Planetons: An example of large amplitude solitary waves

Abstract

The term ‘‘solitary wave'’ is usually used to denote a steadily propagating permanent form solution of a nonlinear wave equation, with the permanency arising from a balance between steepening and dispersive tendencies. It is known that large-scale thermal anomalies in the ocean are subject to a steepening mechanism driven by the beta effect, while at the smaller deformation scale, such phenomena are highly dispersive. It is shown here that the evolution of a physical system subject to both effects is governed by the ‘‘frontal semi-geostrophic equation'’ (FSGE), which is valid for large amplitude thermocline disturbances. Solitary wave solutions of the FSGE (here named planetons) are calculated and their properties are described with a view towards examining the behavior of finite amplitude solitary waves. In contrast, most known solitary wave solutions belong to weakly nonlinear wave equations (e.g., the Korteweg—deVries (KdV) equation).

The FSGE is shown to reduce to the KdV equation at small amplitudes. Classical sech² solitons thus represent a limiting class of solutions to the FSGE. The primary new effect on planetons at finite amplitudes is nonlinear dispersion. It is argued that due to this effect the propagation rates of finite amplitude planetons differ significantly from the ‘‘weak planeton'', or KdV, dispersion relation. Planeton structure is found to be simple and reminiscent of KdV solitons. Numerical evidence is presented which suggests that collisions between finite amplitude solitary waves are weakly inelastic, indicating the loss of true soliton behavior of the FSGE at moderate amplitudes. Lastly, the sensitivity of solitary waves to the existence of a nontrivial far field is demonstrated and the role of this analysis in the interpretation of lab experiments and the evolution of the thermocline is discussed.     

Ocean-Acoustic Solitary Wave Studies and Predictions

Shallow water internal solitary waves have become a major topic ofinterest to oceanographers and acousticians. In this paper we reviewthe cross-disciplinary status of joint ocean-acoustic solitary wavestudies and predictions. We consider the process of acoustical modecoupling in the presence of solitary waves and the correspondingacoustical intensity loss due to increased coupling with the bottom. Astudy of the interaction of an acoustical field with a train ofsolitary waves is undertaken at a range of frequencies. At a resonantfrequency the acoustic field can interact with the solitary wavepacket which results in mode conversions (acoustic energy isredistributed among the modes, often from lower-order to higher-ordermodes). Higher signal losses can occur in the higher order modesthrough increased bottom attenuation and result in an anomalousacoustical intensity loss at the resonant frequency.We present some new results of joint ocean-acoustic research, from adedicated study in the Strait of Messina, where solitary waves aregenerated by semidiurnal tidal flow over topographic variations. TheUniversity of Hamburg weakly nonhydrostatic two layer model is used forsimulating the generation and propagation of solitary waves. In particular, the physicalstates encountered during an October 1995 cruise in the Strait of Messina (betweenItaly and Sicily) are simulated. Various parameter space sensitivity studies, about theexisting cruise conditions, are performed. The modelled solitary wave trains arecompared against conductivity-temperature-depth (CTD) chain measurements interms of amplitudes, wavelengths, phase speeds and correlations with data. Predictedand observed sound speeds are used in acoustical intensity calculations thatare conducted with a parabolic equation (PE) model. The differences in theresultant acoustical intensity fields provide a guide for the tuning of theoceanographic model parameters. The tuned oceanographic model showsagreement with data for the first and second solitary waves in terms ofamplitude, wavelength and phase speed. The calculated available potentialenergy from the simulation results is in the range of the data analogue.     

Nonlinear Shallow Water Theories for Coastal Waves

Ocean waves entering the near-shore zone undergo nonlinear and dispersive processes. This paper reviews nonlinear models, focusing on the so-called Serre equations. Techniques to overcome their limitations with respect to the phase speed are presented. Nonlinear behaviours are compared with theoretical results concerning the properties of Stokes waves. In addition, the models are tested against experiments concerning periodic wave transformation over a bar topography and of the shoaling of solitary waves on a beach.     

