On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

The Effect of Meridional and Vertical Shear on the Interaction of Equatorial Baroclinic and Barotropic Rossby Waves

Simplified asymptotic equations describing the resonant nonlinear interaction of equatorial Rossby waves with barotropic Rossby waves with significant midlatitude projection in the presence of arbitrary vertically and meridionally sheared zonal mean winds are developed. The three mode equations presented here are an extension of the two mode equations derived by Majda and Biello [ 1 ] and arise in the physically relevant regime produced by seasonal heating when the vertical (baroclinic) mean shear has both symmetric and antisymmetric components; the dynamics of the equatorial baroclinic and both symmetric and antisymmetric barotropic waves is developed. The equations described here are novel in several respects and involve a linear dispersive wave system coupled through quadratic nonlinearities. Numerical simulations are used to explore the effect of antisymmetric baroclinic shear on the exchange of energy between equatorial baroclinic and barotropic waves; the main effect of moderate antisymmetric winds is to shift the barotropic waves meridionally. A purely meridionally antisymmetric mean shear yields highly asymmetric waves which often propagate across the equator. The two mode equations appropriate to Ref. [ 1 ] are shown to have analytic solitary wave solutions and some representative examples with their velocity fields are presented.     

地形对正压大气Rossby波非线性相互作用的影响

本文采用弱非线性近似，推导出地形和Ｅｋｍａｎ摩擦共同作用下连续谱正压Ｒｏｓｓｂｙ波的非线性时空演化方程．根据这组方程，我们研究了窄角谱Ｒｏｓｓｂｙ波包的波波相互作用问题，当一个大振幅Ｒｏｓｓｂｙ波包通过大气传播时，如果它的振幅超过某个阈值，非线性相互作用会使一个尺度比它大的Ｒｏｓｓｂｙ波包和一个尺度比它小的Ｒｏｓｓｂｙ波包的振幅随时间指数增长，这两个次级波的本征频率会发生改变，Ｅｋｍａｎ摩擦、频率不匹配、地形坡度以及波包的空间演变共同决定了主波振幅的阈值及次级波本征频率的改变量．     

Periodic structures of oceanic Rossby wave under the influence of wind stress

A simple oceanic barotropic potential vorticity equation on β-plane with the influence of wind stress is applied to investigate the nonlinear Rossby wave in a shear flow. By the reductive perturbation method, we derived the rotational modified KdV (rmKdV for short) equation. And then with the help of Jacobi elliptic functions, we obtain various periodic structures for these equatorial Rossby waves. It is shown that the wind stress is very important for these periodic structures of rational form.     

Periodic structures of oceanic Rossby wave under the influence of wind stress

A simple oceanic barotropic potential vorticity equation on β-plane with the influence of wind stress is applied to investigate the nonlinear Rossby wave in a shear flow. By the reductive perturbation method, we derived the rotational modified KdV (rmKdV for short) equation. And then with the help of Jacobi elliptic functions, we obtain various periodic structures for these equatorial Rossby waves. It is shown that the wind stress is very important for these periodic structures of rational form.     

Ocean-Atmosphere Interaction in the Lifecycle of ENSO： The Coupled Wave Oscillator

To explain the oscillatory nature of E1 Nino/Southern Oscillation （ENSO）, many ENSO theories emphasize the free oceanic equatorial waves propagating/reflecting within the Pacific Ocean, or the discharge/recharge of Pacific-basin-averaged ocean heat content. ENSO signals in the Indian and Atlantic oceans are often considered as remote response to the Pacific SST anomaly through atmospheric teleconnections. This study investigates the ENSO life cycle near the equator using long-term observational datasets. Space-time spectral analysis is used to identify and isolate the dominant interannual oceanic and atmospheric wave modes associated with ENSO. Nino3 SST anomaly is utilized as the ENSO index, and lag-correlation/regression are used to construct the composite ENSO life cycle. The propagation, structure and feedback mechanisms of the dominant wave modes are studied in detail. The results show that the dominant oceanic equatorial wave modes associated with ENSO are not free waves, but are two ocean-atmosphere coupled waves including a coupled Kelvin wave and waves are not confined only to the Pacific a coupled equatorial Rossby （ER） wave. These Ocean, but are of planetary scale with zonal wavenumbers 1-2, and propagate all the way around the equator in more than three years, leading to the longer than 3-year period of ENSO. When passing the continents, they become uncoupled atmospheric waves. The coupled Kelvin wave has larger variance than the coupled ER wave, making the total signals dominated by eastward propagation. Surface zonal wind stress （x） acts to slow down the waves. The two coupled waves interact with each other through boundary reflection and superposition, and they also interact with an off-equatorial Rossby wave in north Pacific along 15N through boundary reflection and wind stress forcing. The precipitation anomalies of the two coupled waves meet in the eastern Pacific shortly after the SST maximum of ENSO and excite a dry atmospheric Kelvin wave which quickly circles the whole equator and leads to a zonally symmetric signal of troposphere temperature. ENSO signals in the Indian and Atlantic oceans are associated with the two coupled waves as well as the fast atmospheric Kelvin wave. The discharge/recharge of Pacific-basin-averaged ocean heat content is also contributed by the two coupled waves. The above results suggest the presence of an alternative coupled wave oscillator mechanism for the oscillatory nature of ENSO.     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

