地球流体中的非线性Rossby波
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摘要
本文主要从准地转位涡方程出发,采用约化摄动法和多重尺度法考虑了正压流体和层结流体中具有推广的β平面近似下Rossby波振幅演变问题(包括地形问题、缓变地形问题、和带有外源、耗散问题、层结效应),以及推广的β平面近似下赤道Rossby包络孤立波和含有地球旋转水平分量的Rossby波等问题进行研究.
球面效应和旋转效应是地球流体的基本特征,1939年Rossby在其发表的论文“Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action"[1]中提出了用球面上的均质流体层作为研究地球流体中观测到的大尺度波动力学简单而恰当的模式,即把地球的球面效应通过另外一个平面上地球行星涡度的局部垂直分量f=2Ωsinφ进行模拟的模式,而且在动力学上起显著作用.把地球的球面效应通过另外平面上f的线性变化进行模拟的模式,称为β平面模式(β平面近似).其中参数f是行星涡度垂直与地面的局部分量,称为柯里奥利参数(柯氏参数),Ω,φ分别是地球旋转角速度的大小和纬度.当时他指出大气长波是由于f随纬度φ变化造成的(即β效应)结论.为了证实其合理性,许多学者[13,21,56-62]研究了β平面近似下采用诸女(?)Korteweg-de Vries(KdV)方程、改进的KdV(mKdV)方程等非线性方程刻画Rossby波振幅,这些研究极大的丰富了地球物理流体动力学理论,并且取得了许多有意义的结果,例如Kartashova(?)(?)L'vov[2]提出了大气季节内振荡的非线性Rossby波共振的物理机制；罗德海[22,23]利用KdV或者mKdV方程解释了大气阻塞现象.
本文在以上结论的基础上,从准地转位涡方程出发,将考虑推广的β平面近似即Rossby参数口随纬度变化下,各种效应作用的非线性Rossby波振幅所满足的非线性方程,以及地球旋转水平分量对Rossby波的影响.
首先,研究了正压流体中非线性Rossby波在推广的β平面近似下的问题.在不考虑地形效应的情况下,利用约化摄动法推导了Rossby波振幅所满足的非线性方程,得到正压流体中Rossby孤立波满足mKdV方程的结果.然后只考虑地形对Rossby波影响,得出线性地形和非线性地形效应下非线性Rossby波振幅满足mKdV方程；进一步,考察了推广的β平面近似下,具有外源、耗散和地形的准地转位涡方程,得出了Rossby孤立波满足非齐次Boussinesq方程与mKdV方程.
由于地球流体具有多时空尺度的特点,我们采用多重尺度法导出了Rossby波满足非线性Schrodinger方程.在考虑地形随时间缓慢变化下,得到了非线性Rossby波振幅演变满足非齐次Benjamin-Davis-Ono(BDO)方程的结论,并且分析了地形强迫和耗散是影响Rossby瓜立波质量、能量变化的原因.
同时本文还研究了层结流体中推广的β平面下,地形效应和层结效应共同作用的Rossby孤立波,应用约化摄动法推导了Rossby波振幅满足mKdV方程,给出了层结流体中地形强迫也是引起Rossby孤立波质量和能量变化的原因.另外,研究了正压流体中赤道Rossby包络孤立波,采用多重尺度法导出来Rossby包络孤立波振幅满足非线性Schrodinger方程.最后,文章还研究了在推广的β平面近似下地球旋转水平分量对Rossby波的影响,从大气方程组出发,应用Taylor级数方法导出Rossby波满足KdV(?)PKdV-mKdV方程的结论.
全文包含八个部分：
1.研究问题的背景和主要结论；
2.正压流体中推广的β平面近似下的非线性Rossby波；
3.正压流体只有地形作用的非线性Rossby波;
4.正压流体中β效应和地形共同作用的非线性Rossby波与Rossby包络孤立波的问题;
5.层结流体中具有β效应、地形效应和层结效应的Rossby孤立波;
6.正压流体中具有缓变地形和耗散作用的非线性Rossby波；
7.推广的β平面近似下赤道Rossby包络孤立波问题;
8.推广的β平面近似下地球旋转水平分量对非线性Rossby波的作用.
In this dissertation, based on the quasi-geostrophic potential vorticity equation, we investigate the evolution of the amplitude of Rossby waves with the generalized beta-plane approximation by employing the reduc-tive perturbation method and multiple scale method (including topography problems; slowly changing topography problems; external source and dissi-pation problems; stratification effect), equatorial envelope Rossby solitons and action of the horizontal component of the earth's rotation on nonlinear Rossby waves in barotropic fluid and stratified fluid.
