Axisymmetric force-free magnetic fields external to a unit sphere are studied as solutions to boundary value problems in an unbounded domain posed by the equilibrium equations. It is well known from virial considerations that stringent global constraints apply for a force-free field to be confined in equilibrium against expansion into the unbounded space. This property as a basic mechanism for solar coronal mass ejections is explored by examining several sequences of axisymmetric force-free fields of an increasing total azimuthal flux with a power-law distribution over the poloidal field. Particular attention is paid to the formation of an azimuthal rope of twisted magnetic field embedded within the global field, and to the energy storage properties associated with such a structure. These sequences of solutions demonstrate (1) the formation of self-similar regions in the far global field where details of the inner boundary conditions are mathematically irrelevant, and (2) the possibility that there is a maximum to the amount of azimuthal magnetic flux confined by a poloidal field of a fixed flux anchored rigidly to the inner boundary. The nonlinear elliptic boundary value problems we treat are mathematically interesting and challenging, requiring a specially designed solver, which is described in the Appendix.