ANALYTIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS IN THE ATMOSPHERE

In this paper,starting with the nonlinear equations of atmospheric motion and using a relatively simple method,we have obtained the periodic solutions to the nonlinear inertia waves,internal gravity waves and Rossby waves.These solutions represent the characteristics of nonlinear waves in the atmosphere.A preliminary analysis reveals that as for the inertia waves and internal gravity waves with finite amplitudes, the larger the amplitudes are,the faster the waves propagate,but for the Rossby waves with finite amplitudes,the larger the amplitudes and wavelengths are,the slower the waves move.The practical senses of the solutions are also discussed in this paper. This paper gives a new way to study the nonlinear waves.This result has certain significance for the weather forecasting and the study of atmospheric turbulence.     

垂直切变流中非线性重力波及其相互作用

利用多重尺度摄动法，推导出斜压大气中(基本风场具有垂直切变)两个非线性重力波相互作用方程，这两个方程联立组合为耦合非线性schrǒdīnger方程组。两个重力波相互作用时可激发出重力驻波。数值计算表明：两个孤立重力波相遇，相追会使波振幅增大，波宽变窄。强烈对流天气突然爆发的可能原因之一是中尺度重力波非线性相互作用的结果。     

切变基本纬向流中赤道Rossby包络孤立波

利用多重尺度摄动法，从描写赤道Rossby波的正压大气位涡度方程中推导出在切变基本纬向流中非线性赤道Rossby波包演变所满足的非线性Schrodinger方程，并得到其单个包络孤立子波解，分析基本流切变对非线性赤道Rossby波动的影响。     

Nonlinear waves in barotropic model

In this paper, from the system of equation describing a barotropic atmosphere using the method of Taylor expansion for the nonlinear terms, the periodic solutions of the nonlinear inertio-surface gravity waves and Rossby waves have been obtained.The finite-amplitude nonlinear inertio-surface gravity waves and Rossby waves with horizontal divergence satisfy all the KdV equation. The solutions are all the cnoidal function, i, e, the cnoidal waves which in-clude the linear waves and form the solitary waves under certain conditions. For the finite-amplitude Rossby waves with horizontal divergence, we find the new dispersive relation including both the wave number and the amplitude parameter. In case of small amplitude it is reduced to the Yeh formula. It is shown that the larger the amplitude and width, the faster the finite-amplitude inertio-surface gravity waves and the slower the finite-amplitude Rossby waves with horizontal divergence propagate. The blocking or cut-off system in which the amplitude and width are large may be considered as Rossby solitary waves.     

MESOSCALE NONLINEAR GRAVITATIONAL WAVES AND THEIR INTERACTION IN VERTICAL SHEAR FLOW*

Interaction equations of two nonlinear gravitational waves in baroclinic atmosphere are presented via multi-scale perturbation method,which can be classified into coupling nonlinear Schrodinger equations.In particular,the interaction course of two nonlinear gravitational waves of basic flow in vertical linear and quadratic shear is illustrated.Numerical calculation displays that wave amplitude enlarges and wave width narrows when two solitary gravitational waves meet and chase;that basic flow with single shear is more beneficial than that with quadratic shear to the interaction of two nonlinear wave packets;and that the interaction of two wave packets makes wave shape change more greatly and energy more dispersive,which contributes to the occurrence of changeable weather.Therefore,one of the probable mechanisms for the appearance of strong convection weather is the interaction between mesoscale nonlinear gravitational waves.     

低纬大气Kelvin波和Rossby波的非线性相互作用

利用多重尺度摄动法,对低纬大气Kelvin波和Rossby波的波包演变进行了分析,得到两波的非线性相互作用方程为耦合的非线性复系数Landau方程组。数值计算表明两波相遇将使波振幅变化,波宽变窄;两波非线性相互作用可能是低纬强烈天气突然爆发的原因之一。     

