are of importance. In this paper, the Mindlin-type model based on the theory of continua is used for describing the microstructure. On the other hand, it is shown how straight-forward numerical methods, like the finite-volume methods, permit to describe the material properties for every discrete element used in numerical simulation. Several phenomena like the existence of solitary waves, the emergence of solitary wave trains, and waves in piece-wise nonlinear laminated materials are briefly discussed and further problems indicated.

Evolution of a shoaling internal solitary wavetrain

Field observations of an internal solitary wavetrain impacting a shoaling bottom are presented. Measurements of the spatio‐temporal characteristics of the shoaling waves are given, as well as estimations of the mixing they may have caused upon impact. The observations are discussed in the context of numerical simulations, laboratory experiments, and hypotheses recently raised on the origin and evolution of internal solitary waves in coastal environments.

Spectral method for solving the equal width equation based on Chebyshev polynomials

A spectral solution of the equal width (EW) equation based on the collocation method using Chebyshev polynomials as a basis for the approximate solution has been studied. Test problems, including the migration of a single solitary wave with different amplitudes are used to validate this algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The interaction of two solitary waves is seen to cause the creation of a source for solitary waves. Usually these are of small magnitude, but when the amplitudes of the two interacting waves are opposite, the source produces trains of solitary waves whose amplitudes are of the same order as those of the initial waves. The three invariants of the motion of the interaction of the three positive solitary waves are computed to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Comparisons are made with the most recent results both for the error norms and the invariant values.

Solitary wave effects north of Strait of Messina

We present new results from a joint ocean-acoustic modeling study of solitary wave generation in the Strait of Messina, their propagation in the Tyrrhenian Sea and subsequent shoaling in the Gulf of Gioia. The nonhydrostatic 3D EULAG model is used for the oceanographic predictions. The simulations are initialized with measured temperature and salinity profiles from an October 1995 survey of the Messina region, and forced with semidiurnal tidal magnitudes predicted by a barotropic tidal model. Parameter sensitivity studies are performed. The predicted solitary wave trains are compared with CTD chain measurements. The model results and data are examined through a wavelet analysis. The wavelengths are tracked by the spines (maximum intensity for each wavelength) at various times. From the slope of the variations, phase speeds are derived as a function of wavelength. For the parameters extracted from CTD measurements and existing tidal conditions, phase speed distribution for wavelengths ranging from about 0.6 m to 1.6 km are obtained. The model predicted phase speed magnitudes range from 0.85 m s −1 to 0.93 m s −1. The phase speeds derived from data range from 0.77 m s −1 to 0.88 m s −1. The model predicted phase speed versus wavelength distribution has similar trends to the phase speed versus wavelength distribution derived from data. The shoaling of the solitary waves in the Gulf of Gioia is studied. Calculations of the acoustical field are conducted, along the solitary wave propagation path, with the parabolic (PE) acoustical model.

General Solution to a Bilinear Reduction of the Higher Order Nonlinear Schrödinger Equation

The general solution is found to a bilinear reduction of the higher order nonlinear Schrodinger equation. Except for the previously known special cases having multisoliton solutions, the solution has the form of a single envelope solitary wave or wave train for all other values of the parameters. We conjecture that the generic solution is limited to such a narrow class of functions because the bilinear system is overdetermined.

Analytical and Observational Studies of Internal Solitary Waves in the Yellow Sea

Abstract This article presents findings regarding the origination and evolution of a large train of internal solitary waves as observed through Synthetic Aperture Radar (SAR). The internal wave train under study propagates into deeper waters away from the coast and, most important, appears to be generated at a well-defined point along a steep topographic variation. The shallow bottom topography revealed itself in the SAR image and provided the opportunity to determine, with a large level of accuracy, the precise area where an internal bore originated. The bore traveled east and broke up into a train of solitary waves. The packet's phase speed, wave amplitude, mixed-layer depth, and density difference for a two-layer structure were remotely estimated based on measurements made from the SAR image, the knowledge of the local bathymetry, historical climatological data, and a two-layer model. Farther along the propagation path the internal waves exhibited an extremely large degree of dispersion, which was rela...

