用Galerkin方法研究地球本征模
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摘要
地球本征模在地球内部物理结构和动力学机制研究中一直是一个重要的课题,与地球物理学、地震学联系紧密。过去采用的方法都是由一组运动方程进行积分,然后在边界上判断是否符合各物理量的约束条件,从而得到地球本征模。提出了一个新的解算此问题的方法,即用Galerkin方法计算本征模和本征函数,相应的边界条件用Tau方法处理。结果表明:在研究地球自由振荡时,Galerkin方法是一种行之有效的方法;随着所采用的地球模型的逐步完善,计算的结果也与观测更加符合。对于三层地球模型,考虑地球自转的结果比不考虑自转的有所改善,但其改善效果不如仅将地幔更细致分层对结果的改善显著。
The study of the Earth's eigen mode is an important way to study physical properties and dynamics in the Earth's interior,and is closely related with geophysics and seismology closely. The Earth's eigen mode is usually solved by classical numerical integration of partial differential equations of motion and boundary conditions.Galerkin method is used in this paper to calculate eigen-periods and eigen-functions of free oscillations of different earth models,while boundary conditions are treated with Tau method.Results show tboldsymbol Galerkin method is an alternative tool for such study.At first Earth models are assumed to be spherical non-rotating elastic isotropic(SNREI).The eigen modes for the free inner core is calculated separately with respect to the other two modes:the 3-layer earth mode including solid inner core,fluid outer core and solid mantle(plus crust),and the 12-layer earth mode in which the mantle(+crust) is separated to 10 layers.Furthermore,the eigen modes of rotating free inner core and 3-layer earth model are calculated respectively.The results are improved and approach more closely to the observation with more realized earth model.For 3-layer Earth,incorporating earth rotation will improve the result of the eigen periods comparing with non-rotating earth model,for example,the difference of the eigen period of _0S_2 with observed one is improved from 3.6%to 3.1%;however,the improvement by separating the mantle to 10 layers is more remarkably than tboldsymbol by introducing earth rotation:changing the earth model from 3 layers to 12 layers(both non-rotating),the difference of the eigen periods with observed ones are reduced significantly from 4.3%(maximum,or 3.6% for _0S_2) to less than 1.6%(maximum,or 0.78%for _0S_2).
The study of the Earth's eigen mode is an important way to study physical properties and dynamics in the Earth's interior,and is closely related with geophysics and seismology closely. The Earth's eigen mode is usually solved by classical numerical integration of partial differential equations of motion and boundary conditions.Galerkin method is used in this paper to calculate eigen-periods and eigen-functions of free oscillations of different earth models,while boundary conditions are treated with Tau method.Results show tboldsymbol Galerkin method is an alternative tool for such study.At first Earth models are assumed to be spherical non-rotating elastic isotropic(SNREI).The eigen modes for the free inner core is calculated separately with respect to the other two modes:the 3-layer earth mode including solid inner core,fluid outer core and solid mantle(plus crust),and the 12-layer earth mode in which the mantle(+crust) is separated to 10 layers.Furthermore,the eigen modes of rotating free inner core and 3-layer earth model are calculated respectively.The results are improved and approach more closely to the observation with more realized earth model.For 3-layer Earth,incorporating earth rotation will improve the result of the eigen periods comparing with non-rotating earth model,for example,the difference of the eigen period of _0S_2 with observed one is improved from 3.6%to 3.1%;however,the improvement by separating the mantle to 10 layers is more remarkably than tboldsymbol by introducing earth rotation:changing the earth model from 3 layers to 12 layers(both non-rotating),the difference of the eigen periods with observed ones are reduced significantly from 4.3%(maximum,or 3.6% for _0S_2) to less than 1.6%(maximum,or 0.78%for _0S_2).
引文
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[4]Alterman Z,Jarosch H,Pekeris C L.Proc.Roy.Soc.A 1959,252:80
[5]Lapwood E R and Usami T.Free Oscillation Of The Earth.Camb.Uni.Press,1981
[6]方明.地球自由振荡(博士论文).中国科学院测量与地球物理研究所,1991
[7]Seyed-Mahmoud B.Wobble/Nutation Of A Rotating Ellipsoidal Earth With Liquid Outer Core,Master Thesis.Memorial University of Newfoundland.1994
[8]李荣华.边值问题的Galerkin有限元法.北京:科学出版社,2005
[9]Rochester M G.NormalModesofRotatingSelf-Gravitating CompressibleStratifiedFluidBodies:the Subseismic Wave Equations.In:Graham G A C,Malik S K.eds.Continuum Mechanics and its applications. Hemisphere:Washington D C,1989:797
[10]郭俊义.地球物理学基础.北京:测绘出版社,2001
[11]Dziewonski A M,Anderson D L.Preliminary reference earth model.Phys.Earth.Planet Int.1981,25: 297
[12]Rochester M G.Rotational Dynamics of the deformable Earth(上海天文台讲义).2006
[13]Dahlen F A,Tromp J.Theoretical global seismology.Princeton:Princeton University Press,1998
[2]Matumoto T and Sato Y.Bull.Earthq.Res.Inst.1954,32:247
[3]Takeuchi H.Geophys.J.1959,2:89
[4]Alterman Z,Jarosch H,Pekeris C L.Proc.Roy.Soc.A 1959,252:80
[5]Lapwood E R and Usami T.Free Oscillation Of The Earth.Camb.Uni.Press,1981
[6]方明.地球自由振荡(博士论文).中国科学院测量与地球物理研究所,1991
[7]Seyed-Mahmoud B.Wobble/Nutation Of A Rotating Ellipsoidal Earth With Liquid Outer Core,Master Thesis.Memorial University of Newfoundland.1994
[8]李荣华.边值问题的Galerkin有限元法.北京:科学出版社,2005
[9]Rochester M G.NormalModesofRotatingSelf-Gravitating CompressibleStratifiedFluidBodies:the Subseismic Wave Equations.In:Graham G A C,Malik S K.eds.Continuum Mechanics and its applications. Hemisphere:Washington D C,1989:797
[10]郭俊义.地球物理学基础.北京:测绘出版社,2001
[11]Dziewonski A M,Anderson D L.Preliminary reference earth model.Phys.Earth.Planet Int.1981,25: 297
[12]Rochester M G.Rotational Dynamics of the deformable Earth(上海天文台讲义).2006
[13]Dahlen F A,Tromp J.Theoretical global seismology.Princeton:Princeton University Press,1998
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