1.1. Ferraro's law of isorotation

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

$$M = \frac{B\_0 L}{\sqrt{\mu\_0 \rho \nu \eta \eta}} = \sqrt{\frac{\sigma}{\rho \nu}} B\_0 L \gg 1 \tag{2}$$

is large. In the above, μ0, r, ν, η, and σ are the magnetic permeability, constant density, viscosity, magnetic diffusivity, and electrical conductivity of the fluid, respectively; B<sup>0</sup> is the typical strength of the external magnetic field; and L is the distance between the boundaries.

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 ℰ in the spherical case and region I in the planar case both bounded by the critical line which grazes the outer/upper boundary must significantly differ from the flow outside those regions. The magnetic lines

Figure 2. The moving electrically conducting boundary drags the field lines with it, but only those lines which experience drag from the top, stationary boundary are tilted (solid lines). The lines within the arcade bounded by the critical C-line are carried with the same velocity as that of the bottom boundary. After Mizerski and Bajer [5].

within the regions ℰ and I do not reach the outer/upper boundary and their both footpoints lie on the inner/bottom boundary, which is moving. Since the moving boundary is assumed electrically conducting with the same conductivity as the fluid, the magnetic field lines within the arcade bounded by the C-line are carried by the fluid without any tilt. Therefore, by the Ferraro's law, the flow within the arcade must be uniform, with the same magnitude as the velocity of the moving boundary. On the contrary, the magnetic lines outside those regions extend from one boundary to the other; therefore, they are tilted due to advection by the inner/ bottom boundary and the drag they experience from the outer/top, stationary boundary. The effect of competition of the moving and the stationary boundary makes the flow vary from one field line to the other. This is illustrated in Figure 2 for the case of planar geometry.

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