Fractal Analysis of the Polarity Reversal of the Earth's Magnetic Field and the Rikitake Self-Reversing Dynamo Model

The Earth’s magnetic field reverses its polarity with durations lasting from 10 Ka to 35 Ma. The data analyzed are a combination of reversal times reported by Kent and Gradstein (1986) and Cande and Kent (1992) from the present to 169 Ma and contains 284 reversal events. The fractal box counting method is used to analyze the clustering of the temporal sequence of reversal events. Results of the analysis show self-similar power scaling for time intervals ranging from 1.2 Ma to 20 Ma with a scaling exponent α ≈ 0.9. The data was split into two parts, one before and one after the Cretaceous quiet zone. Analysis shows self-similar power scaling for both parts with time intervals ranging from 1.2 to 20 Ma and a scaling exponent α ≈ 0.9 (before) and time intervals from 1.2 Ma to 4 Ma with α ≈ 0.9 (after). For all of these data sets, below 1.2 Ma the data roll away from the power function because there are too few intervals less than 1.2 Ma to sustain the scaling. The fractal fragmentation dimension method, also called rank-order, is used here to determine the size versus cumulative number of the lengths of time intervals between reversals (without regard to their position in the sequence). Results of the analysis show self-similar power scaling ranging from 260 Ka to 10 Ma with the scaling exponent α ≈ 1.2. Below 260 Ka the data roll away from the power function because there are too few intervals less than 260 Ka to sustain the scaling. The self-reversing disk dynamo model of Rikitaki (1958) has been run to generate 284 reversal events which have been analyzed with the two fractal methods described above in order to test whether the model exhibits self-similar power scaling. Box counting yields a scaling exponent α ≈ 0.9, the same as observed for the Earth’s magnetic field. The model fails to produce a power fragmentation dimension for the lengths of time intervals between reversals.