  We present a numerical study of dynamo action in a conducting fluid encased
in a metallic spherical shell. Motions in the fluid are driven by differential
rotation of the outer metallic shell, which we refer to as "the wall". The two
hemispheres of the wall are held in counter-rotation, producing a steady,
axisymmetric interior flow consisting of differential rotation and a two-cell
meridional circulation with radial inflow in the equatorial plane. From
previous studies, this type of flow is known to maintain a stationary
equatorial dipole by dynamo action if the magnetic Reynolds number is larger
than about 300 and if the outer boundary is electrically insulating. We vary
independently the thickness, electrical conductivity, and magnetic permeability
of the wall to determine their effect on the dynamo action. The main results
are: (a) Increasing the conductivity of the wall hinders the dynamo by allowing
eddy currents within the wall, which are induced by the relative motion of the
equatorial dipole field and the wall. This processes can be viewed as a skin
effect or, equivalently, as the tearing apart of the dipole by the differential
rotation of the wall, to which the field lines are anchored by high
conductivity. (b) Increasing the magnetic permeability of the wall favors
dynamo action by constraining the magnetic field lines in the fluid to be
normal to the wall, thereby decoupling the fluid from any induction in the
wall. (c) Decreasing the wall thickness limits the amplitude of the eddy
currents, and is therefore favorable for dynamo action, provided that the wall
is thinner than the skin depth. We explicitly demonstrate these effects of the
wall properties on the dynamo field by deriving an effective boundary condition
in the limit of vanishing wall thickness.
