The existence of internal solitary waves in a two-fluid system near the KdV limit

Abstract

Two fluid layers of constant density lying one over the other on top of a rigid horizontal lower boundary with either a free upper surface or a rigid upper boundary can support solitary waves. The existence of a unique branch of such waves emanating from the horizontal flow at a critical speed U _? is demonstrated in both cases by use of the Nash—Moser implicit function theorem. These results complement the global results of Amick and Turner (1986) and are analogous to the work of Friedrichs and Hyers (1954) and Beale (1977) for surface waves. It is also noted that the most obvious variational principle which characterizes these waves as constrained extremals (Benjamin, 1984) is of indefinite type, having a Hessian with infinitely many positive and infinitely many negative eigenvalues.     

Miracles,misconceptions and scotomas in the theory of solitary waves

ABSTRACT

In an age of billion dollar particle accelerators and Mars rovers, it is surprising that solitary waves were first discovered by a man on horseback with no tools but his own eyes. A century and a half later, more complicated patterns of ridges, so-called hyperelliptic two-polycnoidal waves, were discovered in the ocean during a beach vacation. The inverse scattering method, which solves nonlinear partial differential equations through a sequence of solving purely linear equations, is a blend of quantum theory and hydrodynamics that arose from informal, unstructured conversations (i.e. goofing off) among a group of postdocs from different disciplines who were randomly assigned to the same office. The cnoidal wave in the lemniscate case is well-approximated by a nonlinear solitary wave and equally well approximated by a linear sine wave. It is always and exactly the superposition of solitary waves even in the limit in which it is an infinitesimal sine wave. The history and science of solitary waves has the disorienting quality of an M. C. Escher drawing. Here, we cannot give an understanding of these deep subjects in so brief an article; rather we strive to unveil the beauty and unexpectedness of these topics to give the reader a reason to pursue these in the much more comprehensive reviews and books we cite. Further, we stress the “scotomas” (blind spots), misconceptions and surprises, the sociology and epistemology of science. It is true that failed theories, scotomas, serendipity and cognitive saltation (progress in jumps) is characteristic of science. It is also true that the invention of the train was the invention of the train wreck. Engineering learns from each disaster and science should do the same. The highly nonlinear history of nonlinear waves is reported not to disrespect the past but to replace scientific fatalism with a constructive wariness. We are not smarter or more enlightened than Scott Russell or Stokes or Landau, but we can learn from their scotomas and misconceptions as much as from their triumphs.     

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.     

切变基本纬向流中非线性赤道Rossby长波

为了解决观测和理论研究中的一些问题以及更好地了解热带大气动力学 ,有必要进一步研究基本气流的变化对大气中赤道Rossby波动的影响 .本文研究分析基本气流对赤道Rossby长波的影响 ,利用一个简单赤道 β平面浅水模式和摄动法 ,研究纬向基本气流切变中非线性赤道Rossby波 ,推导出在切变基本纬向流中赤道Rossby长波振幅演变所满足的非线性KdV方程并得到其孤立波解 .分析表明 ,孤立波存在的必要条件是基本气流有切变 ,而且基流切变不能太强 ,否则将产生正压不稳定 .     

Method for correction of the prestack migration amplitude of converted waves

The paper presents a method for correction of amplitude of prestack migration using the reflectivity function. The solution of wave equations for heterogeneous media expressed in the form of Neumann series for converted waves was used to find the reflectivity function. The performance of the proposed method was verified using synthetic models of wavefields. The synthetic models of wavefields were also used to compare amplitude correction levels required by the presented method and the standard amplitude correction methods, i.e., the methods using the spherical divergence and extention factor.     

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.     

Finite-amplitude mountain waves in a compressible atmosphere including the effects of dissipation

Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height z_c. The height z_c , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above z_c. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z ≫ z_c is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ^?-8/3 characterizes the marginal state of the wave at z = z_c .     

