Epi-two-dimensional fluid flow: a new topological paradigm for dimensionality

While a variety of fundamental differences are known to separate two-dimensional (2D) and three-dimensional (3D) fluid flows, it is not well understood how they are related. Conventionally, dimensional reduction is justified by an \emph{a priori} geometrical framework; i.e., 2D flows occur under some geometrical constraint such as shallowness. However, deeper inquiry into 3D flow often finds the presence of local 2D-like structures without such a constraint, where 2D-like behavior may be identified by the integrability of vortex lines or vanishing local helicity. Here we propose a new paradigm of flow structure by introducing an intermediate class, termed epi-2-dimensional flow, and thereby build a topological bridge between 2D and 3D flows. The epi-2D property is local, and is preserved in fluid elements obeying ideal (inviscid and barotropic) mechanics; a local epi-2D flow may be regarded as a `particle' carrying a generalized enstrophy as its charge. A finite viscosity may cause `fusion' of two epi-2D particles, generating helicity from their charges giving rise to 3D flow.