The Gravity Effect of Core Modes for a Rotating Earth

Abstract

Previous work has shown that the surface gravity effect associated with core modes (undertones) for a non-rotating Earth model is at or below the nanogal level of detectability for a large earthquake excitation. The assumption in the non-rotating case is that each undertone consists of a single spherical harmonic. In this study we add the Coriolis force to the equations governing the oscillations in a rigid shell with the Boussinesq approximation. By including up to 20 additional harmonics, we find that there are many solutions that are still dominated by low-degree spherical harmonics when judged by the distribution of kinetic energy among the harmonics. For these modes the contribution of higher degrees to the eigenfunctions is not substantial, which suggests that the addition of rotation will probably not significantly change the gravity effects computed without rotation. This conclusion remains tentative until verified in a fuller treatment that explicitly evaluates the excitation for a rotating Earth model.

Influence of liquid core dynamics on rotational modes

“…normalize the eigenfunctions so that W ±1 1 (R) = 1 at the surface r = R, and plot W ±1 1 , 2(U ±1 2 ) 2 + (V ±1 2 ) 2 , W ±1 3 , and 2(U ±1 4 ) 2 + (V ±1 4 ) 2 as a function of r. Many studies, in which it is assumed that the core spectrum contains eigenvalues, showed that the convergence of series (20) of vector spherical harmonics is slow when one tries to compute the hypothetical inertia-gravity modes of an inviscid liquid core (for instance, Johnson & Smylie 1977;Rieutord 1991;Crossley 1993;Wu & Rochester 1993). Therefore, we should not expect that truncated eq.…”

The subseismic approximation in core dynamics-II. Love numbers and surface gravity

“…For example, Pekeris & Accad (1972) found that the tidal Love numbers have resonances at the periods of the internal gravity waves in their model of an earth with a uniformly stable liquid outer core. In previous work (Rochester & Peng 1990, 1993Wu & Rochester 1994) we showed that the Love numbers at the ICB have resonances at periods close to the actual periods of translational oscillations of the solid inner core. It is only when the normal modes of the region in question, considered as an isolated system, are far removed in frequency space from the eigenfrequency of the whole system under discussion that the Love numbers can be treated as 'static'.…”

“…The SSA (Smylie & Rochester 1981) introduces a simplification in the dynamics that is equivalent to dropping the terms Ji2 + y , , 1993) from the right-hand side of (2). Thus eqs (2) become:…”

“…The reason for restricting our test to low-degree modes lies in the indications from previous work (e.g. Crossley et al 1991;Crossley 1993) that the higher-degree modes are probably truncated in the liquid core by effects such as viscosity and in the mantle by geometric spreading (seen in the mantle load Love numbers). Additionally, all undertones are Coriolis coupled down to their lowest azimuthal number m, which, at least for earthquake excitation (m 5 2), means the coupling chains begin with low-degree components.…”

“…Another equally relevant measure, for both core undertones and Slichter modes, is the effect of such vibrations on gravity at the Earth's surface, detectable in principle using a gravimeter or longperiod seismometer (e.g. Hinderer & Crossley 1993). This surface gravity effect can be computed either from arbitrarily normalized eigenfunctions or from a realistic type of physical excitation, such as the release of earthquake energy.…”