  We perform numerical optimization of the axisymmetric flows in a sphere to
minimize the critical magnetic Reynolds number Rm_cr required for dynamo onset.
The optimization is done for the class of laminar incompressible flows of von
Karman type satisfying the steady-state Navier-Stokes equation. Such flows are
determined by equatorially antisymmetric profiles of driving azimuthal
(toroidal) velocity specified at the spherical boundary. The model is relevant
to the Madison plasma dynamo experiment (MPDX), whose spherical boundary is
capable of differential driving of plasma in the azimuthal direction. We show
that the dynamo onset in this system depends strongly on details of the driving
velocity profile and the fluid Reynolds number Re. It is found that the overall
lowest Rm_cr~200 is achieved at Re~240 for the flow, which is hydrodynamically
marginally stable. We also show that the optimized flows can sustain dynamos
only in the range Rm_cr<Rm<Rm_cr2, where Rm_cr2 is the second critical magnetic
Reynolds number, above which the dynamo is quenched. Samples of the optimized
flows and the corresponding dynamo fields are presented.
