Figure 1

Computational domain for simulations of PROMISE. Region I: Inner copper cylinder, angular velocity ${\Omega }_1$. II: Outer copper cylinder and bottom end cap, ${\Omega }_2$. III: Liquid gallium IV: Vacuum. Thick dashed line: insulating upper end cap, $\Omega =0$. Dimensions: $r_1=4.0\mathrm{cm}$; $r_2=8.0\mathrm{cm}$; $h=40.0\mathrm{cm}$; $d_{w\mathrm{I}}=1.0\mathrm{cm}$; $d_{w\mathrm{II}}=1.5\mathrm{cm}$; ${\Omega }_1∕2\pi =3.6\mathrm{rpm}$; ${\Omega }_2∕2\pi =0.972\mathrm{rpm}$. The exact configuration of the toroidal coils being unavailable to us, six coils (black rectangles) with dimensions as shown were used, with 67 turns in the two coils nearest the midplane and 72 in the rest. The currents $I_{\phi }$ were adjusted to reproduce the reported [5] Hartmann numbers $\mathrm{Ha}\equiv B_z^0{r}_1∕\sqrt{\rho {\mu }_0\eta \nu }$.Reuse & Permissions

Figure 2

(Color) Axial velocities $\left(\mathrm{mm}{\mathrm{s}}^{-1}\right)$ versus time and depth sampled $1.5\mathrm{cm}$ from the outer cylinder, for the parameters of the PROMISE [5] with toroidal currents $I_{\phi }$ as marked. Note that height increases upward from the bottom end cap. No-slip velocity boundary conditions are imposed at the rigidly rotating end caps, but the steady part of the resulting Ekman circulation is suppressed in these plots by subtracting the time average at each height. The waves appear to be absorbed near the Ekman jet, at $z\approx 100\mathrm{mm}$.Reuse & Permissions

Figure 3

(Color) (a) Extended version of the case $I_{\phi }=75\mathrm{A}$ shown in Fig. 2 but without subtraction of the time average. The two Ekman cells are visible as the upflow (orange) at $z>100\mathrm{mm}$ and downflow (blue) at $z<100\mathrm{mm}$; these are the expected directions of flow near the outer cylinder. (b) The same case again, except that, after $t=360\mathrm{s}$, the no-slip boundary condition at both endcaps is changed to an ideal Couette profile, i.e., $\Omega \left(r\right)=a+b{r}^{-2}$ with $a$ and $b$ chosen to make $\Omega$ continuous at both cylinders; this eliminates Ekman circulation. Thereafter, the wave seems to be absorbed near the bottom $(z\approx 0\mathrm{mm})$ rather than the jet $(z\approx 100\mathrm{mm})$, which itself dies out after $t\approx 395\mathrm{s}$.Reuse & Permissions