Disc Dynamos and Reversals

What is modelled

A planetary core makes its magnetic field by moving liquid metal, not by a permanent magnet. The smallest working model is a conducting disc spinning in a field: it drives a current, and if that current is sent through a coil that produces the field, the field feeds itself once the disc spins fast enough. Two such discs coupled to each other, the Rikitake dynamo, produce currents that stay one sign for a while and then flip, like the irregular reversals recorded in rocks.

Equations

• Rikitake system x′ = −μx + yz, y′ = −μy + x(z − a), z′ = 1 − xy.
• Fixed points x² = (a + √(a² + 4μ²))/(2μ), y = 1/x, z = μx² and their mirror image.
• Phase-space volume shrinks as e−2μt because the divergence of the flow is −2μ.
• Bullard single disc I′ = (Ω − 1) I, Ω′ = κ(1 − I²) in units of R/L: growth rate Ω − 1, saturation at I = 1, small oscillations of period 2π/√(2κ).

Checks run in the page

• Fourth-order convergence of the solver.
• Fixed point residual, staying put, and phase-volume contraction.
• Bullard small-current growth against its exact value.
• Bullard small-oscillation period against 2π/√(2κ).

Honest simplifications

These are the two classic low-order caricatures, not magnetohydrodynamics: there is no convection, no Coriolis force and no three-dimensional field geometry, and time is dimensionless so the reversal epochs cannot be read as years. The currents are signed, so a flip stands in for a polarity reversal of a dipole field.

• Press Classic reversals and watch the polarity strip: epochs of irregular length.
• Press Slow decay, long epochs to see how a smaller loss rate changes the pattern.
• Press Bullard disc below threshold and compare it with Bullard fast disc.