  A vortex is intuitively recognized as the rotational/swirling motion of the
fluids. However, an unambiguous and universally-accepted definition for vortex
is yet to be achieved in the field of fluid mechanics, which is probably one of
the major obstacles causing considerable confusions and misunderstandings in
turbulence research. In our previous work, a new vector quantity which is
called vortex vector was proposed to accurately describe the local fluid
rotation and clearly display vortical structures. In this paper, the definition
of the vortex vector, named Rortex here, is revisited from the mathematical
perspective. The existence of the rotational axis is proved through real Schur
decomposition. Based on real Schur decomposition, a fast algorithm for
calculating Rortex is also presented. In addition, new vorticity tensor and
vector decompositions are introduced: the vorticity tensor is decomposed to a
rigidly rotational part and an anti-symmetric deformation part, and the
vorticity vector is decomposed to a rigidly rotational vector and a
non-rotational vector. Several cases, including 2D Couette flow, 2D rigid
rotational flow and 3D boundary layer transition on a flat plate, are studied
to demonstrate the justification of the definition of Rortex. It can be
observed that Rortex identifies both the precise swirling strength and the
rotational axis, and thus it can reasonably represent the local fluid rotation
and provide a new powerful tool for vortex dynamics and turbulence research.