Fission of a weakly nonlinear interfacial solitary wave at a step

The transformation of a weakly nonlinear interfacial solitary wave in an ideal two-layer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a weak transformation for propagation into deeper water. Far from the step, the wave dynamics is described by the Korteweg-de Vries equation which is fully integrable. In the vicinity of the step, the reflected and transmitted waves have soliton-like shapes, but their parameters do not satisfy the steady-state soliton solutions. Using the inverse scattering technique it is shown that the reflected wave evolves into a single soliton and dispersing radiation if the wave propagates from deep to shallow water, and only dispersing radiation if the wave propagates from shallow to deep water. The dynamics of the transmitted wave is more complicated. In particular, if the coefficient of the nonlinear quadratic term in the Korteweg-de Vries equation is not changed in sign in the region after the step, the transmitted wave evolves into a group of solitons and radiation, a process similar to soliton fission for surface gravity waves at a step. But if the coefficient of the nonlinear term changes sign, the soliton is destroyed completely and transforms into radiation. The effects of cubic nonlinearity are studied in the framework of the extended Korteweg-de Vries (Gardner) equation which is also integrable. The higher-order nonlinear effects influence the amplitudes of the generated solitons if the amplitude of the transformed wave is comparable with the thickness of lower layer, but otherwise the process of soliton fission is qualitatively the same as in the framework of the Korteweg-de Vries equation.     

固体介质中孤立波的传播及演化特征

由于地球介质中广泛存在断裂、微裂缝等地质现象,实际地震资料中会出现形似孤立波的非线性地震现象。因此,对固体介质中孤立波的研究有利于解释这些非线性地震现象的形成机制。本文基于KdV方程,以雷克子波作为初始条件,采用伸缩子机理构建体力模型,利用有限差分的方法模拟了孤立波的演化过程。理论结果表明,非线性地震纵波可以从雷克子波逐渐演化成孤立波,而且地震波的初始振幅和频散系数对模拟结果也有重要影响。通过与实际资料的对比也能说明这种演化的可能性。同时根据方程系数矩阵中元素带状分布的特征,采用稀疏矩阵的存取方法,可以减小计算内存,提高计算效率。     

Numerical study on transient harbor oscillations induced by successive solitary waves

Tsunamis are traveling waves which are characterized by long wavelengths and large amplitudes close to the shore. Due to the transformation of tsunamis, undular bores have been frequently observed in the coastal zone and can be viewed as a sequence of solitary waves with different wave heights and different separation distances among them. In this article, transient harbor oscillations induced by incident successive solitary waves are first investigated. The transient oscillations are simulated by a fully nonlinear Boussinesq model, FUNWAVE-TVD. The incident successive solitary waves include double solitary waves and triple solitary waves. This paper mainly focuses on the effects of different waveform parameters of the incident successive solitary waves on the relative wave energy distribution inside the harbor. These wave parameters include the incident wave height, the relative separation distance between adjacent crests, and the number of elementary solitary waves in the incident wave train. The relative separation distance between adjacent crests is defined as the ratio of the distance between adjacent crests in the incident wave train to the effective wavelength of the single solitary wave. Maximum oscillations inside the harbor excited by various incident waves are also discussed. For comparison, the transient oscillation excited by the single solitary wave is also considered. The harbor used in this paper is assumed to be long and narrow and has constant depth; the free surface movement inside the harbor is essentially one-dimensional. This study reveals that, for the given harbor and for the variation ranges of all the waveform parameters of the incident successive solitary waves studied in this paper, the larger incident wave heights and the smaller number of elementary solitary waves in the incident tsunami lead to a more uniform relative wave energy distribution inside the harbor. For the successive solitary waves, the larger relative separation distance between adjacent crests can cause more obvious fluctuations of the relative wave energy distribution over different resonant modes. When the wave height of the elementary solitary wave in the successive solitary waves equals to that of the single solitary wave and the relative separation distance between adjacent crests is equal to or greater than 0.6, the maximum oscillation inside the harbor induced by the successive solitary waves is almost identical to that excited by the single solitary wave.     

A reduced model for nonlinear dispersive waves in a rotating environment

Abstract

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.     

Comments on Modelling Short Atmospheric Frontal Waves

The idea of Gerstner's trochoidal waves is used to reconstruct a model of short, finite-amplitude progressive waves on frontal surfaces of the Margules Type. Stable waves associated with a negative (westward) group velocity occur in the model. A wave train can be maintained through the formation of new waves in the rear of the existing waves as a result of the westward energy transport.     

