  Vortex is ubiquitous in nature. However, there is not a consensus on the
vortex definition in fluid dynamics. Lack of mathematical definition has caused
considerable confusions in visualizing and understanding the coherent vortical
structures in turbulence. According to previous study, it is realized that
vortex is not the vorticity tube and vorticity should be decomposed into a
rotational part which is the vortex vector and a non-rotational part which is
the shear. In this paper, several new concepts such as fluid rotation of local
point, the direction of fluid rotation axis and the strength of fluid rotation
are proposed by investigating the kinematics of fluid element in the 2D and 3D
flows. A definition of a new vector quantity called vortex vector is proposed
to describe the local fluid rotation. The direction of the vortex vector is
defined as the direction of local fluid rotation axis. The velocity components
in the plane orthogonal to the vortex vector have zero derivatives along the
vortex vector direction. The magnitude of the vortex vector is defined as the
rotational part of vorticity in the direction of the vortex vector, which is
the twice of the minimum angular velocity of fluid around the point among all
azimuth in the plane perpendicular to vortex vector. According to the
definition of the vortex vector, vortex is defined as a connected flow region
where the magnitude of the vortex vector at each point is larger than zero. The
new definition for the vortex vector and vortex follows three principles: 1.
Local in quantity, 2. Galilean invariant, 3. Unique. The definitions are
carefully checked by DNS and LES examples which clearly show that the new
defined vortex vector and vortex can fully represent the complex structures of
vortices in turbulence.
