Magnetohydrodynamics of a viscous spherical layer rotating in a strong potential field

An analytic solution is given for classical magnetohydrodynamic (MHD) problem of almost rigid-body rotation of a viscous, conducting spherical layer of liquid in an axisymmetric potential magnetic field. Large-scale flows bounded by rigid spheres are described for the first time in a new approximation. Two problems are solved: (1) in which both spheres are insulators and (2) in which the outer sphere is an insulator and the inner sphere a conductor. Axially symmetric flows and azimuthal magnetic fields are maintained by a slightly faster rotation of the inner sphere. The primary regeneration takes place in the boundary and shear MHD layers. The shear layers, described here for the first time, smooth out the large gradients at the boundaries of the MHD structures encompassed by them. There is essentially no azimuthal magnetic field inside these original structures, which are bounded by potential contours tangent to the spheres. An applied constant magnetic field creates a rigid MHD structure outside an axial cylinder tangent to the inner sphere. Inside the cylinder the rotation is faster and the meridional flux depends on height. A magnetic dipole forms a structure tangent to the outer equator. Outside the structure, the rotation is also rigid-body when both spheres are insulators. When a conducting sphere is present, the liquid rotates differentially everywhere, while near the axis and inside the MHD structure, it rotates even faster than the inner sphere. The last example of a general solution is a quadrupole magnetic field. In this case, two equatorially symmetric MHD structures are formed which rotate together with the inner sphere. Outside the structures, as in the most general case, the rotation is differential, the azimuthal magnetic field falls off as the first power of the applied field, and the meridional flux falls off as the square of the field in the first problem, and as the cube in the second.