The study of vortices began in 1858 with Herman von Helmholtz's renowned vorticity theorems,1 followed by contributions by James Clerk Maxwell and William Thomson (Lord Kelvin). In the early 1860s, Maxwell extended Kelvin's fluid flow analogy for electrostatics to model Faraday's lines of force and derive the equations of electromagnetism.2 Maxwell later abandoned this heuristic model but retained the mathematical analogy between the equations of the electromagnetic field and vortex dynamics.3–5 In 1867, building on Helmholtz's observation1 that vortex rings form permanent structures in an incompressible, inviscid fluid, Kelvin proposed that molecules or atoms could be vortex rings in the ether. After Helmholtz's work, the theme of vortices became a significant topic in fluid motion theory. It was the focus of Hicks' report for the British Association for the Advancement of Science6 and the subject of Joseph John Thomson's Adams Prize-winning contribution on two closed vortices in an incompressible fluid.7
Coherent vortical structures are common in nature and appear in planetary atmospheres and oceans in laboratory experiments with rotating fluids and pure electron plasma confined by a magnetic field. The study of vortices is fascinating because similar vortex dynamics might explain phenomena such as Saturn's hexagonal polar vortex,8–10 suction vortices in tornadoes,11 mesovortices in tropical cyclones,12,13 and rotating plumes in volcanic eruptions.14
Coherent vortical structures are a characteristic feature of two-dimensional or quasi-two-dimensional flows.15 These structures last far longer than their rotational period, trapping and transporting particles over distances far exceeding their size. Thus, they are crucial for moving mass, heat, and momentum across large areas. They are produced in laboratory experiments with rotating and stratified fluids.16–18 Additionally, they emerge naturally in two-dimensional turbulence due to self-organization processes typical of two-dimensional flows.19 Notably, many features of two‐dimensional (2D) vortices in an inviscid fluid are found in confined columns of electrons confined by a magnetic field.20
Magnetically confined columns of electrons are sought as one of the best experimental realizations of ideal 2D fluids and two-dimensional vortices in an inviscid fluid because the electron system has no boundary layers to dissipate the vortices and shows little dissipation. The dynamics of two-dimensional vortices in an inviscid, ideal fluid continue to be the subject of theoretical investigation in various settings. For instance, in two-dimensional turbulence, merging extended vortices with the same sign of vorticity has been proposed as a key element in the decay of turbulence in 2D flows.21 In this regard, electron systems offer an excellent setting to study vortex merging and turbulence in 2D flows. The non-neutral plasma system22 also offers experimental insight into shear-induced vortex generation, dynamics, and dissipation.23
This special joint issue in the Physics of Fluids and the Physics of Plasmas highlights recent breakthroughs and emerging trends in coherent vortical structures within fluids and plasmas. This collection includes 17 papers published in Physics of Fluids and Physics of Plasmas, covering topics related to vortices, their decay, and their instability. The coherent vortical structures in quasi-two-dimensional (2D) fluid dynamics and their analog, pure electron plasmas containing columns of electrons confined by a magnetic field, occupy a large part of the Special Issue, indicating that quasi-two-dimensional (2D) fluid dynamics remains a subject of primary interest among the fluid dynamics and plasma physics communities.
The contribution by Dubin et al.24 is an example of the electron system being applied to tackle the phenomena of down scattering, one of the embodiments of self-organization through the axisymmetrization of a single isolated vortex.25–31 Dubin et al. studied the decay of Kelvin/diocotron wave into two waves: a daughter Kelvin/diocotron wave and a “beat wave.” This down scattering involves intriguing nonlinear wave coupling, described by Dubin et al. from a different perspective, based on an eigenmode expansion of the system dynamics. The new theory by Dubin et al. agrees with the experiment, thus enriching the known processes of down scattering through spatial Landau damping (a “direct” resonance).32,33
Ticoş et al.34 explored the issue of dusty plasmas in low-temperature laboratories. They demonstrated that a mono-energetic pulsed e-beam with 13 keV energy and a 30 mA peak current per pulse could create a laminar dust particle flow in a quasi-2D PC within an RF-driven plasma sheath. This flow splits into two counterstreaming branches, forming symmetrical vortices. They initially recorded a peak flow speed of 12 mm/s using PIV, dropping to 5–6 mm/s at steady state. Laminar flow was confirmed through particle tracking velocimetry (PTV) by tracking individual dust trajectories, showing vorticity peaks of 63.8 s−1 on both sides of the e-beam. Molecular dynamics simulations qualitatively supported these findings for similar conditions. These e-beam-induced dust flows can be testbeds for studying energy and vorticity dissipation in charged flows and are relevant to fusion plasmas.
