Direct consequences stem from the close relation between the recently proposed vortex vector (Rortex) and the swirling strength. It is shown that these vortex-identification methods share some relevant properties: (i) both provide the same and, practically, the largest vortex region, (ii) both allow an unlimited uniaxial stretching described by an axisymmetric strain rate, so the strain-rate magnitude inside a vortex may become much larger (without any limitation) than the vorticity magnitude, and (iii) both exhibit a discontinuous outcome, known as the so-called disappearing vortex problem.

There is an explicit close relation between the recently proposed vortex vector (Rortex) and the swirling strength, see Gao *et al.*^{1} In this Letter, we show that these two vortex-identification schemes share some relevant properties as a direct consequence of this close relation.

Let us briefly summarize the two criteria under consideration. Both of them are local (i.e., pointwise) methods based on the velocity-gradient tensor. The swirling strength of Zhou *et al.*^{2} employs the imaginary part *λ*_{ci} of complex conjugate eigenvalues of the velocity-gradient tensor ∇**u**. It is based on the existence of complex eigenvalues of ∇**u** through the Δ-criterion of Chong *et al.*^{3} There are similar papers dealing with complex eigenvalues of ∇**u** by Dallmann,^{4} Vollmers *et al.*,^{5} and Berdahl and Thompson.^{6} Chakraborty *et al.*^{7} further enhanced the swirling-strength criterion. The quantity *λ*_{ci}, which is non-negative by definition, is a measure of the local swirling rate inside a vortex, and the time period for completing one revolution of the streamline is given by^{7} 2*π*/*λ*_{ci}. The Rortex scheme results in a vortex vector endowed with a direction of the real eigenvector of the velocity-gradient tensor with complex eigenvalues.^{8} Its magnitude, determined in the perpendicular plane to the real eigenvector, coincides with the magnitude of planar residual vorticity obtained after the elimination of shearing,^{9} or equivalently, with the planar corotation of line segments near a point.^{10}

The relation between the Rortex vector ** R** and

*λ*

_{ci}reads

^{1}

where ** r** denotes the real unit eigenvector, and

*ω*

_{r}=

**·**

*ω***is the magnitude of vorticity**

*r***in the direction of**

*ω***. Vector**

*r***is required to fulfill the condition**

*r**ω*

_{r}> 0 for uniqueness.

The consequences of relation (1) between Rortex and *λ*_{ci} can be summarized as follows:

Vortex region:

The vortex region according to *λ*_{ci} is for all practical purposes considered as the largest vortex region among the widely used criteria, such as *Q* and *λ*_{2}, as found by Chakraborty *et al.*,^{7} p. 201. The inadequacy of such an oversized vortex region (misrepresenting vortex geometry or providing excessively noisy vortex boundary) was subjected to criticism for a variety of flow examples by Jeong and Hussain,^{11} pp. 81–85. They adopted the discriminant Δ of the characteristic equation for ∇**u** to determine the region of complex eigenvalues. From Eq. (1), it can be directly inferred that *λ*_{ci} > 0 implies |** R**| > 0 and vice versa. Therefore, the vortex region defined by non-zero swirling strength coincides with the vortex region defined by non-zero Rortex, both being given by the region of complex eigenvalues.

Stretching response:

Unlike other vortex-identification schemes, Rortex and *λ*_{ci} are not stretching-sensitive for a uniaxial stretching (coupled with a uniform radial contraction for incompressible flow) and the vorticity vector perfectly aligned with the stretching axis.^{12} For this axisymmetric benchmark input of ∇**u**, the magnitude of *λ*_{ci} directly equals half of the vorticity vector magnitude, and according to Eq. (1), the Rortex vector is identical with the vorticity vector, the direction of which coincides with the real-eigenvector direction. This tensor geometry allows an arbitrary axial strain and does not follow the idea of orbital compactness of the motion inside a vortex formulated by Chakraborty *et al.*^{7} and discussed below. Consequently, the strain-rate magnitude inside a vortex may become much larger (without any limitation) than the vorticity magnitude. Moreover, on closer examination, it can be inferred from Fig. 3(b) of Chakraborty *et al.*^{7} dealing with *λ*_{ci} vortex region, and from the property (1) mentioned above, that this conclusion holds for a more general case of an unlimited uniaxial stretching described by an axisymmetric strain rate for the vorticity vector not necessarily aligned with the stretching axis (excluding the perpendicular case).

