Vortex: Dynamics & Applications

• Vortex is a coherent rotational or phase-singular structure defined by organized circulation, critical in fields such as fluid dynamics, optics, and superconductivity.
• Innovative identification methods—including vortex vector analysis, Liutex techniques, and Rortex criteria—advance our understanding of true rotational motion and instability mechanisms.
• Controlled vortex dynamics enable practical applications in devices like optical elements, memory systems, and astrophysical models, highlighting their interdisciplinary utility.

Searching arXiv for relevant vortex papers across definitions, fluid dynamics, optics, condensates, acoustics, solar MHD, and superconductivity. Vortex denotes a coherent rotational structure, but the term is not uniform across disciplines. In the cited literature it appears as a connected flow region with actual local rotation rather than mere vorticity, as a wavefield phase singularity with azimuthal factor $e^{i l\varphi}$ or $e^{iL\phi}$, as a quantized defect in condensates and superfluids, as a rotating plasma or magnetic structure in the solar atmosphere, and as a nested planar complex of 1-cycles in proximity topology (Tian et al., 2017, Qu et al., 2017, Naify et al., 2016, Ma et al., 2017, Silva et al., 2020, Peters, 2018). The common invariant is organized circulation around a center or axis; the principal differences lie in how that circulation is measured, whether the core is geometric or singular, and whether the relevant state variable is velocity, phase, magnetization, density, or combinatorial incidence.

1. Definitions, kinematics, and objectivity

A central issue in fluid dynamics is that vorticity is not identical to vortex. The paper defining the vortex vector argues that large $\boldsymbol{\omega}=\nabla\times\mathbf{v}$ can occur in pure shear, so vortex identification must isolate the rotational part of the motion. It introduces a local, Galilean-invariant, unique vector quantity

$\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$

where $\mathbf{T}$ is the local fluid rotation axis and $\omega_{\text{rot}}^{(v)}$ is the rotational part of vorticity; a vortex is then defined as a connected region where $|\mathbf{R}|>0$ (Tian et al., 2017). In this formulation, the velocity components in the plane orthogonal to $\mathbf{T}$ have zero derivatives along the vortex-vector direction, and the magnitude of $\mathbf{R}$ is twice the minimum angular velocity among all azimuths in that plane.

A related Liutex-based program defines a vortex core line as the centerline of true rotational motion rather than a vorticity tube or an isosurface artifact. In that framework the Liutex vector is $\mathbf{R}=R\,\mathbf r$, with $e^{iL\phi}$0 the real eigenvector of the velocity-gradient tensor and

$e^{iL\phi}$1

as the core-line condition (Charkrit et al., 2019). The same work uses a modified Omega-Liutex indicator $e^{iL\phi}$2 with empirical visualization threshold $e^{iL\phi}$3, and argues that the ring-like or hairpin vortex is formed separately from the Lambda-vortex through Kelvin–Helmholtz instability rather than self-deformation (Charkrit et al., 2019).

Objective boundary extraction appears in solar MHD through the Instantaneous Vorticity Deviation,

$e^{iL\phi}$4

with vortex boundaries defined as outermost closed contours around local IVD maxima. In three dimensions, horizontal contours are linked through a core line to produce a vortex tube, with convexity deficiency

$e^{iL\phi}$5

used to control admissible contour shape; the reported value is $e^{iL\phi}$6 (Silva et al., 2020). A separate automated solar-detection pipeline uses the SWIRL code with the Rortex criterion

$e^{iL\phi}$7

followed by clustering of connected rotational regions (Pistarini et al., 3 Apr 2025).

Outside continuum mechanics, vortex can also denote a purely topological object. A planar vortex cycle is defined as a collection of non-concentric, nesting 1-cycles with nonempty interiors; overlapping 1-cycles generate an Edelsbrunner–Harer nerve, and overlapping vortex cycles form a vortex nerve complex (Peters, 2018). This usage retains the geometric intuition of a ring-like nested structure while dispensing with any fluid or field interpretation.

