Vortex stretching and reconnection in a compressible fluid

Vortex stretching in a compressible fluid is considered. Two-dimensional (2D) and axisymmetric cases are considered separately. The flows associated with the vortices are perpendicular to the plane of the uniform straining flows. Externally-imposed density build-up near the axis leads to enhanced compactness of the vortices — “dressed" vortices (in analogy to “dressed" charged particles in a dielectric system). The compressible vortex flow solutions in the 2D as well as axisymmetric cases identify a length scale relevant for the compressible case which leads to the Kadomtsev-Petviashvili spectrum for compressible turbulence. Vortex reconnection process in a compressible fluid is shown to be possible even in the inviscid case — compressibility leads to defreezing of vortex lines in the fluid.     

Experimental observations of transverse shear waves in strongly coupled dusty plasmas

We report experimental observations of transverse shear waves in a three-dimensional dusty plasma that is in the strongly coupled fluid regime. These spontaneous oscillations occur when the ambient neutral pressure is reduced below a threshold value and the measured dispersion characteristics of these waves are found to be in good agreement with predictions of a viscoelastic theory of dusty plasmas.     

On the dielectric response function and dispersion relation in strongly coupled magnetized dusty plasmas

Using the generalized viscoelastic fluid model, we derive the dielectric response function in a strongly coupled dusty magnetoplasma which reveals two different dust acoustic(DA) wave modes in the hydrodynamic and kinetic limits. The effects of the strong interaction of dust grains and the external magnetic on these DA modes, as well as on the shear wave are examined. It is found that both the real and imaginary parts of DA waves are significantly modified in strongly coupled dusty magnetoplasmas. The implications of our results to space and laboratory dusty plasmas are briefly discussed.     

Solitary wave complexes in two-component condensates

Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intracomponent but varying intercomponent interaction strengths. Several families of solitary wave complexes are found: (1) vortex rings of various radii in each of the components; (2) a vortex ring in one component coupled to a rarefaction solitary wave of the other component; (3) two coupled rarefaction waves; (4) either a vortex ring or a rarefaction pulse coupled to a localized disturbance of a very low momentum. The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative differences between the chemical potentials of two components. Solitary wave formation, their stability, and solitary wave complexes in two dimensions are discussed.     

Vortex stretching as a mechanism for quantum kinetic energy decay

A pair of perturbed antiparallel quantum vortices, simulated using the three-dimensional Gross-Pitaevskii equations, is shown to be unstable to vortex stretching. This results in kinetic energy K(?ψ) being converted into interaction energy E(I) and eventually local kinetic energy depletion that is similar to energy decay in a classical fluid, even though the governing equations are Hamiltonian and energy conserving. The intermediate stages include the generation of vortex waves, their deepening, multiple reconnections, the emission of vortex rings and phonons, and the creation of an approximately -5/3 kinetic energy spectrum at high wave numbers. All of the wave generation and reconnection steps follow from interactions between the two original vortices. A four vortex example is given to demonstrate that some of these steps might be general.     

Steady vortices in plasmas and geophysical flows

A number of two-dimensional fluid models in geophysical fluid dynamics and plasma physics are examined to find out whether they have steady and localized monopole vortex solutions. A simple and general method that consists of two steps is used. First the dispersion relation is calculated, to find all possible values of the phase velocity of the linear waves. Then an integral relation that determines the center-of-mass velocity of localized structures must be found. The existence condition is that this velocity should be outside the region of linear phase velocities. After a presentation of the method, previous work on the plasma drift wave model and the shallow-water equations is reviewed. In both cases it is found that the center-of-mass velocity is larger than the maximum phase velocity of the linear waves if the amplitude is large enough, and steady localized vortices can therefore exist. New results are then obtained for a number of two-field models. For the coupled ion acoustic-drift modes in plasmas, it is found that the center-of-mass velocity depends on the ratio between the parallel ion velocity component and the electrostatic potential in the vortex. If this ratio is large enough, the vortex can be steady. For the drift-Alfven mode the "center-of-charge" velocity is proportional to the ratio between the parallel current and the total charge in the vortex. It can therefore be steady if this ratio satisfies the appropriate conditions. For the quasigeostrophic two-layer equations, describing stratified flow on a rotating planet, it is found that the center-of-mass velocity is determined by the ratio between the baroclinic and the barotropic components in the vortex. If a baroclinic component with an appropriate sign is added to a barotropic vortex, it propagates faster than the barotropic Rossby waves, and can be steady. Finally, the existence conditions for a vortex in an external zonal flow are examined. It is found that the center-of-mass velocity acquires an additional westward contribution in an anticyclonic shear zone in the framework of the shallow-water equations, and also that an easterly jet south of this shear zone partly shields a vortex situated in the shear zone from the dispersive influence of the fast Rossby waves on the equatorward side.     

