Kinematic Morphology of Large-scale Structure: Evolution from Potential
  to Rotational Flow
  As an alternative way of describing the cosmological velocity field, we
discuss the evolution of rotational invariants constructed from the velocity
gradient tensor. Compared with the traditional divergence-vorticity
decomposition, these invariants, defined as coefficients of characteristic
equation of the velocity gradient tensor, enable a complete classification of
all possible flow patterns in the dark-matter comoving frame, including both
potential and vortical flows. We show that this tool, first introduced in
turbulence two decades ago, proves to be very useful in understanding the
evolution of the cosmic web structure, and in classifying its morphology.
Before shell-crossing, different categories of potential flow are highly
associated with cosmic web structure, because of the coherent evolution of
density and velocity. This correspondence is even preserved at some level when
vorticity is generated after shell-crossing. The evolution from the potential
to vortical flow can be traced continuously by these invariants. With the help
of this tool, we show that the vorticity is generated in a particular way that
is highly correlated with the large-scale structure. This includes a distinct
spatial distribution and different types of alignment between cosmic web and
vorticity direction for various vortical flows. Incorporating shell-crossing
into closed dynamical systems is highly non-trivial, but we propose a possible
statistical explanation for some of these phenomena relating to the internal
structure of the three-dimensional invariants space.
