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Micromagnetic simulations of the vortex switching process of thin ferromagnetic rings under the application of a circular field, as if created from a current-carrying wire passing through the ring center, reveal that for rings with sub-micron dimensions and thicknesses on the order of the exchange length, the vortex to vortex switching process occurs through the nucleation and annihilation of multiple 360° domain walls (DWs). The DWs can be characterized by their circulation relative to the vortex circulation; the DWs form in pairs with opposite topological indices. The DW with the same circulation annihilates first, which has a smaller energy barrier to overcome before annihilating. The contributions from both the exchange energy and demagnetization energy must be considered to predict which DW will annihilate first. Either wall could be annihilated by offsetting the current toward the wall being targeted.