Bifurcations of <i>Q</i>-vortices

We develop a bifurcation theory for three-dimensional vortices defined by the Q-criterion, with time as the parameter. Two types of local bifurcations are identified, denoted punching or pinching, occurring in two settings, either in the flow or close to a no-slip wall. The bifurcations represent the creation, destruction, merging, or splitting of vortices. Furthermore, in a global setting, they can also describe the transition between a spherical vortex and a vortex ring as well as the creation of a hairpin vortex from a wall. Scaling laws for the size of vortices and the distance between vortices close to bifurcation are obtained. The analysis is of a topological and general nature and will also hold for vortices defined by other local vortex criteria.