摘要
本文第一部分(Ⅰ)已给出了线性化旋转二维可压缩流动方程的谱分布,并作了初步分析。本部分(Ⅱ)对这些谱及谱函数作进一步的分析讨论。设基流是低速流,此时可用摄动法求得谱和谱函数,将它们与用差分法所得结果进行比较,表明二者吻合得很好。摄动法中的离散谱初级近似就是无基流时的相应的谱,而连续谱对应的谱函数初级近似则与准地转模式的结果一致。由初级近似和一级修正项可以清楚地说明计算所得的谱和谐函数的许多重要性质。 在低速基流情况下:(1)重力一惯性波为准简谐波,基流和可变的Coriolis参数只给以较小的修正。(2)由于基流在Coriolis力作用下使自由表面有一坡度,Kelvin波必定具有横穿波射线的速度分量,同时顺传和逆传的Kelvin波不再在形态上相似;当基流愈强时上述两性质愈明显;Coriolis参数随空间的变化也改变了顺传和逆传波的相似性,此外,Kelvin波是准非频散的。(3)不为零的基流或科氏参数的变化使慢波离散谱变为非简并的即分立的,它们或者有无穷多个且基流流速为相速的聚点(当基流为常数时),或者只有有限个,甚至不存在;而当基流有切变时则有连续谱。对应于离散谱的谱函数为准简谐波;而对应于连续谱的谐函数则为广义解,但有有限能量。 本问题的谱函数与其伴随算子的谐函数正交。
The distribution of spectra of rotating two-dimensional compressive motion and its preliminary analysis have been given in part I of our paper.The structure of spectral function and further discussion on the spectra is given in part II. In the case of low-speed basic flow it is convenient to apply the perturbation method to solve the spectra and spectral functions and to compare the results with those obtained by using finite difference method described in Part I. The comparison between the results obtained by these two methods shows a very good agreement. Our perturbation method takes the first-order approximation of discrete spectra as the spectra in the case of zero basic flow and the one of spectral functions corresponding to continuous spectrum as the results of quasigeostrophic model.The analyses of the zero-order approximations and the first-order corrections give clear interpretations of many important characteristics of the spectra and spectral functions computed by using finite different method.In the case of low-speed basic flow: (1) The inertia-gravity (characteristic) waves are quasi-harmonic, and the corrections for the influence of the basic flow and the spatial variability of the Coriolis parameter are only small. (2) Due to the permanent slop of the free surface which is accompanied by the nonzero basic flow, the kelvin waves necessarily possess component of velocity perpendicular to the wave ray; and the downwrad and upward propagating Kelvin waves are no longer similar to each other in their shape.The stronger.the basic flow is, the clearer the characters mentioned above appear. Their similarity is also violated by the spatial variability of the Coriolis parameter. Besides, both the two kelvin waves are almost non-dispersive. (3) The nonzero basic flow or the spatial variability makes the discrete spectra corresponding to the slow (characteristic) waves separate from each other, there are either infinitive numbers of such spectra which approach the velocity of the basic flow by their correspondent phase velocity (if the basic flow is constant) or finite numbers, and even no one exists.The continuous spectrum necessarily exists if the basic flow is not a constant.The spectral functions corresponding to the discrete spectra are all quasi-harmonic waves, but every of those corresponding to the continuous spectrum is only a generalized solution with finite energy. Spectral functions of the problem are orthogonal to those of the adjoint operator. Everydisturbance satisfying the same boundary conditions as in the eigenvalue problem can be expanded by using the spectral functions of the adjoint operator.In the case of low-speed basic flow both the operator and its adjoint one can be expanded by using the perturbation method, and they are self-adjoint in the zero-order approximation. The problem with high-speed basic flow will be studied in Part II.
出处
《大气科学》
CSCD
北大核心
1991年第1期1-15,共15页
Chinese Journal of Atmospheric Sciences
关键词
大气
可压缩
流动
谱
函数
分析
Discrete spectrum
Continuous spectrum
Generalized solution
Orthogonality
Expansion theorem.
