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 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-two-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.

Figure 1

An epi-2D flow given by ${\mathit{V}}_{+}$. (Left) The integrable vortex lines. (Right) Contours of ${𝒬}_{+}$ and the flow vector on the surface $y=0$ indicated by the gray cross section in the left figure (the color code ranges from $\text{orange}=2$ to $\text{green}=-2$).

Figure 2

The combination $\mathit{V}={\mathit{V}}_{+}+{\mathit{V}}_{-}$ yields a 3D flow. (Left) The vortex lines become chaotic (nonintegrable). (Right) Contours of ${𝒞}_r$ together with the surface-aligned component of the flow vector on the surface $y=0$ indicated by the gray cross section in the left figure (the color code ranges from $\text{blue}=5$ to $\text{yellow}=-1$).