Dynamo effect in decaying helical turbulence

We show that in decaying hydromagnetic turbulence with initial kinetic helicity, a weak magnetic field eventually becomes fully helical. The sign of magnetic helicity is opposite to that of the kinetic helicity—regardless of whether the initial magnetic field was helical. The magnetic field undergoes inverse cascading with the magnetic energy decaying approximately like $t^{-1/2}$. This is even slower than in the fully helical case, where it decays like $t^{-2/3}$. In this parameter range, the product of magnetic energy and correlation length raised to a certain power slightly larger than unity is approximately constant. This scaling of magnetic energy persists over long timescales. At very late times and for domain sizes large enough to accommodate the growing spatial scales, we expect a crossover to the $t^{-2/3}$ decay law that is commonly observed for fully helical magnetic fields. Regardless of the presence or absence of initial kinetic helicity, the magnetic field experiences exponential growth during the first few turnover times, which is suggestive of small-scale dynamo action. Our results have applications to a wide range of experimental dynamos and astrophysical time-dependent plasmas, including primordial turbulence in the early universe.

Figure 1

Evolution of ${\mathcal{E}}_{\mathrm{K}}$ (blue) and ${\mathcal{E}}_{\mathrm{M}}$ (red) for ${\sigma }_{\mathrm{M}}=0$ (solid), ${\sigma }_{\mathrm{M}}=1$ (dashed), and ${\sigma }_{\mathrm{M}}=-1$ (dotted) for $v_{\mathrm{A}0}/u_0=0.1$ (runs A–C), as well as 0.01 (dot-dashed, run D) and 0.001 (triple dot-dashed, run E) for ${\sigma }_{\mathrm{M}}=0$. The green triple dot-dashed line denotes run ${\mathrm{E}}^{\prime }$, which has zero initial kinetic helicity.

Figure 2

Instantaneous growth rates $\gamma \left(t\right){\tau }_0$ of $B_{\mathrm{rms}}$ for ${\sigma }_{\mathrm{M}}=0$ (solid), ${\sigma }_{\mathrm{M}}=1$ (dashed), and ${\sigma }_{\mathrm{M}}=-1$ (dotted) for $v_{\mathrm{A}0}/u_0=0.1$ (runs A–C), as well as 0.01 (dot-dashed, run D) and 0.001 (triple dot-dashed, run E). The green triple dot-dashed line denotes run ${\mathrm{E}}^{\prime }$ with zero initial kinetic helicity.

Figure 3

$E_{\mathrm{K}}(k,t)$ and $E_{\mathrm{M}}(k,t)$ for $t/\tau =16$, 60, 200, 800, 2000, 8000, and 17 000, for run A. Time is decreasing downward and the last time is shown as fat lines.

Figure 4

$E_{\mathrm{K}}(k,t)$ and $E_{\mathrm{M}}(k,t)$ at $t/\tau =2600$, 9400, 24,000, and 45,000, collapsed spectra using ${\beta }_{\mathrm{M}}=0$ for Run A.

Figure 5

$pq$ diagram for run A for kinetic (blue open symbols) and magnetic (red filled symbols) energy spectra. Near the end of the run (larger symbols), the solution evolves along the $p_{\mathrm{M}}=0.58$ line (dashed) and $q_{\mathrm{M}}\approx 0.55$ with ${\beta }_{\mathrm{M}}=p_{\mathrm{M}}/q_{\mathrm{M}}-1\approx 0.05$ is found at the end of the run. Smaller (larger) symbols denote earlier (later) times.

Figure 6

Same as Fig. 5, but for run C, where the solution evolves along the $p_{\mathrm{M}}=0.57$ line (dashed) and $q_{\mathrm{M}}\approx 0.53$ with ${\beta }_{\mathrm{M}}=p_{\mathrm{M}}/q_{\mathrm{M}}-1\approx 0.08$ is found at the end of the run.

Figure 7

Evolution of ${\beta }_{\mathrm{M}}\left(t\right)$ (black), $p_{\mathrm{M}}\left(t\right)$ (red), and $q_{\mathrm{M}}\left(t\right)$ (blue) for run A (solid lines), run B (dotted lines), and run C (dashed line). Note that ${\beta }_{\mathrm{M}}\left(t\right)$ would reach zero at an extrapolated time of $t_{*}\approx 5300$.

Figure 8

$k^{-1}{H}_{\mathrm{K}}(k,t)$ (red) and $k{H}_{\mathrm{M}}(k,t)$ along with $2E_{\mathrm{K}}(k,t)$ and $2E_{\mathrm{M}}(k,t)$ (black lines) at $t/\tau =0.05$, 0.16, 0.3, 0.6, and 1.7 for run A (red for positive values and blue for negative values). The blue and red arrows indicate the change of $k{H}_{\mathrm{M}}(k,t)$ with time.

Figure 9

$k{H}_{\mathrm{M}}(k,t)$ (red for positive values and blue for negative values) at later times: $t/\tau =5$, 10, and 25 for run A. The arrows indicate the temporal change of $k{H}_{\mathrm{M}}(k,t)$.

Figure 10

Evolution of $\langle {\mathit{B}}^2\rangle {\xi }_{\mathrm{M}}^{\beta }$ with $\beta =0$ for run A (black solid), F (red dotted), G (blue dashed), and H (green dash-dotted). In fully helical turbulence, we expect $\langle {\mathit{B}}^2\rangle {\xi }_{\mathrm{M}}\to \mathrm{const}$, but here $\langle {\mathit{B}}^2\rangle {\xi }_{\mathrm{M}}^{1+{\beta }_{\mathrm{M}}}\approx \mathrm{const}$ with ${\beta }_{\mathrm{M}}=0.05$ at the end of the run.

Figure 11

Evolution of $\langle \mathbf{\omega }·\mathit{u}\rangle$ (blue), $\langle \mathbf{\omega }·\mathit{u}-\mathit{J}·\mathit{B}/{\rho }_0\rangle$ (red), and $\langle \mathit{A}·\mathit{B}\rangle$ (green) for $\mathrm{Re}=160$ (run A).

Figure 12

Similar to Fig. 11, but with different normalizations, using in panels (a)–(c) the initial velocity and the initial peak wave number, in panels (d)–(f) the time-dependent velocity using $k_1$, with a linear time axis for magnetic helicity in panel (f) in separate panels focusing on $\langle \mathbf{\omega }·\mathit{u}\rangle$ (blue), $\langle \mathbf{\omega }·\mathit{u}-\mathit{J}·\mathit{B}/{\rho }_0\rangle$ (red), $-\langle -\mathit{J}·\mathit{B}/{\rho }_0\rangle$ (orange), and $\langle \mathit{A}·\mathit{B}\rangle$ (green) for $\mathrm{Re}=160$ (run A).

Figure 13

$pq$ diagram for ${\mathrm{Pr}}_{\mathrm{M}}=0.1$ (a) and 10 (b). Smaller (larger) symbols denote earlier (later) times. The $p_{\mathrm{M}}=0.58$ line (dashed) is shown for comparison.

Figure 14

$pq$ diagram with constant $\nu =\eta ={10}^{-6}$ for ${\mathrm{Pr}}_{\mathrm{M}}=1$. Again, smaller (larger) symbols denote earlier (later) times and the $p_{\mathrm{M}}=0.58$ line (dashed) is shown for comparison.