Nonhelical turbulence and the inverse transfer of energy: A parameter study

We explore the phenomenon of the recently discovered inverse transfer of energy from small to large scales in decaying magnetohydrodynamical turbulence by Brandenburg et al. [Phys. Rev. Lett. 114, 075001 (2015)], even for nonhelical magnetic fields. For this investigation we mainly employ the Pencil Code performing a parameter study, where we vary the Prandtl number, the kinematic viscosity, and the initial spectrum. We find that to get a decay that exhibits this inverse transfer, large Reynolds numbers ($O\sim {10}^3$) are needed and low Prandtl numbers of the order unity $\text{Pr}=1$ are preferred. Compared to helical MHD turbulence, though, the inverse transfer is much less efficient in transferring magnetic energy to larger scales than the well-known effect of the inverse cascade. Hence, applying the inverse transfer to the magnetic field evolution in the Early Universe, we question whether the nonhelical inverse transfer is effective enough to explain the observed void magnetic fields if a magnetogenesis scenario during the electroweak phase transition is assumed.

Figure 1

Slices of the xy plane of the magnetic field strength $\left|B\right|=\sqrt{B_x^2+B_y^2+B_z^2}$ at the initial time and at $t=100$. The color scale has been adjusted for each panel, the energy has decayed significantly at the later snapshot. (a) Slices of the nonhelical run, ${\mathcal{H}}_0=0$. The size of the eddies grow through the decay of small-scale fluctuations as well as and by the effect of the inverse transfer. (b) Slices of the maximally helical run. The eddies grow through an inverse cascade of magnetic energy. The eddies are now about $1/5$ of the box size.

Figure 2

Magnetic power spectra of a run with maximal helicity. While the magnetic energy decays, the peak of the power spectrum does not, but shifts to lower $k$. This inverse cascade only occurs for maximally helical fields, where $H_k\sim E_k$.

Figure 3

Fitting of a parabola in log-space to determine the value for the wave number $k_I$, which defines the correlation length $L_I=1/k_I$, i.e., the integral scale.

Figure 4

Magnetic power spectra for runs with different viscosities. The hyperviscosity runs are shown in the two upper left panels. High Reynolds numbers, i.e., low viscosities, are needed to observe the effect of nonhelical inverse transfer which dynamically increases the energy on large scales. Spectra are shown for simulation times $t=0,20,2\times {10}^2,\text{and}2\times {10}^3{\tau }_0$, where the green (uppermost) spectrum is the initial condition $t=0$. All runs have the Prandlt number $\mathrm{Pr}=1$.

Figure 5

Time evolution of the energy on large scales for runs with different viscosities. Only certain parameters in the simulation setup give rise to the inverse transfer.

Figure 6

Comparison of the energy on large scales for the runs with different viscosities at time $t=100$. $\mathcal{H}$ denotes the helical comparison run. The values for the viscosities are given in Fig. 4, where $\nu 1$ is the upper left panel and $\nu 6$ the panel in the lower right corner.

Figure 7

Magnetic power spectra of runs with different Prandtl numbers. The spectra are shown for $t=0,20,2\times {10}^2,1\times {10}^3,7\times {10}^3{\tau }_0$. The effect of the inverse transfer of energy is strongest with a Prandtl number of $\mathrm{Pr}=1$ and becomes less pronounced for higher Prandtl numbers, especially the run with $\mathrm{Pr}=1000$ shows very little increase of energy on large scales.

Figure 8

Time evolution of the energy on large scales. The transfer of energy to large scales depends crucially on the Prandtl number where larger Prandtl numbers show less efficient inverse transfer. We also show the evolution of the energy on large scales for the helical case.

Figure 9

Evolution of the integral scale for runs with varying Prandtl numbers. Larger Prandtl numbers lead to a slower increase of the coherence length.

Figure 10

Comparison of the integral scale for the runs with different Prandtl numbers at the time $t=10$. $\mathcal{H}$ denotes the helical case.

Figure 11

Magnetic and kinetic power spectra of the nonhelical simulation with a resolution of $N={1536}^3$. Note at early times the kinetic part can exceed the magnetic part of the spectrum in large scales and might further excite fluctuations in the $B$ field.

Figure 12

Spectral Transfer function $T_{kpq}$ as a function of $k$ and $q$ at $p/k_0=0.0875$, following the analysis of Brandenburg (2001) [38]. Yellow pixels indicate positive values and blue pixels negative values. See also Fig. 11 for the time evolution of the magnetic energy spectrum.

Figure 13

Evolution of total magnetic energy $E_{\mathrm{mag}}$ and total kinetic energy $E_{\mathrm{kin}}$. Starting from nonhelical initial condition with $k_{\mathrm{peak}}=80,N={1536}^3$. The initial setup had zero velocity field; after a short relaxation time the kinetic energy decays as the magnetic energy.

Figure 14

Time evolution of the peak of the spectrum shown in Fig. 11. The solid black line shows the location in $k$-space of the spectrum's peak and the dashed line gives its energy value (right label). One can see that after an initial settling phase the peak shifts to lower $k$ values. Its peak energy follows the same law as the mean magnetic field energy shown in Fig. 13.

Figure 15

Magnetic power spectra of a simulation with energy injected at a single wave number $k=300$. An $E\sim k^4$ spectrum develops very quickly. After this redistribution of energy the spectrum evolves as in the nonhelical case with a continuous inverse transfer of energy to large scales.

Figure 16

Magnetic power spectra of runs with different spectral indices $n$. The inverse transfer of energy can only be observed if there is a steep initial spectrum with $n\ge 3$. If an initial spectrum steeper than $k^4$ is given it will flatten to a causal spectrum, similar to the $\delta$-peak run (see Fig. 15).

Figure 17

Zeus-MP2 run with Pencil Code initial conditions. With the Zeus Code there is a little increase of power at low $k$, indicating it does not have high enough Reynolds numbers.

Figure 18

Temporal evolution of the square of the magnetic vector potential.

Figure 19

Temporal evolution of the total helicity for the large nonhelical run. The magnitude of the helicity is of the order $1\times {10}^{-5}$ compared to ${\mathcal{H}}_{\max }=0.32$ in the maximally helical case.

Figure 20

Spectrum of helicity fluctuations ${\mathcal{H}}_k$.

Figure 21

Spectrum of absolute value of helicity fluctuations ${\mathcal{H}}_k$. A similar behavior as in the energy spectrum can be seen. For the relative fluctuations, see Fig. 22.

Figure 22

Relative error of helicity fluctuations using the data from Figs. 11 and 21.