COMPARISON OF LINEAR AND NONLINEAR WAVES ASSOCIATED WITH STEADY-STATE FORCING SOURCES

Low-frequency phenomena in the atmosphere are intimately related to stationary waves and, in a sense, the former may even be viewed as the time-varying part of the quasi-stationary waves themselves, Much attention has been focused on nonlinear interactions in the conceptual study on stationary waves. Linear and nonlinear primitive-equation baroclinic spectral models are adopted to investigate the response of stationary waves to largescale mechanical forcing and steady-state thermal forcing, both idealized and realistic, followed by calculations of the EP fluxes and three-dimensional wave activity fluxes(Plumb, 1985) for both the linear and nonlinear solutions. Results show that when the forcing source grows intense enough to be comparable to the real one, non-linear interaction becomes very important, especially for the maintenance of tropical and polar stationary waves. Care should be taken, however, in using the EP flux and Plumb's 3-D flux for diagnostic analysis of observational data as they are highly sensitive to nonlinear interaction.     

弱耗散作用下非线性Rossby波的调制不稳定

本文使用WKB方法对弱耗散作用下基本气流具有弱切变的行星大气中非线性Rossby波进行了研究,得到了弱耗散的非线性Rossby波振幅所满足的复系数非线性Schrdinger方程,并分别就无耗散和弱耗散的Rossby波的边带稳定性问题进行了讨论,得到了一些新结果.     

纬向气流对地形Rossby波的影响

本文在半地转近似下,采用相平面方法讨论了纬向基本气流对线性和非线性地形Rossby波的稳定性及解的性质的影响。结果指出:线性和非线性稳定性判据形式一样,纬向气流及其切变对稳定性有明显影响;在非线性近似下,可形成孤立波槽和孤立波脊。     

The role of topography and diabatic heating in the formation of dipole blocking in the atmosphere

In this paper, the nonlinear stationary waves forced by topography and diabatic heating are investigated. It is pointed out that (1) the nonlinear interaction of different stationary waves forced only by topography might form dipole blocking in the atmosphere, this might explain the dipole blocking appeared in the Pacific and Atlantic regions; (2) the dipole blocking could not be caused by the nonlinear interaction of the different stationary waves forced by the diabatic heating alone; (3) the nonlinear interaction of the diffferent stationary waves forced by both topography and diabatic heating could initiate dipole blocking in the atmosphere. In winter, the dipole blocking mainly occurs in the west regions of the Pacific and the Atlantic, and the heat source over the western part of the two oceans is advantageous to the formation of dipole blocking in the west of two oceans. However, in summer, the dipole blocking could be formed in the east part of the two oceans, and the heat source over the eastern part of two continents is favourable for the formation of dipole blocking in the east regions of two oceans.     

半地转近似下的非线性波

本文应用半地转的概念分析和求解了正压和斜压的大气运动。指出:(1)在水平无辐散或准地转条件下求单波解无法表现非线性作用的困难,可以应用半地转的假定来解决。为了避开上述困难,不得已仅保留部分非线性项,但为了准确表征大尺度运动,最好应用半地转假定,保留全部非线性项。(2)准地转的假定可以滤去快速波动,半地转的假定同样也可以滤去快速波动,而且半地转条件下的Rossby波具有充分的非线性特征。所以,它为讨论非线性问题提供了一个途径。     

The Role of Topography and Diabatic Heating in the Formation of Dipole Blocking in the Atmosphere

In this paper, the nonlinear stationary waves forced by topography and diabatic heating are investigated. It is pointed out that (1) the nonlinear interaction of different stationary waves forced only by topography might form dipole blocking in the atmosphere, this might explain the dipole blocking appeared in the Pacific and Atlantic regions; (2) the dipole blocking could not be caused by the nonlinear interaction of the different stationary waves forced by the diabatic heating alone; (3) the nonlinear interaction of the diffferent stationary waves forced by both topography and diabatic heating could initiate dipole blocking in the atmosphere. In winter, the dipole blocking mainly occurs in the west regions of the Pacific and the Atlantic, and the heat source over the western part of the two oceans is advantageous to the formation of dipole blocking in the west of two oceans. However, in summer, the dipole blocking could be formed in the east part of the two oceans, and the heat source over the eastern     

线性和非线性地形罗斯贝波

本文在半地转概念下,讨论了线性和非线性地形罗斯贝波的稳定性及其解。指出:线性和非线性稳定性判据形式一致。在线性时,地形东西向及南北向坡度对稳定波动的周期和传播速度有明显影响,其解为周期函数;在不稳定时,解为非周期函数。二级近似时其解为孤立波形式——不稳定时,在地形不同位置可形成东移或西移的孤立波槽或孤立波脊;而在稳定时只形成孤立波槽。三级近似时解出现间断点。     

Diagnosis of the forcing of inertial-gravity waves in a severe convection system

The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms—linear and nonlinear terms, and diabatic heating—on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.     

