High Magnetic Shear Gain in a Liquid Sodium Stable Couette Flow Experiment: A Prelude to an $\alpha -\Omega$ Dynamo

The $\Omega$ phase of the liquid sodium $\alpha \mathrm{\text{−}}\Omega$ dynamo experiment at New Mexico Institute of Mining and Technology in cooperation with Los Alamos National Laboratory has demonstrated a high toroidal field $B_{\varphi }$ that is $\simeq 8\times B_r$, where $B_r$ is the radial component of an applied poloidal magnetic field. This enhanced toroidal field is produced by the rotational shear in stable Couette flow within liquid sodium at a magnetic Reynolds number $\mathrm{Rm}\simeq 120$. Small turbulence in stable Taylor-Couette flow is caused by Ekman flow at the end walls, which causes an estimated turbulence energy fraction of $(\delta v/v{)}^2\sim {10}^{-3}$.

Figure 1

A schematic drawing of the fluid flow of the experiment shows the inner cylinder of $R_1=15.2\mathrm{cm}$ rotating at ${\Omega }_1/2\pi =68\mathrm{Hz}$ relative to the outer or confining cylinder of $R_2=30.5\mathrm{cm}$ and length $L=30\mathrm{cm}$ and ${\Omega }_2={\Omega }_1/4$ (${\Omega }_2/2\pi =17\mathrm{Hz}$). Liquid sodium in the annular space undergoes stable Couette rotational flow. The fluid Reynolds number is high: $\mathrm{Re}=[({\Omega }_1+{\Omega }_2)/2][(R_2+R_1)/2{]}^2/\nu =1.35\times {10}^7$, $\nu ={10}^{-2}{\mathrm{cm}}^2/\mathrm{s}$. The magnetic Reynolds number becomes $\mathrm{Rm}=\mathrm{Re}(\nu /{\eta }_{\mathrm{Na}})=180$ with ${\eta }_{\mathrm{Na}}\sim 750{\mathrm{cm}}^2/\mathrm{s}$.

Figure 2

(Left) Field-free fluid measurements have an absolute error of the order of 10%. The lower crosses are the measured dc generator torques; the dashed line is the measured bearings torque, and the sum of the two is the top solid curve, the torque due to the Ekman flow. (Right) Shown are the measured and theoretical pressure distributions. The pressure measurement starts at $R_1$ and extends to $R_2$. The finite pressure at $R_1$, $P_1\simeq 100\mathrm{psi}$, is due to the centrifugal pressure of rotating fluid in the narrow gap between the inner cylinder rotating at ${\Omega }_1$ and the end wall rotating at ${\Omega }_2$ (dot-dashed curve). This thin end gap is filled with rotating fluid, which extends from the shaft seal (where $P=1\mathrm{atm}$) at radius $R_3\simeq R_1/3$ to $R_1$, at which radius the Couette flow in the annular volume is initiated. From there, the two curves describe the pressure in the annular volume either extrapolated as $\Omega \propto R^{-2}$ or extrapolated as $R^{-3/2}$. The measurements fall along the $R^{-3/2}$ pressure distribution curve indicating less shear than the ideal Couette flow. The difference is presumed to be due to the torque produced by the Ekman flow.

Figure 3

(Left) The lower curve, left, shows the applied radial magnetic field $B_r\simeq 12\mathrm{G}$, and the upper curve the resulting toroidal field for four overlaid experiments at ${\Omega }_1/2\pi =68\mathrm{Hz}$. The backreaction of this toroidal stress is small: $B_R{B}_{\varphi }/8\pi \simeq 1.7\times {10}^{-2}{\tau }_{\mathrm{Ek}}$, where the stress per unit area in the fluid due to the Ekman flow ${\tau }_{\mathrm{Ek}}$ becomes (measured torque: $2\times {10}^8\mathrm{dyne}\mathrm{cm})/(\mathrm{\text{area}}\times \mathrm{\text{torque}}$ arm). ${\tau }_{\mathrm{Ek}}\simeq 2.6\times {10}^3\mathrm{dyne}/{\mathrm{cm}}^2$. The upper curves show the measured toroidal magnetic field for four overlaid experiments at ${\Omega }_1/2\pi =68\mathrm{Hz}$. The right panel shows the ratio $B_{\varphi }/B_r$ with a maximum of $\times 8$ with an error of $\sim 10\%$ at $\sim (R_2-R_1)/3\simeq 5\mathrm{cm}$ from the inner cylinder wall.