Effects of axial boundary conductivity on a free Stewartson-Shercliff layer

The effects of axial boundary conductivity on the formation and stability of a magnetized free Stewartson-Shercliff layer (SSL) in a short Taylor-Couette device are reported. As the axial field increases with insulating endcaps, hydrodynamic Kelvin-Helmholtz-type instabilities set in at the SSLs of the conducting fluid, resulting in a much reduced flow shear. With conducting endcaps, SSLs respond to an axial field weaker by the square root of the conductivity ratio of endcaps to fluid. Flow shear continuously builds up as the axial field increases despite the local violation of the Rayleigh criterion, leading to a large number of hydrodynamically unstable modes. Numerical simulations of both the mean flow and the instabilities are in agreement with the experimental results.

Figure 1

Schematic of the redesigned Princeton MRI experiment with three independently rotating components; the inner cylinder with rim (${\mathrm{\Omega }}_1$), outer cylinder with ring (${\mathrm{\Omega }}_2$), and inner ring (${\mathrm{\Omega }}_3$).

Figure 2

Measurements of the azimuthal velocity at $r=$ 15.0 cm for experiments with insulating (a) and conducting (b) endcaps. Measurements are at the midplane for a configuration with ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=335\mathrm{rpm}/100$ rpm and $B_z=$ 4200 G. In (c) and (d) are spectrograms of the velocity fluctuations for the data shown in (a) and (b), respectively. The field is on during the interval bounded by dashed vertical lines.

Figure 3

The time averaged mean angular velocity (a), specific angular momentum (b), and flow shear $q$ profiles (c) before (solid) and during (dotted) the application of a magnetic field over a range of field strengths with insulating endcaps. Shown in (d) is a comparison with simulation for $q$ profiles with and without applied field of 4200 G. All data are averaged over 10 s. The red dashed line corresponds to the boundary rotation of the endcaps at the indicated radius. The gray bands indicate the variance of the time average with no magnetic field and are indicative of the measurement error.

Figure 4

Same as in Fig. 3 but for conducting endcaps.

Figure 5

The average $q$ value (top) and the log of the change in total fluctuating power (bottom) at $r=16.0$ cm vs. magnetic field comparing experiments with insulating and conducting endcaps. Vertical lines indicate ${\mathrm{\Lambda }}_{\mathrm{Ga}}=1$ (solid) and ${\mathrm{\Lambda }}_{\mathrm{Cu}}=1$ (dashed).

Figure 6

Contour plot of the normalized radial current for the full device simulations with insulating (left) and conducting (right) endcaps shortly after a magnetic field is applied. The boundary currents remain consistent, but thin current layers in the insulating case are replaced with thick currents in the endcaps, resulting in larger return currents in the fluid. Solid (dashed) contours indicating inward (outward) normalized currents of +/$-$ 0.1 and +/$-$ 0.01. The dotted contour follows $J_r=0$.