The structure and evolution of flow fields and other vector fields

As the field evolves with time, discrete events occur which alter the patterns of folds and cusps. Thom's (1975) list of elementary catastrophes describes these events when the changes in the field are the result of changes of an underlying scalar generating function. In that case the generic events are lips, beak-to-beak, swallowtail, hyperbolic umbilic and elliptic umbilic. If the vector field evolves with time in a more general way, the only generic events are lips, beak-to-beak and swallowtail. Similarly, in a three-dimensional vector field constrained by being derived from an underlying generating function, the generic singularities are the first five in Thom's list: the fold, cusp, swallowtail, elliptic and hyperbolic umbilic; but when the field is not constrained in this way, only the fold, cusp and swallowtail will occur generically. Two geophysical illustrations are used.