The long-term behavior of azimuthally averaged core magnetic fields is examined using numerical calculations of nonlinear, axisymmetric, kinematic α2 and α2ω dynamos. The important elements of the model are convective turbulence, parameterized by an α effect, and a toroidal shear flow, providing the ω effect. Magnetic fields produced by this model exhibit several styles of behavior. For Rα Rω0, we identify three regimes of field behavior corresponding to the magnitude of Rω. Small Rω values produce nonoscillatory (dc) dipolar fields with zero quadrupolar components, while large values of Rω produce ac dipole and ac quadrupole fields. Intermediate values of Rω produce fields with dc dipole components coexisting with poleward propagating αω dynamo waves in the quadrupole family. This regime of field behavior is most consistent with the observed behavior of the Earth's paleomagnetic field on the 103--104 year time scale of paleomagnetic secular variation. ¿ American Geophysical Union 1991<