A new method to calculate normal modes

We developed a new method to calculate normal modes of the Earth and planets. It can treat anelasticity directly as imaginary parts of elastic constants and leaky modes due to the open boundary condition set at the upper atmosphere. The eigenvalue problem is described in complex numbers. It is formulated based on a similar treatment of the global matrix governing the system of oscillations to that in the Henyey-type relaxation method used in solar seismology. In our method, the complex eigenvalue problem of a large system is reduced to an eigenvalue problem of a quite small size matrix. The eigenvalue of the small problem is a correction of an assumed complex eigenfrequency and components of the eigenvector are values of eigenfunctions at the outer boundary. Starting from an arbitrary complex frequency around the eigenfrequency of a target mode, we can arrive there within, at most, a dozen of steps of iterative calculations. We compared the results of our method with those calculated by DISPER80, and found good agreement between them. The rate of convergence of the method depends on the linearity of the correction around the eigenfrequency. A numerical example shows good behaviour of them. Even for a model with an atmosphere in which the fundamental spheroidal mode S-0(29) and the fundamental acoustic mode P-0(29) nearly degenerate, we can easily reach the eigenfrequency of S-0(29) and distinguish it from that of P-0(29) without any confusion. And we found that the eigenfrequency of P-0(29) calculated for a realistic atmospheric model which varies annually, most approaches the solid mode in August. The behaviour of P-0(29) can be interpreted with the aid of an acoustic potential which characterizes vertical propagation of sound waves. In addition to the efficiency in the convergence to the eigenfrequencies, numerical tests show strong numerical stability of the method. It stems from the stability in the relaxation method because of the similarity in algebraic structure. For those reasons, we propose the method as an efficient way in calculating synthetic sesimograms, barograms and ionograms for recently observed phenomena relating with coupling between the solid earth, the oceans and the atmosphere.