Driven transverse shear waves in a strongly coupled dusty plasma

The linear dispersion properties of transverse shear waves in a strongly coupled dusty plasma are experimentally studied in a DC discharge device by exciting them in a controlled manner with a variable frequency external source. The dusty plasma is maintained in the strongly coupled fluid regime with (1<Γ?Γc) where Γ is the Coulomb coupling parameter and Γc is the crystallization limit. A dispersion relation for the transverse waves is experimentally obtained over a frequency range of 0.1 Hz to 2 Hz and found to show good agreement with viscoelastic theoretical results.     

强耦合尘埃等离子体中正电扰动下的尘埃声激波

研究了强耦合尘埃等离子体的尘埃声波的线性色散关系和尘埃声孤波的非线性传播。考虑一个包含电子、离子、正电扰动尘埃颗粒的完全电离的三成分模型等离子体。假定其电子、离子数密度服从玻尔兹曼分布,而大质量的尘埃成分用一组经典流体方程描述,对系统方程进行线性化,得到了尘埃声波的线性色散关系,发现离子的集中参数对色散关系的影响很大。用约化摄动法对系统方程进行展开,得到了描述小振幅孤波的伯格斯方程。基于伯格斯方程研究了尘埃声孤波的基本特性,发现尘埃颗粒的强耦合效应对尘埃声孤波有很大的修正作用。该研究结果有助于理解尘埃空间等离子体中局域波的一些特性。     

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研究了强耦合尘埃等离子体的尘埃声波的线性色散关系和尘埃声孤波的非线性传播。考虑一个包含电子、离子、正电扰动尘埃颗粒的完全电离的三成分模型等离子体。假定其电子、离子数密度服从玻尔兹曼分布,而大质量的尘埃成分用一组经典流体方程描述,对系统方程进行线性化,得到了尘埃声波的线性色散关系,发现离子的集中参数对色散关系的影响很大。用约化摄动法对系统方程进行展开,得到了描述小振幅孤波的伯格斯方程。基于伯格斯方程研究了尘埃声孤波的基本特性,发现尘埃颗粒的强耦合效应对尘埃声孤波有很大的修正作用。该研究结果有助于理解尘埃空间等离子体中局域波的一些特性。     

Nonlinear dynamics of surface dust vortex and dust zonal flow systems

We present analytical and simulation studies of highly resolved dust fluid flows involving nonlinearly coupled incompressible surface dust vortex modes (SDVMs) and dust zonal flows (DZFs) in nonuniform unmagnetized dusty plasmas. For this purpose, we use the hydrodynamic equations for the dust fluid and Boltzmann distributed electrons and ions and obtain a set of equations that exhibit nonlinear couplings between the SDVMs and DZFs. The nonlinear equations are then used to investigate the parametric excitation of DZFs by the Reynolds stresses of the SDVMs. Large scale SDVMs emerge through nonlinear interactions with DZFs, and they suppress the dust particle transport across the density gradient. In contrast, DZFs possess short scale vortices with a higher turbulent transport. The relevance of our investigation into the role of coherent structures in a nonuniform dusty plasma is discussed.     

Compressible turbulence: Multi-fractal scaling in the transition to the dissipative regime

Multi-fractal scaling in the transition to the dissipative regime for fully-developed compressible turbulence is considered. The multi-fractal power law scaling behavior breaks down for very small length scales thanks to viscous effects. However, the effect of compressibility is found to extend the single-scaling multi-fractal regime further into the dissipative range. In the ultimate compressibility limit, thanks to the shock waves which are the appropriate dissipative structures, the single-scaling regime is found to extend indeed all the way into the full viscous regime. This result appears to be consistent with the physical fact that vortices become more resilient and stretch stronger in a compressible fluid hence postponing viscous intervention. The consequent generation of enhanced velocity gradients in a compressible fluid appears to provide an underlying physical basis for the previous results indicating that fully-developed compressible turbulence is effectively more dissipative than its incompressible counterpart.     

