Subcritical dynamos in rapidly rotating planar convection

We study dynamo action using numerical simulations of planar Boussinesq convection at rapid rotation (low Ekman numbers, $\text{Ek}$), focusing on subcritical dynamo action in which the dynamo is sustained for Rayleigh numbers, $\text{Ra}$, below the critical Rayleigh number for the onset of non-magnetic convection, ${\text{Ra}}_c$. These solutions are found by first investigating the supercritical regime, in which the dynamo is able to generate a large-scale magnetic field that significantly influences the convective motions, with an associated Elsasser number of order ${\text{Ek}}^{1/3}$. Subcritical solutions are then found by tracking this solution branch into the subcritical regime, taking a supercritical solution and then gradually lowering the corresponding Rayleigh number. We show that decreasing the Ekman number leads to an extension of the subcritical range of $\text{Ra}/{\text{Ra}}_c$, down to an optimal value of $\text{Ek}={10}^{-5}$. For magnetic Prandtl numbers of order unity, subcriticality is then hampered by the emergence of a large-scale mode at lower Ekman numbers when the dynamo driven by the smaller scale convection generates relatively stronger large-scale magnetic field. The inability of the large-scale mode to sustain dynamo action leads to an intermittent behavior that appears to inhibit subcriticality. The subcritical solutions are also sensitive to the value of the magnetic Reynolds number (or equivalently, the magnetic Prandtl number, $\text{Pm}$), as values of the magnetic Reynolds number greater than 70 are required to produce dynamo action, but large values lead to fluctuations that are able to push the system too far from the subcritical branch and towards the trivial conducting state.

Figure 1

Location of the simulations in the parameter space $\text{Ek}\text{−}\text{Ra}/{\text{Ra}}_c$. Filled symbols represent cases where a dynamo is sustained and the open symbols represent cases where the dynamo fails. The marker color denotes $\text{Ek}$ and the marker shape $\text{Pm}$. For $\text{Ek}={10}^{-5}$ and $\text{Ek}=5\times {10}^{-6}$, points with different $\text{Pm}$ have been shifted vertically for visibility. It should be noted that an infinite time series would be required to conclusively state the existence of a subcritical dynamo. The majority of the cases presented here persist for long integrations (for a period of time much greater than $t_e$), however cases at $\text{Ek}=5\times {10}^{-7}$ were only run for a comparatively short time (see Table 1).

Figure 2

Time series of the Elsasser number $\mathrm{\Lambda }$ and magnetic Reynolds number $\text{Rm}$ for Case I1. The gray area indicates the super-exponential growth phase.

Figure 3

Snapshots in the (a–c) kinematic phase and (d–f) saturated phase of (a, d) $u_z$ in a horizontal slice, (b, e) 3D isosurfaces of $u_z$, and (c, f) the absolute value of the magnetic field, $\left|\mathit{B}\right|$, in a horizontal slice for Case I1. The horizontal slices are taken at $z=0.92$. The isosurfaces correspond to $\pm 50\%$ of the maximum at that time.

Figure 4

Power spectra (calculated from representative snapshots) of the magnetic energy and kinetic energy as a function of the horizontal wave number, $k_h={(k_x^2+k_y^2)}^{1/2}$, during (a) the kinematic and (b) the saturated phases for Case I1. To include the contribution from the mean magnetic field (for which $k_h=0$) on the logarithmic scale, the horizontal wave numbers are shifted to $k_h+1$.

Figure 5

(a) Time series of the $x$ and $y$ components of the mean magnetic field at $z=0.85$ and (b) snapshots of their vertical profile during the saturated phase for Case I1.

Figure 6

Spectral contributions to the depth-averaged horizontal mean e.m.f., ${\langle {\overline{\mathcal{E}}}_h\rangle }_z={\langle {({\overline{\mathcal{E}}}_x^2+{\overline{\mathcal{E}}}_y^2)}^{1/2}\rangle }_z$, taken from representative snapshots in (a) the kinematic phase and (b) the saturated phase of Case I1. Here ${\langle ...\rangle }_z$ denotes averaging in $z$.

Figure 7

Vertical profile of the unfiltered and filtered versions of $\overline{{\mathcal{E}}_x}$ in (a) the kinematic phase and (b) the saturated phase (snapshot) for Case I1 where contributions from $k_h=k_f$ are kept and other wave numbers are filtered out.

