Induction, helicity, and alpha effect in a toroidal screw flow of liquid gallium

We investigate experimentally induction mechanisms in a screw flow of gallium in a toroidal channel. The flow is nonstationary and operated in a spin-down regime: the channel (and fluid) are initially set into solid body rotation; as the channel is stopped the fluid is set into strong helical motion by diverters located inside the channel. In this study, we put a particular emphasis on the induction generated by these helical motions, which are expected to develop over the entire range of turbulent scales. We apply an external magnetic field either perpendicular to the channel axis parallel to it. At large scales the nonlinear induction mechanisms are associated with the Parker stretch and twist effect and with the expulsion due to overall rotation. Induction mechanisms can also originate in the small scale helicity as in the alpha induction effect of mean-field magnetohydrodynamics. Our measurements yield an upper bound for the alpha coefficient, significantly lower than estimates based on dimensional analysis. We discuss the consequences of our observations for the engineering of homogeneous dynamos in the laboratory.

Figure 1

(Color online) Principle of the Perm screw flow. (a) skematics of the experiments and components (1; toroidal channel, 2; diverter fixed in channel with left-hand or right-hand rotation, 3; magnetic sensor, 4; steel cover, 5; electromotor, 7; brake); (b) large scale flow vizualized by the trajectory of polystyrene particles ($1–2\mathrm{mm}$ in diameter); (c) photograph of the screw flow in the torus seeded with air bubbles (illustration of the modes $m=3–5$); (d) small-scale structure of the screw flow in the torus, visualized by kalliroscope particles. The ratio $R∕r_0$ is 3.89 for the gallium channel and 2.75 for the water channel, the corresponding Reynolds numbers are $1.3\times {10}^6$ and $1.4\times {10}^6$.

Figure 2

(Color online) Magnetic induction measurements. (a) Case 1: the external magnetic field is applied transverse to the torus, ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, with $B_0=45\mathrm{G}$. The Hall probes are either stationary, enclosed in the steel casing (static probe), or moving with the torus at location $mr$ and $mz$, $5\mathrm{mm}$ away from the channel. At location $mr$ the probe is set to measure the radial field (along ${\mathbf{e}}_r$), while at location $mz$ it measures the axial field (along ${\mathbf{e}}_z$). (b) Case 2: the external field is toroidal: the Fluxgate probes are set at locations ($r=8.7\mathrm{cm}$, $\theta =0$, $z=6.5\mathrm{cm}$) and ($r=8.7\mathrm{cm}$, $\theta =90°$, $z=6.5\mathrm{cm}$) where the applied field is respectively ($B_{0,r}=-1.2\mathrm{G}$, $B_{0,\theta }=27\mathrm{G}$, $B_{0,z}=3.9\mathrm{G}$) and ($B_{0,r}=3.6\mathrm{G}$, $B_{0,\theta }=33.5\mathrm{G}$, $B_{0,z}=-0.8\mathrm{G}$).

Figure 3

(Color online) Transverse applied field, ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$. Local measurement with a Hall probe located at position (2) (cf. Fig. 2). (left): channel equipped with left diverters, rotated in either direction [(a) and (c)]; (right): similar measurement for the channel equipped with right-handed diverters [(b) and (d)]. The experimental data have been low-pass filtered with a corner frequency equal to $40\mathrm{Hz}$; many measurements are averaged.

Figure 4

(Color online) Transverse applied field, ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$. Schematics of the induction created by advection of the applied field lines: (a) action of the poloidal velocity component; (b) action of the toroidal velocity component. The drawings corresponds to the situation with right diverters and a positive value of the initial rotation of the channel.

Figure 5

(Color online) Nonlinear expulsion effect, transverse applied field. (a) and (b) sketch of the mechanism of expulsion of the applied field lines; (c) measurements of the maximum of the radial induced field $B_r$ at location $mr$ (diamonds) and of the maximum of the axial induced field $B_z$ at location $mz$. The magnetic Reynolds number has been corrected to include the slight variation of the braking time of the channel as its initial speed increases. The applied field is $B_{0,z}=45\mathrm{G}$.

Figure 6

(Color online) Applied toroidal field, ${\mathbf{B}}_0=35\mathrm{G}$. Measurement using an induction coil, as shown in Fig. 2b: the data corresponds to the average field ${\overline{B}}_z\left(t\right)$ across the coil. (left): raw data for positive and negative rotation rates; (right): corresponding odd and even parts.

Figure 7

(Color online) Applied toroidal field, ${\mathbf{B}}_0=35\mathrm{G}$. Odd (a) and even (b) parts of the measurement with the induction coil. The data corresponds to Figs. 6d, 6f.

Figure 8

Contribution from inhomgeneities. (a) Local cylindrical coordinates; (b) orientations of the vectors $\mathit{g}$, $\mathit{W}$, $\mathbf{\Omega }$ which enter in the mean electromotive force [Eqs. (8)].