2. Phase singularities, orbital angular momentum, and wave vortices

In optics, an optical vortex is a light wave with a twisting wavefront around its propagation axis and null intensity in the beam center. Such beams carry orbital angular momentum, and their canonical phase factor is azimuthal: $e^{iL\phi}$8 A $e^{iL\phi}$9-plate implements the map

$\boldsymbol{\omega}=\nabla\times\mathbf{v}$0

so the beam flips circular polarization and acquires a vortex phase (Qu et al., 2017). The plasma $\boldsymbol{\omega}=\nabla\times\mathbf{v}$1-plate extends this mechanism to high intensities by replacing a birefringent solid with a suitably magnetized plasma. Its birefringence follows from

$\boldsymbol{\omega}=\nabla\times\mathbf{v}$2

with $\boldsymbol{\omega}=\nabla\times\mathbf{v}$3, and the phase retardation

$\boldsymbol{\omega}=\nabla\times\mathbf{v}$4

For $\boldsymbol{\omega}=\nabla\times\mathbf{v}$5 the conversion is complete, giving SAM-to-OAM transfer with opposite output helicity. Three-dimensional PIC simulations with EPOCH for a $\boldsymbol{\omega}=\nabla\times\mathbf{v}$6 Gaussian input report conversion into a Laguerre-Gaussian-like ring beam with a central null, a strong $\boldsymbol{\omega}=\nabla\times\mathbf{v}$7 component, saturation around $\boldsymbol{\omega}=\nabla\times\mathbf{v}$8, and power conversion efficiency reaching $\boldsymbol{\omega}=\nabla\times\mathbf{v}$9; the same design is described as tunable from terahertz to infrared/optical frequencies (Qu et al., 2017).

Structured optical vortices can also be built on caustic fields. The vortex Pearcey-Gauss beam is

$\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$0

while the vector vortex Pearcey-Gauss beam is a non-separable superposition of two orthogonally polarized vortex components,

$\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$1

Experimentally, an SLM generates the scalar Pearcey-Gauss field and a $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$2-plate converts it into either a scalar VPeG or a VVPeG depending on the input polarization. Stokes polarimetry shows that the VVPeG beam evolves from a pure vector beam into a vector mode of quasi-homogeneous polarisation distribution (Rodríguez-Fajardo et al., 2024).

Acoustic vortices are the corresponding helical-phase solutions in sound fields. A compact metamaterial aperture based on a wrapped acoustic leaky-wave antenna produces

$\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$3

and supports both integer and non-integer modes. The reported experimental modes span $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$4, with fractional states such as $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$5 and $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$6 appearing as wavefront dislocations and paired singular structures (Naify et al., 2016). Selective transmission can then be imposed by an acoustic vortex filter composed of an upper and lower acoustic metasurface. Rotating the upper layer changes the intrinsic topological charge $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$7, and the diffraction law

$\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$8

determines whether a given incident mode remains propagating or becomes evanescent in the waveguide. The demonstrated device filters opposite topological charges by changing the UAM orientation; in the asymmetric-transmission configuration the reported transmission efficiencies are about $\mathbf{R}=\omega_{\text{rot}}^{(v)}\mathbf{T},$9 for left incidence and about $\mathbf{T}$0 for right incidence, corresponding to a transmission-loss difference of $\mathbf{T}$1 (Li et al., 2024).

3. Quantized vortices in condensates and superfluids

In condensates, vortex is a quantized phase defect. For exciton-polariton condensates, the phase winding is an integer multiple of $\mathbf{T}$2, and the stable charges used for information storage are

$\mathbf{T}$3

Off-resonant ring-shaped pumps create robust vortex condensates, and phase locking between nearby rings constrains the charge of later-formed vortices. The reported control primitives are inversion, by turning one pump off and on so that a neighbor re-imprints opposite circulation, and copying, by using a triangular $\mathbf{T}$4-locked geometry so that a reference vortex’s topological charge is replicated in another site. A four-vortex chain storing the binary state $\mathbf{T}$5 is explicitly demonstrated (Ma et al., 2017).