Figure 8

Spectral distributions of the terms generating the (left) horizontal and (right) vertical vorticity as a function of the horizontal wave number for representative snapshots of Case I1 in the (a, b) kinematic and (c, d) saturated phases. The nonlinear inertial terms are denoted by $\mathcal{I}=(\mathit{u}·\mathbf{\nabla })\mathit{\omega }-(\mathit{\omega }·\mathbf{\nabla })\mathit{u}$, the Coriolis term by $\mathcal{C}=(\text{Pr}/\text{Ek}){\partial }_z\mathit{u}$, the viscous term by $\mathcal{V}=\text{Pr}{\nabla }^2\mathit{\omega }$, the buoyancy term by $\mathcal{B}=\text{Pr}\text{Ra}\mathbf{\nabla }\times \left(\theta {\mathit{e}}_z\right)$, the Lorentz force term involving the mean field by $\overline{\mathcal{L}}$, and the Lorentz force term involving the fluctuating field only by ${\mathcal{L}}^{\prime }$.

Figure 9

(a) Time series of the $x$ and $y$ components of the mean magnetic field at $z=0.85$ and (b) snapshot of its vertical profile in the Case G5 at $\text{Ra}/{\text{Ra}}_c=0.91$.

Figure 10

Time series of the Elsasser number, $\mathrm{\Lambda }$, and magnetic Reynolds number, $\text{Rm}$, for (a) Case H5 ($\text{Ra}=0.89{\text{Ra}}_c$) and (b) H6 ($\text{Ra}=0.87{\text{Ra}}_c$).

Figure 11

The dependence of some of the global output quantities as a function of $\text{Ra}/{\text{Ra}}_c$: (a) $\text{Rm}$, (b) $\mathrm{\Lambda }$, (c) $\mathrm{\Lambda }{\text{Ek}}^{-1/3}$, and (d) $\overline{\mathrm{\Lambda }}{\text{Ek}}^{-1/3}$. The horizontal dotted line in (c, d) represents the switch-over value ${\mathrm{\Lambda }}_{\text{sw}}{\text{Ek}}^{-1/3}=1.14$. As in Fig. 1, each color corresponds to a different value of $\text{Ek}$, as indicated in (a), while the different symbols correspond to different choices of $\text{Pm}$, as indicated in (c).

Figure 12

Critical Rayleigh number at the linear onset of rotating magnetoconvection [from Ref. 21] normalized by the critical Rayleigh number in the nonmagnetic case as a function of ${\mathrm{\Lambda }}_0{\text{Ek}}^{-1/3}$, where ${\mathrm{\Lambda }}_0$ is the Elsasser number based on the imposed field strength, in the case where the rotation axis is aligned with gravity and the imposed field is uniform and horizontal. The boundary conditions are the same as in equation (7). The vertical dashed line corresponds to the switch-over value ${\mathrm{\Lambda }}_{\text{sw}}{\text{Ek}}^{-1/3}=1.14$. The red area corresponds to the approximate values of $\overline{\mathrm{\Lambda }}{\text{Ek}}^{-1/3}$ in our subcritical simulations.

Figure 13

Time series of the Elsasser number $\mathrm{\Lambda }$ and magnetic Reynolds number $\text{Rm}$ for Case J1 ($\text{Ra}/{\text{Ra}}_c=1.18$) and Case J2 ($\text{Ra}/{\text{Ra}}_c=0.98$). The gray lines indicate the locations of the snapshots presented in Fig. 14.

Figure 14

Snapshots of the horizontal distribution of the vertical velocity (at $z=0.92$) for Case J1 in (a) the kinematic phase, (b) near the peak of $\mathrm{\Lambda },$ and (c) during the decaying phase. The bottom row shows the corresponding power spectra for the magnetic energy (blue) and the kinetic energy (red), as a function of the horizontal wave number for (d) the kinematic phase, (e) near the peak in $\mathrm{\Lambda },$ and (f) during the decaying phase. The locations of each of these snapshots are represented by gray lines in Fig. 13. The box-size mode corresponds to $k_h=4$ for this aspect ratio.