VLBI observations of free core nutations and viscosity at the top of the core

Abstract

Very long baseline interferometry (VLBI) nutation measurement series, both in excess of 23 years length, from Goddard Space Flight Center (GSFC) and the United States Naval Observatory (USNO), have been analyzed for free core nutation resonances. VLBI nutation observations can only be made when the radio sources being used are visible, rendering the data sequence inherently non-equispaced. This poses the problem of the approach to be taken with unevenly spaced sampling. Both the conventional Discrete Fourier Transform (DFT) and the Fast Fourier Transform Algorithm (FFT) for its computation strictly require a fixed sampling interval. Our approach is to find the Discrete Fourier Transform of the non-equispaced record by minimizing an objective function which weights the error between the DFT representation and the measured values in inverse proportion to the square of their standard errors. The resulting conditional equations have a coefficient matrix of Toeplitz form but the recursive Levinson algorithm has been found inadequate for their solution, even when implemented in double quad precision. Instead, we employ the Singular Value Decomposition technique to solve the least squares problem of fitting the Discrete Fourier Transform to the non-equispaced VLBI nutation observations. A novel feature of our procedure is to use the Parseval relation to determine the number of singular values of the coefficient matrix to be eliminated. We report the observation for the first time of the prograde mode predicted by Jiang [Jiang, X., 1993. Wobble–nutation modes of the earth, Ph.D. thesis, York University, Toronto, Canada]. The long series of observations allow the determination of the time evolution of the free core nutations. We observe both the prograde and the retrograde modes to be in free decay. In addition to providing measures of the viscosity just below the core–mantle boundary (CMB), the free decays suggest impulsive excitations rather than continuous excitation by electromagnetic core–mantle coupling or the atmosphere. The average recovered viscosity at the top of the core is of the order 615 Pa s in contrast to the value of 8 × 1 0 − 3 Pa s found by Gans [Gans, R., 1972. Viscosity of the Earth’s core, J. Geophys. Res. 77, 360–366] from the extrapolation of laboratory measurements, and commonly used by dynamo theorists.