A different driven-dissipative polariton platform uses a homogeneous pump with intensity grooves. A narrow ring groove supports a dark-ring state with a $\mathbf{T}$6-phase jump, but when the surrounding condensate density becomes too large the ring undergoes snake instability and reduces into vortex–antivortex pairs. Multiple vortex-pair states can be stable in the same dark ring; for example, at $\mathbf{T}$7 both the $\mathbf{T}$8 and $\mathbf{T}$9 states are stable, and at $\omega_{\text{rot}}^{(v)}$0 three distinct vortex-pair states coexist (2206.12157). Broader grooves support higher-order dark states that act as vortex waveguides. In a U-shaped guide, an imprinted vortex with $\omega_{\text{rot}}^{(v)}$1 travels counter-clockwise and exits after about $\omega_{\text{rot}}^{(v)}$2, whereas a vortex with $\omega_{\text{rot}}^{(v)}$3 follows the opposite route and requires more than $\omega_{\text{rot}}^{(v)}$4 to reach the left corner before exiting in less than $\omega_{\text{rot}}^{(v)}$5 (2206.12157).

In multicomponent Bose–Einstein condensates, the dominant organizing principle can be the effective pairwise interaction energy $\omega_{\text{rot}}^{(v)}$6 between fixed vortices. Within a constrained Gross–Pitaevskii treatment, attractive inter-component coupling can favor overlapping vortices, mass imbalance can produce a finite-distance dimer minimum at around $\omega_{\text{rot}}^{(v)}$7 and giant-vortex collapse near $\omega_{\text{rot}}^{(v)}$8, and coherent Rabi coupling can stabilize both dimers and trimers (Dantas et al., 2015). For three-component systems with symmetric couplings, the trimer energy $\omega_{\text{rot}}^{(v)}$9 develops a minimum at finite $|\mathbf{R}|>0$0; frustrated Rabi couplings such as $|\mathbf{R}|>0$1 can stabilize bound molecules even without inter-component contact repulsion (Dantas et al., 2015).

The holographic superfluid provides a nonperturbative finite-temperature setting for vortex–anti-vortex annihilation. In a $|\mathbf{R}|>0$2 periodic box, the separation obeys

$|\mathbf{R}|>0$3

with two scaling regimes cleanly separated by the vortex size $|\mathbf{R}|>0$4. For $|\mathbf{R}|>0$5, $|\mathbf{R}|>0$6; for $|\mathbf{R}|>0$7, $|\mathbf{R}|>0$8. At chemical potential $|\mathbf{R}|>0$9 the reported core size is $\mathbf{T}$0, so $\mathbf{T}$1, and the representative annihilation time is $\mathbf{T}$2 (Lan et al., 2018). From these fits the inferred attractive forces are $\mathbf{T}$3 in the first stage and $\mathbf{T}$4 in the second, leading to a two-body decay law $\mathbf{T}$5 at low vortex density (Lan et al., 2018).

4. Formation, instability, and numerical treatment in classical fluids

In wall-bounded flow, vortex organization can be selected by boundary geometry. A laminar vortex ring of diameter $\mathbf{T}$6 and $\mathbf{T}$7 impacting a fakir-like wall of posts first develops competing azimuthal instabilities with $\mathbf{T}$8. On a hexagonal lattice, the impact locks onto a regular $\mathbf{T}$9 mode: a weaker secondary ring is pushed radially outward rather than wrapping around the primary ring, six lobes develop at the outer edge, and counter-rotating vortex pairs at the lobe tips generate six radial wall jets (Li et al., 2018). Rotating the surface rotates the jet orientation, whereas a random lattice with the same porosity fails to produce the same coherent secondary structure (Li et al., 2018).