The basic features of geophysical fluid are spherical effect and ro-tational effect of the earth's. In1939, Rossby in his article "Relation between variations in the intensity of the zonal circulation of the atmo-sphere and the displacements of the semi-permanent centers of action"[1] suggested the study of a homogeneous sheet of fluid on a sphere as the simplest relevant model for the dynamics of the observed large-scale waves in the earth's atmosphere, namely that the effect of spherical earth so that only the local vertical component of the earth's planetary vorticity another plane f=2Ω sin φ simulation mode, but it is dynamically significant. Such the model, in which the effect of earth's sphericity is modeled by a linear variation of f in an otherwise planar geometry, is called the β model (β plane approximation), where the parameter f is the local component of the planetary vorticity normal to the earth's surface and is called the Cori-olis parameter, Ω, φ is the earth rotation angular velocity and latitude. He pointed out that the atmospheric long wave is due to f with latitude φ change(i.e. β effect). In order to confirm Rossby's this argument is rationali-ty, Many scholars[13,21,56-60]have studied the Rossby waves problems in β plane approximation, they give Rossby waves amplitude satisfy Korteweg-de Vries(KdV)equation, the modified Korteweg-de Vries(mKdV) equation for a class of nonlinear equations. These research greatly enriched the geo-physical fluid dynamics theory, and have achieved many significant results. For example, Kartashova and L'vov[2] proposed the physical mechanism of resonance of the nonlinear Rossby waves intraseasonal oscillation; Luo De-hai[22,23]used KdV or mKdV equation explains the atmospheric blocking phenomenon.
In this dissertation, based on the results mentioned above, using the quasi-geostrophic vorticity equation, we will consider the approximate Rossby parameter β vary with latitude β plane under the generalized non-linear equations, nonlinear Rossby wave amplitude of the various effect, and influence of Rossby waves under the horizontal component of the earth's rotation.
Firstly, we consider nonlinear Rossby waves under the generalized β plane approximation in barotropic fluids problem. Without considering the effect of topography condition, by using the reductive perturbation method for nonlinear equations is derived for nonlinear Rossby waves, we obtain Rossby solitary waves satisfy mKdV equation in barotropic fluids. Then only consider the influence of topography on Rossby waves, getting Rossby waves amplitude satisfies the mKdV equation under the effect of linear and nonlinear topography. Further, under the generalized β plane approximation condition, we investigated the quasi-geostrophic vorticity equation with external source, dissipation and topography, obtaining Ross-by solitary waves satisfied inhomogeneous Boussinesq equation dan mKdV equation.
Because of the characteristics of the geophysical fluid with multi tem-poral and spatial scales, hence we use the method of multiple scales is derived for the Rossby waves satisfy nonlinear Schrodinger equation. By considering topography varies slowly with time, we give the amplitude non-linear Rossby waves satisfy inhomogeneous Benjamin-Davis-Ono(BDO) e-quation results, and analysis of the topographic forcing and dissipation effect Rossby solitary wave quality, energy change.
At the same time, under the generalized β plane approximation con-dition, the dissertation also studied Rossby solitary wave interaction effect of topography and stratification effect in stratified fluid, using the pertur-bation method is used to deduce the Rossby wave amplitude satisfies the mKdV equation, we give topograph force in stratified fluid also caused the Rossby solitary waves of mass and energy change. In addition, we consid-ered equatorial envelope solitary Rossby wave, in barotropic fluids, based on the equatorial potential vorticity equation, the nonlinear equatorial en-velope solitary Rossby wave are investigated by the asymptotic method of multiple scales. The inhomogeneous Schrodinger equation, satisfied by Rossby solitary waves packet is derived.
Finally, this dissertation also studies the influence of the horizontal component of earth rotation approximation of Rossby wave in β plane promotion, starting from the atmospheric equations, using Taylor series method to derive the Rossby wave to satisfy KdV and KdV-mKdV equa-tions of the conclusion.
