2. Mathematical formulation

We study two types of stationary, magnetohydrodynamic Couette flow, that is, a flow between two parallel boundaries one of which is moving with a constant velocity: plane and spherical. The flow interacts with a strong (large Hartmann numbers) force-free magnetic field tangent to the boundaries at some isolated points. In the spherical case, the external field is a dipole field with a source at the center of the system and in the plane case, it is harmonic with oscillatory dependence in the direction perpendicular to the velocity of the moving boundary with an arbitrary period <sup>2</sup> k <sup>π</sup>. Figure 1 illustrates both the spherical and the plane cases.

We focus here on the phenomenon of super velocities in the regions of singularity of the Hartmann boundary layers which are present in this problem, that is, in the vicinity of points, where the magnetic field becomes tangent to the stationary boundary. In those regions, the fluid's velocity exceeds the velocity of the moving boundary. The aim of this chapter is to review the influence of conductivity of the upper/outer boundary on the enhancement of the super-velocity magnitude and explain why the super velocities are larger in the case when the stationary boundary is conducting when compared to the case where it is insulating. As mentioned in the introduction, this fact was proved numerically by several authors. We adopt here the analytic approach and notations of Dormy et al. [4] and Mizerski and Bajer [5]. Majority of the analysis will be done in the simpler and therefore more transparent planar geometry.

We consider here a stationary state in which the velocity of the fluid and the induced magnetic field have only one component, the same as the velocity of the moving boundary, axisymmetric for the spherical case and translationally invariant in the direction of the flow for the flat case. Small differential rotation/motion of the boundaries is assumed for the Couette flow dominated by the magnetic forces, that is, the magnetic Reynolds number is assumed small,

$$Re\_M = \frac{L\mu\_0}{\eta} \ll 1\tag{3}$$

where u<sup>0</sup> is the velocity of the moving boundary and the Hartmann number (2) is large. The above assumption of small ReM implies that the flow-induced component of the magnetic field is of the order ReM, and thus the magnetic field is decomposed in the following way

$$\mathbf{B} = \mathbf{B}\_{\mathbf{0}}(\mathbf{x}, z) + \mathrm{Re}\_{M} b(\mathbf{x}, z) \hat{\mathbf{e}}\_{y} \quad \text{for the planar case} \tag{4}$$

$$\mathbf{B} = \mathbf{B}\_0(s, z) + \mathrm{Re}\_M b(s, z)\widehat{\mathbf{e}}\_{\psi} \quad \text{for the spherical case} \tag{5}$$

where ðs;φ; zÞ are the cylindrical polar coordinates, B0 is the external potential field, and ReMb is the perturbation magnetic field generated by the flow. Indeed, in the numerical simulations of Hollerbach and Skinner [16] for infinitesimally small rotation rate, the flow was axisymmetric with only the azimuthal components of the velocity and the induced magnetic field present. The assumption of the small magnetic Reynolds number is crucial for the spherical case to neglect the nonlinear term which does not vanish because of the curvature effects. In the flat case, however, this assumption is not necessary to simplify the equations, because the nonlinear term vanishes due to the translational symmetry in the direction of the flow. Nevertheless, we keep the Reynolds numbers small even in the plane flow, since for high Rm, the unidirectional solutions are most probably unstable.

Furthermore, the solution for the plane flow is also valid when both boundaries are moving with different velocities since it is just a matter of changing the frame of reference to one moving at the same constant velocity as one of the boundaries. In the spherical case, however, when both boundaries rotate at different angular velocities, the Coriolis force substantially modifies the solution even in the case of small differential rotation unless the flow is strongly dominated by the magnetic force. The problem of MHD Couette flow with Coriolis force was investigated numerically by Hollerbach [17] and Dormy et al. [1] and analytically, for small Elsasser numbers, by N. Kleeorin et al. [18]. As remarked in the introduction, Brito et al. [8] demonstrated experimentally the detrimental effect of the Coriolis force on superrotation.

## 2.1. The equations and the main flow solution

As mentioned, we present the analysis for the flat case illustrated on the left panel of Figure 1. In Cartesian coordinates ðx; y; zÞ, the lower boundary is moving in the "y" direction and the "z" axis is perpendicular to both parallel boundaries. The dimensionless external magnetic field is given by

$$\mathbf{B}\_{\mathbf{0}}(\mathbf{x}, \mathbf{z}) = \nabla A \times \hat{\mathbf{e}}\_{y} = \nabla \Phi = \left[ e^{-\lambda z} \sin \lambda \mathbf{x}, 0, e^{-\lambda z} \cos \lambda \mathbf{x} \right]\_{\prime} \tag{6}$$

where A ¼ expð�λzÞsinλx=λ, Φ ¼ �expð�λzÞcosλx=λ, and 2π=λ is the arbitrary period of oscillation of the external field in the "x" direction.

