Small-scale turbulent dynamo for low-Prandtl number fluid: comparison of the theory with results of numerical simulations

Abstract

Context: During the last decades, significant progress has been made in both numerical simulations of turbulent dynamo and theoretical understanding of turbulence. However, there is still lack of quantitative comparison between the simulations and the theory of the dynamo. Results: We study the critical magnetic Reynolds number ($Rm_c$) and the growth rate near the threshold both in the limit of very high and in the case of moderate Reynolds numbers. We argue that in Kazantsev equation for magnetic field generation, one should use the quasi-Lagrangian correlator of velocities instead of Eulerian, as usually implied when comparing theory and simulations. The theoretical results obtained with this correlator agree well with numerical results. We also propose the explanation of the decrease of $Rm_c$ as a function of Reynolds number ($Re$) at intermediate-high $Re$. It is probably due to Reynolds-dependent intermittency of the velocity structure function: we show that the scaling exponent of this function in the inertial range affects strongly the magnetic field generation, and it is known to be an increasing function of the Reynolds number. Conclusions: Use of quasi-Lagrangian correlator in the Kazantsev theory gives good accordance with numerical simulations. An ideal way to compare them should be to find the correlator substituted to the Kazantsev equation and the generation properties in the same simulation. At least one has to use universal parameters independent of the properties of pumping scale. Reynolds-dependent intermittency can explain recently observed decrease of the critical magnetic Reynolds number at small Prandtl numbers.

Figures (3)

• Figure 1: Shape of $b(\rho)$ for the Sharp (Eq. \ref{['b-Sharp']}) and the Smooth (Eq. \ref{['b-Smooth']}) models, for the same values of the parameters $s$, $b_{\infty}$ and $\Lambda$.
• Figure 2: Effective potential $U(\rho)$ (Eq. \ref{['U-pot']}) that corresponds to the generation threshold for the two 'Sharp' and two 'Smooth' models. The length scale is normalized by the diffusion scale $r_d$, which is taken the same for both models. The vertical arrows correspond to the delta functions in the 'Sharp' model's potential.
• Figure 3: Logarithmic derivative $\sigma (\rho)$, (\ref{['sigma']}), for the four models. The length scale is normalized by same $r_d$. The parameters of each model correspond to the generation threshold.