Discontinuity:

The so-called “*disappearing vortex problem*” has been described in detail by Chakraborty *et al.*,^{7} pp. 199–200. For a wide range of parameters determining the ∇**u** tensor geometry, a discontinuous “clearly counterintuitive” behavior of *λ*_{ci} has been found. Expressed more explicitly, for a fixed tensor geometry, fixed strain rate, and the only variable being vorticity-vector magnitude (see Fig. 1), with increasing relative vorticity magnitude the vortex disappears and reappears again above a certain relative vorticity magnitude. The same anomalous behavior is described below for Rortex to show that the disappearing vortex problem occurs for both schemes as a direct consequence of their identical yes/no vortex-identification outcome given by Eq. (1).

Following the work of Chakraborty *et al.*,^{7} to express all the possible configurations of ∇**u**, the velocity-gradient tensor can be written in the system of strain-rate principal axes (*p*_{1}, *p*_{2}, and *p*_{3}), according to Fig. 1, as (the tensor is deviatoric without loss of generality as explained below)

where *ξ* is a strain-field parameter, *θ* and *ϕ* are vorticity-vector configuration angles relative to strain-rate principal axes *p*_{1} and *p*_{2}, respectively, principal strain rates are ordered *σ*_{1} ≥ *σ*_{2} ≥ *σ*_{3}, *σ*_{1} is taken as a scaling factor, and *a* = |** ω**|/2

*σ*

_{1}. The relative vorticity tensor magnitude $\Vert $

**Ω**$\Vert $ compared to the strain-rate tensor magnitude $\Vert $

**$\Vert $ is then given by**

*S*^{7}$\Vert \Omega \Vert /\Vert S\Vert =2a/\xi 2/2\u2212\xi +2$. The flow depicted in Fig. 1 is converging for

*σ*

_{1}≥ −

*σ*

_{3}and 0 ≤

*ξ*≤ 1. Note that the diverging flow for

*σ*

_{1}< −

*σ*

_{3}and −1 ≤

*ξ*< 0 does not need to be explicitly considered as analyzing the ∇

**u**configurations for the converging case is sufficient.

^{7}Regarding compressibility, a uniform dilatation (or compression) can be removed prior to the determination of eigenvalues and eigenvectors and added to the resulting eigenvalues changing the real parts of them. In the case of complex conjugate eigenvalues, the imaginary part

*λ*

_{ci}and all the three eigenvectors (the pair of complex conjugate eigenvectors and the real one) remain unaffected by a uniform dilatation or compression.

From Eq. (2), consequently, the eigenvalues and eigenvectors are functions of the second and third invariants, which themselves are (up to a scaling factor *σ*_{1}) functions of *a*, *ξ*, and, in case of the third invariant, also a function of a specific quantity *ψ* combining the configuration angles *θ* and *ϕ* as^{7}

The complex conjugate eigenvalues of ∇**u** read *λ*_{cr} ± i*λ*_{ci}, where *λ*_{cr} (not discussed up to now) denotes the real part of the eigenvalues. The instantaneous streamline pattern is spiraling or closed on a plane spanned by the associated complex eigenvectors, the so-called swirl plane. The sign of *λ*_{cr} distinguishes between outward or inward spiraling of streamlines in the swirl plane.

Chakraborty *et al.*^{7} have shown the contour lines of regions of complex eigenvalues (*λ*_{ci} > 0) in their Figs. 3(b)–3(d)^{7} for the strain-field parameter *ξ* = 1 (axisymmetric strain rate), 0.5, and 0 (planar strain rate), respectively. They discussed in detail the disappearing vortex problem on pages 199–200^{7} and eliminated the problem by introducing the idea of orbital compactness. The orbital compactness requires that the separation of swirling material points inside a vortex is bounded and remains small. Therefore, they imposed a restriction on the ratio *λ*_{cr}/*λ*_{ci} (the so-called inverse spiraling compactness). The local parameter *λ*_{cr}/*λ*_{ci}, which approximates the measure of the non-local orbital compactness according to Chakraborty *et al.*,^{7} was motivated by the earlier non-local vortex criterion by Cucitore *et al.*^{13} (see also the review of vortex-identification methods by Epps).^{14} The ratio *λ*_{cr}/*λ*_{ci} = 0 represents a closed elliptical orbit, while a positive or negative value indicates outward or inward spiraling, respectively.

