层结大气中非线性椭圆余弦波和孤立波的研究

在前文采用半地转近似和行波法来研究层结大气中的非线性波动的基础上,进一步探讨非线性方程及其级数近似方程的解的稳定性,波动解的存在条件和波动的一些特征。研究指出,在平衡点附近,将非线性方程用其级数近似方程代替是可行的、合理的。在初始拟能满足一定的条件下,椭圆余弦波和孤立波是存在的。文中还求得了椭圆余弦波解的周期、ｘ向波长和振幅以及孤立波的宽度和振幅。并指出,孤立波的宽度和振幅不但与波速有关,还与β因子和层结稳定度有关,而且在相同的某条件下,西行的混合Ｒｏｓｓｂｙ－重力孤立波比起东行的惯性重力孤立波来,宽度要小但振幅却大。     

正压大气中的代数Rossby孤立波

本文研究了基本气流具有线性弱切变的非线性Rossby波,得到了一个非线性发展方程为Kubota方程,在一定的条件下,可变为Benjamin-Ono方程,并指出当代数Rossby孤立波的振幅越大时,代数Rossby孤立波的传播速度越小,基流切变越强,代数Rossby孤立波越慢,同时我们还指出在代数Rossby孤立波的振幅满足一定的条件下,代数Rossby孤立波才随纬度的增高(β减小)而减慢。并且代数Rossby孤立波的结构与大气中的偶极子阻塞是一致的。     

非线性正压Rossby波的精确解

本文从半地转近似下的正压大气运动方程组出发,导出了非线性正压Rossby波所精确满足的KdV方程,求得了椭圆余弦波解和孤立波解,并讨论了非线性波动的波速公式及其振幅与其它波参数间的关系。同时指出,在对大气模式进行数学简化时必须尽可能保证原模式所具有的一些守恒关系,否则有可能使原问题的解受到歪曲。     

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。     

A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear

In this paper an evolution equation in integral-differential form for finite amplitude Rossby waves on a weak shear is presented and an efficient method for its numerical solution is set up. It is shown that a propa-gation of solitary wave is possible whenever a proper weak shear in basic flows acts with the nonlinear effects and dispersion of the media, both in the atmosphere and in the ocean. To test the numerical method for solving the evolution equation, a series of experiments are carried out. The results indicate that the solitary solutions do exist and interact with each other in quite a succinct, manner. Therefore the method is successful and efficient for solving initial value problems of the above equation. The time decoupling problem arising in the numerical scheme and the related filtering technique are discussed. A variety of interesting phenomena such as the interaction of solitary Rossby waves, damping, dispersion and the development of nonlinear wave train are numerically studied.     

Haurwitz solutions of the nonlinear shallow-water equations for small froude number

Summary In this note we find special solutions of the nonlinear shallow-water equations. From the first integrals of the potential vorticity and energy equations for steady flow we derive a single equation in the streamfunction. In the limiting case of very small Froude number, that equation has no solutions corresponding to gravity waves. Under a nonlinear transformation of dependent variable, it becomes a linear equation whose solutions are related to Haurwitz's solutions for nondivergent barotropic flow in spherical surfaces. The distinguishing feature of these solutions is that the streamlines coincide with contours of the free surface: thus, although the height of the free surface varies, the motions of the fluid are horizontal, and the flow is nondivergent.The solutions are easily modified to correspond to Rossby waves propagating eastward or westward without change of shape.The National Center for Atmospheric Research is sponsored by the National Science Foundation.This paper is dedicated to the memory of my scientific mentor and old friend, Bernhard Haurwitz.     

N层模式大气中的斜压孤立Rossby波

文中导得了N层模式的KdV方程,作为例子计算了三层模式的孤立Rossby波流型。基本气流的垂直切变、水平切变和Froude数对孤波流型都有重要影响;不同模态的孤波具有不同流型,不同高度位面上的孤波流型一般是不同的。     

大气非线性波动方程的解

本文从非线性大气运动方程出发,用比较简洁的方法,求得了大气非线性惯性波、非线性重力内波和非线性Rossby波的周期解,这些解反映了非线性大气波动的特色。分析指出:对有限振幅的惯性波和重力内波,振幅大的波传播越快,而有限振福的非线性Rossby波,振幅大、波长长的波传播越慢;本文还分析了这些解的某些可能的实际意义。 这些非线性波的研究提供了解非线性方程的新的途径,而且,它可以与大家熟悉的线性波动进行比较。对于天气预报和大气湍流的研究都有一定的意义。     

Equatorial solitary waves Part 4. Kelvin solitons in a shear flow

It is shown that a mean flow with shear makes the Kelvin wave dispersive. This in turn modifies its nonlinear behavior and makes it necessary to replace the one-dimensional advection equation derived in an earlier work of the author's by the Korteweg-deVries equation instead. The frontogenesis predicted in the earlier paper will still occur, but the wave breaking will not. Instead, once a steep front has formed, it will disintegrate into a train of solitary waves. These then propagate towards the east at a faster-than-linear rate. It is also shown that Kelvin solitary waves will have much smaller zonal widths than Rossby solitons of the same height; “round” Kelvin solitary waves (equal zonal and latitudinal width) are to be expected, and are fully consistent with the small amplitude, weak dispersion theory. An interesting implication of the Korteweg-deVries model is that the peak signal from a nonlinear Kelvin wave packet may be roughly double that of a linear Kelvin wavetrain.     