A Long Wave Approximation for Capillary-Gravity Waves and an Effect of the Bottom

The Korteweg–de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the 2-dimensional water wave problem, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. In this article, we consider the validity of this approximation in the case of the presence of the surface tension. Moreover, we consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation. We derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval. We also prove that if the initial data and the bottom decay at infinity in a suitable sense, then the KdV equation takes the place of the coupled equations.     

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.     

Propagation of the nonlinear damped Korteweg‐de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods

In this article, we consider the problem formulation of dust plasmas with positively charge, cold dust fluid with negatively charge, thermal electrons, ionized electrons, and immovable background neutral particles. We obtain the dust‐ion‐acoustic solitary waves (DIASWs) under nonmagnetized collision dusty plasma. By using the reductive perturbation technique, the nonlinear damped Korteweg‐de Vries (D‐KdV) equation is formulated. We found the solutions for nonlinear D‐KdV equation. The constructed solutions represent as bright solitons, dark solitons, kink wave and antikinks wave solitons, and periodic traveling waves. The physical interpretation of constructed solutions is represented by two‐ and three‐dimensional graphically models to understand the physical aspects of various behavior for DIASWs. These investigation prove that proposed techniques are more helpful, fruitful, powerful, and efficient to study analytically the other nonlinear nonlinear partial differential equations (PDEs) that arise in engineering, plasma physics, mathematical physics, and many other branches of applied sciences.     

Linear Waves and Nonlinear Wave Interactions in a Bounded Three‐Layer Fluid System

We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior.     

Numerical solutions and stability analysis for solitary waves of complex modified Korteweg–de Vries equation using Chebyshev pseudospectral methods

In this research article, the authors investigate the interaction of solitary waves for complex modified Korteweg–de Vries (CMKdV) equations using Chebyshev pseudospectral methods. The proposed method is established in both time and space to approximate the solutions and to prove the stability analysis for the equations. The derivative matrices are defined at Chebyshev–Gauss–Lobbato points and the problem is reduced to a diagonally block system of coupled nonlinear equations. For numerical experiments, the method is tested on a number of different examples to study the behavior of interaction of two and more than two solitary waves, single solitary wave at different amplitude parameters and different polarization angles. Numerical results support the theoretical results. A comprehensive comparison of numerical results with the exact solutions and other numerical methods are presented. The rate of convergence of the proposed method is obtained up to seventh-order.     

两层流体中具有β变化和地形影响的Rossby波

本文研究了两层流体中具有变化的Rossby参数和地形Rossby波的问题.利用行波法和摄动的方法,获得了Rossby波振幅满足齐次KdV方程和齐次mKdV方程,推广了Rossby参数和地形对Rossby孤立波的影响.     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

An approach to the analysis of the interaction of surface waves with depth-varying current fields

An approach is developed for the analysis of the interactions of surface waves with depth-varying current fields. The approach is based on an approximate dispersion relation for wave-current interactions derived from the governing equations of the problem (the inviscid Orr-Sommerfeld equations) coupled with the wave kinematic-wave action formulation of surface wave propagation. The approach is particularly useful in that it focuses directly on the wave parameters of interest (the amplitude, frequency, direction, and wavelength of the wave) and eliminates the requirement to solve the inviscid Orr-Sommerfeld equations to derive these parameters. The validity of the approach is demonstrated by comparisons with exact Orr-Sommerfeld solutions.     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.     

完整Coriolis力作用下带有外源强迫的非线性KdV方程

利用摄动方法,从描写既有Coriolis力垂直分量又含有水平分量的位涡方程出发,给出了近赤道非线性Rossby波所满足的具有外源强迫的非线性KdV方程,并利用Jacobi椭圆函数展开法,求解了改进后的非线性KdV方程的行波解及孤立波解.通过分析KdV方程的行波解,指出Coriolis力的水平分量和外源对Rossby波动的影响.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.