Shell-to-shell energy transfer in magnetohydrodynamics. II. Kinematic dynamo

We study the transfer of energy between different scales for forced three-dimensional magnetohydrodynamics turbulent flows in the kinematic dynamo regime. Two different forces are examined: a nonhelical Taylor-Green flow with magnetic Prandtl number and a helical ABC flow with . This analysis allows us to examine which scales of the velocity flow are responsible for dynamo action and identify which scales of the magnetic field receive energy directly from the velocity field and which scales receive magnetic energy through the cascade of the magnetic field from large to small scales. Our results show that the turbulent velocity fluctuations in the inertial range are responsible for the magnetic field amplification at small scales (small-scale dynamo) while the large-scale field is amplified mostly due to the large-scale flow. A direct cascade of the magnetic field energy from large to small scales is also presented and is a complementary mechanism for the increase of the magnetic field at small scales. The input of energy from the inertial range velocity field into the small magnetic scales dominates over the energy cascade up to the wave number where the magnetic energy spectrum peaks. At even smaller scales, most of the magnetic energy input is from the cascading process.

Figure 1

Spectra of kinetic energy (thick solid line) and magnetic energy (thin solid line) scaled up by a factor ${10}^6$ for the Taylor-Green runs during the kinematic dynamo regime. The dashed line indicates the Kolmogorov slope as a reference. Note that during this stage, all the magnetic modes grow with approximately the same rate.

Figure 2

Spectra of kinetic energy (thick solid line) and magnetic energy (thin solid line) scaled up by a factor ${10}^2$ for the ABC runs. The dashed line indicates the Kolmogorov slope as a reference.

Figure 3

Transfer functions (a) ${\mathcal{T}}_{ub}$, and (b) ${\mathcal{T}}_{bb}$ [see Eqs. (4, 5)] as a function of $Q$ and $K$ in the TG run during the kinematic regime. ${\mathcal{T}}_{ub}$ is positive for all values of $Q$ and $K$ shown. Each point $(Q,K)$ in panel (a) represents the rate of transfer of energy from velocity mode $Q$ to magnetic mode $K$. Each point $(Q,K)$ in panel (b) represents the rate of transfer of energy from magnetic mode $Q$ to magnetic mode $K$. The dashed line indicates the diagonal $K=Q$.

Figure 4

Transfer function ${\mathcal{T}}_{ub}(Q,K)$ (from the kinetic energy at $Q$ to the magnetic energy at $K$) for fixed values of $Q$ during the kinematic regime of the TG run.

Figure 5

Transfer function ${\mathcal{T}}_{ub}(Q,K)$ for fixed values of $Q$ during the kinematic regime of the ABC run.

Figure 6

Transfer function ${\mathcal{T}}_{ub}(Q,K)$ for fixed values of $K$ during the kinematic regime of the TG run.

Figure 7

Transfer function ${\mathcal{T}}_{ub}(Q,K)$ for fixed values of $K$ during the kinematic regime of the ABC run.

Figure 8

Transfer function ${\mathcal{T}}_{bb}(Q,K)$ (from magnetic energy at shell $Q$ to magnetic energy at shell $K$) at fixed values of $Q$ during the kinematic regime of the TG and ABC runs.

Figure 9

Ratio ${\mathcal{R}}_{\mathit{LS}}$ of energy received by the magnetic field at wave number $K$ from the forced wave numbers against all wave numbers and ratio ${\mathcal{R}}_C$ of energy received by the magnetic field at wave number $K$ from the magnetic field at larger scales through a cascade process against energy received from the velocity field. The solid lines correspond to the TG run while the dashed lines correspond to the ABC run, both in the kinematic regime.

Figure 10

Energy received by the magnetic field at scales larger than the forcing band ($K=1$ and 2) from the velocity field at wave numbers $Q$.

Figure 11

Terms determining the growth rate of magnetic energy [see Eq. (6)] as a function of wave number $K$ for the TG run during the kinematic regime. The dashed line is the energy received by the magnetic field from the velocity field, and the dash-dotted line is the cascade of magnetic energy. The solid line is the total energy received by the magnetic field, while the dotted line is the Ohmic dissipation. The difference between the last two curves gives the growth rate.

Figure 12

The budget of magnetic energy giving rise to the growth rate for the ABC run during the kinematic regime. Labels are as in Fig. 11.

Figure 13

Transfer function from the kinetic energy at $Q$ to the magnetic energy at $K=20$ for three different times as the magnetic field approaches saturation in the TG run.

Figure 14

Transfer function from the kinetic energy at $Q=16$ to the magnetic energy at $K$ for three different times as the magnetic field approaches saturation in the TG run.

Figure 15

Transfer of magnetic energy at shell $Q$ to shell $K$ (for $K=10$ and $K=20$) for three different times as the magnetic field approaches saturation in the TG run.