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.     

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.     

mKdV Equation for Solitary Rossby Waves with Linear Topography Effect in Barotropic Fluids

The modify Korteweg-de Vries (mKdV) equations, governing the evolution of the amplitude of solitary Rossby waves, are derived from quasi-geostrophic vorticity equation by using the perturbation method. The result manifests that the linear topography effect with the change of latitude can induce solitary Rossby wave.     

STUDY OF NONLINEAR ROSSBY WAVES IN THE ATMOSPHERE UNDER SEMI-GEOSTROPHIC APPROXIMATION

Under semi-geostrophic approximation the nonlinear ordinary differential equations are obtained for the motion inthe barotropic and baroclinic atmospheres with the effects of zonal shear basic flow and topographic forcing included.Two constraints are acquired of finite-amplitude periodic and solitary waves in the original model with the aid of thephase-plane geometric qualitative theory of a dynamic system defined by the differential equation.The explicit solutionof the nonlinear waves is found by means of the approximation method and some significant results are achieved.     

STUDY OF NONLINEAR ROSSBY WAVES IN THE ATMOSPHERE UNDER SEMI-GEOSTROPHIC APPROXIMATION*

Under semi-geostrophic approximation the nonlinear ordinary differential equations are obtained for the motion in the barotropic and baroclinic atmospheres with the effects of zonal shear basic flow and topographic forcing included.Two constraints are acquired of finite-amplitude periodic and solitary waves in the original model with the aid of the phase-plane geometric qualitative theory of a dynamic system defined by the differential equation.The explicit solution of the nonlinear waves is found by means of the approximation method and some significant results are achieved.     

The effects of zonal flow on nonlinear Rossby waves

In this paper, we using phase plane method have derived the stability criteria of linear and nonlinear Rossby waves under the conditions of semi-geostrophic approximation and have gotten the solutions and geostrophic vorticity of corresponding solitary Rossby waves. It is pointed out that the wave stability is connected with the distri-bution of zonal flow and when the zonal flow is different the solitary wave trough or ridge is formed.     

均匀基流中非线性行波解的稳定性及其存在条件

本文利用地转动量近似＾「１」并结合行波解的方法,研究了均匀基本气流中的层结大气非线性波动,首先由绝热无磨擦的闭合方程组导得关于垂直Ｐ速度的单一变量的非线性方程,然后讨论了非线性方程解的稳定性,并求得近似的非线性方程的椭圆余弦波和孤立波解及其存在条件。     

热力强迫的非线性奇异惯性重力内波与高原低涡的联系

利用相平面分析法,由非绝热大气运动方程组导出了与非线性惯性重力内波有关的KdV方程,然后用直接积分法得到两类有天气意义的孤立波解,重点分析了与青藏高原暖性低涡有联系的一类具有间断点的奇异孤立波解的特征,进而讨论了高原非绝热加热对高原低涡生成、移动及高原低涡暖心结构的作用。     

热带大气运动的Rossby孤波

广东热带海洋气象研究所     

The domain of validity of the KdV-type solitary rossby waves in the shallow water β-plane model

The Domain, where the necessary and sufficient conditions for the existence of the KdV-type solitary Rossby waves are satisfied is defined in the shallow water β-plane model. The KdV-type solitary Rossby waves are the Rossby waves whose time-longitude dependence is determined by the KdV equation. As far as an appropriate amplitude and an appropriate ratio of the scales of the east-west and north-south directions are given, the KdV-type solitary Rossby waves can exist for every basic zonal flow. This result suggests the large validity of the soliton model in geophysical fluid dynamics. The KdV-type solitary Rossby waves are classified into four categories: (1) shear solitons studied by Long, Larsen, Benny, Redekop, and Hukuda, (2) β-divergent solitons studied by Clarke, Yamagata, and Nogami, (3) β-solitons found in the case of the strong stratification, and (4) divergent solitons which exist in the planetary-geostrophic-scale zonal flow. A remarkable result is that, in addition to the conventional east-west elongated solitons, the north-south elongated solitons can also exist for the case of the divergent solitons.