Horizontal advection of temperature and salinity by Rossby waves in the North Pacific

The Aquarius satellite has been used for the first time to characterize Rossby waves in sea surface salinity (SSS) measurements for the North Pacific Ocean. Westward propagating wave signals are delineated by the SSS zonal salinity gradients. The phase velocities and spectral properties obtained from zonal salinity gradients are closely correlated with corresponding values obtained from the sea surface temperature (SST) zonal gradient and the altimetry-derived meridional velocity. The westward propagating SSS signals are consistent with Rossby wave advection across the strong meridional gradients of water characteristics. Following Killworth, we attempted to provide satellite-based estimates of the contribution of horizontal Rossby wave advection to the surface transfer of temperature and salinity in the North Pacific Ocean. Westward propagating signals in the SST and SSS zonal gradient fields show that the observed intensity of meridional advection by the ambient gradients of SST and SSS is less than the intensity predicted by an analytical solution of the transfer equation for Rossby waves. Our results extend the previous studies of physical mechanisms of Rossby wave manifestation at the sea surface and we demonstrate that Rossby waves are responsible for low-frequency oscillations in SST and SSS concentration in the North Pacific.     

Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations

In the present paper, a difference scheme on a non-uniform grid is constructed for the stationary propagating localized waves of the 2D Boussinesq equation in an infinite region. Using an argument stemming form a perturbation expansion for small wave phase speeds, the asymptotic decay of the wave profile is identified as second-order algebraic. For algebraically decaying solution a new kind of nonlocal boundary condition is derived, which allows to rigorously project the asymptotic boundary condition at the boundary of a finite-size computational box. The difference approximation of this condition together with the bifurcation condition complete the algorithm. Numerous numerical validations are performed and it is shown that the results comply with the second-order estimate for the truncation error even at the boundary lines of the grid. Results are obtained for different values of the so-called ‘rotational inertia’ and for different subcritical phase speeds. It is found that the limits of existence of the 2D solution roughly correspond to the similar limits on the phase speed that ensure the existence of subcritical 1D stationary propagating waves of the Boussinesq equation.     

基于多重反射法对周期结构的频率响应研究

采用多重反射法对受到外扰的二组元周期梁结构的频率响应进行了研究．施加至Ⅱ周期梁结构上的外部扰动被假定为一入射波，传播波入射到不连续处会产生反射波和透射波，进而在周期结构中会产生多重的反射和透射．首先，基于波的多重反射，考虑施加扰动的组元上的波场；其次，由于波的透射，分别考虑两个传播方向上的其他组元的波场，作为初始波场；最后，可先考虑某个组元右侧的所有组元上的向左传播的波在其上的叠加，作为一次迭代波场；再考虑某个组元左侧的所有组元上的向右传播的波在其上的叠加，作为二次迭代波场．依次类推，基于多重反射法，叠加了入射波引起的多重反射和透射，得到了所有组元的波场．给出了周期梁结构中任一点的波幅与入射波幅之间的函数关系，确定了受外扰的周期梁结构的传播常数及相应的波场的迭代次数．     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

非均匀缓变折射率平板波导放大器中的畸形波

以缓变波导中光束传播的非线性传输方程为研究对象,研究了非均匀缓变折射率平板波导放大器中畸形波的非线性动力学性质.通过相似变换和直接假设,构建出带有自由函数的一阶精确畸形波解.在此基础上,针对不同类型的自由函数,通过数值模拟得到了不同畸形波的波形图,对于描述光纤中出现的一些物理现象具有重要的意义.     

微结构固体中孤立波的动力学稳定性

描述微结构固体中波传播的一种KdV类方程作为控制方程并利用积分因子方法,对微结构固体中传播的孤立波的动力学稳定性进行了数值模拟研究。主要以高斯波、Ricker子波以及双曲正割波作为初始扰动,考察了不同小扰动下孤立波能否较长时间保持波形结构和传播速度而稳定传播问题。模拟结果表明,不同的小扰动对孤立波的影响不同,孤立波的稳定传播与扰动幅度和宽度都有关系,只有受到幅度和宽度都非常小的扰动下在微结构固体中传播的孤立波才能显现出一定程度的抗干扰性和动力学稳定性,可在微结构固体中较长时间稳定传播。     

微结构固体中孤立波的动力稳定性

描述微结构固体中波传播的一种KdV类方程作为控制方程并利用积分因子方法,对微结构固体中传播孤立波的动力学稳定性进行了数值模拟研究.主要以高斯波、Ricker子波以及双曲正割波扰动作为初始扰动,考察了不同小扰动下孤立波能否较长时间保持波形结构和传播速度而稳定传播问题.结果表明,不同的小扰动对孤立波的影响不同,孤立波的稳定传播与扰动幅度和宽度都有关系,只有受到幅度和宽度都非常小的扰动下在弱微尺度非线性效应的微结构固体中传播的孤立波才能显现出一定程度的抗干扰性和动力学稳定性,能够在微结构固体中较长时间稳定传播.     

Numerical scheme based on similarity reductions for the regularized long wave equation

Numerical solution based on similarity reductions for partial differential equations used to get the numerical scheme for the regularized long wave (RLW) equation. The similarity reductions for RLW equation are obtained locally on subdomains defined by the classical three-point stencil. The ordinary differential equation, which deduced from the similarity reduction can be linearized, integrated analytically and then obtain the solution. This approch eliminates the difficulties associated with boundary conditions for the similarity reduction over the whole solution domain. Numerical results are obtained for test problem. The computed results using our scheme confirm the accuracy of our scheme.     