The lower moving boundary is assumed to have the same conductivity as the fluid, while the conductivity of the upper one, which is at rest,

$$
\epsilon = \frac{\sigma\_u}{\sigma\_f} \tag{7}
$$

can vary from zero to infinity, where σ<sup>u</sup> and σ<sup>f</sup> are the electrical conductivities of the upper boundary and the fluid, respectively. The magnetic permeabilities of boundaries and the fluid are assumed to be same.

The general set of equations for the analyzed stationary state is obtained by taking the "y" components of the induction and the Navier-Stokes equations

$$\begin{aligned} \mathbf{B}\_0 \cdot \nabla u + \nabla^2 b &= 0 \\ \mathbf{B}\_0 \cdot \nabla b + \frac{1}{M^2} \nabla^2 u &= 0 \end{aligned} \quad \text{for} \quad 0 < z < 1 \tag{8}$$

with M ≫ 1 and no-slip boundary conditions for the velocity field u xð ; zÞ

$$\begin{aligned} \mu(\mathbf{x}, 1) &= \mathbf{0} \\ \mu(\mathbf{x}, 0) &= 1. \end{aligned} \tag{9}$$

Inside the rigid conductors, the magnetic field b has to satisfy Laplace equations:

$$\begin{aligned} \nabla^2 b\_1 &= 0 \quad \text{for} \quad 1 < z < 1 + \delta\_1\\ \nabla^2 b\_2 &= 0 \quad \text{for} \quad -\delta\_2 < z < 0 \end{aligned} \tag{10}$$

where bi and δ<sup>i</sup> are the perturbation magnetic fields inside the rigid conductors and their dimensionless thickness (with i ¼ 1 for the upper conductor and i ¼ 2 for the lower one). At the boundaries with the insulator at z ¼ 1 þ δ<sup>1</sup> and at z ¼ �δ2, the perturbation magnetic field must vanish since there is no imposed magnetic field in the "y" direction

$$\begin{aligned} b\_1(\mathbf{x}, 1 + \delta\_1) &= 0 \\ b\_2(\mathbf{x}, -\delta\_2) &= 0 \end{aligned} \tag{11}$$

Finally, the conditions at z ¼ 0 and at z ¼ 1 for the magnetic field can be written in the following form

$$\begin{cases} b(\mathbf{x},0) = b\_2(\mathbf{x},0) \\ \frac{\partial b}{\partial z}|\_{z=0} = \frac{\partial b\_2}{\partial z}|\_{z=0} \end{cases} \text{and} \begin{cases} b(\mathbf{x},1) = b\_1(\mathbf{x},1) \\ \epsilon \frac{\partial b}{\partial z}|\_{z=1} = \frac{\partial b\_1}{\partial z}|\_{z=1} \end{cases} \tag{12}$$

To understand the structure of the flow, it is very important to note the symmetries in the <sup>π</sup> system with respect to planes defined by <sup>x</sup> <sup>¼</sup> <sup>n</sup> <sup>2</sup><sup>λ</sup> for <sup>n</sup><sup>∈</sup> <sup>N</sup> (see Figure <sup>1</sup>). Since the problem has to be periodic in the "x" direction with the period of the applied field <sup>2</sup> λ <sup>π</sup>, it is enough to <sup>2</sup><sup>π</sup> analyze only the region where 0 < x < <sup>λ</sup> . The symmetry of the external field B0, the symmetric boundary conditions on u xð ; zÞ, and the boundary conditions on b xð ; zÞ listed above imply that

$$\begin{cases} \begin{aligned} u\left(\frac{\pi}{2\lambda} - a, z\right) &= u\left(\frac{\pi}{2\lambda} + a, z\right) \\ u\left(\frac{3\pi}{2\lambda} - a, z\right) &= u\left(\frac{3\pi}{2\lambda} + a, z\right) \end{aligned} \\\begin{aligned} u\left(\frac{3\pi}{2\lambda} - a, z\right) &= u\left(\frac{\pi}{\lambda} + a, z\right) \\\ u\left(\frac{\pi}{\lambda} - a, z\right) &= u\left(\frac{\pi}{\lambda} + a, z\right) \end{aligned} \\\end{cases} \\\begin{aligned} b\left(\frac{3\pi}{\lambda} - a, z\right) &= b\left(\frac{\pi}{\lambda} + a, z\right) = b\left(\frac{3\pi}{2\lambda} + a, z\right) \end{aligned} \tag{13}$$

for any α∈ R. One can see now the precise analogy between the flat and spherical cases. In the spherical problem, the meridional angle "ϑ" corresponds to the "x" coordinate in the flat case, with the equatorial plane corresponding to the planes <sup>x</sup> <sup>¼</sup> <sup>π</sup> and <sup>x</sup> <sup>¼</sup> <sup>3</sup> 2 π <sup>λ</sup>. The azimuthal angle <sup>2</sup><sup>λ</sup> "φ" and the radial coordinate "r" are of course analogous to "y" and "z," respectively.