In transitional boundary layers, Liutex core lines and POD are used to argue against the classical self-deformation picture of hairpin formation. The reported interpretation is that the Lambda-vortex and the ring-like or hairpin vortex are distinct structures: the latter is generated separately by Kelvin–Helmholtz instability in the shear layer above the A-vortex. POD mode 1 contains the streamwise mean structure, while higher modes are spanwise fluctuation modes that occur in pairs with similar amplitudes, shapes, and time evolution, consistent with K–H roll-up (Charkrit et al., 2019).

The mathematical reduction of vorticity dynamics to point-vortex dynamics remains central for ideal two-dimensional flow. In the full plane, the vortex method approximates the vorticity by

$\mathbf{R}$0

and the point positions satisfy an ODE system induced by the Biot–Savart law. In an exterior domain, impermeability and circulation around the obstacle require an additional harmonic correction, and the boundary is discretized by point vortices with strengths chosen so that the flow remains tangent at collocation points and the total vorticity around the obstacle is conserved (Arsénio et al., 2017). The same work gives a rigorous justification for smooth exterior domains and introduces the fluid charge method as an alternative boundary discretization that is better conditioned numerically (Arsénio et al., 2017).

These examples show that vortex dynamics in classical fluids are governed simultaneously by local rotation, instability mechanisms, and nonlocal boundary or geometry constraints. The same object can therefore appear as a coherent ring, a secondary wall-induced structure, a K–H-generated hairpin head, or a singular measure in a numerical discretization, depending on the level of description.

5. Solar, rotating-fluid, and astrophysical vortex systems

Solar vortex tubes are detected as three-dimensional, coherent, coned structures concentrated in intergranular lanes. In a MURaM plage simulation with initial mean field $\mathbf{R}$1, 17 vortices persisting for $\mathbf{R}$2 are reported. Their average radius increases from about $\mathbf{R}$3 at $\mathbf{R}$4 to about $\mathbf{R}$5 at $\mathbf{R}$6, their tangential velocity profile is better fit by a cubic than by a linear law, the tangential speed decreases inward often by about $\mathbf{R}$7, and the magnetic field is stronger at the center than at the boundary by about $\mathbf{R}$8 to $\mathbf{R}$9 (Silva et al., 2020). The same study distinguishes kinematic vortices, magnetic vortices, and cases where rotating plasma only bends or drags field lines; among the three highlighted examples, vortex #12 is the only one that forms and sustains a twisted magnetic flux tube (Silva et al., 2020).

A broader statistical solar survey with Mancha3D and SWIRL examines SSD, unipolar $\mathbf{R}=R\,\mathbf r$0, and unipolar $\mathbf{R}=R\,\mathbf r$1 configurations at resolutions down to $\mathbf{R}=R\,\mathbf r$2. The mean vortex radii are reported in the range $\mathbf{R}=R\,\mathbf r$3 to $\mathbf{R}=R\,\mathbf r$4, the number of detections has a minimum around $\mathbf{R}=R\,\mathbf r$5 height, stronger magnetic fields and higher resolutions produce more vortices, and the area fraction occupied by vortices increases with height up to about $\mathbf{R}=R\,\mathbf r$6–$\mathbf{R}=R\,\mathbf r$7 in the highest-resolution SSD runs and up to $\mathbf{R}=R\,\mathbf r$8–$\mathbf{R}=R\,\mathbf r$9 in the unipolar cases (Pistarini et al., 3 Apr 2025). Vortices are systematically hotter than the domain mean; heating is dominated by viscosity, and vortex heating exceeds the domain mean mainly below about $e^{iL\phi}$00, which the authors interpret as lower-atmosphere dissipation followed by upward transport of heated plasma (Pistarini et al., 3 Apr 2025).

In rotating laboratory fluid, an axisymmetric inertial-wave attractor can feed a slow vortex manifold. In an annular domain with conical bottom and oscillating lid forcing, the early state is a quasi-linear axisymmetric $e^{iL\phi}$01 inertial-wave attractor established after roughly $e^{iL\phi}$02, but after times of order $e^{iL\phi}$03 a saturated nonlinear regime emerges in which the slow manifold organizes into eight cyclonic vortices in a regular polygon (Boury et al., 2020). The cluster precesses cyclonically around the inner cylinder, with one full turn taking about $e^{iL\phi}$04, and the saturated regime exhibits a strong cyclonic–anticyclonic asymmetry in the probability density function of vertical vorticity (Boury et al., 2020).