This dissertation contains eight parts:
1. The background and the main results of study problem;
2. Nonlinear Rossby waves under the generalized beta-plane approxima- tion in barotropic fluid;
3. Nonlinear Rossby waves with topography effect in barotropic fluid;
4. Nolinear Rossby waves and envelop Rossby solitary waves problems with beta effect and topography synergism in barotropic;
5. Rossby solitary waves with beta effect, topography effect and stratifica-tion effect in stratified fluid;
6. Nonlinear Rossby waves problem excited slowly changing topography and dissipation in barotropic;
7. Equatorial envelope solitary Rossby waves under the generalized beta-plane approximation;
8. Action of the horizontal component of the earth's rotation on nonlinear Rossby waves.
球面效应和旋转效应是地球流体的基本特征,1939年Rossby在其发表的论文“Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action"[1]中提出了用球面上的均质流体层作为研究地球流体中观测到的大尺度波动力学简单而恰当的模式,即把地球的球面效应通过另外一个平面上地球行星涡度的局部垂直分量f=2Ωsinφ进行模拟的模式,而且在动力学上起显著作用.把地球的球面效应通过另外平面上f的线性变化进行模拟的模式,称为β平面模式(β平面近似).其中参数f是行星涡度垂直与地面的局部分量,称为柯里奥利参数(柯氏参数),Ω,φ分别是地球旋转角速度的大小和纬度.当时他指出大气长波是由于f随纬度φ变化造成的(即β效应)结论.为了证实其合理性,许多学者[13,21,56-62]研究了β平面近似下采用诸女(?)Korteweg-de Vries(KdV)方程、改进的KdV(mKdV)方程等非线性方程刻画Rossby波振幅,这些研究极大的丰富了地球物理流体动力学理论,并且取得了许多有意义的结果,例如Kartashova(?)(?)L'vov[2]提出了大气季节内振荡的非线性Rossby波共振的物理机制；罗德海[22,23]利用KdV或者mKdV方程解释了大气阻塞现象.
本文在以上结论的基础上,从准地转位涡方程出发,将考虑推广的β平面近似即Rossby参数口随纬度变化下,各种效应作用的非线性Rossby波振幅所满足的非线性方程,以及地球旋转水平分量对Rossby波的影响.
首先,研究了正压流体中非线性Rossby波在推广的β平面近似下的问题.在不考虑地形效应的情况下,利用约化摄动法推导了Rossby波振幅所满足的非线性方程,得到正压流体中Rossby孤立波满足mKdV方程的结果.然后只考虑地形对Rossby波影响,得出线性地形和非线性地形效应下非线性Rossby波振幅满足mKdV方程；进一步,考察了推广的β平面近似下,具有外源、耗散和地形的准地转位涡方程,得出了Rossby孤立波满足非齐次Boussinesq方程与mKdV方程.
由于地球流体具有多时空尺度的特点,我们采用多重尺度法导出了Rossby波满足非线性Schrodinger方程.在考虑地形随时间缓慢变化下,得到了非线性Rossby波振幅演变满足非齐次Benjamin-Davis-Ono(BDO)方程的结论,并且分析了地形强迫和耗散是影响Rossby瓜立波质量、能量变化的原因.
同时本文还研究了层结流体中推广的β平面下,地形效应和层结效应共同作用的Rossby孤立波,应用约化摄动法推导了Rossby波振幅满足mKdV方程,给出了层结流体中地形强迫也是引起Rossby孤立波质量和能量变化的原因.另外,研究了正压流体中赤道Rossby包络孤立波,采用多重尺度法导出来Rossby包络孤立波振幅满足非线性Schrodinger方程.最后,文章还研究了在推广的β平面近似下地球旋转水平分量对Rossby波的影响,从大气方程组出发,应用Taylor级数方法导出Rossby波满足KdV(?)PKdV-mKdV方程的结论.
全文包含八个部分：
1.研究问题的背景和主要结论；
2.正压流体中推广的β平面近似下的非线性Rossby波；
3.正压流体只有地形作用的非线性Rossby波;
4.正压流体中β效应和地形共同作用的非线性Rossby波与Rossby包络孤立波的问题;
5.层结流体中具有β效应、地形效应和层结效应的Rossby孤立波;
6.正压流体中具有缓变地形和耗散作用的非线性Rossby波；
7.推广的β平面近似下赤道Rossby包络孤立波问题;
8.推广的β平面近似下地球旋转水平分量对非线性Rossby波的作用.
In this dissertation, based on the quasi-geostrophic potential vorticity equation, we investigate the evolution of the amplitude of Rossby waves with the generalized beta-plane approximation by employing the reduc-tive perturbation method and multiple scale method (including topography problems; slowly changing topography problems; external source and dissi-pation problems; stratification effect), equatorial envelope Rossby solitons and action of the horizontal component of the earth's rotation on nonlinear Rossby waves in barotropic fluid and stratified fluid.
