S. Zatman$^{a}$ and J. Bloxham$^{b}$
$^{a}$ Department of Earth and Planetary Sciences, Washington University, Saint Louis, Missouri, 63130, USA $^{b}$ Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts, 02138, USA
We examine a high frequency model of the flow at the surface of the core for the presence of torsional oscillations. The flow model is constructed from a model of the geomagnetic field from 1950-2001 up to spherical harmonic degree 14, using observatory and satellite data, parameterised in time using cubic b splines with knots at annual intervals. The flow model consists of a steady flow up to spherical harmonic degree 20, plus a time dependent flow consisting only of the odd axizonal toroidal flow coefficients ($t_l^0$ with $l$ odd), parameterised in time using cubic b splines with knots at annual intervals. No temporal smoothing is applied in the flow inversion. The time dependent portion of the flow is therefore consistent with the time dependent part of the signal being the surface reflection of the rotation of coaxial cylinders in the core, as is expected if the flow on these timescales consists primarily of torsional oscillations. We then examine the time dependent portion of the flow for the presence of torsional oscillations, by non-linear fitting of harmonic waves with complex frequency and amplitude as a function of latitude (so that the waves may either grow or decay with time, and may vary in phase between the pole and equator), plus a steadily accelerating flow to account for long-timescale behaviour. While the first two or three waves account for the vast bulk of the signal (95.2\% or 98.4\% of the variance respectively), we find that the flow model suggests the presence of perhaps 6 waves of this form, as evidenced in two ways: there is significant variance reduction of the residual to the flow model after fitting each additional wave, and the residuals after fitting each successive wave visually appears wave-like rather than random. In a similar previous study (Zatman and Bloxham, 1997, {\it Nature} {\bf 388} 760-763) using a longer time interval (1900-1990) with coarser time resolution (5 year knots in the flow model), we found evidence for two waves (with the second wave being poorly resolved), both decaying. Here, most of the waves that we find are growing. We speculate that this is because torsional oscillations were being damped in the first half of the twentieth century, but excited in the second half. The improvement of data quality and quantity greatly aids the resolution of oscillations in this study. The periods of the fitted waves in this study are 44.9, 19.8, 12.7, 9.9, 8.2 and 7.0 years. We suspect that the longest period of these waves is related to the poorly-resolved wave B of Zatman and Bloxham, 1997 (wave A from that study being too long in period to be resolved in this study). We expect shorter period torsional oscillations to have higher spatial complexity. Therefore, it is not clear that the very short period oscillations are well resolved in space, even if they are well resolved in time.