Critical Magnetic Prandtl Number for Small-Scale Dynamo

We report a series of numerical simulations showing that the critical magnetic Reynolds number ${\mathrm{R}\mathrm{m}}_{\mathrm{c}}$ for the nonhelical small-scale dynamo depends on the Reynolds number $\mathrm{R}\mathrm{e}$. Namely, the dynamo is shut down if the magnetic Prandtl number ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}=\mathrm{R}\mathrm{m}/\mathrm{R}\mathrm{e}$ is less than some critical value ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}\lesssim 1$ even for $\mathrm{R}\mathrm{m}$ for which dynamo exists at ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}\ge 1$. We argue that, in the limit of $\mathrm{R}\mathrm{e}\to \infty$, a finite ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}$ may exist. The second possibility is that ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}\to 0$ as $\mathrm{R}\mathrm{e}\to \infty$, while ${\mathrm{R}\mathrm{m}}_{\mathrm{c}}$ tends to a very large constant value inaccessible at current resolutions. If there is a finite ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}$, the dynamo is sustainable only if magnetic fields can exist at scales smaller than the flow scale, i.e., it is always effectively a large-${\mathrm{P}\mathrm{r}}_{\mathrm{m}}$ dynamo. If there is a finite ${\mathrm{R}\mathrm{m}}_{\mathrm{c}}$, our results provide a lower bound: ${\mathrm{R}\mathrm{m}}_{\mathrm{c}}\gtrsim 220$ for ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}\le 1/8$. This is larger than $\mathrm{R}\mathrm{m}$ in many planets and in all liquid-metal experiments.

Figure 1

(a) Evolution of magnetic energy $⟨B^2⟩(t)$ in all runs. Also shown are the versions of runs A3 and B4 that started from the saturated state of run A1 (at $t=75$) and run B1 (at $t=50$), respectively. Runs A1, A2, and B1 have resolution ${128}^3$, runs A3, B2, B3, and B4 are ${256}^3$. (b) Growth rates vs ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}$ (at fixed $\eta$) for the same runs plus for 7 runs (${128}^3$) extending series B to ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}>1$. The dynamo is again shut down at ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}\sim 100$ because velocity is strongly damped by large viscosity and $\mathrm{R}\mathrm{m}$ drops below critical. We show $\mathrm{R}\mathrm{m}$ (divided by its value for run B1, $\mathrm{R}\mathrm{m}\simeq 210$) in the same plot.

Figure 2

Magnetic-energy spectra normalized by $⟨B^2⟩/2$ and averaged over time: (a) series A, (b) series B. Also given are velocity spectra multiplied by $k^2$ and the reference Kolmogorov slope $k^{1/3}$. The time intervals used for averaging are for runs A1, A2: $5\le t\le 40$; for run B1: $2.5\le t\le 17.5$; for runs A3, B2, B3, B4: $10\le t\le 25$.

Figure 3

Characteristic scales for series B [cf. Fig. 1b]: $k_{\mathrm{r}\mathrm{m}\mathrm{s}}=(⟨|\nabla \mathbf{B}{|}^2⟩/⟨B^2⟩{)}^{1/2}$ is (roughly) the inverse direction-reversal scale, $k_{\parallel }=(⟨|\mathbf{B}·\nabla \mathbf{B}{|}^2⟩/⟨B^4⟩{)}^{1/2}$ is the inverse fold length, and $k_{\lambda }=(⟨|\nabla \mathbf{u}{|}^2⟩/⟨u^2⟩{)}^{1/2}$ is the inverse Taylor microscale (times $\sqrt{5}$) (see Refs. 11, 12 for discussion of these quantities).