## Abstract

A magnetohydrodynamic flow of a viscous and conducting fluid confined between two parallel differentially moving boundaries is considered. The whole system is in a strong magnetic field chosen in such a way that the Hartmann boundary layers which form in this problem become singular at the points where the magnetic field becomes tangent to the boundary. Two geometries are taken into account: plane and spherical. Within the class of such configurations, the velocity field of the fluid and the influence of the conductivity of the boundaries on the fluid’s motion are reviewed here. In the region of singularity, where the magnetic field is tangent to the boundary, the fluid’s velocity exceeds that of the moving boundary. The effect of nonzero conductivity of the boundaries on the super-speeding jets is vital and has been enlightened in a series of papers, including experimental and theoretical findings. The mechanism of the formation of super-speeding jets in the considered configurations has been explained, which is based on strong Hartmann currents allowed to enter the boundary layer due to the singularity. In the case of both perfectly conducting boundaries, the super velocity was shown to be as strong as to scale with the Hartmann number as O M 1 / 2 .

### Keywords

- super rotation
- magnetohydrodynamics
- MHD boundary layers
- Hartmann layer singularity
- nonzero conductivity

## 1. Introduction

Super-speeding jets in the geometry of magnetohydrodynamic (MHD) spherical Couette flow have been first noticed in the numerical simulations of Dormy et al. [1]. They have analyzed a flow of an electrically conducting fluid in a spherical gap between concentric spherical shells, rapidly and differentially rotating about a common axis in a centered axial dipolar magnetic field. The solid inner sphere, which had the same conductivity as the fluid, was spinning slightly faster than the insulating outer shell. The stationary flow obtained via DNS exhibited a super-rotating structure near the region, where the critical magnetic field line, henceforth denoted by

Hollerbach [2] for the first time investigated numerically the effect of nonzero conductivity of the outer shell in the same spherical geometry but the outer boundary was held motionless, thus eliminating the Coriolis force from the problem. He reported that the super rotation in the singular region was greatly enhanced and scaled with the value of the Hartmann number
^{1} Hollerbach [3] studied the MHD spherical Couette flow for several different topologies of the external field lines and also observed that singular points of isolated contact of the magnetic lines with boundaries result in the formation of jets.

In a following sequence of three theoretical papers, the mechanism of super-velocity formation and the effect of nonzero conductivity of the boundaries have been explained. Dormy et al. [4] performed a joint analytical and numerical study of the system analyzed previously by Hollerbach [2], where they have described the super rotating shear layer along the critical magnetic line

The magnitude of the super rotation

The phenomenon of super rotation was also observed in the experimental setup called “Derviche Tourneur Sodium” (DTS) located in Grenoble at the Université Joseph-Fourier. Nataf et al. [7] conducted experiments on the spherical Couette flow of liquid sodium in an external, centered axial dipolar field, with both boundaries differentially rotating. The outer shell was only 5 mm thick, about

Wei and Hollerbach [9] investigated numerically the effect of strong inertia, that is, large Reynolds number, on the spherical Couette flow configuration with the outer shell stationary. Three configurations of the external magnetic field were chosen, which resulted from a combination of dipolar and axial fields. The super-speeding jets have been destabilized by increasing the Reynolds number, whereas strengthening the filed had the opposite effect. Most recently, Hollerbach and Hulot [10] performed numerical analysis of a similar problem in cylindrical geometry, putting an emphasis on the role of conductivity of the boundaries. The field configurations were also chosen so as to create singularities in the flow. When the boundaries were electrically conducting, super-speeding jets were reported on the contrary to the case with insulating boundaries, when simply shear layers were observed in the singular regions. A curious observation is made by the introduction of a nonzero azimuthal component of the external field in which case the conductivity of the boundaries has the opposite effect to the previous case, greatly suppressing the magnitude of super rotation.

The motivation for some of the aforementioned studies was justified on geophysical grounds. The investigations of the Earth’s interior reveal differential rotation of the inner core (cf. [11, 12]) and that the electrical conductivity of the lower mantle is nonnegligible [13]. Moreover, some evidence can be found for the existence of a very thin layer of anomalously high conductivity at the base of the mantle [14, 15]. It must be said, however, that the model of MHD spherical Couette flow is so idealized with respect to the true dynamics of the core, neglecting thermal and compositional driving, turbulence, the solidification processes at the inner core, etc., that no direct comparisons with the flow at the core-mantle boundary can be made. Nevertheless, it might be possible that the effect of super rotation manifests itself on the field zero isolines locally at the core mantle boundary.

### 1.1. Ferraro’s law of isorotation

Throughout this chapter, we will assume that the Hartmann number,

is large. In the above,

In such a case, the Ferraro’s law of isorotation states that for a steady azimuthal motion about an axis of symmetry of an electrically conducting fluid, the magnitude of the angular velocity is predominantly constant along a magnetic field line. This means that in the studied configurations presented in Figure 1, the flow in the equatorial region

In the following, we review the analytic approach and most important results for the two cases introduced in Figure 1.

## 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

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,

where

where

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

where

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,

can vary from zero to infinity, where

The general set of equations for the analyzed stationary state is obtained by taking the “

with

Inside the rigid conductors, the magnetic field

where

Finally, the conditions at

To understand the structure of the flow, it is very important to note the symmetries in the system with respect to planes defined by

for any

It is also clear from (13) that the “

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

where

It is clear now that the magnetic field lines which are tangent to the upper boundary, referred to as the

thus, the fluid in region

Since the

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

The obvious conclusion of the above analysis is that the acceleration of the fluid at

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.

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

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.

## 3. The shear layer along the
C
-line

We will now briefly introduce the reader into the mathematical approach to the analysis of the shear layer structure, which is based on the singular perturbation theory. To take into account of the curvature of the

and the point

and letting

where

For the simplest case of poorly conducting or insulating upper/outer boundary, with the use of the shear layer coordinates

where

In the spherical geometry, the analogous formulation leads to the coupling of the two equations for

The coupling term, however, may be neglected if the narrow gap limit is assumed. More importantly, however, the two equations, in both—planar and spherical configurations, are coupled through the boundary conditions at

However, when the boundaries are perfectly conducting, the problem becomes more complicated. The same equations as (20) and (22) are obtained for

(cf. Eq. (3.20) in [6]), but the problem becomes analytically intractable due to the complications arising from vanishing of the current component parallel to the boundary at

## 4. Summary

The plane and spherical magnetohydrodynamic Couette flow with an applied strong external magnetic field creating Hartmann layer singularities on a boundary is a setting where fastly moving jets form, with the magnitude of the flow exceeding that of the moving boundary, which drives the entire flow. These are the so-called super velocities (super rotation in the spherical case). We have concentrated here on the review of the results and analytic approach to the problem of the formation of super velocities in strong, potential fields, with particular emphasis on the enhancement of super velocities by the conductivity of the resting boundary.

As found by Soward and Dormy [6], the conductivity of the resting (upper/outer) boundary

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## Notes

- He fitted an exponent of M 0.6 to his numerical results, which however, was later shown not to be the true asymptotic scaling law.