Magnetized Ekman layer and Stewartson layer in a magnetized Taylor-Couette flow

In this paper we present axisymmetric nonlinear simulations of magnetized Ekman and Stewartson layers in a magnetized Taylor-Couette flow with a centrifugally stable angular-momentum profile and with a magnetic Reynolds number below the threshold of magnetorotational instability. The magnetic field is found to inhibit the Ekman suction. The width of the Ekman layer is reduced with increased magnetic field normal to the end plate. A uniformly rotating region forms near the outer cylinder. A strong magnetic field leads to a steady Stewartson layer emanating from the junction between differentially rotating rings at the endcaps. The Stewartson layer becomes thinner with larger Reynolds number and penetrates deeper into the bulk flow with stronger magnetic field and larger Reynolds number. However, at Reynolds number larger than a critical value $\sim 600$, axisymmetric, and perhaps also nonaxisymmetric, instabilities occur and result in a less prominent Stewartson layer that extends less far from the boundary.

Figure 1

Computational domain for studies of magnetic Ekman layer. Region (I) Perfect conducting inner cylinder, angular velocity ${\Omega }_1$, infinitely long. (II) Liquid metal. (III) Perfectly insulating inner ring ${\Omega }_3$ extending to infinity. (IV) Perfectly insulating outer ring ${\Omega }_4$ extending to infinity. (V) Perfectly conducting outer cylinder ${\Omega }_2$ infinitely long. Thin dash line: the midplane. $B_z$ is the initial background vertical uniform magnetic field.

Figure 2

MRI linear growth rates $\left(\gamma {\text{s}}^{-1}\right)$ as a function of magnetic Reynolds number ${\text{Re}}_m$ and Lundquist number $S$ with dimensions as in Table : $r_1=7.1\text{cm}$, $r_2=20.3\text{cm}$, and $h=27.9\text{cm}$ and $\mu ={\Omega }_2/{\Omega }_1=0.13325$. The system is MRI stable if ${\text{Re}}_m\lesssim 10$ regardless of $S$.

Figure 3

The thickness of the Ekman layer $\delta$ versus Elsasser number $\Lambda$ for $\text{Re}=1600$, ${\text{Re}}_m=5$. ${\Omega }_1/2\pi =1000\text{rpm}$, ${\Omega }_2/2\pi =1000\text{rpm}$, ${\Omega }_3/2\pi =1010\text{rpm}$, ${\Omega }_4/2\pi =1010\text{rpm}$. $r_1=15\text{cm}$, $r_2=35\text{cm}$, and $h=20\text{cm}$. The data is measured at $r=\left(r_1+r_2\right)/2=20\text{cm}$. The dashed line is the theoretical result. The solid line is the one obtained from modified ZEUS-2D simulations. We have chosen larger $r_1$ and $r_2$ than the ones of the Princeton MRI experiment to minimize the curvilinear effects and larger $h⪢10{\delta }_E$ to minimize the influence of the top endcap.

Figure 4

The thickness of the Ekman layer $\delta$ versus Elssaser number $\Lambda$ for $\text{Re}=6400$, ${\text{Re}}_m=2.5$. Parameters as in Table . The data are measured at $r=\left(r_1+r_2\right)/2=13.7\text{cm}$. The dashed line is from the linear analysis [Eq. (11)]. The solid line is obtained from modified ZEUS-2D simulations.

Figure 5

Contour plots of final-state velocities and fields with uniformly rotating endcaps. $\text{Re}=6400$, ${\text{Re}}_m=2.5$ with $B_{z0}=1500\text{G}$ $\left(\Lambda =1.09\right)$. Parameters as in Table . (a) Poloidal flux function $\Psi \left(\text{G}{\text{cm}}^2\right)$ (b) Poloidal stream function $\Phi \left({\text{cm}}^2{\text{s}}^{-1}\right)$ (c) toroidal field $B_{\phi }\left(\text{G}\right)$ (d) angular velocity $\Omega \equiv r^{-1}{V}_{\phi }\left(\text{rad}{\text{s}}^{-1}\right)$.

Figure 6

$\Delta R/\left(r_2-r_1\right)$ vs $\Lambda$. $\Delta R$ is the radial width of the dynamically active region. ${\text{Re}}_m=2.5$, $\text{Re}=6400$ with end plates corotating with outer cylinder. Parameters as in Table . Note that in the simulations the magnetic diffusivity $\eta$ is fixed to $\eta \sim 2000{\text{cm}}^2{\text{s}}^{-1}$, however, the imposed axial magnetic field $B_{z0}$ is varied.

Figure 7

Similar to Fig. 5, but with two differential rotating rings and $\Lambda =1.5$. Parameters as in Table . The Stewartson layer is located between the rings at $r_d=\left(r_1+r_2\right)/2=13.7\text{cm}$ and breaks the two big Ekman cells into eight cells.

Figure 8

Two independently rotating rings generate eight cells. Solid line, ideal Couette state; dashed line, rotation profile at the endcaps. Arrows indicate the radial flow directions near the endcaps.

Figure 9

Azimuthal velocity $v_{\phi }\text{cm}{\text{s}}^{-1}$ versus radius $r$ at different heights with ${\text{Re}}_m=2.5$, $\text{Re}=6400$, and the endcaps divided into two rings at $r_d=\left(r_1+r_2\right)/2=13.7\text{cm}$. Parameters as in Table . Solid line, ideal Couette state; long dashes, $z=1.33\text{cm}$ (relative to the bottom endcap); dash dot, $z=2.79\text{cm}$; short dashes, $z=13.95\text{cm}$. (a) $\Lambda =0.38$; (b) $\Lambda =1.5$.

Figure 10

(a) $\left|\partial \Omega /\partial r\right|$ of the Stewartson layer at $z=1.33\text{cm}$ vs Reynolds number $\text{Re}$. (b) $\left|\left(\Omega -{\Omega }_0\right)/{\Omega }_0\right|$ with $r_d=\left(r_1+r_2\right)/2=13.7\text{cm}$ at the midplane vs $\text{Re}$. ${\text{Re}}_m=2.5$, $\Lambda =1.5$, and the endcaps divided into two rings at $r_d=\left(r_1+r_2\right)/2=13.7\text{cm}$. Parameters as in Table . Note that in the simulations the rotation speed profile is fixed to ${\Omega }_1/2\pi =500\text{rpm}$, ${\Omega }_2/2\pi =66.625\text{rpm}$, ${\Omega }_3/2\pi =227.5\text{rpm}$, ${\Omega }_4/2\pi =81.25\text{rpm}$, however, the kinematic viscosity $\nu$ is changeable.

Figure 11

Growth rates $\left(\gamma {\text{s}}^{-1}\right)$ predicted by Eq. (13) as a function of Elsasser number $\Lambda$ and vorticity parameter $\zeta$ at the junction of rings, $r_d=\left(r_1+r_2\right)/2$ with $\text{Re}=6400$. Parameters as in Table . (a) the vertical mode number $n=1$; (b) $n=2$.

Figure 12

MRI linear growth rates $\left(\gamma {\text{s}}^{-1}\right)$ as a function of Reynolds number $\text{Re}$ and vorticity parameter $\zeta$ at the junction between rings, $r_d=\left(r_1+r_2\right)/2$ with $\Lambda =1.5$. Parameters as in Table . (a) the vertical mode number $n=0.25$; (b) $n=1$.