Behavior of hydrodynamic and magnetohydrodynamic turbulence in a rotating sphere with precession and dynamo action

The effect of precession in a rotating sphere filled with fluid was studied with direct numerical simulations, in both incompressible hydrodynamics (HD) and magnetohydrodynamics (MHD) scenarios. In both cases the asymptotic state and its dependence with both rotating and precession frequency was analyzed. For the MHD case no self-sustaining dynamos were found for the prograde precession case, whereas on the other hand a critical retrograde precession frequency was found above which dynamo action is self-sustained. It was also found that these correspond to small-scale dynamos with a developed turbulent regime. Furthermore, the presence of reversals of the magnetic dipole moment is observed, with greater waiting times between reversals for smaller precession frequencies.

Figure 1

Energy $E_k$ (left) and averaged enstrophy $\langle {\omega }^2\rangle$ (right) as a function of time for the runs HD04, HD05, and HD06.

Figure 2

Mean kinetic energy $\langle E^v\rangle$ of all the runs in Table 1 as a function of $\gamma {\mathrm{\Omega }}_0$. The dashed line corresponds to the scaling of the proposed model, and filed and open symbols represent the runs with $\gamma \ge 1$ and $\gamma =0.1$, respectively.

Figure 3

Kinetic power spectra $E$ as a function of the CK wave number $k$ for different runs with varying ${\mathrm{\Omega }}_0$. The reference line corresponds to the Kolmogorov spectrum $k^{-5/3}$.

Figure 4

Magnetic and kinetic energy as a function of time for MHD01 (left), MHD02 (middle), and MHD03 (right), which differ in the precession frequency, $\gamma =0.1,1,10$, respectively, and have the same value of ${\mathrm{\Omega }}_0=10$. It can be observed that these runs do not generate dynamos, except for very high values of ${\mathrm{\Omega }}_0$ and $\gamma$, as is the case for the MHD03 run (rightmost panel).

Figure 5

Magnetic energy vs time for the prograde and retrograde cases (left); normalized kinetic energy spectra for both cases (right). The spectrum at different times is shown in colors, and the time average is shown as a dashed line for the prograde case and a solid line for the retrograde case.

Figure 6

Comparison of the time evolution of the magnetic energy for different $\gamma$ values. Each panel is distinguished by the corresponding value of ${\mathrm{\Omega }}_0$. On the left with ${\mathrm{\Omega }}_0=1$ are the simulations MHD04, MHD05, and MHD06; in the middle with ${\mathrm{\Omega }}_0=8$ are MHD07, MHD08, and MHD12; and finally on the right with ${\mathrm{\Omega }}_0=16$ are the runs MHD13, MHD14, and MHD18.

Figure 7

Total energy as a function of the quantity $\epsilon /\sqrt{\gamma {\mathrm{\Omega }}_0}$ for different runs (MHD08–MHD12 and MHD14–MHD18). The proposed scaling is indicated with a dashed line.

Figure 8

Magnetic energy spatial spectra for runs with ${\mathrm{\Omega }}_0=8$ (MHD08–MHD12) on the left and with ${\mathrm{\Omega }}_0=16$ (MHD14–MHD18) on the right; all of them are for the time $t=150$. A predominance in the high values of $k$ can be observed for all of the runs. This suggests that these are small-scale dynamos.

Figure 9

Time evolution of the normalized $z$ component of the magnetic dipole moment $m_z/\left|m\right|$ for a time window between $t$=100 and $t$=120, for all the dynamo runs with ${\mathrm{\Omega }}_0=8$ (left) and values of $\gamma =-3,-3.5,-4,-4.5,-5$, from top to bottom. Normalized histograms of the time $\tau$ between reversals (waiting times) are shown on the right. The simulations that are shown in this figure are MHD08–MHD12.

Figure 10

(Top panel) Magnetic energy evolution during a reversal, at time $t\approx 140.4$, of the different orders of the magnetic multipole expansion ($E_l^B$) labeled with the subscript $l$, in different symbols/colors. The dashed line indicates the evolution of the magnetic dipole latitude ($\alpha$). (Bottom panel) Total magnetic energy ($E^B$) as a function of time during the same interval. This plot corresponds to the run MHD08.