The computation of seismic normal modes with rotation as a quadratic eigenvalue problem

TL;DR

This paper introduces a novel computational method for seismic normal modes in rotating, heterogeneous planets, combining finite-element, multipole, and Lanczos techniques to improve accuracy and efficiency.

Contribution

It presents a new quadratic eigenvalue problem approach that effectively incorporates rotation and heterogeneity in seismic mode calculations.

Findings

Accurate computation of seismic modes for rotating planets.

Effective separation of the essential spectrum in fluid outer cores.

Application to Mars models demonstrating heterogeneity effects.

Abstract

A new approach is presented to compute the seismic normal modes of a fully heterogeneous, rotating planet. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method (FMM). To solve the resulting quadratic eigenvalue problem (QEP), the approach utilizes extended Lanczos vectors forming a subspace computed from a non-rotating planet -- with the shape of boundaries of a rotating planet and accounting for the centrifugal potential -- to reduce the dimension of the original problem significantly. The subspace is guaranteed to be contained in the space of functions to which the seismic normal modes belong. The reduced system can further be solved with a standard eigensolver. The computational accuracy is illustrated using all the modes with relative small meshes and also tested against standard perturbation calculations relative to a standard Earth model. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Mars models studying the effects of heterogeneity on a large range of scales.