By subjecting a model axisymmetric magnetic field to a nonaxisymmetric perturbation, or instability, linear stability analyses can provide an indirect measure of the toroidal field strength in the Earth's core. It is thought that such field strengths have an upper bound of the order of 50 Gauss. We have constructed a model of the Earth's core to investigate how magnetic field instabilities evolve nonlinearly and find that at reasonably low viscosities, the instability evolves to a finite amplitude and rotates rigidly, as predicted from the theory of bifurcations in rotating systems. As the field strength is increased the solution adopts a completely different spatial configuration and has a different characteristic frequency to the branch found at the lower field strength. The primary Hopf bifurcation is thought to be a subcritical nature which accords with a previous weakly nonlinear analysis. We conjecture that such an instability could provide the geometric field with a mechanism for evolution to a different state at field strengths lower than the strength at the point of marginal stability. ¿ American Geophysical Union 1995