3D scattering of waves by a cylindrical irregular cavity of infinite length in a homogeneous elastic medium

The three-dimensional (3D) wave field scattered by an irregular, cylindrical cavity of infinite length contained in a homogeneous elastic medium illuminated by a dilatational point load is obtained. This model is used to evaluate the effect of the cross-sectional geometry of the cavity on the waves propagating in its vicinity. It particularly highlights the identification of the normal modes excited both in the frequency and time domain. The solution is formulated using the boundary element method for a wide range of frequencies and spatially harmonic line loads, which are then synthesized to obtain the time responses. The 3D solution is obtained as a summation of two-dimensional responses for different axial wavenumbers.The responses in the frequency vs. axial-wavenumber domains are presented, allowing the recognition, identification, and physical interpretation of the variation of the wave field when five irregular cross-sections are used, namely a circle, an oval, a thin oval, a kidney and a boomerang.     

The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas

This paper investigates the solitary wave solutions of the two-dimensional regularized long-wave equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas. The main idea behind the numerical solution is to use a combination of boundary knot method and the analog equation method. The boundary knot method is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution, the boundary knot method uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to method of fundamental solution, the radial basis function is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method. According to the analog equation method, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Furthermore, in order to show the efficiency and accuracy of the proposed method, the present work is compared with finite difference scheme. The new method is analyzed for the local truncation error and the conservation properties. The results of several numerical experiments are given for both the single and double-soliton waves.     

Error approximation in the solution of a linear PDE initial-valued problem

This article establishes an error approximation of a semidiscretized scheme in the solution of a linear partial differential equation initial-valued problem. The error approximation relates the error to both equations and conditions in a problem and serves as a problem-oriented error bound. Two examples of a continuous and a discontinuous wave propagating with a speed varying in space have been studied. The results show that the propagating wave suffers distortion, and the numerical wave is not in synchronization with the exact one. This contributes to that a pointwise error measurement misleads. The discrete Fourier transform can be defined on a finite set of unequally spaced mesh points. Errors of various sources interact and cancel each other; therefore, a solution accuracy improvement can be achieved by utilizing error interaction.     

Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.     

Tides as phase-modulated waves inducing periodic groundwater flow in coastal aquifers overlaying a sloping impervious base

This paper presents procedures for analysing the interactions between tidal waves and coastal aquifers overlaying a sloping impervious bed, using Boussinesq equation and wave equation. Fourier series solutions of a linearised Boussinesq equation are presented subject to a periodic boundary condition (BC). The periodic BC is a phase-modulated periodic solution of the wave equation which has been shown to satisfactorily simulate uneven twin peaks of semi-diurnal tides as observed in a study area on the east coast of Queensland, Australia. Numerical analyses show that the Fourier series solutions of Boussinesq equation subject to periodic BCs reveal two important features of the tidal waves. First, the tidal waves damp towards landward, and second, the half amplitude of the tide above the mean sea level is greater than that below it. While the first feature is clearly expected, the second feature is physically more meaningful and important, and is confirmed by the field data.     

Numerical solution of GRLW equation using Sinc-collocation method

In this paper, we present a meshfree technique for the numerical solution of the generalized regularized long wave (GRLW) equation. This approach is based on a global collocation method using Sinc basis functions. The propagation of single solitons and the interaction of two solitary waves are used to validate the method which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the method.     

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor-corrector finite-difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.     

Two-dimensional wave propagation in a nonlinear elastic half-space

Finite amplitude wave propagation in an elastic, isotropic half-space is investigated. A numerical scheme that was previously developed and shown to yield satisfactory accurate results whenever smooth solutions occur is modified here for the cases in which steep solutions are obtained. The stability analysis of the proposed numerical procedure is carried out, and the stability criteria are given in terms of the spectral radii of the matrices involved in the equations of motion. The hyperbolicity conditions of the equations of motion are derived and shown to impose restrictions on the possible values of displacement gradients so that the range of variation of the strength of the applied load is limited. As a first chek of the accuracy of the numerical results, a propagating shock wave is produced numerically and compared with the analytical solution. In a second check, propagating circularly polarized waves are numerically simulated and compared with the corresponding analytical solution. In each case good agreement is obtained. For the “quadratic material” adopted in this paper, it is shown that a compressive normal line force yields propagating pulses having larger amplitudes, broader widths and larger arrival times, as compared with those caused by a tensile one. The linear response is also shown for comparison.     

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

(2+1)-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.     

Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails

The propagation of stationary solitary waves on an infinite elastic rod on elastic foundation equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solution of type of solitary wave, which is a bifurcation problem.     

Simulated wave-driven ANN model for typhoon waves

This paper investigates an artificial neural network (ANN) model for typhoon waves used to modify poor calculations of the numerical model in special cases. Two key factors, local winds and simulated waves produced by the numerical model, were used as input parameters of the proposed ANN model. The waves were simulated by the numerical model from a wave action equation indicating the physical processes of energy transfer and wave propagation. Simulated wave input is a very important parameter for the proposed ANN model, allowing for the accurate calculation of water waves in the sea. The applicable Mike21_SW model was chosen to provide an accurate calculation. Through model verification, the proposed ANN model has a particularly accurate calculation at the peak of each typhoon and at its occurrence time. The computed waves of each typhoon were examined to be consistent with the observed waves.