Forced solitary Rossby waves under the influence of slowly varying topography with time

By using a weakly nonlinear and perturbation method, the generalized inhomogeneous Korteweg-de Vries (KdV)-Burgers equation is derived, which governs the evolution of the amplitude of Rossby waves under the influence of dissipation and slowly varying topography with time. The analysis indicates that dissipation and slowly varying topography with time are important factors in causing variation in the mass and energy of solitary waves.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

耗散对孤波与局地地形相互作用的影响

利用扰动法，由包括耗散和地形的准地转位涡度方程导出了强迫mKdV-Burgers方程，求得了小耗散情形下mKdV-Burgers方程的近似分析解，分析了mKdV孤波质量和能量的时间演变特性。对给定的局地地形，利用拟谱法对强迫mKdV-Burgers方程进行了数值求解。结果显示，小耗散的存在使弧波的振幅和移速随时间缓慢地减小，孤波宽度则随时间缓慢增大；在耗散和地形强迫的非线性系统中，在孤波与地形的相互作用中，耗散的存在使孤波在强迫区附近振荡传播，这有利于大振幅扰动的形成。     

Berry phase in open quantum systems: a quantum Langevin equation approach

The evolution of a two level system with a slowly varying Hamiltonian, modeled as a spin 1/2 in a slowly varying magnetic field, and interacting with a quantum environment, modeled as a bath of harmonic oscillators is analyzed using a quantum Langevin approach. This allows to easily obtain the dissipation time and the correction to the Berry phase in the case of an adiabatic cyclic evolution.     

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.     

缓变地形下Rossby波振幅演变满足的带有强迫项的KdV方程

运用非线性方法与摄动法,讨论了当地形随时间缓变时Rossby波振幅的演变问题.从均值流体准地转涡度方程推导,得到Rossby波振幅演变满足带有强迫项的KdV方程的结论,而地形随时间的缓慢变化诱导了该方程的强迫项. 关键词： 非线性Rossby波 带有强迫项的KdV方程 摄动法 缓变地形     

Numerical wave propagation on non-uniform one-dimensional staggered grids

The wave propagation behaviour of centered difference schemes on one-dimensional non-uniform staggered grids is investigated. Previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids. For waves of a given frequency, the wave field is decomposed into right- and left-propagating components, and a wave energy conservation law is derived in terms of these components. For slowly varying grids, separate evolution equations for the right- and left-propagating components are derived, leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid, provided that waves of that frequency are resolvable. However, there will be reflection from any location at which the wave group velocity goes to zero. The possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted. Grids with an abrupt jump in resolution are also investigated. It is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump. However, it does not appear possible to avoid complete reflection, except by introducing extra dissipation terms, if the waves are not resolvable on one side of the jump. An example is presented of a second-order accurate scheme that spontaneously radiates waves from the resolution jump.     

缓变下垫面和耗散作用的非线性Rossby波

从准地转位涡方程出发,采用摄动方法和时空伸长变换推导了在缓变下垫面和耗散共同作用的Rossby代数孤立波方程,得到Rossby波振幅满足带有缓变下垫面的非齐次Benjamin-Davis-Ono-Burgers(BDOBurgers)方程的结论.指出地形效应和耗散是诱导非线性Rossby波产生的重要因素,说明了在缓变下垫面强迫效应和非线性作用相平衡的假定下,Rossby孤立波振幅的演变满足非齐次BDO-Burgers方程,给出在切变基本气流下缓变下垫面和正压流体中Rossby波的相互作用.     

Role of equilibrium plasma flow on damping of slow MHD waves

In the solar corona waves and oscillatory activities are observed with modern imaging and spectral instruments. These oscillations are interpreted as slow magneto-acoustic waves excited impulsively in coronal loops. This study explores the effect of steady plasma flow on the dissipation of slow magneto-acoustic waves in the solar coronal loops permeated by uniform magnetic field. We have investigated the damping of slow waves in the coronal plasma taking into account viscosity and thermal conductivity as dissipative processes. On solving the dispersion relation it is found that the presence of plasma flow influences the characteristics of wave propagation and dissipation. We have shown that the time damping of slow waves exhibits varying behavior depending upon the physical parameters of the loop. The wave energy flux associated with slow magnetoacoustic waves turns out to be of the order of 10⁶ erg cm⁻² s⁻¹ which is high enough to replace the energy lost through optically thin coronal emission and the thermal conduction below to the transition region.     