Effect of Second Nonthermal Ion on the Characteristics of Dust Acoustic Solitary Waves in a Magnetized Dusty Plasma with Variable Dust Charge

The nonlinear dust acoustic solitary waves in a magnetized dusty plasma with nonthermal ions and variable dust electric charge is studied analytically. Using reductive perturbation method the Zakharov‐Kuznetsov (ZK) equation is derived and effect of nonthermal coefficient, external magnetic field, and variable dust electric charge on the amplitude and width of soliton in dusty plasma is investigated. With increasing the rate of dust charge variation with respect of plasma potential, the amplitude of generated solitary waves in magnetized dusty plasma increases to a constant magnitude while its width decreases. Increasing the nonthermal ions coefficient leads to a noticeable decrease in the amplitude of solitons while the width of soliton increases. The amplitude of generated solitary waves in such a dusty plasma is independent of applied external magnetic field but we will have more localized solitons with increasing the external magnetic field strength. It is found that solitons are strongly influenced by the direction of external magnetic field. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

入射和反射激波诱导重气泡变形和失稳的三维数值研究

 采用大涡模拟方法,对入射激波及其反射激波诱导球形重气泡的变形失稳过程进行了三维数值模拟,利用已有实验验证了计算模型的可靠性,重点考察了反射激波与已经失稳的气泡界面的再次作用,讨论了涡环的形成及其三维失稳的过程。研究结果显示：入射和反射激波与球形重气泡作用产生斜压效应,会在流场中产生旋转方向截然相反的多个涡环;反射激波诱导的涡环具有较小的强度,故更加容易失稳,甚至能完全形成具有流向涡量的复杂小尺度涡结构。     

Numerical Simulations of Two Coaxial Vortex Rings Head-on Collision

Vortex rings have been a subject of interest in vortex dynamics due to a plethora of physical phenomena revealed by their motions and interactions within a boundary. The present paper is devoted to physics of a head-on collision of two vortex rings in three dimensional space, simulated with a second order finite volume scheme and compressible. The scheme combines non-iterative approximate Riemann-solver and piecewise-parabolic reconstruction used in inviscid flux evaluation procedure. The computational results of vortex ring collisions capture several distinctive phenomena. In the early stages of the simulation, the rings propagate under their own self-induced motion. As the rings approach each other, their radii increase, followed by stretching and merging during the collision. Later, the two rings have merged into a single doughnut-shaped structure. This structure continues to extend in the radial direction, leaving a web of particles around the centers. At a later time, the formation of ringlets propagate radially away from the center of collision, and then the effects of instability involved leads to a reconnection in which small-scale ringlets are generated. In addition, it is shown that the scheme captures several experimentally observed features of the ring collisions, including a turbulent breakdown into small-scale structures and the generation of small-scale radially propagating vortex rings, due to the modification of the vorticity distribution, as a result of the entrainment of background vorticity and helicity by the vortex core, and their subsequent interaction.     

Propagation of orifice- and nozzle-generated vortex rings in air

The effect of the exit geometry of a vortex ring generator was studied experimentally. Two types of exit geometries were chosen: an orifice and a nozzle. Vortex rings were generated by pushing a solenoid-valve-controlled, pressurized-air jet through the circular opening of the orifice or nozzle. Experiments were performed over a wide range of initial Reynolds number (450≤Re≤4580) and length-to-diameter ratio (0.7≤L/D≤7.0) of the air jet. The exit geometry was found to significantly influence the entire course of propagation of the vortex ring. The orifice-generated vortex ring had superior characteristics to that produced by the nozzle under the same conditions. The vorticity generated along the wall in the orifice exit plane had a negligible effect on the circulation of the vortex ring within the specified range of Reynolds number. Compared to the nozzle-generated vortex ring, the orifice-generated ring showed reduced initial vorticity losses and less diffusive entrainment of ambient fluid. The vortex rings produced by the orifice attained more circulation, less entrainment of ambient fluid and hence rapidly propagated through longer distances in comparison to the nozzle-generated rings.     

