Caustic conditions for d-dimensional Lagrangian fluids

Caustics form in Lagrangian fluids when fluid elements cross and multi-stream regions form. For low-dimensional Lagrangian fluids, the caustics have been classified by catastrophe theory. In the case of potential flow, for oneand two-dimensional fluids, Arnol’d et al. (1982) related this classification to conditions on the displacement field of the fluid. These conditions are called the caustic conditions. This paper concerns nothing less than an entirely new proof for the classification of Lagrangian catastrophes. We provide an alternative, more direct and transparent, path towards the classification of singularities as proposed by Arnol’d. In this study we give a novel derivation of the caustic conditions for the explicit situation of three-dimensional fluids, following up on those for oneand twodimensional fluids. Moreover, our derivation scheme allows us to extend these to general Lagrangian fluids in spaces of arbitrary dimensions. Applications to cosmic structure formation and to turbulence are indicated, and shortly discussed. In an accompanying publication we apply this towards a full three-dimensional study of caustics in the context of the formation of the cosmic web.