Not much helicity is needed to drive large-scale dynamos

Understanding the in situ amplification of large-scale magnetic fields in turbulent astrophysical rotators has been a core subject of dynamo theory. When turbulent velocities are helical, large-scale dynamos that substantially amplify fields on scales that exceed the turbulent forcing scale arise, but the minimum sufficient fractional kinetic helicity $f_{h,C}$ has not been previously well quantified. Using direct numerical simulations for a simple helical dynamo, we show that $f_{h,C}$ decreases as the ratio of forcing to large-scale wave numbers $k_F/k_{\min }$ increases. From the condition that a large-scale helical dynamo must overcome the back reaction from any nonhelical field on the large scales, we develop a theory that can explain the simulations. For $k_F/k_{\min }⩾8$ we find $f_{h,C}\lesssim 3\%$, implying that very small helicity fractions strongly influence magnetic spectra for even moderate-scale separation.

Figure 1

Redimensionalized magnetic (solid) and kinetic (dashed) energy spectra after 90$\tau$ for run 4-60 (thick black) forced at $k_F=4$ and for run 3-60 (thin red/light gray) with $k_F=3$. At a fixed $f_h=60\%$ here, increased scale separation provides for the transition between SSD and LSD.

Figure 2

Magnetic energy density $E_b\left(k\right)$ for $k=1$ (solid line), $k=6$ (dashed), and total (dotted) versus time for run 3-80. The gray line indicates the fit $\gamma =(8.6\pm 1.4){10}^{-3}$ to the solid curve for growth of the $k=1$ mode after SSD saturation.

Figure 3

Growth rate ${\gamma }_{k=1}$ of $E_b(k=1)$ versus fractional helicity. From Table , $\times$($k_F=2$), $+(k_F=3$), $\square$($k_F=4$), $▵$($k_F=5$), $\diamond$($k_F=6$), $*$($k_F=8$), and least-squares linear fits (dashed).

Figure 4

$f_{h,C}$ from least-squares fits versus $k_F/k_{\min }$ (symbols as in Fig. 3). Dashed line is best fit to Eq. (2), giving $C=0.21$ and $\xi =0.46$. Dotted line is kinematic MFT prediction $f_{h,C}=\beta k_{\min }/\left(\right|{\alpha }_0\left|k_F\right)$.