It is also clear from (13) that the "z" component of the currents <sup>j</sup> <sup>¼</sup> <sup>∂</sup><sup>b</sup> has to be symmetric with <sup>z</sup> <sup>∂</sup><sup>x</sup> <sup>3</sup><sup>π</sup> respect to the planes <sup>x</sup> <sup>¼</sup> <sup>π</sup> while the "x" component <sup>j</sup> ¼ � <sup>∂</sup><sup>b</sup> remains antisymmetric. This <sup>2</sup><sup>λ</sup> , <sup>2</sup><sup>λ</sup> <sup>x</sup> <sup>∂</sup><sup>z</sup> means <sup>3</sup><sup>π</sup> that <sup>j</sup> must have an external value and <sup>j</sup> must vanish at <sup>x</sup> <sup>¼</sup> <sup>π</sup> <sup>z</sup> <sup>x</sup> <sup>2</sup><sup>λ</sup> , <sup>2</sup>λ.

� � � � The main flow is defined as the flow outside all boundary and internal layers in the problem. When the upper boundary is insulating or only weakly conducting, the problem is greatly simplified since the magnetic coupling of the fluid with the lower conductor, in the limit of the large Hartmann number, is much stronger than with the upper one. The fluid therefore should lock on to the lower boundary generating large shear in a Hartmann boundary layer adjacent to the upper conductor, where the velocity decreases to zero on a distance in the order of M�<sup>1</sup> . This allows to deduce that the electrical currents in the system, generated by the flow and circulating through the boundary layers and both boundaries, in particular through the upper �<sup>1</sup> poor conductor, should scale as O M everywhere except for the boundary layers where the shear is large. A schematic picture of the current circulation when the outer/upper boundary is poorly conducting or insulating for both geometries is provided in Figure 3. It follows, that �<sup>1</sup> everywhere the perturbation magnetic field is weak, that is b x<sup>ð</sup> ; <sup>z</sup>Þ ¼ O M . Therefore, from (8), we infer that the main flow for the case of poorly conducting or insulating boundary is determined by

$$\begin{cases} \mathbf{B\_o} \cdot \nabla u = 0 + \mathcal{O}(M^{-1}) \\ \mathbf{B\_o} \cdot \nabla b = 0 + \mathcal{O}(M^{-2}) \end{cases} \Rightarrow \begin{cases} u = \mathcal{F}(A) + \mathcal{O}(M^{-1}) \\ b = \mathcal{G}(A) + \mathcal{O}(M^{-2}) \end{cases} \tag{14}$$

Figure 3. A schematic picture of the current circulation in the planar (left panel) and spherical (right panel) configurations. In the planar case, the direction of the external field oscillates in the x direction and so does the direction of currents, which are strongest in a shear layer along the critical C-line. In the spherical case, strong currents flow from the inner sphere to the outer shell in the shear layer along C and return in polar regions.

where F and G depend on A alone, thus the flow and the induced magnetic field are constant on the field lines. The second equation in (14) means of course, that at the leading order, the Lorentz force vanishes everywhere in the main flow, which in turn implies that the currents are parallel to the external field B0.

It is clear now that the magnetic field lines which are tangent to the upper boundary, referred to as the C lines, divide the flow into three regions I, II, and III (see Figure 1), and the properties of the solution for each region are somewhat different since in regions II and III, the external field lines intersect with both boundaries and in region I, only with the lower one. Regions II and III are, therefore, very similar and the only difference between them is the sign of the perturbation magnetic field since it is antisymmetric with respect <sup>3</sup><sup>π</sup> to the planes <sup>x</sup> <sup>¼</sup> <sup>2</sup> π <sup>λ</sup> , <sup>2</sup>λ. This antisymmetry of b and symmetry of u, together with (14) results in

$$\begin{cases} u \equiv 1 + O\left(M^{-1}\right) \\ b \equiv 0 + O\left(M^{-2}\right) \end{cases} \text{ in region I} \tag{15}$$

