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 ε greatly influences the current leakage from the shear layer to that boundary near the point of its tangent contact with the critical C-line. In the case of weakly conducting boundary ˜ ° <sup>e</sup>M<sup>3</sup>=<sup>4</sup> <sup>≪</sup> 1, the current leakage is of the order <sup>O</sup> <sup>e</sup>M<sup>3</sup>=<sup>4</sup> and it increases with <sup>ε</sup> to become order unity when eM<sup>3</sup>=<sup>4</sup> ≫ 1. This strong current is perpendicular to the external field in the singular region, and thus a strong Lorentz force j ˜ B<sup>0</sup> is created, which accelerates the flow (or decelerates in some cases as shown by [16], depending on the symmetries of the applied field). In the case of interest when the moving boundary is strongly conducting and M ≫ 1, the ˜ ° ˜ ° M<sup>1</sup>=<sup>2</sup> <sup>2</sup>=<sup>3</sup>M<sup>1</sup>=<sup>2</sup> resulting super-velocity scales like O when e ≫ 1 is of the order O e when 1 ≫ e ≫ M°3=<sup>4</sup> and becomes order unity when e ≪ M°3=<sup>4</sup> .
