Gravitational oscillations in the length of day

Gravitational interactions between mass anomalies in the core and the mantle cause rotational oscillations of the Earth that can be detected as fluctuations in the length of day. A simple theoretical expression for the frequency of these oscillations is derived using a Lagrangian formulation. The value of the frequency depends on the aspherical distribution of mass, providing a useful constraint on the structure of the Earth. Numerical estimates are obtained using two different models of density, both of which are based on seismically inferred density anomalies. The predicted period of the gravitational oscillation is 2 to 3 years, depending on the choice of density model. Longer period modes of oscillation can also arise from the smaller convective fluctuations in the core about the hydrostatic state. Qualitative estimates for these low frequency oscillations depend on the size of the density fluctuations, but typical values yield a period of several hundred years.