A numerical study of solitary waves in a non-linear Bragg reflector

A two-dimensional time domain finite difference beam propagation method, based on the slowly varying envelope approximation is presented. Expressions to correct this approximation and to include third-order non-linear effects are given. The method is applied on solitary waves in a non-linear Bragg reflector, assuming realistic materials parameters.     

Analogien zwischen außerordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen

Within the limits of Linear Optics we treat analogies between ordinary and extraordinary waves in uniaxial media which become conspicuous through a nonorthogonal transformation of coordinates. To any ordinary wave solution in unbounded uniaxial media we can construct a corresponding extraordinary wave solution by interchanging electrical and magnetical field components. Boundary conditions for instance for ideal conducting plane surfaces approximately preserve their original form, if the optical axis or the middle wave vector are normal to the surface. The parabolic approximative equations for slowly varying amplitudes are derived, the polarisation of these waves being considered as a slowly varying quantity. Further these approximative equations are expanded to include frequency dispersion. Through the specified transformation we can simplify problems with extraordinary waves.     

Magnetostatic waves in a time-dependent medium

Propagation of magnetostatic waves (MSW) in planar magnetic structures with time-dependent parameters is studied theoretically and experimentally for the case where the structure is based on ferrite films. The time-dependent parameter is chosen to be the magnetizing field, which is taken to be uniform in space and slowly varying in time. Asymptotic perturbation theory is used to obtain an equation for the slowly varying amplitude and a relation for the phase, which describes the time behavior of the frequency and envelope of an MSW pulse. The time behavior of the frequency spectrum of an MSW propagating in an iron-yttrium garnet film is studied experimentally for different forms of modulation of the magnetic field. Space-time focusing of MSW pulses is studied for the case of a time-dependent field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 54–66, November, 1988.     

Comments on Surface Plasma Wave Solitons

We discuss surface waves propagating on a semi-infinite cold plasma with slowly varying density. Previous results are modified.     

Autoresonance in a ferromagnetic plate with a stripe domain structure

Nonlinear dynamics of domain walls in a ferromagnetic plate with a stripe domain structure in an external field with slowly varying amplitude and frequency has been theoretically investigated. The conditions for guided autoresonant excitation of nonlinear domain-wall oscillations in the presence of dissipation have been determined.     

宽度缓慢均匀变化水槽中的非传播表面孤波

用多重尺度微扰理论导出了宽度均匀缓变水糟中非传播孤波所服从的非线性方程及其解析解。结果指出，孤波恒向宽度较窄的一端近加速移动。加速度正比于宽度的变化率．本文所用的方法原则上可以推广于其他均匀缓变波导中线性波和非线性波的理论研究。 关键词：     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry–Perot effect.     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry-Perot effect.     

Generation of Solitary Rossby Waves by Unstable Topography

The effect of topography on generation of the solitary Rossby waves is researched.Here,the topography,as a forcing for waves generation,is taken as a function of longitude variable x and time variable t,which is called unstable topography.With the help of a perturbation expansion method,a forced mKdv equation governing the evolution of amplitude of the solitary Rossby waves is derived from quasi-geostrophic vorticity equation and is solved by the pseudospectral method.Basing on the waterfall plots,the generational features of the solitary Rossby waves under the influence of unstable topography and stable topography are compared and some conclusions are obtained.     

Dispersion Relations for Drift-Alfvén and Drift-Acoustic Waves in Arbitrary Directions

A dispersion equation for low-frequency waves in a fully ionized plasma with slowly varying density and magnetic field is derived from the two-fluid equations. The solutions are discussed in several limiting cases. Phase-velocity and refractive index surfaces are presented for the fast and slow magneto-acoustic waves and for the shear-Alfvén wave, influenced by the inhomogeneity drifts.     

Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation

Following the ideas of Howard and Kopell [9] a perturbation theory is developed for slowly varying fully nonlinear wavetrains (i.e. solutions which appear locally as travelling waves, but with frequencies and wavelengths which may vary widely on long length and time scales). This perturbation theory is applied to the Ginzburg-Landau equation. The motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber. Numerical results supporting this theory are presented. A shock structure is proposed and numerically verified. These results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains. The evolution of a marginally diffusively stable wavetrain is also examined; it is argued that the evolution is governed by a perturbed Korteweg-de Vries equation.