Inertial modes of an Earth model with a compressible fluid core and elastic mantle and inner core

We use the linear momentum description (LMD) of the dynamics of the Earth in order to investigate the effects of mantle and inner core elasticity on the frequencies of some of the inertial modes of a spherical Earth model with a liquid core. Traditionally, a liquid core with rigid boundaries is considered to study these modes. A Galerkin method is applied to solve the linear momentum and the Poisson’s equations with the relevant boundary conditions at the interfaces. To test the validity of our method, we compute the periods of some of the Earth’s other normal modes such as the Slichter modes and the spheroidal modes and compare the results with the predicted and observed (when available) values in the literature. We show that the computed dimensionless frequencies [ω/(2Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega /(2\varOmega $$\end{document})] of the inertial modes may be significantly affected by the elasticity of the mantle and inner core. For example, the frequencies of the (2,1,1), also known as the spin-over mode (SOM), (4,1,1), (4,2,1) and (4,3,1) modes are changed from 0.5000, -0.4100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 0.4100$$\end{document}, 0.3060 and 0.8540 for a Poincaré model to 0.4995, -0.4208\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 0.4208$$\end{document}, 0.3150 and 0.8587, respectively. The change in the frequency of the SOM may seem small, but it is consistent with the change in the frequency of the free-core nutation, which is the same mode as the SOM of a wobbling Earth, which changes from ≈0.50144\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 0.50144$$\end{document} for an Earth model with rigid mantle and inner core to ≈0.50116\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 0.50116$$\end{document} for an elastic Earth model. We will show that a great advantage of this method is that we ensure that the frequencies are converged and that it may be generalized to solve other problems in geodynamics including the study of the Earth’s free and forced nutation/wobble.