Simplification of strong ground motion considering inelastic responses of structures

Simplification of strong ground motions to 1 cycle sine waves was investigated from the elastic and inelastic earthquake response analyses and response analyses under sine wave input using single‐degree‐of‐freedom systems. Strong ground motions could be simplified to 1 cycle sine waves if large plastic deformations, with ductility factor more than 2, were assumed. This is because the approximate maximum responses from input sine waves are determined by the initial response cycle, due to period elongation and plastic energy dissipation of the systems. A sine wave whose acceleration amplitude is the peak ground acceleration (PGA) and whose period is that of an equivalent 1 cycle sine wave is proposed. The period of an equivalent sine wave is easily obtained from the elastic response acceleration spectrum of a seismic record. This means that the inelastic responses are approximately determined by the PGA and an equivalent 1 cycle sine wave period. Therefore, an equivalent 1 cycle sine wave period provides a single index to express the frequency characteristics of a strong ground motion. Copyright © 2000 John Wiley & Sons, Ltd.     

具有非线性地形的正压流体中孤立Rossby波的mKdV方程

正压流体中,采用摄动方法将准地转位涡方程推导出地形效应的mKdV方程,得到Rossby波振幅的演变满足地形效应的mKdV方程的结论,说明地形效应是诱导Rossby孤立波的重要因素.     

Dynamics of the large-scale ULF electromagnetic wave structures in the ionosphere

The present article displays the results of theoretical investigation of the planetary ultra-low-frequency (ULF) electromagnetic wave structure, generation and propagation dynamics in the dissipative ionosphere. These waves are stipulated by a spatial inhomogeneous geomagnetic field. The waves propagate in different ionospheric layers along the parallels to the east as well as to the west and their frequencies vary in the range of (10–10⁻⁶) s⁻¹ with a wavelength of order 10³ km. The fast disturbances are associated with oscillations of the ionospheric electrons frozen in the geomagnetic field. The large-scale waves are weakly damped. They generate the geomagnetic field adding up to several tens of nanotesla (nT) near the Earth's surface. It is prescribed that the planetary ULF electromagnetic waves preceding their nonlinear interaction with the local shear winds can self-localize in the form of nonlinear long-living solitary vortices, moving along the latitude circles westward as well as eastward with a velocity different from the phase velocity of the corresponding linear waves. The vortex structures transfer the trapped particles of medium, as well as energy and heat. That is why such nonlinear vortex structures can be the structural elements of the ionospheric strong macro-turbulences.     

固体介质中非线性波的能量初探

本文从能源问题与勘探问题的密切关系出发,提出进行深入的非线性波研究的必要性——进一步提供可靠而详实的地质资料,以解决实际地质问题.说明了非线性科学与非线性波动的关系和特点,非线性波动是非线性科学的一个重要的分支,而地球本身的特点也决定了非线性科学是解决地质问题的重要基础.回顾了波动理论的研究历程及当今国内外非线性研究的进展情况.通过上述认识,提出了非线性波动问题中能量的重要性,然后分别从震源与冲击波的形成,界面处的能量分配,介质中能量的衰减,波的相互作用四个方面讨论了非线性波的能量问题.通过对能量问题的讨论,进一步讨论了影响非线性波能量的因素.从能量的角度出发,探讨了以后进行固体中非线性波研究的重点.最后对非线性波在地学应用中的可能性进行了探讨.     

Modelling Internal Solitary Waves in the Coastal Ocean

In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography can generate large-amplitude, horizontally propagating internal solitary waves. These waves often occur in regions where the waveguide properties vary in the direction of propagation. We consider the modelling of these waves by nonlinear evolution equations of the Korteweg–de Vries type with variable coefficients, and we describe how these models are used to describe the shoaling of internal solitary waves over the continental shelf and slope. The theories are compared with various numerical simulations.     

Interpretation of the precursor to the 1960 great chilean earthquake as a seismic solitary wave

Seismic solitary waves may exist. The situation envisaged is that of an elastic layer over a half space where transverse, linearly polarized surface waves—Love waves—propagate. In the long wavelength limit, nonlinearities may compensate dispersion so that localized wave packets can propagate without distortion. A model equation is derived that explicitly exhibits this phenomenon, having among its solutions a strain solitary wave. It is predicted that strains of seismic interest should propagate with the shear wave velocity of the underlying half space, and they should have a wavelength about three orders of magnitude greater than the depth of the top layer. In light of the above considerations the precursor to the 1960 great chilean earthquake recorded with the Pasadena strain seismogram is interpreted as a seismic solitary wave produced by the foreshock that preceded the main shock by about 15 min. It traveled as a surface wave guided by the oceanic crust.     