Nonlinear wave propagation on a sphere: Interaction between Rossby waves and gravity waves; stability of the Rossby waves

Abstract

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the coefficients of nonlinear interaction are obtained in the asymptotic case of a slowly rotating sphere, i.e. when the normal modes can be expressed as single spherical harmonics.     

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.     

低频大气声重波的非线性Schrdinger方程

本文采用多重时空尺度展开的方法,对于不可压缩等温大气,导出了二维情况下低频声重波的非线性Schrdinger方程,讨论了声重波的非线性性质.在垂直方向上,声重波具有可传播性和幅度随高度增长的特性;在水平方向上,声重波能表现为孤子包络调制的波列.     

中美洲海域第二模态内孤立波的地震海洋学研究

以往利用地震海洋学方法发现的内孤立波大多是第一模态内孤立波,本文利用地震海洋学方法首次在中美洲太平洋沿岸海域发现了第二模态内孤立波群.通过叠前偏移观察该第二模态内孤立波群细结构的变化,发现位于陆坡上的第二模态内孤立波ISW4在大约50 s的采集过程中,出现反射同相轴分叉、合并以及密跃层中心深度变化.利用共偏移距道集叠前偏移剖面计算的这些第二模态内孤立波视相速度在0.5 m·s^-1左右,其视传播方向都是沿地震测线从SW到NE(44°N方向,0°指向北).通过对视相速度误差相对较小的三个第二模态内孤立波(ISW1、ISW3和ISW5)的视相速度进行分析,发现总体上第二模态内孤立波视相速度随着水深的增加而增加,另外一般情况下具有较大最大振幅的第二模态内孤立波的视相速度较大.     

Some interactions between small numbers of baroclinic,geostrophic vortices

Abstract

Various interactions between small numbers (two and four) of baroclinic, geostrophic point vortices in a two-layer system are studied with attention to the qualitative changes in behavior which occur as size of the deformation radius is varied.

A particularly interesting interaction, which illustrates the richness of baroclinic vortex dynamics, is a collision between two hetons. (A heton is a vortex pair in which the constituent vortices have opposite signs and are in opposite layers. The “breadth” of a heton is the distance between its constituent vortices. A translating heton transports heat.) When two hetons, which initially have different breadths, collide, the result is either an exchange of partners, or a “slip-through” collision in which the initial structures are preserved. It is shown here that the outcome is always an exchange, provided the deformation radius is sufficiently small. This strongly contrasts with a collision between pairs of classical, one-layer vortices in which no exchange occurs if the initial ratio of the breadths is sufficiently extreme.

Finally the transport of passive fluid by a translating baroclinic pair is investigated. A pair of vortices in the top layer transports no lower layer fluid if the distance between the vortices is less than 1.72 deformation radii. By contrast, the size of the region trapped by a heton increases without bound as the spacing between the vortices increases.     

恒春海脊附近第二模态内孤立波的特征研究

以往通过地震海洋学方法发现的大多是第一模态内孤立波,本文在台湾岛以南恒春海脊处捕捉到了两个大振幅凸型第二模态内孤立波.利用叠前偏移剖面得到内孤立波的视相速度约为0.7 m·s^-1,其视传播方向沿地震测线从S到N,总体上满足较大最大振幅的波视相速度也较大的规律.地震海洋学剖面得到的振幅曲线观测值和理论方程计算出的第二模态内波振幅垂向变化理论值之间具有较好的一致性.通过叠前偏移剖面观察该内孤立波细结构的变化,发现在采集过程中,出现反射同相轴后翼变陡、分叉、合并的现象.由于密跃层中心偏离水层中心以及背景剪切流的影响,两个内孤立波尾部发育了高频内波,使得能量耗散增强.此外,第二模态内孤立波的生成和传播特征受到水体层结、黑潮、背景剪切流以及海底地形等多个因素的影响.

     

Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L _b , π/L _b ], where L _b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.     

