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 ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}\ensuremath{\lesssim}1$ even for $\mathrm{R}\mathrm{m}$ for which dynamo exists at ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}\ensuremath{\ge}1$. We argue that, in the limit of $\mathrm{R}\mathrm{e}\ensuremath{\rightarrow}\ensuremath{\infty}$, a finite ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}$ may exist. The second possibility is that ${\mathrm{P}\mathrm{r}}_{\mathrm{m}\mathrm{,}\mathrm{c}}\ensuremath{\rightarrow}0$ as $\mathrm{R}\mathrm{e}\ensuremath{\rightarrow}\ensuremath{\infty}$, 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: ${\mathrm{R}\mathrm{m}}_{\mathrm{c}}\ensuremath{\gtrsim}220$ for ${\mathrm{P}\mathrm{r}}_{\mathrm{m}}\ensuremath{\le}1/8$. This is larger than $\mathrm{R}\mathrm{m}$ in many planets and in all liquid-metal experiments.