Computer Algebra for Automatically Solving Kinematic Dynamo Problems

抄録

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 S₂^2c 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.