Self-Similar Turbulent Dynamo

The amplification of magnetic fields in a highly conducting fluid is studied numerically. During growth, the magnetic field is spatially intermittent: it does not uniformly fill the volume, but is concentrated in long thin folded structures. Contrary to a commonly held view, intermittency of the folded field does not increase indefinitely throughout the growth stage if diffusion is present. Instead, as we show, the probability-density function (PDF) of the field-strength becomes self-similar. The normalized moments increase with magnetic Prandtl number in a powerlike fashion. We argue that the self-similarity is to be expected with a finite flow scale and system size. In the nonlinear saturated state, intermittency is reduced and the PDF is exponential. Parallels are noted with self-similar behavior recently observed for passive-scalar mixing and for map dynamos.

Figure 1

Evolution of normalized moments of the magnetic-field strength. The (square root of) kurtosis of the velocity field is given for comparison. The kinematic (growth) and nonlinear (saturation) stages are demarcated in the plot.

Figure 2

(a) Evolution of the PDF of the magnetic-field strength. (b) The PDFs of $B/B_{\mathrm{r}\mathrm{m}\mathrm{s}}$ in the kinematic regime: collapse onto a self-similar profile. The PDF in the saturated state is also shown.

Figure 3

The PDF of $B/B_{\mathrm{r}\mathrm{m}\mathrm{s}}$ averaged over the kinematic diffusive stage and the lognormal profile (5) with the same $D$. Also given is the PDF in the saturated state.

Figure 4

The kurtosis $⟨B^4⟩/⟨B^2{⟩}^2$ during the diffusive kinematic and the nonlinear stages of a sequence of runs with $\nu =5\times {10}^{-2}$ (low $\mathrm{R}\mathrm{e}$) and $25\le {\mathrm{P}\mathrm{r}}_{\mathrm{m}}\le 2500$. All averages are over 20 time units ($\sim 20$ turnover times).