The Earth's free core nutation: Formulation of dynamics and estimation of eigenperiod from the very-long-baseline interferometry data

The free-core nutation (FCN) is a rotational normal mode of the Earth's outer core. We derive the equations of motion for FCN w.r.t. both the inertia space F0 and the uniformly rotating frame FΩ, and show that the two sets of equations are invariant in form under the reference frame transformation, as required by physics. The frequency-domain formulation describes the FCN resonance (to nearby tidal signals), which has been exploited to estimate the complex eigenfrequency of FCN, or its eigenperiod P and quality factor Q. On the other hand, our time-domain formulation in terms of temporal convolution describes the response of the free FCN under a (continual) excitation. The convolution well explains the dynamic behaviors of FCN in the observed very-long-baseline interferometry (VLBI) data (in F0), including the undulation of the FCN amplitude and the apparent fluctuations in the period and phase over time, as well as the temporal concurrence of a large phase jump with the near-zero amplitude during ∼1998-2000, in complete analogy to the observed behavior of the Chandler wobble (in FΩ). The reverse, deconvolution process is further exploited to yield optimal estimates for FCN's eigenfrequency using the VLBI data, following the approach of Furuya and Chao (1996) of locating minimum excitation power. While this method is found to be insensitive to Q owing to the short timespan of the data, we obtain the estimate of P=441±4.5 sidereal days (sd) where the 1-sigma uncertainty is assessed via extensive Monte Carlo simulations. This value is closer to the theoretical value of ∼460 sd predicted by Earth models assuming hydrostatic equilibrium than do the prior estimates (425-435 sd) by the resonance method. The deconvolution process also yields the excitation function as a by-product, the physical sources of which await further studies.