Domino model for geomagnetic field reversals

We solve the equations of motion of a one-dimensional planar Heisenberg (or Vaks-Larkin) model consisting of a system of interacting macrospins aligned along a ring. Each spin has unit length and is described by its angle with respect to the rotational axis. The orientation of the spins can vary in time due to spin-spin interaction and random forcing. We statistically describe the behavior of the sum of all spins for different parameters. The term “domino model” in the title refers to the interaction among the spins. We compare the model results with geomagnetic field reversals and dynamo simulations and find strikingly similar behavior. The aggregate of all spins keeps the same direction for a long time and, once in a while, begins flipping to change the orientation by almost 180 degrees (mimicking a geomagnetic reversal) or to move back to the original direction (mimicking an excursion). Most of the time the spins are aligned or antialigned and deviate only slightly with respect to the rotational axis (mimicking the secular variation of the geomagnetic pole with respect to the geographic pole). Reversals are fast compared to the times in between and they occur at random times, both in the model and in the case of the Earth's magnetic field.

Figure 1

Sketch of the domino model.

Figure 2

The magnetization of the standard run as a function of time. At the top, the times of all zero-crossings are indicated (all reversals), while at the bottom, only those where a central band of $M=[-0.5,0.5]$ is crossed are displayed (true reversals).

Figure 3

(a) Log-log plot of the power spectrum of the magnetization. (b) Normalized distribution of the magnetization.

Figure 4

(a) Log-log plot of the number of reversals as a function of the times between reversals. (b) Normalized distribution of the times between reversals for short duration chrons.

Figure 5

The total magnetization of all spins, ${\sum }_{i=1}^N\cos {\theta }_i/N$ (top panel), and the magnetization of the individual spins, $\cos {\theta }_i$, before and after a reversal at $t\approx 3450$.

Figure 6

Examples of a true reversal (top panel), an aborted reversal (middle panel), and two excursions (bottom panel).

Figure 7

The mean time between reversals depending on the number of spins $N$. The symbol for the standard model of Sec. 3 is plotted larger. The values of all other parameters are as in our standard model.

Figure 8

The mean time between reversals depending on (a) $\gamma$, (b) $\lambda$, (c) $\kappa$, and (d) $\epsilon$. The symbol for the standard model of Sec. 3 is plotted larger. All other parameters are as in our standard model.

Figure 9

Dipole tilt angle in a MHD dynamo simulation with Ekman number $E=2·{10}^{-2}$, Rayleigh number $\mathrm{Ra}=300$, Prandtl number $P=1$, and magnetic Prandtl number $\mathrm{Pm}=10$ (adapted from Ref. [20]).

Figure 10

Geomagnetic reversal record from present to 166 Myr BP. The record comprises 332 reversals. A magnetization of $+1$ is assigned for chrons with normal polarity and of $-1$ for chrons with reversed polarity.

Figure 11

(a) Log-log plot of the number of reversals as a function of times between reversals of the geomagnetic reversal record shown in Fig. 10. (b) Normalized distribution of the times between reversals for short duration chrons of less than 1 Myr. These make up 298 of the 332 reversals. The bin size here is four times the interval size. The dashed line is the expected probability density function in the case of a Poissonian process with a mean polarity residence time of 300 kyr. (c) Power spectrum of the geomagnetic reversal record.

Figure 12

(a) Variability of the virtual axial dipole moment (VADM) during the past 2 Myr, the SINT-2000 data set. The absolute values of the VADM are given, disregarding the five reversals during this period. (b) Power spectrum of the VADM time series. (c) Distribution of the VADM.