Role of Mass-Transfer Rate and Surface Inertia on Solitary Wave Trains at Liquid Surface

We report several findings about solitary wave trains in a liquid surface, which is produced by the Marangoni instability. The wave trains were induced by adsorption and absorption of the volatile species into another miscible liquid. Although the present system has been investigated focusing on the interfacial fluid dynamics in detail, our special interest in this paper is a mapping of the wave-mode on a certain diagram. First of all, we reported a kind of conservation law between the number of wave crests and the traveling speed. As the next, we mentioned that a liquid layer composed of a volatile species was formed on the less-volatile liquid. On the basis of these results, we successfully performed the mode-mapping on a diagram, the abscissa and ordinate of which represent the mass-transfer rate and the thickness of the layer, respectively. Both axes, respectively, corresponded to the strength of the driving force and the inertia against the movement.

Yellow Sea ocean‐acoustic solitary wave modeling studies

This study is in an area south of the Shandong peninsula, near the region where Zhou et al. (1991) observed anomalous drops in acoustical intensity. Solitary wave generation and propagation simulations are performed using the Lamb (1994) nonhydrostatic model. The model simulations show that, for summer conditions, the existing semi‐diurnal tidal flow over the topographic variations formed internal bores and solitary waves. For the Shandong area, we analyzed summer observations from Synthetic Aperture Radar (SAR) that tracked solitary wave trains from their surface roughness signatures. The images contained seven events consisting of internal bores and solitary waves that traveled in a well‐defined direction for 2.5 days. The origin of the trains appeared at a well‐defined point along a steep topographic drop. The SAR observations guided and tuned the model simulations, by comparing spectra of observed and modeled wavelengths. The tuned model yields wavelengths within factors of 2, or less, of those derived from SAR data. Wavelength and amplitude dispersion analysis showed two dispersion regimes. Modeled phase speeds were at the lower limit of phase speeds deduced from SAR data, from about 0.8 to 1.0 m/s. Acoustical intensity calculations in the presence of solitary wave trains will be undertaken in a subsequent paper using a parabolic equation acoustical model along the path of solitary wave train propagation.

Internal waves and undular bores in mesospheric inversion layers

The linear theory of internal gravity waves and the nonlinear development of undular bores supported by mesospheric thermal inversion layers is presented. Whenever the buoyancy frequency has a local maximum as a result of a local increase in the potential temperature gradient, internal waves having a frequency between the maximum and minimum values of the buoyancy frequency can exist. Two‐dimensional nonlinear solutions of a simplified model show that long‐wavelength initial disturbances of finite amplitude, nonlinearly steepen, and form undulatory bores and that the analysis and numerical solutions presented show that mesospheric bores are consistent with nonlinear internal gravity waves trapped within a thermal inversion layer. The implications of linear theory and numerical solutions to the observations of mesospheric bores are discussed. In particular, the simulations agree with the observed separation of wave crests, which is in consistent with the notion that a mesospheric bore is composed of a train of solitary waves each with an amplitude‐dependent speed.

Solitary Waves on Vortex Lines in Ginzburg-Landau Models for the Example of Bose-Einstein Condensates

Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (p)-substitution energy (Epsilon) plane with p-->0.09 rho kappa(3)/c(2), Epsilon-->0.091 rho kappa(3)/c as U-->c, where rho is the density, c is the speed of sound, kappa is the quantum of circulation, and U is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.

Transient flow and heat transfer phenomena in inclined wavy films

A finite-element numerical scheme is used to study rigorously the flow of an inclined liquid film and the heat transfer from the constant-temperature wall. Regular inlet disturbances are predicted to evolve into periodic or solitary waves depending on the frequency of the forcing. At very low disturbance frequencies parasitic crests appear and the regularity of the wavetrain is lost. The effect of a solitary wavetrain on heat transfer from the wall is studied, and it is predicted that a stationary temperature distribution develops with periodic flux variation that follows the waves. The thinning of the substrate between successive humps combines with the effect of convection at the crest and tail of the solitary humps to produce heat transfer enhancement significantly above the conduction limit.