The anomalous behavior of Rortex, representing the disappearing vortex problem, is shown in Fig. 2 for six representative tensor configurations given by the strain-field parameter *ξ* and vorticity-vector configuration angles *θ* and *ϕ*, according to Fig. 1. The independent variable of relative vorticity is given by the ratio $\Vert $**Ω**$\Vert $/$\Vert $** S**$\Vert $. Positive criterion values in Fig. 2 identify a vortex. The values of parameter

*ψ*given by (3) are also stated in Fig. 2 for a closer examination and comparison with Fig. 3 of the work of Chakraborty

*et al.*

^{7}depicting the vortex region in the space of relative vorticity and

*ψ*. Figure 2 indicates not only the disappearing vortex problem for Rortex and

*λ*

_{ci}but also the performance of the well-known vortex-identification criteria

*Q*and

*λ*

_{2}for comparison purposes.

The crucial aspect of both Rortex and *λ*_{ci} distributions is the “bubble” indicating the disappearing vortex at lower relative vorticity, say below 1, for which there is no physically rational explanation in the frame of vortex identification as the growth of vorticity (for a fixed tensor configuration) eliminates vortex. At the same time, note that values of $\Vert $**Ω**$\Vert $/$\Vert $** S**$\Vert $ around and below 1 are interesting from the vortex-identification viewpoint since this is near the vortex boundary where discrepancies among individual schemes occur, whereas for higher values of $\Vert $

**Ω**$\Vert $/$\Vert $

**$\Vert $, the criteria generally agree on classifying the point as a part of a vortex. The plots in Fig. 2 suggest that although the magnitude of Rortex in the bubble tends to be smaller than**

*S**λ*

_{ci}for most of the affected configurations, the boundaries of these anomalous bubbles remain identical. This phenomenon has been found (for the vorticity vector in the first octant) for a limited range of

*θ*, ∼57° ≤

*θ*< 90°, while there is no restriction for

*ϕ*. Considering the strain field, the disappearing vortex problem extends from

*ξ*= 1 (axisymmetric case) beyond

*ξ*= 0.5 as found earlier for

*λ*

_{ci}by Chakraborty

*et al.*

^{7}Note that the given triplet of input parameters

*θ*,

*ϕ*, and

*ξ*results in a fixed value of

*ψ*, although this value is shared by a set of configuration triplets following the nonlinear relation (3).

Figure 3 shows two examples of fixed configuration angles *θ* and *ϕ* and variable strain-field parameter *ξ*. The aim is to depict a well-pronounced transformation of the closed “bubble” into a non-monotonic hump of the Rortex outcome. Although the disappearing vortex problem is of primary interest, the described non-monotonic response of Rortex (as the vortex-identification criterion) to increasing relative vorticity magnitude is also hard to explain as the growth of vorticity (for a fixed tensor configuration and fixed strain rate) reduces vortex intensity.

Rortex is a mathematically rigorous tool suitable for vortex characterization. However, from the vortex-identification point of view, some features of Rortex remain questionable. These aspects should be eliminated, for example, in the manner of the work of Chakraborty *et al.*^{7} by imposing an additional restriction on the orbital compactness of the motion inside a vortex in terms of a limited value of *λ*_{cr}/*λ*_{ci}.

## DATA AVAILABILITY

Presented values of all criteria have been computed—and the results can be reproduced—using subroutines from our open-source *Vortex Analysis Library (VALIB, version 1.1)*.^{15}

This work was supported by the Czech Science Foundation through Grant No. 18-09628S and the Czech Academy of Sciences through Nos. RVO:67985874 and RVO:67985840.