Nested polyhedra model of isotropic magnetohydrodynamic turbulence

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary $k^{-5/3}$ spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional $k$ spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a $k^{-1/2}$ scaling.

Figure 1

Reference case with ${\text{Pr}}_m=1,\nu ={10}^{-9},N=60,$ and $h_f={10}^{-3}$ kinetic (solid line) and magnetic (dotted line) energy spectra, which follow perfectly the Kolmogorov $k^{-5/3}$ spectrum. The case shown here is from a run up to $t=500$, and the result is averaged over the polyhedra nodes and from $t=460$ to $t=500$. In fact, even an instantaneous spectrum is not so different when averaged over the polyhedra nodes as shown in Ref. [26] for Navier-Stokes.

Figure 2

The high-resolution case with ${\text{Pr}}_m=1.0,\nu ={10}^{-10}$, $N=60$, and $h_f={10}^{-3}$ kinetic (solid blue line) and magnetic (dotted blue line) energies, together with the low-resolution case with $N=30$ and $\nu ={10}^{-6}$ kinetic (solid red line) and magnetic (dotted red line) energies. The results are averaged over the nodes and from $t=460$ to $t=500$.

Figure 3

The case with ${\text{Pr}}_m={10}^{-2}$ (red) and ${\text{Pr}}_m={10}^{-4}$ (blue) with $\nu ={10}^{-10},N=60$, and $h_f={10}^{-3}$ kinetic (solid line) and magnetic (dotted line) energy spectra. Note that the kinetic energy spectrum for ${\text{Pr}}_m={10}^{-2}$ seems to follow a $k^{-8/3}$ power law in the high-$k$ range where the magnetic energy spectrum becomes dissipative. However, when the Prandtl number is decreased further, this is shown to be a nonuniversal feature. The result is averaged over the nodes and from $t=80$ to $t=100$.

Figure 4

Different modes of the system as a function of time showing that the dynamo effect kicks in some time after the large scales are saturated. Here we can see the effects of random forcing on ${\left|u_{32}^i\right|}^2$, which then couples to other nodes.

Figure 5

Kinetic (top) and magnetic (bottom) energy averaged over the polyhedra nodes as a function of the polyhedra number $n$ (i.e., scale) and time. Injection scale can be distinguished clearly at the top figure around $t=0$.

Figure 6

Equipartition time $\delta {\tau }_k={\tau }_k-{\tau }_0$ with respect to the equipartition time of the smallest scales (i.e., ${\tau }_0$) seems to roughly follow a $\delta {\tau }_k\propto k^{-1/2}$ scaling.