On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

Tsunamis and Barge Canals

Russell’s velocity formula was at the center of the controversy over the existence of the solitary wave; but today the topic is rarely mentioned. It is an immediate corollary of modern bifurcation theory; and it is fundamental to modelling waves in deep water. A tsunami 60 cm high in an ocean 4 km deep is 377 km long, travels with a velocity of 713 km/hr, and carries a transverse energy density of 2 × 10⁹ joules/meter, yet the maximum speed of the surface current is only 2.97 cm/s.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

Evolvement of tsunami waves on the continental shelves with gentle slope in the China Seas

Potential tsunami generated in the Okinawa Trench or the Manila Trench may attack the southeast coast of China. The continental shelves with extremely gentle slope in the China Seas affect the evolvement of tsunami waves. In this paper, we carry out the simulation of tsunami propagation based on the fully nonlinear and highly dispersive Boussinesq model, which could describe the nonlinearity and dispersion of water waves quite well. So the undulation characters could be well presented. In terms of the real topographies of the East China Sea and the South China Sea, we take some typical profiles to simulate the hypothetical tsunamis generated in the Okinawa Trench and the Manila Trench. Different waveforms in the near shore regions are obtained. The N-shape tsunami waves will evolve into long wave trains, undular bores or solitons near the coastal area. The numerical results of the near shore waveform provide essential conditions for the further studies of tsunami runup and inundation.     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation

The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength ℓ, approximating the kinetic energy for the case of “rather long and rather low” waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ℓ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied.     

Solitary waves at the interface of a two-layer fluid

In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account. First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries (KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion. It is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients and of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind (=0), and conclude that solitary waves cannot exist in the case considered as >0, but may still occur as <0, being in the form other than that of the KdV solitary wave.As for the critical case of second kind (=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.     

The Benjamin–Lighthill Conjecture for Steady Water Waves (revisited)

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.     

Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of , ^5/2 and ^−3/2, respectively, where the small parameter is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.     

Depth‐integrated,non‐hydrostatic model with grid nesting for tsunami generation,propagation, and run‐up

Tsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock‐capturing dispersive wave model in the spherical coordinate system for basin‐wide evolution and coastal run‐up of tsunamis and discusses the implementation of a two‐way grid‐nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth‐integrated model describes dispersive waves through the non‐hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi‐implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum‐conserved advection scheme with an upwind flux approximation. The two‐way grid‐nesting scheme utilizes the Dirichlet condition of the non‐hydrostatic pressure and both the horizontal velocity and surface elevation at the inter‐grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter‐grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth‐dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run‐up. Numerical experiments of solitary wave propagation and N‐wave run‐up verify the implementation of the grid‐nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment. Copyright © 2010 John Wiley & Sons, Ltd.     

基于势流理论的内孤立波与立管作用数值研究

内孤立波是一种发生在水面以下的在世界各个海域广泛存在的大幅波浪, 其剧烈的波面起伏所携带的巨大能量对以海洋立管为代表的海洋结构物产生严重威胁, 分析其传播演化过程的流场特征及立管在内孤立波作用下的动力响应规律对于海洋立管的设计具有重要意义. 本文基于分层流体的非线性势流理论, 采用高效率的多域边界单元法, 建立了内孤立波流场分析计算的数值模型, 可以实时获得内孤立波的流场特征. 根据获得的流场信息, 采用莫里森方程计算内孤立波对海洋立管作用的载荷分布. 将内孤立波流场非线性势流计算模型与动力学有限元模型结合来求解内孤立波作用下海洋立管的动力响应特征, 讨论了内孤立波参数、顶张力大小以及内部流体密度对立管动力响应的影响. 发现随着内孤立波波幅的增大, 海洋立管的流向位移和应力明显增大. 由于上层流体速度明显大于下层, 且在所研究问题中拖曳力远大于惯性力, 因此管道顺流向的最大位移发生在上层区域. 顶张力通过改变几何刚度阵的值进而对立管的响应产生明显影响. 对于弱约束立管, 内部流体的密度对管道的流向位移影响较小.      

Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 2: Hybrid spatial-spectral implementations

In Part 1 (van Groesen and Andonowati [1]), we derived models for the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. In this paper, we will describe hybrid spatial-spectral implementations of the models that retain the basic variational formulation of irrotational surface waves that underlays the derivation of the continuous models. It will be shown that the numerical codes are robust and efficient from results of simulations of two test cases of waves above a 1:20 sloping bottom from 30 m to 15 m depth: one simulation of a bichromatic wave train, and one of irregular waves of JONSWAP type. Measurements of scaled experiments at MARIN hydrodynamic laboratory and simulations with two other numerical codes will be used to test the performance. At the end of the full time trace of 3.5 h details of the irregular waves that travelled over more than 5000 m are clearly resolved with a correlation of more than 90%, in calculation times of less than 5% of the physical time. Also freak-like waves that appear in the irregular wave are shown to be modelled to a high degree of accuracy.     

Asymmetrical solitary waves in coupled nonlinear transmission lines

Asymmetrical solitary waves are characterized in symmetrical, capacitively coupled transmission lines, called coupled nonlinear transmission lines (NLTLs), which are periodically loaded with Schottky varactors, in order to amplify short electrical pulses. In this work, the transmission equations of coupled NLTLs are numerically solved, using two symmetrical Korteweg–de Vries (KdV) systems with linear dispersionless coupling, to show that the lines exhibit symmetry breaking in order to support asymmetrical solitary waves. In particular, two interacting solitary waves with different polarities are investigated. The KdV systems show that the polarities of the preceding and the subsequent waves in coupled NLTLs are interchanged through colliding interactions. The process is quantified in the framework of reduction theory, assuming small velocity differences and interaction strength between the two waves. In addition, results show that the leading wave gains amplitude via collision, which means that coupled NLTLs provide good platforms for amplifying short electrical pulses.     

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

分层流体中内孤立波在台阶上的反射和透射

基于匹配渐近展开和格林函数的方法，研究了两层流体系统中内孤立波在台阶地形上透射、 反射及其分裂的演化特征. 通过保角变换和求解奇异Fredholm积分方程，获得了反映地形 效应对Boussinesq方程影响的约化边界条件，藉此建立了KdV演化方程的``初值'问题， 根据散射反演理论获得了反射波和透射波的解析表达式. 分析结果表明：上下流体层的厚度 比、密度比以及台阶高度对于反射和透射波振幅及其分裂具有显著的影响. 尤其当上层流体 厚度小于下层厚度时，由于存在临界点，在其附近反射波的幅值随台阶高度的演化由单调增 变为单调减，透射波的幅值由单调减变为单调增；上台阶的反射波与入射波反相，其最大幅 值可达到入射波的数倍；此外，下台阶反射波也可发展为单支孤立波，它区别于单层流体中 反射波仅为衰减的振荡波列.