非定常扰动下非线性Rossby孤立波动力学研究
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摘要
自从J.S.Russell首次观察到孤立波以来,孤立波的研究一直是力学、物理学、应用数学、大气和海洋科学等学科交叉的前沿性课题,Rossby孤立波的研究是这一主题的一个重要分支。
本论文采用理论分析、符号运算和数值模拟相结合的方法,研究了不同深度正压、层结流体中非定常扰动激发的Rossby孤立波的生成、演化以及相互作用,特别考虑了耗散等因素的影响。前人的研究通常针对定常扰动对Rossby孤立波的影响展开,本文主要考虑非定常扰动对Rossby孤立波生成、演化的影响以及与Rossby孤立波的相互作用,具有重要的理论意义与应用价值。
首先针对无粘、不可压缩的正压流体,从包含耗散、地形的准地转位涡方程出发,运用扰动和时空伸长变换方法,推导出控制Rossby孤立波振幅的一维模型。利用推广的雅可比椭圆函数法以及扰动方法,获得了非受迫模型的解析解,分析了耗散对Rossby孤立波速度、宽度的影响,并得到了Rossby孤立波的守恒律。利用拟谱方法对模型进行数值求解,讨论了各种状态(共振、超临界、亚临界)下不同地形(不稳定地形,波动地形)对Rossby孤立波的影响,解释了波动地形能在一定程度上阻止孤立波破碎等海洋现象。
其次针对无粘、不可压缩的层结流体,采用相似步骤得到层结流体Rossby孤立波的控制模型,特别是获得了新的代数Rossby孤立波控制模型,推广了通常描述代数Rossby孤立波的BO模型和ILW模型。通过数值模拟,讨论了非定常外源对Rossby孤立波的影响,解释了大气和海洋中的阻塞和振荡等现象。
最后研究了正压流体中二维Rossby孤立波,推导出带耗散的二维经血Rossby孤立波控制模型,讨论了二维Rossby孤立波的守恒律,并运用正弦和余弦函数方法获得了模型的周期解与类孤立波解,讨论了耗散对Rossby孤立波振幅、移动速度以及宽度的影响。同时首次研究了二维代数Rossby孤立波,推导出了控制二维代数Rossby孤立波振幅的修正二维BO-Burgers模型,修正二维ILW-Burgers模型,扩大了二维Rossby孤立波的研究范围。本部分的研究是对非线性Rossby孤立波动力学研究的一个有益补充。
Since J.S.Russell observed the solitary waves firstly, the research on solitarywaves has become an advanced subject in mechanics, physics, applied mathemat-ics, atmospheric and oceanic sciences and so on, and has significant theoreticaland real meaning. The research on Rossby solitary waves is an important branchof the theme.
In this thesis, the generation, evolution, interaction of Rossby solitary waveswhich excited by unsteady perturbation in barotropic and stratified fluid of difer-ent depth are investigated by employing the method of combination of theoreticalanalysis, symbolical computation and numerical simulation, especially here dissi-pation efect is considered. The former research commonly focused on the efectof steady perturbation on the Rossby solitary waves, while in the paper we con-sider the efect of unsteady perturbation on the generation, evolution, interactionof Rossby solitary waves, which has important theoretical and application value.
Firstly, for the inviscid, incompressible barotropic fluid of diferent depth,from the quasi-geostrophic vorticity equation including topography and dissipa-tion, with the help of the perturbation method and stretching transform of timeand space, one-dimensional models governing the amplitude of Rossby solitarywaves are derived. By using the Jacobi elliptic function expansion method andperturbation method, the analytical solutions of corresponding homogeneous e-quations of the above models are obtained, the speed, dissipation efect and so onof Rossby solitary waves are discussed and the conservation laws of Rossby soli-tary waves are derived. By virtue of the pseudo-spectral method, the numericalsimulation of models are carried out. With the help of these water-fall plots, theefects of topography(unsteady topography, wavy topography) on Rossby solitarywaves are discussed under diferent states(resonant state, supercritical state, sub-critical state). The discussion can be used to explain the ocean phenomenon thatthe wavy bottom forcing can prevent wave breaking to some extent.
Next, the inviscid, incompressible stratified fluid of diferent depth is con- sidered. By using the same method and procedure, the models governing theamplitude of Rossby solitary waves in the stratified fluid are derived. Comparingwith the former models, the nonlinearity is stronger in these models. Some ofthem are the generalization of the former models. By numerical simulation, theefect of unsteady external source on Rossby solitary waves are discussed and theblock and oscillation phenomenon which happen in ocean and atmosphere areexplained.
Finally, the two-dimensional Rossby solitary waves in barotropic fluid arestudied. Similar to the previous chapters, the two-dimensional classical Rossbysolitary waves model with dissipation efect is derived. The conservation laws oftwo-dimensional Rossby solitary waves are analyzed. By employing the sin andcos function method, the periodical solutions and the solitary-wave-like solutionsof model are obtained. By virtue of the analytic solution, the efect of dissipa-tion on the amplitude, speed and width of Rossby solitary waves is researched.Meanwhile, the research of two-dimensional algebraic Rossby solitary waves isfirst carried out, and the modified two-dimensional BO-Burgers model, modifiedtwo-dimensional ILW-Burgers model are derived. It enlarges the research scopeof two-dimensional Rossby solitary waves model. The research in the paper is abeneficial supplement for nonlinear Rossby waves dynamics.