Modulation of electron-acoustic waves

FAST observations have indicated signatures of large amplitude solitary waves in the auroral zone of the earth's ionosphere. Our objective here is to propose a model for the generation of density cavities by the ponderomotive force of electron-acoustic waves. For this purpose, we derive a nonlinear Schrödinger equation for the electron-acoustic wave envelope as well as a driven (by the electron-acoustic wave ponderomotive force) ion-acoustic wave equation. Possible stationary solutions of our coupled equations are obtained.     

Nonlinear internal waves in ideal rotating basins

Abstract

Starting from Euler's equations of motion a nonlinear model for internal waves in fluids is developed by an appropriate scaling and a vertical integration over two layers of different but constant density. The model allows the barotropic and the first baroclinic mode to be calculated. In addition to the nonlinear advective terms dispersion and Coriolis force due to the Earth's rotation are taken into account. The model equations are solved numerically by an implicit finite difference scheme. In this paper we discuss the results for ideal basins: the effects of nonlinear terms, dispersion and Coriolis force, the mechanism of wind forcing, the evolution of Kelvin waves and the corresponding transport of particles and, finally, wave propagation over variable topography. First applications to Lake Constance are shown, but a detailed analysis is deferred to a second paper [Bauer et al. (1994)].     

Nonlinear waves in rotating magnetohydrodynamics

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.     

Generation and evolution of internal solitary waves in the southern Taiwan Strait

ABSTRACT

The generation processes and potential energy sources of internal solitary waves (ISWs) in the southern Taiwan Strait are investigated by driving a high resolution non-hydrostatic numerical model with realistic background conditions. Two main types of ISWs are clarified according to their different energy sources. One is generated by the nonlinear disintegration of remote internal tides emanating from Luzon Strait, and the other type is generated by local tide-topography interaction at the continental slope. The basic properties and evolution processes differ between these two kinds of ISWs. The waves originated from the remote internal tides at Luzon Strait have amplitudes comparable to previous field observations. In contrast, the ISWs generated locally are much weaker than observed waves, even in the presence of a steady offshore background current, which intensifies the generation of onshore ISWs. The ISWs induced by remotely generated M₂ internal tides are stronger than those induced by K₁ internal tides, and the fraction of internal wave energy transmitted onto the shelf is not significantly influenced by the intensity of remotely generated internal tides.     

High-frequency vertical current observations in stratified seas

Although large-scale tidal and inertial motions dominate the kinetic energy and vertical current shear in shelf seas and ocean, short-scale internal waves at higher frequencies close to the local buoyancy frequency are of some interest for studying internal wave breaking and associated diapycnal mixing. Such waves near the upper limit of the inertio-gravity wave band are thought to have relatively short O (10²–10³ m) horizontal scales and to show mainly up- and downward motions, which contrasts with generally low aspect ratio large-scale ocean currents. Here, short-term vertical current (w) observations using moored acoustic Doppler current profiler (ADCP) are presented from a shelf sea, above a continental slope and from the open ocean. The observed w, with amplitudes between 0.015 and 0.05 m s⁻¹, all span a considerable part of the water column, which is not a small vertical scale O(water depth) or O (100–500 m, the maximum range of observations), with either 0 or π phase change. This implies that they actually represent internal waves of low vertical modes 1 or 2. Maximum amplitudes are found in layers of largest stratification, some in the main pycnocline bordering the frictional bottom boundary layer, suggesting a tidal source. These ‘pycnocline-w’ compose a regular train of (solitary) internal waves and linearly decrease to small values near surface and bottom.     

电子束流激发的电子声波数值模拟

通过一维静电粒子模拟程序研究了电子束流不稳定性,其中束流电子的温度远大于背景电子的温度.结果发现,所激发的波动主要是电子声波,波动的演化经历了线性增长和非线性饱和两个阶段.在非线性饱和阶段,由于电子声波相速度随频率是变化的,它可以通过非线性相互作用将背景比较冷的电子加速到很高的能量.此外,还研究了束流电子的温度、束流电子和背景电子的相对密度以及束流电子的漂移速度对电子束流不稳定性的影响.