Kinetic helicity needed to drive large-scale dynamos

Magnetic field generation on scales that are large compared with the scale of the turbulent eddies is known to be possible via the so-called $\alpha$ effect when the turbulence is helical and if the domain is large enough for the $\alpha$ effect to dominate over turbulent diffusion. Using three-dimensional turbulence simulations, we show that the energy of the resulting mean magnetic field of the saturated state increases linearly with the product of normalized helicity and the ratio of domain scale to eddy scale, provided this product exceeds a critical value of around unity. This implies that large-scale dynamo action commences when the normalized helicity is larger than the inverse scale ratio. Our results show that the emergence of small-scale dynamo action does not have any noticeable effect on the large-scale dynamo. Recent findings by Pietarila Graham et al. [Phys. Rev. E 85, 066406 (2012)] of a smaller minimal helicity may be an artifact due to the onset of small-scale dynamo action at large magnetic Reynolds numbers. However, the onset of large-scale dynamo action is difficult to establish when the kinetic helicity is small. Instead of random forcing, they used an ABC flow with time-dependent phases. We show that such dynamos saturate prematurely in a way that is reminiscent of inhomogeneous dynamos with internal magnetic helicity fluxes. Furthermore, even for very low fractional helicities, such dynamos display large-scale fields that change direction, which is uncharacteristic of turbulent dynamos.

Figure 1

Example showing the evolution of the normalized $\langle {\overline{\mathit{B}}}^2\rangle$ (dashed) and that of $\langle {\overline{\mathit{B}}}^2\rangle +\tau d\langle {\overline{\mathit{B}}}^2\rangle /dt$ (dotted), compared with its average in the interval $1.2⩽2\eta k_1^{2}t⩽3.5$ (horizontal blue solid line), as well as averages over three subintervals (horizontal red dashed lines). Here, $\overline{\mathit{B}}$ is evaluated as an $xz$ average, ${\langle \mathbf{B}\rangle }_{xz}$. For comparison, we also show the other two averages, ${\langle \mathbf{B}\rangle }_{xy}$ (solid) and ${\langle \mathbf{B}\rangle }_{yz}$ (dash-dotted), but their values are very small.

Figure 2

Dependence of relative kinetic helicity ${\tilde{\epsilon }}_f$ (a) and normalized kinetic helicity ${\epsilon }_f$ (b) on the helicity parameter $\sigma$ of the forcing function Eq. (8) together with the analytical expression $2\sigma /(1+{\sigma }^2)$ (solid line).

Figure 3

Dependence of $k_{\omega }/k_f$ on $\text{Re}$. The open and closed circles correspond to runs with ${\text{Pr}}_M=1$ without and with magnetic field, respectively, while squares correspond to runs with ${\text{Pr}}_M=100$ and $\text{Re}={\text{Re}}_M/{\text{Pr}}_M$ is small. Triangles denote the results for $k_f/k_1=1.5$ of Ref. [36] (BP12).

Figure 4

Steady state values of $\langle {\overline{\mathit{B}}}^2\rangle /B_{\mathrm{eq}}^2$ as a function of $C_{\alpha }$ together with the theoretical prediction from Eq. (16) (dashed line) and a linear fit (dotted line).

Figure 5

Steady state values of $\langle {\overline{\mathit{B}}}^2\rangle /B_{\mathrm{eq}}^2$ as a function of ${\epsilon }_f$ for various scale separation values $k_f/k_1$ together with linear fits.

Figure 6

Critical value for the normalized kinetic helicity ${\epsilon }_f$ for which LSD action occurs for different scale separations.

Figure 7

Steady-state values of $\langle {\mathit{b}}^2\rangle /B_{\mathrm{eq}}^2$ as a function of $C_{\alpha }$ together with the fit formula from Eq. (23) with $n=4$, compared with $n=1$ (dotted) and $n=2$ (dashed). Different symbols denote different values of $k_f/k_1$.

Figure 8

Steady-state values of $\langle {\overline{\mathit{B}}}^2\rangle /B_{\mathrm{eq}}^2$ as a function of $C_{\alpha }$ for ${\text{Pr}}_M=100$ and ${\text{Pr}}_M=1$ for $k_f/k_1=5$ and different values of ${\text{Re}}_M$ (different symbols), compared with the theoretical prediction (dotted line).

Figure 9

Comparison of kinetic and magnetic energy spectra for ${\text{Pr}}_M=100$ (upper panel) and ${\text{Pr}}_M=1$ (lower panel) for $\sigma =0.2$ (solid lines) and $0.12$ (dashed lines). Magnetic energy spectra are shown as thick red lines, while kinetic energy spectra are shown as thin blue lines.

Figure 10

Visualization of $B_x$ on the periphery of the domain for ${\text{Pr}}_M=100$ after resistive saturation.

Figure 11

Evolution of total magnetic field ($B_{\mathrm{rms}}$, upper black line), small-scale magnetic field ($b_{\mathrm{rms}}$, blue in the middle), and large-scale magnetic field (${\overline{B}}_{\mathrm{rms}}$, lower red line) for three values of ${\text{Re}}_M$ over a time stretch of 160 turnover times.

Figure 12

Similar to Fig. 1, but for time-dependent ABC-flow driving. As in Fig. 1, we have here $k_f/k_1=15$ and ${\text{Re}}_M\approx 6$.

Figure 13

Dependence of the normalized $\langle {\overline{\mathit{B}}}^2\rangle$ for different planar averages: $yz$ (black), $xz$ (red, dotted), and $xy$ (blue, dashed), for $\sigma =0.1$ (upper panel) and $\sigma =0.01$ (lower panel).