本论文采用理论分析、符号运算和数值模拟相结合的方法,研究了不同深度正压、层结流体中非定常扰动激发的Rossby孤立波的生成、演化以及相互作用,特别考虑了耗散等因素的影响。前人的研究通常针对定常扰动对Rossby孤立波的影响展开,本文主要考虑非定常扰动对Rossby孤立波生成、演化的影响以及与Rossby孤立波的相互作用,具有重要的理论意义与应用价值。
首先针对无粘、不可压缩的正压流体,从包含耗散、地形的准地转位涡方程出发,运用扰动和时空伸长变换方法,推导出控制Rossby孤立波振幅的一维模型。利用推广的雅可比椭圆函数法以及扰动方法,获得了非受迫模型的解析解,分析了耗散对Rossby孤立波速度、宽度的影响,并得到了Rossby孤立波的守恒律。利用拟谱方法对模型进行数值求解,讨论了各种状态(共振、超临界、亚临界)下不同地形(不稳定地形,波动地形)对Rossby孤立波的影响,解释了波动地形能在一定程度上阻止孤立波破碎等海洋现象。
其次针对无粘、不可压缩的层结流体,采用相似步骤得到层结流体Rossby孤立波的控制模型,特别是获得了新的代数Rossby孤立波控制模型,推广了通常描述代数Rossby孤立波的BO模型和ILW模型。通过数值模拟,讨论了非定常外源对Rossby孤立波的影响,解释了大气和海洋中的阻塞和振荡等现象。
最后研究了正压流体中二维Rossby孤立波,推导出带耗散的二维经血Rossby孤立波控制模型,讨论了二维Rossby孤立波的守恒律,并运用正弦和余弦函数方法获得了模型的周期解与类孤立波解,讨论了耗散对Rossby孤立波振幅、移动速度以及宽度的影响。同时首次研究了二维代数Rossby孤立波,推导出了控制二维代数Rossby孤立波振幅的修正二维BO-Burgers模型,修正二维ILW-Burgers模型,扩大了二维Rossby孤立波的研究范围。本部分的研究是对非线性Rossby孤立波动力学研究的一个有益补充。
Since J.S.Russell observed the solitary waves firstly, the research on solitarywaves has become an advanced subject in mechanics, physics, applied mathemat-ics, atmospheric and oceanic sciences and so on, and has significant theoreticaland real meaning. The research on Rossby solitary waves is an important branchof the theme.
In this thesis, the generation, evolution, interaction of Rossby solitary waveswhich excited by unsteady perturbation in barotropic and stratified fluid of difer-ent depth are investigated by employing the method of combination of theoreticalanalysis, symbolical computation and numerical simulation, especially here dissi-pation efect is considered. The former research commonly focused on the efectof steady perturbation on the Rossby solitary waves, while in the paper we con-sider the efect of unsteady perturbation on the generation, evolution, interactionof Rossby solitary waves, which has important theoretical and application value.
Firstly, for the inviscid, incompressible barotropic fluid of diferent depth,from the quasi-geostrophic vorticity equation including topography and dissipa-tion, with the help of the perturbation method and stretching transform of timeand space, one-dimensional models governing the amplitude of Rossby solitarywaves are derived. By using the Jacobi elliptic function expansion method andperturbation method, the analytical solutions of corresponding homogeneous e-quations of the above models are obtained, the speed, dissipation efect and so onof Rossby solitary waves are discussed and the conservation laws of Rossby soli-tary waves are derived. By virtue of the pseudo-spectral method, the numericalsimulation of models are carried out. With the help of these water-fall plots, theefects of topography(unsteady topography, wavy topography) on Rossby solitarywaves are discussed under diferent states(resonant state, supercritical state, sub-critical state). The discussion can be used to explain the ocean phenomenon thatthe wavy bottom forcing can prevent wave breaking to some extent.
Next, the inviscid, incompressible stratified fluid of diferent depth is con- sidered. By using the same method and procedure, the models governing theamplitude of Rossby solitary waves in the stratified fluid are derived. Comparingwith the former models, the nonlinearity is stronger in these models. Some ofthem are the generalization of the former models. By numerical simulation, theefect of unsteady external source on Rossby solitary waves are discussed and theblock and oscillation phenomenon which happen in ocean and atmosphere areexplained.
Finally, the two-dimensional Rossby solitary waves in barotropic fluid arestudied. Similar to the previous chapters, the two-dimensional classical Rossbysolitary waves model with dissipation efect is derived. The conservation laws oftwo-dimensional Rossby solitary waves are analyzed. By employing the sin andcos function method, the periodical solutions and the solitary-wave-like solutionsof model are obtained. By virtue of the analytic solution, the efect of dissipa-tion on the amplitude, speed and width of Rossby solitary waves is researched.Meanwhile, the research of two-dimensional algebraic Rossby solitary waves isfirst carried out, and the modified two-dimensional BO-Burgers model, modifiedtwo-dimensional ILW-Burgers model are derived. It enlarges the research scopeof two-dimensional Rossby solitary waves model. The research in the paper is abeneficial supplement for nonlinear Rossby waves dynamics.
引文
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