The paper by Voitiv et al.35 on optical vortices with hollow intensity distribution is similar to fluid vortices with a central flow singularity.36 It falls under the analogy between laser physics and superfluid, which followed the reduction of laser equation into complex Ginzburg–Landau equations.37 This equation describes pattern formation in superconductivity, superfluidity, and Bose–Einstein condensation.38 Voitiv and co-workers expanded the scope and implications of trapping optical vortices in Bessel beams39 and have demonstrated how to generate arbitrary vortex configurations and trap them in the divergence-free nature of Bessel beams. The authors presented experimental demonstrations with the theoretical proposal to validate the potential application of Bessel hard-wall traps as testing grounds for engineering few-body vortex interactions within trapped, two-dimensional compressible fluids. Thanks to the analogies mentioned earlier, the Bessel hard trap dynamics proposed by Voitiv et al. might overcome the limitation of the harmonic trap predictions, which allow nucleation of other vortices because of the hard-wall boundary and steep phase gradients. The paper by Voitiv et al. points out the potential application of hard-wall traps as testing grounds for engineering few-body vortex interactions within trapped, two-dimensional compressible fluids.
The paper by McQueen II and Lim40 used the vortex gas approximation to study low positive Onsager temperature states for the vortex gas model on a 2D annular domain. They found evidence for edge modes. The vorticity in vortex gas is approximated by a collection of interacting point particles forming a Hamiltonian system. This model uses concepts of statistical mechanics to explain the macroscopic behavior of gases. The study may benefit research on turbulence in confined plasma systems and atmospheric or astrophysical phenomena involving multiple vortices. Additionally, exploring the relationship between vortex models and Chern numbers in different scenarios could be valuable.
Wadas et al.41 studied the interaction between a shock wave and a heavy gaseous cylinder, which results in clumps of the cylinder material. The interaction initially causes the formation of a vortex dipole and the growth of three-dimensional perturbations along the core due to the Crow instability. The authors aimed to assess the feasibility of an experiment using a laser-driven shock wave to observe the growth of the Crow instability along a heavy cylinder, like the setup at the Omega EP laser facility. They conducted a linear stability analysis based on geometric parameters from two-dimensional simulations and verified their findings with three-dimensional simulations. The study confirmed the development of the instability and the pinch-off of the vortex dipole into isolated vortex rings, visible as clumps of the original cylinder material. The authors also performed a scaling analysis to determine the relevant spatiotemporal scales of the instability development, showing that a shocked cylinder with specific initial parameters is expected to form clumps due to the Crow instability approximately 40 ns after being shocked, which can be observed at the Omega EP laser facility.
The paper by Hurst et al.42 presents an example of using the similarities between the Drift–Poisson equations, which describe plasmas, and the Euler equations, which describe ideal fluids. The experiments involved using pure electron plasmas in a Penning–Malmberg trap to study the issue of elliptical, quasi-two-dimensional (2D) fluid vortices splitting. The authors showed that the aspect ratio threshold for vortex splitting is significantly higher for vortices with realistic, smooth edges than that predicted by a simple “vortex patch” model, where the vorticity is treated as a piecewise constant inside a deformable boundary. The authors used a particle-in-cell method to perform numerical simulations, model the evolution of point vortices, and conduct experiments. Hurst et al. discovered that the aspect ratio splitting threshold could be up to twice the one predicted for the vortex patch model and is influenced by the edge vorticity gradient. The authors think spatial Landau damping plays a key role in the splitting mechanic through stabilizing Love modes on the vortex edge, resulting in a decreased vortex aspect ratio over time. The study by Hurst et al. will find a wide range of applications in quasi-2D fluid systems, including geophysical fluids, astrophysical disks, and drift-wave eddies in tokamak plasmas.
