Computer Algebra for Automatically Solving Kinematic Dynamo Problems
Computer Algebra for Automatically Solving Kinematic Dynamo Problems
复制标题
自动解决运动发电机问题的计算机代数
DOI：
10.5636/jgg.42.35
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
M. Kono
中科院分区：
文献类型：
--
作者：
M. Kono
开发了一种计算机算法，通过该算法可以自动解决运动发电机问题的不同组合的速度场，速度径向函数，启动磁场，最大程度，和在半径的分割数。分析方法是基于最初由Bullard和Gellman开发的方案，并进行了微小的修改。在前半部分，该程序进行了代数处理的感应方程，并获得适当的特定的速度场和球谐函数的最大程度的扩展到极向和环形模式扩展的方程。前半部分的结果被提供给程序的后半部分，在后半部分中，公式被数字取代，并且针对所获得的矩阵求解特征值问题。对于非常少的给定参数（通常为3到8），程序将感应方程展开到环形和极向模式，形成差分方程的矩阵，在半径上具有给定的分割数，求解本征值问题，并获得包括本征值和本征向量的结果，以及其它信息，例如各个谐波中的能量和不同模式之间的能量传递。将该程序应用于T1和S22 c的Bullard-Gellman速度场，并将结果与以前报道的值进行了比较。本文得到的本征值与BULLARD和GELLMAN（1954），LILLEY（1970），PEKERIS等（1973）的结果很好地一致，但与吉布森和ROBERTS（1969）的结果不一致。结果表明，本文所编制的程序是正确的，计算机代数方法是解决地磁发电机问题的有力手段。
A computer algorithm was developed by which kinematic dynamo problems can be automatically solved for different combinations of velocity fields, velocity radial functions, starting magnetic fields, maximum degree, and number of divisions in the radius. The method of analysis is based on the scheme originally developed by Bullard and Gellman, with minor modifications. In the first half, the program performs an algebraic treatment of the induction equation and obtains the equations expanded into poloidal and toroidal modes appropriate to the particular velocity field and the maximum degree of expansion in spherical harmonics. The result from the first half is supplied to the second half of the program, in which formulas are replaced by numbers, and the eigenvalue problem is solved for the obtained matrix. For a very small number of given parameters (usually 3 to 8), the program expands the induction equation to toroidal and poloidal modes, forms the matrix for the difference equation with a given number of divisions in the radius, solves the eigenvalue problem, and obtains the results including eigenvalues and eigenvectors, as well as other information such as the energy in the individual harmonics and energy transfer between different modes. The program was applied to the Bullard-Gellman velocity field of T1 and S22c and the results were compared with previously reported values. The eigenvalues obtained in this paper agree well with the previous results of BULLARD and GELLMAN (1954), LILLEY (1970), and PEKERIS et al. (1973), but not with those of GIBSON and ROBERTS (1969). It is concluded that the present programs work correctly, and that the method of computer algebra is a powerful approach to the geomagnetic dynamo problem.