负荷勒夫数渐近值解析表达式

Asymptotic load Love numbers in exact closed-form

地表质量负荷引起的地球变形问题是大地测量学和地球物理学中的一个重要的研究内容.负荷勒夫数的计算和收敛的负荷格林函数的构建是模拟地球负荷形变的基础,其中当球谐阶数趋于无穷大时负荷勒夫数的渐近值,在格林函数计算中发挥重要的作用.本文首先介绍了地表质量负荷的边值问题,然后简要回顾了目前已有的计算负荷勒夫数渐近值三种方法的推导过程,最后给出推导渐近值解析表达式的一种新方法.新方法运用求解地球变形边值问题的解析解方法,以及一种高效传播矩阵方法,建立地表边界条件与高阶负荷勒夫数之间的直接联系,解出了负荷勒夫数渐近值解析表达式.本文的解析表达式更准确地描述了负荷勒夫数的渐近特征.新方法仅涉及常系数常微分方程组的基解矩阵及基础的矩阵运算,原理简单易懂.

The Earth′s deformation due to surface mass loading is an important issue in geodesy and geophysics.Computing load Love numbers and constructing the convergent load Green′s functions are the fundamentals in simulating the Earth′s deformation due to surface mass loading,in which the analytical asymptotic load Love numbers for the harmonic degree at infinity play an important role.In this paper,the boundary value problem for the surface mass loading is introduced first.The three methods proposed before by other scholars to derive the asymptotic values are then briefly reviewed.Finally a new novel approach is proposed.The new way applies the analytical method,which is used to solve the boundary value problem for Earth′s deformation,and an efficient and powerful propagator matrix method to link the boundary conditions on the Earth′s surface and the high-degree load Love numbers.In this way,new exact closed-form asymptotic expressions of the load Love numbers are obtained.The analytical asymptotic expressions of the load Love numbers are able to more accurately describe the asymptotic characteristics of the load Love numbers.The new approach is simple in theory and is easy to understand,which only involves the solution base of the ordinary differential equations with constant coefficients and some primary matrix algebras.