Magnetic field reversals and long-time memory in conducting flows

Employing a simple ideal magnetohydrodynamic model in spherical geometry, we show that the presence of either rotation or finite magnetic helicity is sufficient to induce dynamical reversals of the magnetic dipole moment. The statistical character of the model is similar to that of terrestrial magnetic field reversals, with the similarity being stronger when rotation is present. The connection between long-time correlations, $1/f$ noise, and statistics of reversals is supported, consistent with earlier suggestions.

Figure 1

The ratio of kinetic energy vs magnetic energy $E_k/E_m$ as a function of time, for three runs with different initial conditions and same $H_m=0.03,\Omega =16$. Thicker line (black online) $E_k/E_m(t=0)=1$, intermediate thick line (red online) $E_k/E_m(t=0)=0.5$, and thin line (blue online) $E_k/E_m(t=0)=2$.

Figure 2

Velocity (top) and magnetic (bottom) field lines in the run with $\Omega =16,H_m=0.03$. The field lines change color according to the intensity of the field, from red to yellow, blue, and magenta. The red, green, and blue arrows indicate, respectively, the $x,y,z$ axis, with $\Omega$ in the $z$ direction.

Figure 3

Time series of the normalized magnetic dipole moment for four different values of the angular velocity of rotation $\mathit{\Omega }$ and magnetic helicity $H_m$. Time is measured in units of Ma (mega anni), 1 Ma = ${10}^6$ years, as indicated in the text.

Figure 4

Frequency spectra of the magnetic dipole moment for four different values of the angular velocity of rotation $\mathit{\Omega }$ and magnetic helicity $H_m$. A $f^{-1}$ power-law spectrum is indicated as a reference in each plot. Also, insets show the compensated spectra $fP\left(f\right)$ for each case. Units of frequency are 1/Ma, 1 Ma = ${10}^6$ years.

Figure 5

Distribution function of the waiting time between reversals, for four different values of the angular velocity of rotation $\mathit{\Omega }$ and magnetic helicity $H_m$ (dashed lines). The continuous (red online) line corresponds to the distribution function for the observational Cande and Kent 1995 data [1].

Figure 6

Autocorrelations functions vs time-lag $\delta$.

Figure 7

Hurst analysis for $H_m>0,\Omega >0$. The structure functions, computed up to the fourth moment, are represented with open symbols, while the fits, from Eq. (10), are reported with lines. In the legend of the plot, the results of the fit $H_q$ are reported as well.

Figure 8

Hurst analysis for HD model. (Top) time series; (middle) power spectrum; (bottom) structure function analysis.