Model multi-cloud parameterizations for convectively coupled waves: Detailed nonlinear wave evolution

Recent observational analysis reveals the central role of three cloud types, congestus, stratiform, and deep-convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating 2-day waves, and the Madden–Julian oscillation. Recently, a systematic model convective parametrization highlighting the dynamic role of the three cloud types has been developed by the authors involving two baroclinic modes of vertical structure: a deep-convective heating mode and a second mode with low level heating and cooling corresponding, respectively, to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep-convective precipitation and also a nonlinear switch which favors either deep or congestus convection depending on the relative dryness of the middle troposphere. The detailed nonlinear evolution of large scale convectively coupled waves in the model parametrization is studied here in a chaotic intermittent regime of the nonlinear dynamics associated with weaker mean radiative cooling where such waves are isolated in space and time. This regime is utilized to elucidate in a clean fashion several novel features of the model parametrization. In particular, four stages of nonlinear wave evolution occur: in the preconditioning and birth stages, the role of congestus moistening and second baroclinic convergence are crucial while in the dying stage of the large scale convectively coupled wave, the role of the nonlinear switch, and the drying of the troposphere are essential. In the mature phase, the large scale features of the convectively coupled waves resemble those in observations of convectively coupled Kelvin waves including the propagation speed, wave tilt, temperature, heating, and velocity structure.     

LINEAR AND NONLINEAR BAROTROPIC INSTABILITY

Based on a non-frictional and non-divergent nonlinear barotropic vorticity equation and its solutions oftravelling waves,the criteria for linear and nonlinear barotropic instability are gained respectively at an equilibriumpoint of the equation on a phase plane.The linear and nonlinear analytical solutions to instability waves arealso found.The computational results show that if their amplitudes are equal at the initial time,the amplitudeincrements of nonlinear instable barotropic wave are always less than those of linear instable barotropic wave.The nonlinear effects can slow down the exponential growth of linear instability.The time needed for makingthe amplitude double that of initial time by instabilities,is about 6h for linear instability and about 18h fornonlinear instability,the latter is in agreement with the observations in the real atmosphere.     

Equatorial solitary waves of tropical atmospheric motion in shear flow

Starting from the primary equations, the author derives the KdV equation which describes solitary Rossby waves in the tropical atmosphere, and indicates that, because these waves are ageostrophic, they differ from the quasigeostrophic solitary Rossby waves studied by Redekopp et al. Owing to nonlinear action, these waves are also different from traditional linear waves of the tropical atmosphere. The author believes that the stationary tropical atmospheric waves reflect the characteristics of solitary waves in that the energy does not disperse.     

Plumb三维波作用通量和EP剖面图——对定常波模式输出结果的诊断分析

本文利用Plumb(1985年)所导出的线性、准地转定常波的三维波作用通量和Eliassen-Palm(EP)剖面图,对于一个线性和一个非线性的关于大气定常波的斜压初始方程三维谱模式的若干输出结果进行了诊断分析。分析结果表明,Plumb通量和EP剖面图,不仅可以用来诊断波作用在三维空间的传播、波动与纬向平均气流之间的相互作用,而且也可以用来诊断非线性作用的显著性,在利用它们对于观测资料进行诊断分析时,应当充分注意到非线性作用的影响。     

100 hPa极地涡旋分裂和重建的能量机制

对于100 hPa极涡分裂过程,对流层2波能量的大量上传是主要因素,其次是波与波非线性相互作用的影响。而对极涡重建过程,主要因素是波与波非线性相互作用,其次是对流层能量上传变化的影响。