Hydromagnetic waves in a differentially rotating,stratified spherical shell

Abstract

Investigations of an earlier paper (Friedlander 1987a) are extended to include the effect of an azimuthal shear flow on the small amplitude oscillations of a rotating, density stratified, electrically conducting, non-dissipative fluid in the geometry of a spherical shell. The basic state mean fields are taken to be arbitrary toroidal axisymmetric functions of space that are consistent with the constraint of the ‘‘magnetic thermal wind equation''. The problem is formulated to emphasize the similarities between the magnetic and the non-magnetic internal wave problem. Energy integrals are constructed and the stabilizing/destabilizing roles of the shears in the basic state functions are examined. Effects of curvature and sphericity are studied for the eigenvalue problem. This is given by a partial differential equation (P.D.E.) of mixed type with, in general, a complex pattern of turning surfaces delineating the hyperbolic and elliptic regimes. Further mathematical complexities arise from a distribution of the magnetic analogue of critical latitudes. The magnetic extension of Laplace's tidal equations are discussed. It is observed that the magnetic analogue of planetary waves may propagate to the east and to the west.     

Variation of Froude number versus depth during the passage of internal solitary waves from the in-situ observation and a numerical model

There are few studies on the variation of Froude number versus depth during the passage of internal solitary waves (ISWs). In this paper, according to the limited observational data of the ISWs near the Dongsha Islands in the northern South China Sea (SCS), it is found that, during the passage of an ISW, the local Froude number near the main thermocline gets maximum and the flow is supercritical, with the Froude number decreasing gradually upward and downward, whilst before and after the passage of the ISW, the local Froude number is less than 1 and the flow is subcritical in the entire water column. Since the observational current data are only available in the upper ocean, a two-dimensional continuously stratified nonlinear model is employed here to study the variation of Froude number versus depth during the passage of an ISW. The modeled ISW induced current field agrees well with that observed during the passage of the observational ISW. It is found that, even during the passage of the ISW, although the Froude number is larger than 1 in the upper layer, it is less than 1 in the lower layer, and the duration when the Froude number is larger than 1 also changes with depth. Thus, it is suggested that, once the Froude number near the main thermocline is larger than 1, the passage of ISWs could be identified. Moreover, it would be better to choose a sampling interval of no more than 1/4 passage period of the ISW during an in-situ observation of ISWs.     

基于COSMIC卫星观测数据的平流层重力波的全球分布特征研究

本文利用2007年1月至2012年12月的COSMIC卫星温度剖线,从中提取了垂直波长在3~10 km的重力波扰动信息,进而分析了全球平流层大气重力波的分布特征.赤道地区低平流层重力波表现出明显的准两年变化,这种变化与风场的准两年变化具有明显的相关性,向下发展速度约为1 km/月;赤道地区高平流层(35 km以上区域)的重力波活动则存在明显的半年变化.中高纬度重力波活动主要表现为冬季强夏季弱.在南极地区存在着与急流的时间、空间以及强度变化密切相关的重力波分布特征,这说明在南极极夜急流是非常重要的一个重力波源;而在北极极夜急流的作用则没有那么强.此外,通过考察不同高度的重力波活动特征,我们发现:30 km以下重力波活动较强区域主要在赤道地区且与强对流区分布基本吻合,地形诱发的以及与天气系统相关的强重力波活动在该高度范围内同样出现;而在30 km以上的区域重力波活动强度分布则会出现与平流层爆发性增温以及极夜急流有关的变化.     

Scattering of plane P waves by circular-arc layered alluvial valleys: An analytical solution

IntroductionThe site effect of local inhomogeneity and irregularity on seismic wave propagation is one of the most attractive topics in seismology. The problem may be resolved by either analytical method or numerical method. Here, the analytical method is the wave function expansion method; numerical methods include the finite difference, finite element, boundary integration equation, and discrete wave number method, etc. Although these numerical methods can be applied for sites of arbitrary s…