A New Analytical Model for Internal Solitons in the Ocean*

Abstract A new model for tidally generated internal solitons in the ocean is advanced, based on a little-known asymptotic analytical solution to the weakly nonlinear Korteweg–de Vries equation. The solution has many of the properties observed in oceanic internal solitary waves (solitons). It has been adapted to simulate the dynamics of stratified fluid flow driven by tidal forcing at continental shelves and sills, and the adaptation describes the canonical behavior of such solitary wave trains. It is shown that the solitons are positioned at the leading edge of an internal tidal bore on the shelf and that the distance between successive packets of solitons constitutes the internal tidal wavelength. A future paper will apply the model to field observations from four geographical regions.

Orientation effects in shocked nickel single crystals via molecular dynamics

Shock wave propagation and crystal deformation features were examined in nickel single crystals using nonequilibrium molecular-dynamics simulations for three crystallographic orientations: 〈100〉, 〈110〉, and 〈111〉. Two Morse-type pair potentials and one embedded-atom method (EAM) empirical potential were used to describe bond forces for single crystal nickel atoms. The two Morse-type pair potentials yield more accurate elastic moduli of nickel than does the EAM potential. The dissipation mechanism leading to shear stress relaxation was investigated at the piston/particle speed up=2.5(ε/m)1/2, where ε is an energy parameter and m is the atomic mass. For shock propagation along the 〈100〉 orientation, the plastic waves overtook almost all elastic precursors for all three potentials. However, for wave propagation along both 〈110〉 and 〈111〉 orientations, elastic precursors followed by steady plastic waves were observed. A clearly discernible leading solitary wave train followed by an extended elastically deformed region was observed in the 〈110〉 orientation. The EAM potential and the pair potentials showed some differences in shock wave propagation. In terms of crystal deformation, the EAM potential showed a greater degree of plasticity than the pair potentials. The two Morse-type pair potentials showed comparable results with respect to wave propagation features and crystal deformation. The details of the atomistic changes associated with shock-induced elastic-plastic deformation are presented for all three potentials.

Ocean-Acoustic Solitary Wave Studies and Predictions

Shallow water internal solitary waves have become a major topic of interest to ocean- ographers and acousticians. In this paper we review the cross-disciplinary status of joint ocean- acoustic solitary wave studies and predictions. We consider the process of acoustical mode coupling in the presence of solitary waves and the corresponding acoustical intensity loss due to increased coupling with the bottom. A study of the interaction of an acoustical field with a train of solitary waves is undertaken at a range of frequencies. At a resonant frequency the acoustic field can interact with the solitary wave packet which results in mode conversions (acoustic energy is redistributed among the modes, often from lower-order to higher-order modes). Higher signal losses can occur in the higher order modes through increased bottom attenuation and result in an anomalous acoustical intensity loss at the resonant frequency. We present some new results of joint ocean-acoustic research, from a dedicated study in the Strait of Messina, where solitary waves are generated by semidiurnal tidal flow over topographic variations. The University of Hamburg weakly nonhydrostatic two layer model is used for simulat- ing the generation and propagation of solitary waves. In particular, the physical states encountered during an October 1995 cruise in the Strait of Messina (between Italy and Sicily) are simulated. Various parameter space sensitivity studies, about the existing cruise conditions, are performed. The modelled solitary wave trains are compared against conductivity-temperature-depth (CTD) chain measurements in terms of amplitudes, wavelengths, phase speeds and correlations with data. Pre- dicted and observed sound speeds are used in acoustical intensity calculations that are conducted with a parabolic equation (PE) model. The differences in the resultant acoustical intensity fields provide a guide for the tuning of the oceanographic model parameters. The tuned oceanographic model shows agreement with data for the first and second solitary waves in terms of amplitude, wavelength and phase speed. The calculated available potential energy from the simulation results is in the range of the data analogue.