层结流体中Rossby波振幅的mKdV方程
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摘要
本文在层结流体准地转位涡方程的基础上，用弱非线性方法得到了层结流体中Rossby波的振幅满足mKdV方程的结论。
首先将涡度方程进行无量纲化，引进缓变量坐标X，T滤掉快变量x，t；用小参数展开，得到各阶摄动问题的方程，再用分离变量法将变量y分离出来，最后利用消奇异条件讨论弱非线性Rossby孤立波的振幅A(X，T)在慢变量X，T下的演化规律。具体作了如下几方面的工作：
(1)β效应是y的非线性函数，基本流无切变，N为常数。
(2)β效应是y的非线性函数，切变基本流存在，N为常数。
(3)β效应是y的非线性函数，基本流无切变，N是z的函数。
(4)β效应是y的非线性函数，切变基本流存在，N是z的函数。
In this thesic the mKdV equation for the amplitude of Rossby wave in stratified fluid are derived from stratified quasi-geostrophic vorticity equation by using wealky nonlinear method .First ,the vorticity equation is converted into non-dimerisional form.Then,long time and space scales are introduced.At last,the amplitude of Rossby wave is discussed under the long time and space variable X,T by using perturbation method.
We deal mainly with four problems
(1) The effect of β is nonlinear function of latitude, the basic flow is not a shear flow ,and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt remains a constant.
(2)The effect of β is nonlinear function of latitude with a shear flow ,and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt remains a constant.
(3)The effect of β is a nonlinear function of latitude ,the basic flow is not a shear flow, and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt is a function of z.
(4)The effect of β is a nonlinear function of latitude ,with a shear flow and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt is a function of z.
Owing to the balance of nonlinear effect and dispersion effection,the nonlinear mKdV equation controlling the amplitude of nonlinear Rossby waves is derived in stratified fluid.
首先将涡度方程进行无量纲化，引进缓变量坐标X，T滤掉快变量x，t；用小参数展开，得到各阶摄动问题的方程，再用分离变量法将变量y分离出来，最后利用消奇异条件讨论弱非线性Rossby孤立波的振幅A(X，T)在慢变量X，T下的演化规律。具体作了如下几方面的工作：
(1)β效应是y的非线性函数，基本流无切变，N为常数。
(2)β效应是y的非线性函数，切变基本流存在，N为常数。
(3)β效应是y的非线性函数，基本流无切变，N是z的函数。
(4)β效应是y的非线性函数，切变基本流存在，N是z的函数。
In this thesic the mKdV equation for the amplitude of Rossby wave in stratified fluid are derived from stratified quasi-geostrophic vorticity equation by using wealky nonlinear method .First ,the vorticity equation is converted into non-dimerisional form.Then,long time and space scales are introduced.At last,the amplitude of Rossby wave is discussed under the long time and space variable X,T by using perturbation method.
We deal mainly with four problems
(1) The effect of β is nonlinear function of latitude, the basic flow is not a shear flow ,and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt remains a constant.
(2)The effect of β is nonlinear function of latitude with a shear flow ,and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt remains a constant.
(3)The effect of β is a nonlinear function of latitude ,the basic flow is not a shear flow, and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt is a function of z.
(4)The effect of β is a nonlinear function of latitude ,with a shear flow and the frequency of V(a|¨)is(a|¨)l(a|¨)—Brunt is a function of z.
Owing to the balance of nonlinear effect and dispersion effection,the nonlinear mKdV equation controlling the amplitude of nonlinear Rossby waves is derived in stratified fluid.
引文
[1] 郭秉荣，线性波与非线性波导论，气象出版社，1990
[2] 王斌，翁衡毅，地球物理流体动力学导论，海洋出版社，1981
[3] 余志豪，杨大升等，地球物理流体动力学，气象出版社，1996
[4] 朱抱真，金飞飞，刘征宇，大气和海洋的非线性动力学概论，海洋出版社，1991
[5] S. Friedlandder, An Introduction to the mathemathical theory of geophysical fluid dynamics North-Hdland publishing company, 1980
[6] 谢定裕，渐进方法—在流体力学中应用，友谊出版社，1983
[7] 钱伟长，奇异摄动理论及其在力学中的应用，科学出版社，1981
[8] 达朝究，正压流体中Rossby波振幅的KdV方程，硕士论文，内蒙古大学，2005
[9] Howard, L. N., fandamentals of the theory of rotating fluid, J. Appl Mech. 1963
[10] Yih, C. S., On the flow of a stratified fluid, Proc 3dus. Nath. congr. Appl mech, 1958
[11] Redekopp, L. G. On the theory of solitay Rossby waves, J. Fluid Mech. 82, 725-745, 1977
[12] Long, R. R, 1953a: Some Aspects of the flow of stratified fluids, P. t. I, A theoretical Investigation Tellas
[13] 刘式适，刘式达，大气动力学，北京出版社，1991
[14] G. B. Whitham, Linear and nonlinear waves, Jonhn Wiely and Sons, Inc. 1974
[15] Rossby, G. G Relation between variations in the intensity of the zonal circulation and displavement of the semipermanent centers of action, J. Marine. Res. 2, 38-55. 1939
[16] L. G. Redekopp and P. D. Weidman, Solitary Rossby Waves in Zonal Shear Flows and Their Interactions, Joyrnal of The Atmospheric Sciences, 1978(35): 790-804
[17] 易家训，分层流，科学出版社，1983，
[18] 范红，缪锦海，杨玉峰，副高与阻高形成的Rossby孤立波理论，气象学报，1996 54(2)：186-194
[2] 王斌，翁衡毅，地球物理流体动力学导论，海洋出版社，1981
[3] 余志豪，杨大升等，地球物理流体动力学，气象出版社，1996
[4] 朱抱真，金飞飞，刘征宇，大气和海洋的非线性动力学概论，海洋出版社，1991
[5] S. Friedlandder, An Introduction to the mathemathical theory of geophysical fluid dynamics North-Hdland publishing company, 1980
[6] 谢定裕，渐进方法—在流体力学中应用，友谊出版社，1983
[7] 钱伟长，奇异摄动理论及其在力学中的应用，科学出版社，1981
[8] 达朝究，正压流体中Rossby波振幅的KdV方程，硕士论文，内蒙古大学，2005
[9] Howard, L. N., fandamentals of the theory of rotating fluid, J. Appl Mech. 1963
[10] Yih, C. S., On the flow of a stratified fluid, Proc 3dus. Nath. congr. Appl mech, 1958
[11] Redekopp, L. G. On the theory of solitay Rossby waves, J. Fluid Mech. 82, 725-745, 1977
[12] Long, R. R, 1953a: Some Aspects of the flow of stratified fluids, P. t. I, A theoretical Investigation Tellas
[13] 刘式适，刘式达，大气动力学，北京出版社，1991
[14] G. B. Whitham, Linear and nonlinear waves, Jonhn Wiely and Sons, Inc. 1974
[15] Rossby, G. G Relation between variations in the intensity of the zonal circulation and displavement of the semipermanent centers of action, J. Marine. Res. 2, 38-55. 1939
[16] L. G. Redekopp and P. D. Weidman, Solitary Rossby Waves in Zonal Shear Flows and Their Interactions, Joyrnal of The Atmospheric Sciences, 1978(35): 790-804
[17] 易家训，分层流，科学出版社，1983，
[18] 范红，缪锦海，杨玉峰，副高与阻高形成的Rossby孤立波理论，气象学报，1996 54(2)：186-194