Dharodi and Kostadinova43 studied the classic problem of vortex merging, which is relevant to various fields such as hydrodynamic fluids, geophysical flows, plasma flows, astrophysical systems, and aeronautics within the context of a strongly coupled dusty plasma medium. The authors modeled the problem as a two-dimensional merging between two Lamb–Oseen co-rotating vortices in a viscoelastic fluid. The authors considered several parameters influencing the merging phenomenon, such as the aspect ratio (core size/separation distance), the relative circulation strengths of each vortex, and the coupling strength of the medium. Researchers discovered that shear waves in viscoelastic fluids help merge two vortices, even when they are far apart. They also found that the merging process is accelerated in media with higher coupling strengths, and the resulting vortex decays rapidly. The authors reported that scale and vortex circulation do not affect the merging process.
Wani et al.44 investigated turbulence mixing caused by the Rayleigh–Taylor instability in a two-dimensional, strongly coupled dusty plasma system through classical molecular dynamics simulation. The authors examined the entire evolution cycle of the thermalization process, encompassing the initial equilibrium, instability, turbulent mixing, and a new equilibrium through energy spectra. They identified 2D buoyancy-driven turbulent features at smaller wavenumbers using the fully developed spectrum based on Bolgiano–Obukhov scaling. At higher wavenumbers, the energy spectrum representing the system's thermalization was proportional to the wave number, a characteristic feature of 2D Euler turbulence. Wani et al. reported that the system energy spectrum exhibits the Kolmogorov scale at longer timescales. The authors discovered that strong coupling in Yukawa fluids and similar plasmas hinders thermalization and turbulent mixing in fluids with low Reynolds numbers.
Lu et al.45 used a large eddy simulation method to study Taylor–Couette–Poiseuille flow in an annular gap with supercritical carbon dioxide as the working fluid. They observed that, initially, the flow showed small vortices, spiral ring vortices, and annular vortices at the inlet region along the flow direction. As the flow progressed, these small vortices transformed into hairpin swirl vortices before being disrupted by turbulent flow disturbances and merging into spiral and annular vortices. This resulted in a flow field characterized by high-frequency hairpin swirl vortices and small vortices with strong randomness. Additionally, they noted that increasing the swirl number shifted the Taylor vortex toward the inlet and increased the turbulent kinetic energy at the outer wall side rather than at the inner wall side. Furthermore, vortices undergo stabilization, diffusion, and mixing as they move downward in the flow direction. The radius ratio affects the strength of vorticity, reduces velocity fluctuations consistently, and changes the distribution of helicity bands from wide and scattered to compact and densely grouped. As the axial Reynolds number increases, the strength of the vortices increases, leading to more pronounced velocity fluctuations and the transformation of the helicity bands from a consistent ring pattern to fluctuating bands of vortices, accompanied by a decrease in helicity.
Davies et al.46 studied the dynamics of vortex dipoles, which are pairs of opposite-sign vortices, and their long-term evolution. Vortex dipoles transport fluid trapped inside their cores across large distances, and they are isolated and long-lived coherent vortical structures found in oceans and atmospheres. Davies et al. used high-resolution numerical simulations to shed light on the self-organization and rich dynamics of dipolar vortices through the evolution of Larichev–Reznik dipoles in an equivalent-barotropic quasi-geostrophic beta-plane model. Davies et al. have discovered new mechanisms for the breakdown of dipole evolution caused by Rossby wave radiation. This breakdown occurs when the initial tilt of the dipole toward the zonal direction is too large or when the dipole is too weak. Otherwise, dipoles adjust to different states by drifting eastward via damped oscillations. The authors discussed the prevalence of two mechanisms they discovered: (1) spontaneous dipole instability due to a growing critical linear mode and (2) meridional separation of dipole partners that accumulates over the adjustment period and prevents the above instability. The prevalence depends on the initial tilt and dipole strength.
The study by Fang et al.47 used a combination of large eddy simulation and the dynamic mode decomposition (DMD) method to analyze the cavitation cloud structures and shedding frequencies of vortex structures in jets from Helmholtz, organ pipe, and Venturi nozzles. The goal of the study is to provide guidance for using jets with different properties in various engineering applications. According to the authors, turbulent kinetic energy convergence sustains the coherent structure and the formation of cavitation bubbles in three distinct stages: priming, expansion, and collapse. The nozzle type determines the shape of the primary cavitation bubbles. The study reported that organ pipe nozzle jets exhibit a high peak velocity at the center axis, and the vortex structures are stable and only stretch downstream before collapsing, like vortex structures from other nozzles. The authors also found that the jets generated by the three nozzles have similar static DMD modes. Still, Helmholtz nozzles produce jets with higher energy and periodically shed small-scale vortex structural modes. These modes are coupled to the static or base flow field, resulting in quasiperiodic oscillations of the Helmholtz nozzle jets. The periodic oscillation effect of the Helmholtz nozzle jets is higher than that of the other nozzle jets. Additionally, the study reported that high-energy modes of Venturi nozzle jets exhibit anisotropic and small-scale vortex structures, and Venturi nozzle jets demonstrate good dispersion and cavitation properties.
