Numerical calculation of modes of oscillation of the Earth's core

SUMMARY This paper describes the numerical implementation of a variational principle for the calculation of the very low‐frequency (< 300 μHz) modes of oscillation of the fluid outer core using realistic models of Earth structure. The scalar, generalized displacement potential is represented by a new set of local, polynomial basis functions which are either purely even or purely odd in the equatorial plane, depending on the parity of the mode being computed. Elastic boundary conditions are matched through the use of Love numbers for the shell and inner core and by the imposition of linear constraints on the bilinear symmetric functional. The observed compressibility of the outer core is incorporated in the calculation and small, non‐neutral stratification is allowed for the through the introduction of ω 2 v , the signed square of the Väisälä angular frequency, as a parameter. The resulting eigenvalue problem involves polynomial band matrices for which a special iterative procedure for the calculation of eigenvectors and corrections to eigenvalues is developed. Modes are ordered in terms of increasing spatial complexity by using an equivalent average wavenumber. A selection of numerical examples, including plots of the generalized displacement potential and the displacement fields themselves, are presented to illustrate the method of calculation. The work is expected to provide the basis for identification of anomalies in gravimeter signals and nutation series, and possibly, with the recent discovery (Beroza & Jordan 1990) of unexpectedly high levels of seismic energy at long time‐scales in ‘slow’and ‘silent’earthquakes, the detection of core modes excited by earthquakes.