The basic features of geophysical fluid are spherical effect and ro-tational effect of the earth's. In1939, Rossby in his article "Relation between variations in the intensity of the zonal circulation of the atmo-sphere and the displacements of the semi-permanent centers of action"[1] suggested the study of a homogeneous sheet of fluid on a sphere as the simplest relevant model for the dynamics of the observed large-scale waves in the earth's atmosphere, namely that the effect of spherical earth so that only the local vertical component of the earth's planetary vorticity another plane f=2Ω sin φ simulation mode, but it is dynamically significant. Such the model, in which the effect of earth's sphericity is modeled by a linear variation of f in an otherwise planar geometry, is called the β model (β plane approximation), where the parameter f is the local component of the planetary vorticity normal to the earth's surface and is called the Cori-olis parameter, Ω, φ is the earth rotation angular velocity and latitude. He pointed out that the atmospheric long wave is due to f with latitude φ change(i.e. β effect). In order to confirm Rossby's this argument is rationali-ty, Many scholars[13,21,56-60]have studied the Rossby waves problems in β plane approximation, they give Rossby waves amplitude satisfy Korteweg-de Vries(KdV)equation, the modified Korteweg-de Vries(mKdV) equation for a class of nonlinear equations. These research greatly enriched the geo-physical fluid dynamics theory, and have achieved many significant results. For example, Kartashova and L'vov[2] proposed the physical mechanism of resonance of the nonlinear Rossby waves intraseasonal oscillation; Luo De-hai[22,23]used KdV or mKdV equation explains the atmospheric blocking phenomenon.
In this dissertation, based on the results mentioned above, using the quasi-geostrophic vorticity equation, we will consider the approximate Rossby parameter β vary with latitude β plane under the generalized non-linear equations, nonlinear Rossby wave amplitude of the various effect, and influence of Rossby waves under the horizontal component of the earth's rotation.
Firstly, we consider nonlinear Rossby waves under the generalized β plane approximation in barotropic fluids problem. Without considering the effect of topography condition, by using the reductive perturbation method for nonlinear equations is derived for nonlinear Rossby waves, we obtain Rossby solitary waves satisfy mKdV equation in barotropic fluids. Then only consider the influence of topography on Rossby waves, getting Rossby waves amplitude satisfies the mKdV equation under the effect of linear and nonlinear topography. Further, under the generalized β plane approximation condition, we investigated the quasi-geostrophic vorticity equation with external source, dissipation and topography, obtaining Ross-by solitary waves satisfied inhomogeneous Boussinesq equation dan mKdV equation.
Because of the characteristics of the geophysical fluid with multi tem-poral and spatial scales, hence we use the method of multiple scales is derived for the Rossby waves satisfy nonlinear Schrodinger equation. By considering topography varies slowly with time, we give the amplitude non-linear Rossby waves satisfy inhomogeneous Benjamin-Davis-Ono(BDO) e-quation results, and analysis of the topographic forcing and dissipation effect Rossby solitary wave quality, energy change.
At the same time, under the generalized β plane approximation con-dition, the dissertation also studied Rossby solitary wave interaction effect of topography and stratification effect in stratified fluid, using the pertur-bation method is used to deduce the Rossby wave amplitude satisfies the mKdV equation, we give topograph force in stratified fluid also caused the Rossby solitary waves of mass and energy change. In addition, we consid-ered equatorial envelope solitary Rossby wave, in barotropic fluids, based on the equatorial potential vorticity equation, the nonlinear equatorial en-velope solitary Rossby wave are investigated by the asymptotic method of multiple scales. The inhomogeneous Schrodinger equation, satisfied by Rossby solitary waves packet is derived.
Finally, this dissertation also studies the influence of the horizontal component of earth rotation approximation of Rossby wave in β plane promotion, starting from the atmospheric equations, using Taylor series method to derive the Rossby wave to satisfy KdV and KdV-mKdV equa-tions of the conclusion.
This dissertation contains eight parts:
1. The background and the main results of study problem;
2. Nonlinear Rossby waves under the generalized beta-plane approxima- tion in barotropic fluid;
3. Nonlinear Rossby waves with topography effect in barotropic fluid;
4. Nolinear Rossby waves and envelop Rossby solitary waves problems with beta effect and topography synergism in barotropic;
5. Rossby solitary waves with beta effect, topography effect and stratifica-tion effect in stratified fluid;
6. Nonlinear Rossby waves problem excited slowly changing topography and dissipation in barotropic;
7. Equatorial envelope solitary Rossby waves under the generalized beta-plane approximation;
8. Action of the horizontal component of the earth's rotation on nonlinear Rossby waves.
引文
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