Hysteresis of dynamos in rotating spherical shell convection

Bifurcations of dynamos in rotating and buoyancy-driven spherical Rayleigh-Bénard convection in an electrically conducting fluid are investigated numerically. Both nonmagnetic and magnetic solution branches comprised of rotating waves are traced by path-following techniques, and their bifurcations and interconnections for different Ekman numbers are determined. In particular, the question of whether the dynamo branches bifurcate super- or subcritically and whether a direct link to the primary pure convective states exists is answered.

Figure 1

Magnetic RWs for $\mathrm{Ek}=0.001$ of the $m=3$, 4, and 5 dynamos. The star marks the benchmark example at the $m=4$ branch and open circles mark supercritical Hopf bifurcations. Solid (dashed) lines denote stable (unstable) solutions.

Figure 2

Contour plots of the radial velocity for the nonmagnetic solution (a) in the equatorial plane and (c) in the middle of the spherical gap at $\text{Ek}=0.001$ and $\text{Ra}=100$. (b) and (d) Same for the dynamo solution. In each plot the color is normalized to the maximum (red) and minimum (blue) of the presented field component.

Figure 3

Convective (primary) and magnetic (secondary) bifurcations of RWs for $\mathrm{Ek}=0.00164$. On the vertical axis, $E=E_{\mathrm{kin}}+0.25E_{\mathrm{mag}}$. Thick lines mark the dynamo branches. The inset shows a close-up of the $m=4$ bifurcating magnetic branch.

Figure 4

The $m=4$ dynamos for various Ekman numbers, which are labeled at the corresponding branches. Thick (thin) lines mark stable (unstable) solutions.

Figure 5

The $m=3$ dynamo branches for the same Ekman numbers as in Fig. 4.