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《Chinese Journal of Atmospheric Sciences》 1991-01
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SPECTRA AND SPECTRAL FUNCTIONS OF ROTATING TWO DIMENSIONAL COMPRESSIVE MOTION PART( II): STRUCTURE OF SPECTRAL FUNCTIONS AND FURTHER DISCUSSION ON SPECTRA

Zeng Qingcun, Li Rongfeng and Zhang Ming(LASG, Institute of Atmospheric Physics)  
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.
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