Scale dependence of kinetic helicity and selection of the axial dipole in rapidly rotating dynamos

The dominant polarity of the magnetic field in rapidly rotating spherical dynamos is the axial dipole. Studies of the onset of magnetoconvection in the limit of vanishing Ekman number have proposed that the dipole is favored over other polarities because its equatorial symmetry generates kinetic helicity that would be otherwise absent in nonmagnetic convection. This study explores the effect of the magnetic field in the selection of the axial dipole in rapidly rotating, supercritical dynamos. The strength of convection is such that the axial dipole grows from a starting seed field in the nonlinear dynamo but fails to grow in the kinematic dynamo at the same parameters. The magnetic field is shown to excite convection over a range of length scales larger than the energy injection scale and at the same time extract energy from smaller scales through the Lorentz force in order to feed itself. This leads to substantial helicity generation in a range of length scales and helicity loss in smaller scales, both relative to nonmagnetic convection. The crossover point between the helicity surplus and deficit regions is displaced to smaller length scales as the rotation rate is increased (by decreasing the Ekman number). The helicity deficit occurs in the region of the spectrum where the Lorentz force approximately balances the Coriolis force in the vorticity equation, so that energy may be drawn from these scales via vortex stretching. The timescale for the increase in convection intensity relative to the nonmagnetic state approximately coincides with the timescale for the formation of the axial dipole, which indicates that the Lorentz force has an important role in polarity selection. Crucially, the dipole forms from a chaotic state well before the saturation of the dynamo, implying that planetary dynamos choose their polarity during their nonlinear growth phase. The generation of the toroidal part of the dipole field is primarily through the classical $\mathrm{\Omega }$ effect, although the pattern of the zonal flow switches from spherical harmonic degree $l=1$ to $l=3$ in the presence of the magnetic backreaction.

Figure 1

Time-averaged energy spectra for the nonlinear dynamo simulations, showing the variation of kinetic ($E_k$) and magnetic ($E_m$) energies with spherical harmonic degree $l$. The red lines show $E_m$ and the blue lines show $E_k$.

Figure 2

(a) Evolution in time (measured in units of the magnetic diffusion time) of the axial dipole (${\mathit{B}}_{10}^P$) Elsasser number in nonlinear (red line) and kinematic (blue line) dynamo simulations at $E=1.2\times {10}^{-5}$, $\text{Pr}=\text{Pm}=5,$ and two Rayleigh numbers.

Figure 3

(a) Evolution in time (measured in units of the magnetic diffusion time) of the Elsasser number in nonlinear dynamo simulations at $E=1.2\times {10}^{-5}$, $\text{Pr}=\text{Pm}=5$, $\text{Ra}=220$ (red line) and $E=1.2\times {10}^{-6}$, $\text{Pr}=\text{Pm}=1$, $\text{Ra}=400$ (blue line). The alphabet labels represent the snapshots in time given in Fig. 4 below. (b) Comparison of the axial dipole (${\mathit{B}}_{10}^P$) Elsasser numbers for the nonlinear (red line) and kinematic (blue line) dynamo runs at $\text{Ra}=1.2\times {10}^{-5}$, $\text{Ra}=220$, $\text{Pr}=\text{Pm}=5$. (c) Comparison of the axial dipole (${\mathit{B}}_{10}^P$) Elsasser numbers for the nonlinear (red line) and kinematic (blue run) dynamo runs at $\text{Ra}=1.2\times {10}^{-6}$, $\text{Ra}=400$, $\text{Pr}=\text{Pm}=1$.

Figure 4

Shaded contours of the radial magnetic field at the outer boundary in nonlinear dynamo simulations at two parameter regimes. The times, measured in units of the magnetic diffusion time, are taken from points on the Elsasser number plots in Fig. 3.

Figure 5

(a) & (c): Variation over spherical harmonic degree $l$ of the time-averaged kinetic energy density in the non-magnetic (dashed lines) and nonlinear dynamo (solid lines) calculations. The $z$-energy is shown in red and the $s$-energy is shown in blue. (b) & (d): The difference in $z$-energy density between the dynamo and non-magnetic runs (red) and the respective difference in $s$ energy density (blue). The parameter regime ($E,Ra$) considered is given above the panels.

Figure 6

Top panels [(a), (c), (e)]: Snapshots in time (measured in units of the magnetic diffusion time) of the isosurfaces of $u_z$ (contour level $\pm 80$) for $l\le 30$ in the nonlinear dynamo simulation. Bottom panels [(b), (d), (f)]: $u_z$ (contour level $\pm 150$) at the same times, but for $l>30$. The parameters used are $E=1.2\times {10}^{-6},\text{Pr}=\text{Pm}=1,\text{Ra}=400$.

