Rikitake Dynamo Attractor

Overview

In 1958, Japanese geophysicist Tsuneji Rikitake formulated a two-disk dynamo model to explain one of geophysics’ deepest mysteries: why the Earth’s magnetic field periodically and unpredictably flips polarity over geological epochs.

Paleomagnetic rock records showed that Earth’s north and south magnetic poles have reversed hundreds of times at irregular intervals. Rikitake conceived a mechanical model consisting of two electrically coupled Faraday disks driven by constant external torques. He showed that this coupled system produces a natural strange attractor where electrical currents oscillate chaotically and spontaneously reverse direction.

The Mathematical Model

The dimensionless equations governing currents $x$ and $y$ in the two dynamos, and angular velocity $z$, are:

$\frac{dx}{dt} = -\mu x + zy$

$\frac{dy}{dt} = -\mu y + x(z - a)$

$\frac{dz}{dt} = 1 - xy$

Geophysical Interpretation

• $x, y$: Electric currents circulating through the two disk circuits. A positive sign represents current flowing in one direction (normal polarity); a negative sign indicates reversed polarity.
• $z$: Angular velocity of the rotating conductive disks driven by convection within Earth’s molten iron outer core.

The Reversal Mechanism

The Rikitake system features two unstable focus points corresponding to steady dynamo states with opposite magnetic polarities:

• Trajectories oscillate with growing amplitude around one polarity for dozens of cycles.
• Suddenly, without any external trigger, the trajectory shoots past a critical tipping point into the basin of the opposite polarity.
• This reproduces the statistical distribution of geomagnetic reversals observed in Earth’s oceanic crustal stripes.