thus, the fluid in region I flows with the same uniform velocity of the bottom boundary and the perturbation magnetic field vanishes at leading order. Therefore, in region I, the currents also must vanish. However, in the area of singularity of the Hartmann layer, namely at <sup>x</sup> <sup>¼</sup> <sup>π</sup> <sup>3</sup><sup>π</sup> <sup>2</sup><sup>λ</sup> , and at z ¼ 1, the "z" component of the currents, as it was stated earlier, has an <sup>2</sup><sup>λ</sup> external value (while the "x" component is zero) and interacting with the external magnetic field creates a Lorentz force which has to either accelerate or decelerate the fluid depending on whether the signs of j and B0<sup>x</sup> are the same or opposite. <sup>z</sup>

Since the C lines that create the singularity of the upper Hartmann layer connect regions of different flow characteristics, a thin area along the lines has to be treated differently and the dissipation must play an important role in this region. Those are shear layers, for which the precise analysis allows to compute the magnitude of super velocity.

The situation is more complicated when the upper boundary is strongly conducting. According to Soward and Dormy [6], the Ferraro's law still holds in region I (equatorial region E for the spherical case) where the fluid locks on to the moving boundary, however, in regions II and III (equivalently in polar regions P), the Ferraro's law is violated by the influence of ohmic diffusion. This happens, because the induced magnetic field b is no longer small, but of comparable magnitude with the velocity field. Nevertheless, the shear layer along the critical C-line still forms and strong currents enter the upper boundary in the region of tangent contact between the line C and the boundary, thus creating strong Lorentz force, which accelerates the flow. The results of numerical simulations of Mizerski and Bajer [5] are recalled here on Figure 4 to demonstrate the enhancement of super velocities with the increasing conductivity of the upper boundary.

The obvious conclusion of the above analysis is that the acceleration of the fluid at x ¼ π=2λ and z ≈ 1 is due to the curvature of the applied field B0 generating singularity at this point, and the antisymmetry of external field's "z" component with respect to the plane x ¼ π=2λ which is responsible for the direction of the currents and therefore also the Lorentz force at z ¼ 1. At the singular point, the intensity of the currents entering the boundary layer and the upper boundary increases with the conductivity of this boundary because its interaction with the conducting fluid strengthens. This also implies the increase of the magnitude of the super velocities with e.

These conclusions are also true for the spherical case for which the whole analysis differs only with slightly more complicated boundary conditions and diffusive terms. This complication, however, at the leading order affects mainly the analysis of the shear layer presented in the next section but does not make the main flow analysis more difficult in any way.

Figure 4. Velocity profiles at x ¼ π=2 for four different values of the conductivity ratio for the upper boundary e ¼ 0, 0:01, 0:1, and 1. The magnitude of super velocities is the highest near the upper boundary (in the region of singularity of the Hartmann layer) and significantly increases with e. After Mizerski and Bajer [5].

It may also be interesting to make a comment on a similar problem studied numerically by Hollerbach & Skinner [16] of spherical Couette flow with axial magnetic field aligned with the axis of rotation in terms of the singular perturbation method for large Hartmann numbers, infinitesimal rotation and conductivity of the inner sphere. In this case, the Hartmann layers also become singular at the equator where the external filed becomes tangent to the boundaries. This time, however, only the singularity at the inner sphere is important since the field lines tangent to the outer shell leave the fluid and do not couple it to the boundary. Outside a cylinder tangent to the inner sphere and aligned with the axis of rotation the fluid must be at rest, since the velocity field must be constant on the magnetic field lines and the outer stationary sphere has the same conductivity as the fluid, thus the fluid is locked on to it. In such a case, the currents leaving the inner boundary layer at <sup>ϑ</sup> <sup>¼</sup> <sup>π</sup> and interacting with the external <sup>2</sup> magnetic field create a Lorentz force which decelerates the fluid and produce a counterrotating jet as found by Hollerbach and Skinner [16].

A simple conclusion which can be stated now is that super- and counter-rotating jets in such MHD systems as considered above are, in general, the outcome of three major features of these systems: the presence of isolated singular points where the external magnetic field is tangent to the boundary, the symmetries of the external magnetic field in respect to planes containing the singular points and perpendicular to the boundaries (namely, antisymmetry of the component perpendicular to the boundary and symmetry of the parallel component) and the symmetric boundary conditions for the velocity field. However, as observed by [10] the singular points can also be created in side the domain (away from the boundaries) by a magnetic field configuration with X-type null points (see field configuration 4 in [10]); also in this case the presence of super-rotation depends on the conductivity of boundaries.