Figure 7

[(a), (c)] Variation over the spherical harmonic degree $l$ of the time-averaged kinetic helicity in the lower hemisphere in the nonmagnetic (dashed lines) and nonlinear dynamo (solid lines) calculations. The $z$ helicity is shown in red and $s$ helicity is shown in blue. [(b), (d)] The difference in $z$ helicity between the dynamo and nonmagnetic runs (red) and the respective difference in $s$ helicity (blue). The parameter regime ($E,\text{Ra}$) considered is given above the panels.

Figure 8

Time-averaged $z$ helicity on the section $z=0.3$ parallel to the equator. The parameters used in the dynamo model are $E=3\times {10}^{-7}$, $\text{Ra}=540,$ and $\text{Pr}=\text{Pm}=1$. Panels (a) and (b) compare the respective helicities in the nonmagnetic and dynamo calculations for spherical harmonic degrees $l\le 46$; panels (c) and (d) give the same comparison for $l>46$.

Figure 9

Plots of the magnetic (M), buoyancy (A), and Coriolis (C) terms in the $z$-vorticity equation as a function of spherical harmonic degree $l$ for the saturated nonlinear dynamo at $E=1.2\times {10}^{-6}$, $\text{Ra}=400,$ and $\text{Pr}=\text{Pm}=1$. The line styles used are the following: C (red circles), A (green line), M based on the axial dipole field only (black line), and M based on the nondipole field (blue line).

Figure 10

Contour plots of the time-averaged Coriolis (C) term in the $z$-vorticity equation shown on a cylinder of radius $s=1.1$. The parameters used in the dynamo simulation are $E=1.2\times {10}^{-6}$, $\text{Ra}=400$, $\text{Pr}=\text{Pm}=1$. Two ranges of the spherical harmonic degree are considered: [(a), (b)] $l\le 30$; [(c), (d)] $l>30$. The upper panels (a) and (c) represent nonmagnetic convection and the lower panels (b) and (d) represent the saturated dynamo.

Figure 11

Contour plots of the time-averaged Lorentz (M) term in the $z$-vorticity equation shown on a cylinder of radius $s=1.1$. The parameters used in the dynamo simulation are $E=1.2\times {10}^{-6}$, $\text{Ra}=220$, $\text{Pr}=\text{Pm}=1$. Panels (a) and (b) show the M term based on the axial dipole and nondipole parts of the field respectively.

Figure 12

Upper panels [(a), (c), (e), (g)] show isosurfaces of the time-averaged anticyclonic (A) and cyclonic (C) $z$ helicity for nonmagnetic convection. Lower panels [(b), (d), (f), (h)] show the respective plots for the saturated (nonlinear) dynamo. Two ranges of the spherical harmonic degree are considered: (a)–(d) $l\le 30$, shown at contour levels $\pm 2\times {10}^4$, and (e)–(h) $l>30$, shown at contour levels $\pm 2\times {10}^5$. The parameters used in the dynamo model are $E=1.2\times {10}^{-6}$, $\text{Pr}=\text{Pm}=1$, and $\text{Ra}=400$.

Figure 13

(a) Volume integral of induction term dotted with ${\mathit{B}}_{20}^T$. The line colors are red, $\mathbf{\nabla }\times ({\mathit{u}}^T\times {\mathit{B}}^P)$; blue, $\mathbf{\nabla }\times ({\mathit{u}}^P\times {\mathit{B}}^T)$; black, $\mathbf{\nabla }\times ({\mathit{u}}^P\times {\mathit{B}}^P)$; magenta, $\mathbf{\nabla }\times ({\mathit{u}}^T\times {\mathit{B}}^T)$; and green, $\mathbf{\nabla }\times ({\mathit{u}}_{m=0}^T\times {\mathit{B}}_{m=0}^P)$. Panels (b) and (c) give the energy matrix for $\mathbf{\nabla }\times ({\mathit{u}}_{m=0}^T\times {\mathit{B}}_{m=0}^P)$ at two different times, $t_d=0.414$ and $t_d=0.522$ respectively, shown by dashed vertical lines in panel (a). The parameters used in the simulation are $E=1.2\times {10}^{-5}$, $\text{Ra}=220$, and $\text{Pr}=\text{Pm}=5$.

Figure 14

Meridional plots of time and azimuthally averaged $\varphi$ velocity in the calculation at $E=1.2\times {10}^{-5}$, $\text{Ra}=220$, and $\text{Pr}=\text{Pm}=5$. (a) Nonmagnetic state; $\left(\text{b}\right)$ saturated nonlinear dynamo.