Why dynamos are prone to reversals

Abstract

In a recent paper [1] it was shown that a simple mean-field dynamo model with a spherically symmetric helical turbulence parameter α can exhibit a number of features which are typical for Earth's magnetic field reversals. In particular, the model produces asymmetric reversals (with a slow decay of the dipole of one polarity and a fast recreation of the dipole with opposite polarity), a positive correlation of field strength and interval length, and a bimodal field distribution. All these features are attributable to the magnetic field dynamics in the vicinity of an exceptional point of the spectrum of the non-selfadjoint dynamo operator where two real eigenvalues coalesce and continue as a complex conjugated pair of eigenvalues. Usually, this exceptional point is associated with a nearby local maximum of the growth rate dependence on the magnetic Reynolds number. The negative slope of this curve between the local maximum and the exceptional point makes the system unstable and drives it to the exceptional point and beyond into the oscillatory branch where the sign change happens. A weakness of this reversal model is the apparent necessity to fine-tune the magnetic Reynolds number and/or the radial profile of α in order to adjust the operator spectrum in an appropriate way. In the present paper, it is shown that this fine-tuning is not necessary in the case of higher supercriticality of the dynamo. Numerical examples and physical arguments are compiled to show that, with increasing magnetic Reynolds number, there is strong tendency for the exceptional point and the associated local maximum to move close to the zero growth rate line where the indicated reversal scenario can be actualized. Although exemplified again by the spherically symmetric α 2 dynamo model, the main idea of this "self-tuning" mechanism of saturated dynamos into a reversal-prone state seems well transferable to other dynamos. As a consequence, reversing dynamos might be much more typical and may occur much more frequently in nature than what could be expected from a purely kinematic perspective.

A statistical analysis of polarity reversals of the geomagnetic field

“…Actually, it is the negative slope of this curve between the local maximum and the exceptional point that makes the system unstable and drives it to the exceptional point and beyond into the oscillatory branch where the sign change happens. The apparent weakness of this reversal model, namely the necessity to fine-tune the radial profile of α in order to adjust the operator spectrum in an appropriate way, was overcome in a follow-up paper (Stefani et al, 2006a). For highly supercritical dynamos the exceptional point and the associated local growth rate maximum were demonstrated to tend towards the zero growth rate line where the indicated reversal scenario can be actual-on statistical properties of reversals is left for future works.…”

Inferring basic parameters of the geodynamo from sequences of polarity reversals

“…The correlation time T c has always been set to 0.005 · T c which would correspond to 1 kyr in case that the diffusion time is set to 200 kyr. The resulting time series show reversal sequences quite similar to those of the geodynamo [23,26,29]. For the sake of simplicity, we define a reversal as the sign change of the poloidal field component s(1, τ ) at the outer radius r = 1.…”

“…This simple model had turned out to be quite helpful for understanding the basic principle of the reversal process as a noise-induced relaxation-oscillation in the vicinity of an exceptional point of the spectrum of the non-selfadjoint dynamo operator [23,25,26]. This exceptional point, at which two real eigenvalues coalesce and continue as a complex conjugated pair of eigenvalues, is associated with a nearby local maximum of the growth rate situated at a slightly lower magnetic Reynolds number.…”

“…Actually, a few dependencies on individual parameters were already published in preceeding papers. In [23] the dependence of typical time scales on the supercriticality of the dynamo was studied, in [20] some dependencies of the clustering property on the supercriticality and the noise level were shown, and in [24] the influence of the diffusion time scale on the RTD between reversals was touched upon. What is new in the present paper is that we take all three reversal features together and try to infer from them some essential parameters of the geodynamo.…”

Forward and inverse problems in fundamental and applied magnetohydrodynamics

“…Solving such inverse spectral dynamo problems by means of an evolutionary strategy it was possible to obtain such α(r) profiles that lead to oscillatory dynamo solutions [58]. This model was later used to develop simple models of geomagnetic reversals that are just based on noise triggered jumps and relaxation oscillations in the vicinity of spectral exceptional points [59,60,61]. On this basis, a first attempt was further made to infer some of the most essential parameters of the geodynamo, such as its overcriticality and the turbulent resistivity from the temporal behavior and the statistical properties of field reversals [62].…”