Physics of multiple level hairpin vortex structures in turbulence

Previous experimental and numerical studies have revealed that the hairpin vortex is a basic flow element of transitional boundary layer. The hairpin vortex is believed to have legs, necks and a ring head. Based on our DNS study, the legs and the ring head are generated separately by different mechanisms. The legs function like an engine to generate low speed zones by rotation, create shear layers with surrounding high speed neighbor fluids, and further cause vortex ring formation through shear layer instability. In addition, the ring head is ?-shaped and separated from quasi-streamwise legs from the beginning. Contrary to the classical concept of "vortex breakdown", we believe transition from laminar flow to turbulence is a "buildup" process of multiple level vortical structures. The vortex rings of first level hairpins are mostly responsible for positive spikes, which cause new vorticity rollup, second level vortex leg formation and finally smaller second level vortex ring generation. The third and lower level vortices are generated following the same mechanism. In this paper, the physical process from ?-vortex to multi-level hairpin vortices is described in detail.     

Experimental observation of dust circulation in unmagnetized cogenerated dusty plasma

The vortex motion of a dust cloud was experimentally observed in unmagnetized cogenerated dusty plasma in different experimental parameters. Particle image velocimetry analysis demonstrated that several vortex zones exist in the dust cloud at relatively low pressures (0.06 mbar (or 6 Pa)–0.08 mbar (or 8 Pa)) and low discharge voltages (peak‐to‐peak voltage 540–560 V), whereas in relatively high pressure (0.4 mbar (or 40 Pa)–0.7 mbar (or 70 Pa)) and high discharge voltage (peak‐to‐peak voltage 690–740 V), dust vortices formed in dense dust cloud with background plasma fluctuation.     

Observation of vector solitons with hidden vorticity

This letter reports the first experimental observation, to our knowledge, of optical vector solitons composed of two incoherently coupled vortex components. We employ nematic liquid crystal to generate stable vector solitons with counterrotating vortices and hidden vorticity. In contrast, the solitons with explicit vorticity and corotating vortex components show azimuthal splitting.     

Self-consistent turbulence in the two-dimensional nonlinear Schrödinger equation with a repulsive potential

The dynamics of dark solitons (vortices) with the same topological charge (vorticity) in the two-dimensional nonlinear Schr?dinger (NLS) equation in a defocusing medium is studied. The dynamics differ from those in incompressible media due to the possibility of energy and angular momentum radiation. The problem of the breakup of a multicharged dark soliton, which is a local decrease of the wave function intensity, into a number of chaotically moving vortices with single charge, is studied both analytically and numerically. After an initial period of intensive wave radiation, there emerges a nonuniform, steady turbulent self-organized motion of these vortices which is restricted in space by the size of the potential well of the initial multicharged dark soliton. Separate orbits of finite widths arise in this turbulent motion. That is, the statistical probability to observe a vortex in a given point has maxima near certain points (orbit positions). In spite of the fact that numerical calculations were performed in a finite region, the turbulent distributions of the vortices do not depend on the size of the container when its radius is larger than the size of the potential well of the primary multicharged dark soliton. The steady turbulent distribution of vortices on these orbits can be obtained as the extremal of the Lyapunov functional of the NLS equation, and obeys some simple rules. The first is the absence of Cherenkov resonance with linear (sound) waves. The second is the condition of a potential energy maximum in the region of vortex motion. These conditions give an approximately equidistant disposition of orbits of the same number of vortices on each orbit, which corresponds to a constant rotating velocity. The magnitude of this velocity is mainly determined by the sound velocity. An integral estimation of the self-consistent rotation of the vortex zone is given.     

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called Oseens vortex. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.Acknowledgement The first author is indebted to J. Dolbeault and, especially, to C. Villani for suggesting the beautiful idea of using the Boltzmann entropy functional in the context of the two-dimensional Navier-Stokes equation. The research of C.E.W. is supported in part by the NSF under grant DMS-0103915.