In protoplanetary discs, a long-lived anticyclonic vortex acts as a dynamical perturber as well as a dust trap. Hydrodynamical simulations with a Gaussian pressure bump that triggers the Rossby Wave Instability show that low-mass planets migrating toward the vortex are either locked to it or stopped farther away when the vortex is stronger (Ataiee et al., 2014). In the standard case the planet becomes trapped at a constant angular separation typically about $e^{iL\phi}$05 from the vortex center; even embryos with mass less than $e^{iL\phi}$06 can be expelled from inside the vortex and then settle into the same locked configuration (Ataiee et al., 2014). This suggests a dual role for the vortex as embryo birthplace and orbital shepherd.

6. Magnetic vortices, clustered matter, and device-oriented implementations

Ferromagnetic superconductors modify vortex physics by coupling vortices to magnetic moments. In the London description with Zeeman term $e^{iL\phi}$07, magnetic polarization generates a long-range attractive contribution

$e^{iL\phi}$08

which competes with short-range London repulsion and long-range Pearl repulsion (Lin et al., 2012). The resulting nonmonotonic interaction stabilizes circular vortex clusters at low density, and the Meissner-to-vortex-cluster transition at $e^{iL\phi}$09 becomes first order (Lin et al., 2012). Under Lorentz drive, moving clusters excite magnons; beyond the threshold

$e^{iL\phi}$10

the magnetic response becomes unstable, domain walls are generated, and the vortex configuration evolves into stripe-like modulated states with enhanced viscosity and clear transport signatures in the I–V curve (Lin et al., 2012).

Confined magnetic vortices in Permalloy dots exhibit a different form of symmetry-sensitive dynamics. A centered vortex has near cylindrical symmetry, but an in-plane bias field displaces the core and breaks that symmetry. Broadband ferromagnetic resonance and OOMMF simulations show two distinct regimes: a stable vortex state for $e^{iL\phi}$11 and a metastable vortex state for $e^{iL\phi}$12 (0812.4954). In the stable state, the $e^{iL\phi}$13 and $e^{iL\phi}$14 doublets split in frequency; in the metastable state, new eigenmodes appear and depend strongly on whether the rf drive is parallel or perpendicular to the bias field (0812.4954). The response is therefore controlled by confinement, broken cylindrical symmetry, and pumping geometry rather than by topology alone.

Rapidly rotating gas vortices can be used directly as optical elements. Interferometric measurements on compressible, viscous gas vortices show that a short $e^{iL\phi}$15–$e^{iL\phi}$16 aperture acts as a gas lens, while a longer $e^{iL\phi}$17 structure functions as a guiding channel prototype (Kaganovich et al., 2020). Near the axis, the Burgers-vortex-like core has azimuthal velocity

$e^{iL\phi}$18

and the density profile is approximately quadratic in $e^{iL\phi}$19, producing an approximately parabolic phase front and a negative lens (Kaganovich et al., 2020). The focal length is extracted from the defocus Zernike coefficient through

$e^{iL\phi}$20

the useful core radius in the reported example is about $e^{iL\phi}$21, and choking occurs below roughly $e^{iL\phi}$22–$e^{iL\phi}$23, where both mass flow rate and core optical parameters saturate (Kaganovich et al., 2020). The calibrated density profiles can be tuned into the $e^{iL\phi}$24 range relevant to plasma wakefield and laser-guiding applications (Kaganovich et al., 2020).

Taken together, these systems show that vortex is not merely a descriptor of swirling motion. It is also a design principle for memory elements, wave converters, filters, resonant magnetic media, and gaseous optical components. The cited work supports a unified but technically strict view: a vortex is a structured rotational or phase-singular entity whose identity depends on the governing field, while its utility depends on how precisely its core, charge, symmetry, and interactions can be controlled.