Magnetic structure, dipole reversals, and $1/f$ noise in resistive MHD spherical dynamos

A parametric study of the magnetic dipole behavior in resistive incompressible magnetohydrodynamics inside a rotating sphere is performed, using direct numerical simulations and considering Reynolds and Ekman numbers as controlling parameters. The tendency is to obtain geodynamolike magnetic dipole reversal regimes for sufficiently small Ekman and large Reynolds numbers. The typical dipole latitude obtained in the reversal regime is around ${40}^{\circ }$ (with respect to the rotation axis of the sphere). A statistical analysis of waiting times between dipole reversals is also performed, obtaining a non-Poissonian distribution of waiting times, indicating long-term memory effects. We also report the presence of a $1/f$ frequency power spectrum in the magnetic dipole time series, which also shows a tendency to grow toward lower frequencies as the Ekman number is decreased.

Figure 1

Magnetic field lines (left) and streamlines (right) for simulation REV13 at $t=1980$. Both sets of lines are colored according to the local intensity of the respective field and the magnetic dipole moment's direction is marked with a red arrow. The angular speed $\mathbf{\Omega }$ and the orientation of unit vectors $\widehat{\mathit{x}},\widehat{\mathit{y}}$, and $\widehat{\mathit{z}}$ are also shown.

Figure 2

Numerical simulations performed and their classification depending on the type of dynamo solution, as a function of Ekman $\left(\text{Ek}\right)$ and magnetic Reynolds $\left(\text{Rm}\right)$ numbers. Below the markers, the corresponding time-averaged dipolarity percentages are specified.

Figure 3

Magnetic energy per spherical harmonic degree $l$, integrated over the entire sphere, for times between 0 and 200. For clarity, only three modes are displayed, namely, $l=1$ (dipolar energy), $l=3$, and $l=6$. The plots correspond to simulations STAT01 (top), REV12 (middle), and SS03 (bottom).

Figure 4

Kinetic (blue), magnetic (orange), and total (green) energy spectra for run REV11, averaged between $t=300$ and $t=400$. A $k^{-5/3}$ power law is also shown for comparison.

Figure 5

Scatter plot indicating the average absolute latitude of the magnetic dipole moment as a function of the operating magnetic Reynolds and Ekman numbers. Text color denotes the regime associated with that simulation, using the same labels as in Fig. 2.

Figure 6

Normalized $z$ component of the magnetic dipole moment as a function of time for runs REV13 (top), REV12 (middle), and REV14 (bottom), with decreasing Ekman number. Polarities are marked with cyan when positive and red when negative. Corresponding Ekman numbers are tagged on the right.

Figure 7

Shown on top are the magnetic dipole latitude $\alpha$ (solid blue line), dipolar energy $E_1^b$ (orange dashed line), and quadrupolar energy $E_2^b$ (dotted green line) as a function of time for simulation REV13 during the reversal at $t\approx 108$. The bottom shows the total magnetic energy $E^b$ as a function of time for the same run. For convenience, the magnetic energy was normalized to its maximum during the displayed interval.

Figure 8

Distribution of the waiting times between reversals for selected simulations (blue bars); the runs are the same as in Fig. 6. The Ekman number of each run is indicated on the axes. The corresponding mean trend is indicated by the red line. Also, the geomagnetic reversal distribution is shown as a reference by the yellow circles.

Figure 9

Compensated power spectrum $fP\left(f\right)$ for the $z$ component of the magnetic dipole moment as a function of the frequency $f$, for the same runs as in Fig. 6. Both raw (blue) and smoothed (orange) spectra are shown. Regions compatible with $1/f$ behavior are also indicated (green line). Vertical dotted lines mark the energy-containing nonlinear frequency $f_{\text{nl}}$. Inset axes display the compensated power spectrum only for the $1/f$ interval. Corresponding Ekman numbers for each run are tagged on the right.