Low magnetic Prandtl number dynamos with helical forcing

We present direct numerical simulations of dynamo action in a forced Roberts flow. The behavior of the dynamo is followed as the mechanical Reynolds number is increased, starting from the laminar case until a turbulent regime is reached. The critical magnetic Reynolds for dynamo action is found, and in the turbulent flow it is observed to be nearly independent on the magnetic Prandtl number in the range from $\sim 0.3$ to $\sim 0.1$. Also the dependence of this threshold with the amount of mechanical helicity in the flow is studied. For the different regimes found, the configuration of the magnetic and velocity fields in the saturated steady state are discussed.

Figure 1

(Color online) Critical magnetic Reynolds $R_M^c$ as a function of $R_V$ for different Roberts flows (thick lines), $f=g$ (◇), $f=g∕\sqrt{2}$ (◻), $f=g∕0.77$ (▵). The dark (red) area corresponds to the region where the Roberts flow is hydrodynamically stable. For a comparison with Ref. 8, our Reynolds numbers should be divided by $2\pi$. The light (orange) area corresponds to the region of hydrodynamic oscillations, while the white area corresponds to the turbulent regime. The thin lines connected with crosses are shown as a reference and correspond to the threshold for dynamo instability in Taylor-Green flow [1], DNS (solid line) and $\alpha$ model (dashed line).

Figure 2

Critical magnetic Reynolds $R_M^c$ as a function of $P_M^{-1}$ for the Roberts flow with $f=g$ (thick lines). The solid line corresponds to the laminar regime $(R_V\lesssim 100)$, the dashed line to the periodic flow $(100\lesssim R_V\lesssim 1000)$, and the dotted line to the turbulent regime $(R_V\gtrsim 1000)$. The double-valuedness results from the effects of two different values of $R_V$.

Figure 3

Kinetic energy spectra as a function of $R_V$. The Kolmogorov’s spectrum is shown as a reference.

Figure 4

Magnetic energy spectra during the kinematic regime, for different values of $R_V$. The values of $R_M$ for each curve correspond to the smallest value for which dynamo action was observed (see Fig. 5).

Figure 5

Growth rates as a function of $R_M$. Each line corresponds to several simulations at constant $R_V$ (fixed $\nu$), and each point in the line indicates the exponential growth (or decay) rate at a fixed value of $R_M$. The point where each curve crosses $\sigma =0$ gives the threshold $R_M^c$ for dynamo instability. $R_V=63$ (◻), $R_V=130$ (×), $R_V=420$ (▵), $R_V=970$ (◇), $R_V=1100$ (∗), $R_V=1300$ (+), $R_V=1900$ (엯), and $R_V=3300$ (▿).

Figure 6

(Color online) Time history of the total kinetic [thick (blue) lines] and magnetic energy (thin lines) in dynamo simulations. The dashed lines correspond to $R_V=63$ and $R_M=79$ (laminar flow), while the solid lines are for $R_V=R_M=420$. The shaded region indicates the period of time when the flow is oscillating in this simulation. The inset shows the time history for a turbulent run with $R_V=3300$ and $R_M=1100$.

Figure 7

Time history of the total energy in the dynamo simulation with $R_V=R_M=420$. The shaded area is a blow up of the shaded region in Fig. 6 and corresponds to the hydrodynamic oscillations.

Figure 8

Evolution of the magnetic energy in different shells in Fourier space: (a) $R_V=63$ and $R_M=78$ (laminar flow), (b) $R_V=R_M=420$ (periodic case), (c) $R_V=3300$ and $R_M=1100$ (turbulent regime). The dotted line corresponds to $k=1$, solid line to $k=2$, and the dashed lines to $k=9,10,11,12$.

Figure 9

(Color online) Plots of the kinetic and magnetic fields for the saturated regime of the run with $R_V=63$ and $R_M=78$, (a) cut at $z=0$, $vz$ in color and $vx$, $vy$ indicated by arrows, (b) same as in (a) for the magnetic field, (c) cut at $y=0$, $vy$ in color and $vx$, $vz$ indicated by arrows, (d) same as in (c) for the magnetic field, (e) same as in (b) but for a cut at $y=\pi ∕4$, and (f) same as in (e) for the magnetic field.

Figure 10

(Color online) Plots of the kinetic and magnetic fields for the saturated regime of the run with $R_V=R_M=420$. Labels and fields are as in Fig. 9.

Figure 11

(Color online) Plots of the kinetic and magnetic fields for the saturated regime of the run with $R_V=3300$ and $R_M=1100$. Labels and fields are as in Fig. 9.

Figure 12

(Color online) Visualization of the kinetic (left) and magnetic energy density (right) for the saturated regime of the run with $R_V=63$ and $R_M=78$. Velocity field lines are indicated in black.

Figure 13

(Color online) Visualization of the kinetic (left) and magnetic energy density (right) for the saturated regime of the run with $R_V=3300$ and $R_M=1100$. Velocity field lines are indicated in black.

Figure 14

(Color online) Kinetic [thick (blue) lines] and magnetic energy spectra (thin lines) for different times for the simulation with $R_V=3300$ and $R_M=1100$.

Figure 15

Normalized transfer functions (a) $-T_L\left(k\right)$, and (b) $T_M\left(k\right)$ for different runs in the kinematic regime. Labels of the curves are as in Figs. 3, 4.