The Earth's Free Oscillations and

The Differential Rotation of the Inner Core

Gabi Laske and Guy Masters

Abstract and Introduction

Differential rotation of the inner core has been inferred by several body-wave studies with most agreeing that a superrotation may exist with a rate between 0.2 and 3 degress per year. The wide range of inferred rotation rates is caused by the sensitivity of such studies to local complexities in structure which have been demonstrated to exist. Free-oscillation "splitting functions" are insensitive to local structure and are therefore better candidates for estimating differential inner core rotation.
The exact cause for the signal in our splitting functions (3D heterogeneity or anisotropy) is irrelevant in our study. All we are interested in is whether the patters change with time.

We use a recently developed method ( AR technique) for analyzing free oscillations which is insensitive to earthquake source, location and mechanism to constrain this differential rotation. In a prior study, we found that inner core differential rotation has been essentially zero over the last 20 years. We have revisited this issue, including additional earthquakes and modes in our analysis. The normal modes investigated here are also quite sensitive to mantle structure (see also comments). It has therefore been suggested that our results may be influenced by the mantle corrections we apply in our analysis. We show that this is not the case but that fluctuations in rotation rates are mode specific.

Spectra, Receiver Strips and Splitting Functions

Spectra from vertical recordings of of individual stations show anomalously split inner-core sensitive modes (here 13S2). Due to 3D structure of the Earth, the ''spectral peak'' exhibits fine-scale splitting, within a band defined as the splitting width (grey area). Some spectra even exhibit clearly split peaks (ANMO, CCM). Rotation and hydrostatic ellipticity of the Earth cause a splitting of only 6.7mHz (black bar). The smaller peaks at 4.87 mHz are the faster decaying mode 9S7. For each of the 45 earthquakes in our database (spread over 24 years), we collapse the typically 100 spectra into receiver strips, without loss of information. See background information for details. Even though the Bonin Islands earthquake was 20 times smaller than the Bolivia event, it excited the mode well enough to produce high signal-to-noise strips. On the other hand, the greater Indian Ocean event on June 19, 2000 did not excite this particular mode sufficiently well to be considered in this analysis.
A selection of high signal-to-noise receiver strips is then used to determine the splitting matrix, and ultimately the splitting function. The receiver strips are autoregressive. We use this property to formulate a linear inverse problem for the propagator matrix P(t)= exp[i(H+Iw)t], solve for the splitting matrix, H, and ultimately determine the elastic splitting function f(q,f).
An advantage of the autoregressive technique over other techniques is that it does not use any earthquake-related terms (the a(0) cancel out). By using events from different time periods (e.g. 1977-1985 vs. 1994-2001) we can, in principle, investigate the time-dependence of the splitting function. We show a comparison of splitting functions for mode 13S2, corrected for crust and mantle structure.
The correlation plot shows the correlation at degrees 2 and 4 (and their sum) of the non-zonal patterns as a function of rotation angle about the rotation axis (relative longitude). harmonic degrees 2 and 4 require different angles for the highest correlation which is inconsistent with the inner core rotating as a ridig body.
We suspect that this inconsistency originates from errors in the poorly determined old splitting functions that lack recordings of suitable earthquakes. We therefore abandon this type of comparison and adopt a forward approach which is described in the next section.

The Hypothesis Test

In a forward modelling approach we test the hypothesis that the inner core is rotating at a certain rate. We then test whether the receiver strips over time are consistent with this rate.

Using only modern data we determine the current splitting function of a mode. We subtract the mantle signal from this splitting function ( corrected splitting function ).
For a given earthquake, we rotate the remaining "core-splitting function" using an assumed rotation rate. The mantle signal is then added back to the splitting function and the propagator matrix, P, is reconstructed. We then determine how well P maps a receiver strip bn into a lagged one, bn+1. By iterating this procedure we find the best rotation rate for each earthquake and a best fitting rate is determined assuming that the rotation rate does not change over time. This is repeated for each mode, and the average over all modes gives our final inner core rotation rate.

Determining the Rotation Rates for Each Mode

Rotation angles and inferred differential rotation rates are shown here for the "best nine modes" (i.e. high number of events). The solid line marks the best fitting straight line that has zero angle at the median time for which the "recent" splitting function was determined. Grey dashed lines mark assumed rotation rates of 1 and 3 per year. For most modes, these high rates are inconsistent with the measured angles. the number of events for each mode is given in the upper right corner.

Determining the Final Inner Core Rotation Rate

This figure shows inner core rotation rates obtained for 13 inner core-sensitive modes, using our preferred mantle model SB10L18. Also shown are the results obtained using other mantle models. The results using different models are remarkably consistent and variations seem to be mode specific, not model specific. Possible reasons are given in the next section.

The least squares fitting rotation rate, using all modes, is 0.13 +/- 0.11/yr (light grey area). A rotation rate of 0.25/yr is marginally consistent with our data.

Though slightly higher than our earlier measured rate of 0.01 +/- 0.21/yr (see the "Letter to Nature"), the rotation rate inferred here is still consistent with the idea that the inner core is gravitationally locked to the mantle.

Some cases of modes, that we exclude in our study

Some l=2 modes may couple strongly to radial modes which is not accounted for in our study (the strongest coupling mode 7S2 was excluded here). It is not clear how this effect biases our results. If we ignore l=2 modes, our final rotation rate is 0.34+/-0.13/yr.

The splitting functions of l=1 modes at almost entirely zonal after a mantle correction. These modes are not considered here.

Some modes overlap in frequency with other modes of relatively high l (e.g.2S3 with 0t7, 0S7). Even without significant coupling, the older earthquakes do not have sufficient records to reliably determine receiver strips (in this case, all modes have to be included to calculate the strips).


Our initial results on inner core rotation:
Laske, G. and G. Masters, Limits on differential rotation of the inner core from an analysis of the Earth's free oscillations. Nature, 402, 66-68, 1999. pdf

A description of the Autoregressive Technique to determine splitting matrices:
Masters, G., Laske, G. and F. Gilbert, Autoregressive Estimation of the Splitting Matrix of Free-Oscillation Multiplets. Geophys. J. Int., 141, 25-42, 2000. pdf

Examples for applications to un-coupled and coupled modes:
Masters, G., Laske, G. and Gilbert, F., Matrix autoregressive analysis of free-oscillation coupling and splitting. Geophys. J. Int. (Knopoff Festschrift), 143, 478-489, 2000. pdf

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