The study by Choi48 is part of the research on the multi-scale interaction among different collective structures in magnetized fusion plasmas across diverse spatiotemporal scales. Choi48 utilized gyrokinetic theory to examine the long-term collisionless damping of a self-generated monopolar E × B vortex flow in a tokamak magnetic island. The study showed that the magnetic precession-induced coupling of the monopolar vortex to the island geodesic acoustic mode (IGAM) results in long-term oscillatory damping. Furthermore, it demonstrated that the IGAM signal is distinct from GAM and sound wave signals and can indicate turbulence invasion into the tokamak magnetic island.
El-Tantawy and his team offered insights on nonlinear ion-acoustic solitary waves (IASWs). While this may seem unrelated to coherent vortical structures in fluids and plasma, readers might still be interested in the single- and multi-soliton solutions they presented. For instance, Almas et al.49 studied the oblique propagation of ion-acoustic solitary waves (IASWs) with arbitrary amplitude in a magnetoplasma. The magnetoplasma consisted of inertialess non-Maxwellian (nonthermal) electrons, inertialess Maxwellian positrons, and inertial adiabatically heated ions. The authors showed the simultaneous coexistence of compressive and rarefactive IASWs, significantly influenced by positrons and nonthermal electron parameters. In their study, Alyousef et al.50 investigated the characteristics of nonlinear large-amplitude ion-acoustic waves in a non-Maxwellian plasma. They specifically looked at the impact of inertialess electron concentration, electron beam concentration, and electron beam velocity on the existence of solitary waves (SWs). The authors also examined the properties of ion-acoustic SWs (IASWs). They demonstrated that the Cairns distribution of electrons could significantly influence the characteristics of large-amplitude IASWs, depending on the parameter controlling the distribution (inertialess electron nonthermal). Furthermore, they discussed the unique features of solitary waves and their existence in relation to relevant plasma parameters. Shohai and colleagues51 studied how trapping relativistically degenerate electrons influences the creation and interaction of nonlinear ion-acoustic solitary waves (IASWs) in quantum plasmas. The authors provided the formula for the number density of electrons in a state of relativistic degeneracy and an analysis of the non-relativistic and ultra-relativistic scenarios. Using the reductive perturbation, the authors obtained the Korteweg–de Vries (KdV) equation to analyze the properties of the IASWs and obtain single- and multi-soliton solutions for the KdV equation with Hirota bilinear formalism. The amplitude of the IASWs is found to be maximum for the non-relativistic, intermediate for the ultra-relativistic, and minimum for the fully relativistic limit. Most importantly, it is found that the fastest interaction occurs in the non-relativistic limit and the slowest in the fully relativistic limit.
The resemblance between magnetically confined electron columns, ideal 2D fluids, and vortices in a fluid supports integrating the Physics of Plasmas and Physics of Fluids on vortical structures. This Special Issue aims to enable both communities to collaborate and deepen their understanding of vortex stability and decay. The Special Issue indicates a need for experimental confirmation of the fundamental findings reported in this Special Issue, and the bridges between the two communities need to be built. The critical question is what lessons fluid submissions offer to the plasma-physics community and vice versa. While various insights have emerged from combining these subfields, the guest editors encourage readers to find further opportunities for mutual learning.
The guest editors thank all contributors for making this Special Issue a rich and compelling collection of articles. The article collection offers a snapshot of the current research structure in the two fields of physics and hints at future developments in these active areas.
The guest editors extend their gratitude to all the authors, the editorial boards of Physics of Fluids and Physics of Plasmas, particularly to Editor-in-Chiefs Professor Alan Jeffrey Giacomin and Professor Michael E. Mauel, and journal managers Mark Paglia and Brian Solis for their support and efforts.
