Study of the Earth's Eigen Mode by Galerkin Method

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).