Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence

We investigate the transfer of energy from large scales to small scales in fully developed forced three-dimensional magnetohydrodynamics (MHD) turbulence by analyzing the results of direct numerical simulations in the absence of an externally imposed uniform magnetic field. Our results show that the transfer of kinetic energy from large scales to kinetic energy at smaller scales and the transfer of magnetic energy from large scales to magnetic energy at smaller scales are local, as is also found in the case of neutral fluids and in a way that is compatible with the Kolmogorov theory of turbulence. However, the transfer of energy from the velocity field to the magnetic field is a highly nonlocal process in Fourier space. Energy from the velocity field at large scales can be transferred directly into small-scale magnetic fields without the participation of intermediate scales. Some implications of our results to MHD turbulence modeling are also discussed.

Figure 1

Spectra of kinetic energy (solid line) and magnetic energy (dashed line) of the ABC and Taylor-Green runs, where the Taylor-Green spectra have been shifted down by a factor of 20 for clarity. The Kolmogorov slope is showed as a reference. Note that the magnetic Prandtl number $P_M\equiv \nu ∕\eta$ differs for the two runs.

Figure 2

The transfer of energy ${\mathcal{T}}_{uu}(Q,K)$ for the Taylor-Green run. The figure shows the rate that energy is transferred from modes $Q=3,10,20,30$ to all other modes $K$.

Figure 3

The transfer of energy ${\mathcal{T}}_{uu}(Q,K)$ for the ABC run. The figure shows the rate that energy is transferred from modes $Q=3,10,20,30$ to all other modes $K$.

Figure 4

Top panel: the transfer of energy ${\mathcal{T}}_{uu}(Q,K)$ for the Taylor-Green run. The figure shows the rate that kinetic energy is transferred from modes $Q=3,10,20,30$ to kinetic energy to all other modes $K$. Bottom panel: the transfer of energy ${\mathcal{T}}_{bb}(Q,K)$ for the same flow. The figure shows the rate that magnetic energy is transferred from modes $Q=3,10,20,30$ to magnetic energy to all other modes $K$.

Figure 5

Same as Fig. 4 for the ABC run.

Figure 6

The transfer of kinetic energy to magnetic energy ${\mathcal{T}}_{ub}(Q,K)$ for the Taylor-Green run. The figure shows the rate that kinetic energy is transferred from modes $Q=3,4,5$ (inset modes $Q=15,17,20$) to magnetic energy in modes $K$.

Figure 7

The transfer of kinetic energy to magnetic energy ${\mathcal{T}}_{ub}(Q,K)$ for the ABC run. The figure shows the rate that kinetic energy is transferred from modes $Q=3,4,5$ (inset modes $Q=15,17,20$) to magnetic energy in modes $K$.

Figure 8

The transfer of kinetic energy to magnetic energy ${\mathcal{T}}_{ub}(Q,K,)$ for the Taylor-Green run. The figure shows the rate that kinetic energy is transferred from modes $Q$ ($x$ axis) to magnetic energy in modes $Q=10$ (top panel), $Q=20$ (middle panel), and $Q=30$ (bottom panel).

Figure 9

The transfer of kinetic energy to magnetic energy ${\mathcal{T}}_{ub}(Q,K,)$ for the ABC run. The figure shows the rate that kinetic energy is transferred from modes $Q$ ($x$ axis) to magnetic energy in modes $Q=10$ (top panel), $Q=20$ (middle panel), and $Q=30$ (bottom panel).

Figure 10

A comparison of the transfers ${\mathcal{T}}_{uu}(Q,K)$, ${\mathcal{T}}_{bb}(Q,K)$, ${\mathcal{T}}_{ub}(Q,K)$, and ${\mathcal{T}}_{bu}(Q,K)$ for $Q=15$ for the Taylor-Green flow.

Figure 11

A comparison of the transfers ${\mathcal{T}}_{uu}(Q,K)$, ${\mathcal{T}}_{bb}(Q,K)$, ${\mathcal{T}}_{ub}(Q,K)$, and ${\mathcal{T}}_{bu}(Q,K)$ for $Q=15$ for the ABC flow.

Figure 12

A comparison of the large $K$ tails of transfers ${\mathcal{T}}_{uu}(Q,K)$, ${\mathcal{T}}_{bb}(Q,K)$, and ${\mathcal{T}}_{ub}(Q,K)$ in a log-linear plot for $Q=10$ for the Taylor-Green flow flow.

Figure 13

The ratio $NL∕L$ of energy received through the nonlocal transfer ${\mathcal{T}}_{ub}$ to local ${\mathcal{T}}_{bb}$ for the ABC and TG simulations. The small scales receive 20% of their energy through the nonlocal transfer ${\mathcal{T}}_{ub}$.

Figure 14

A sketch of the energy transfer between different scales and different fields. The thickness of the lines is an indication of the magnitude of the transfers. The figure illustrates how energy is transferred to magnetic modes with wave number $k=q$ in the inertial range. The transfer between same fields is always local and direct. Each magnetic mode receives energy from all larger-scale velocity modes and gives to slightly smaller-scale velocity modes.

Figure 15

A comparison of the Elsässer energy transfer ${\mathcal{T}}_{z^{+}{z}^{+}}(Q,K)$ with the transfers from the local ${\mathcal{T}}_{uu}+{\mathcal{T}}_{bb}$ and the nonlocal ${\mathcal{T}}_{ub}+{\mathcal{T}}_{bu}$ contributions. The inset shows a blowup of the small-$K$ tail where the nonlocal interactions are more dominant.