  Most of the currently popular Eulerian vortex identification criteria,
including the Q criterion, the Delta criterion and the Lambda_ci criterion, are
based on the analysis of the velocity gradient tensor. More specifically, these
criteria are exclusively determined by the eigenvalues of the velocity gradient
tensor or the related invariants and thereby can be regarded as
eigenvalue-based criteria. However, these criteria have been found to be
plagued with two shortcomings: (1) these criteria fail to identify the swirl
axis or orientation; (2) these criteria are prone to contamination by shearing.
In this paper, an alternative eigenvector-based definition of Rortex is
introduced. The real eigenvector of the velocity gradient tensor is used to
define the direction of Rortex as the possible axis of the local fluid
rotation, and the rotational strength obtained in the plane perpendicular to
the possible local axis is defined as the magnitude of Rortex. This alternative
definition is mathematically equivalent to our previous one but allows a much
more efficient implementation. Furthermore, a complete and systematic
interpretation of scalar, vector and tensor versions of Rortex is presented to
provide a unified and clear characterization of the instantaneous local rigidly
rotation. By relying on the tensor interpretation of Rortex, a new
decomposition of the velocity gradient tensor is proposed to shed light on the
analytical relations between Rortex and eigenvalue-based criteria. It can be
observed that shearing always manifests its effect on the imaginary part of the
complex eigenvalues and consequently contaminates eigenvalue-based criteria,
while Rortex can exclude the shearing contamination and accurately quantify the
local rotational strength.
