Straight Vortex Line Research Articles

Turbulent transport of a tracer: An electromagnetic formulation

We present a formulation of the problem of advection and diffusion of a passive tracer by an arbitrary, incompressible velocity field. A Wiener path integral is employed to prove that the problem is identical to the diffusive dynamics of a charged particle in electromagnetic fields constructed from the velocity field. The case of zero diffusion has characteristics that coincide with the integral curves of the velocity field. This case is, of course, structurally unstable, and the limit of small diffusion is correctly described by the Wenzel-Kramers-Brillouin limit of our path-integral principle, wherein the tracer dynamics equals the orbits of point charges in electromagnetic fields. To lowest order, diffusive effects are accounted for within a Hamiltonian framework, and the limit of zero diffusion emerges as an unstable submanifold embedded in a six-dimensional phase space. We illustrate these ideas by considering the simple case of tracer advection-diffusion in the flow field of a time-independent, straight vortex line. We also briefly discuss generalization of the path-integral principle for the case where tracer sources and/or sinks are included. When the velocity field obeys the Navier-Stokes equation, the associated electromagnetic fields satisfy the equations of magnetohydrodynamics for a fluid with resistivity that is equal to the viscosity of the (real) fluid.

Dynamics of vortex lines in the three-dimensional complex Ginzburg-Landau equation: Instability, stretching, entanglement, and helices

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginzburg-Landau equation (CGLE). It is shown that a straight vortex line is unstable with respect to spontaneous stretching and bending in a substantial range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The boundary of the three-dimensional instability in parameter space is determined. Near the stability boundary, the supercritical saturation of the instability is found, resulting in the formation of stable helicoidal vortices.

THE VORTEX LABILITY IN A SUPERSATURATED SUPERFLUID 3HE-4HE MIXTURE AND THEQUANTUM PHASE-SEPARATION KINETICS

The supersaturated superfluid3He-4He mixture in the presence of quantized vortices is considered. A specific type of the lability against the vortex core expansion arises from concentrating3He atoms onto the vortex core. This affects significantly the region of the3He concentrations at which the rate of nucleating stable3He-concentrated phase can be noticeable. The most clear case of the vortex structure is a straight-line vortex with the uniformly distributed 3He atoms along the core. Such lability leads to some features in the thermal activation and quantum tunneling nucleation, resulting, in particular, in a relatively small supersaturation of about 1% which can be achieved. The critical value of the supersaturations corresponding to the vortex lability is dispersed because of the finite size of the3He-concentrated phase along the vortex core. We analyze this dispersion and its effect on the quantum separation in supersaturated3He-4He mixtures. The value of the supersaturation conserves its order of the magnitude, changing only by a numerical factor. We also emphasize an important difference of the coefficient in the term responsible for the lability from the superfluid density ρs. The supersaturation observed in experiments lies within the range of the vortex lability and, probably, is connected with this lability.

Instability and Stretching of Vortex Lines in the Three-Dimensional Complex Ginzburg-Landau Equation

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginsburg-Landau equation (CGLE). It is proved that a straight vortex line is unstable with respect to spontaneous stretching and bending in a certain range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The analysis shows that the standard approach relating the velocity of the filament with the local curvature is insufficient to describe the instability and stretching of vortex lines.

Dynamics of flux penetration and critical currents in type-II superconductors

We study the dynamics of flux penetration and vortex pinning in a type-II superconductor by numerical simulations with forces appropriate in the London regime. Straight vortex lines at zero temperature are studied in a slab geometry, and, unlike previous studies, full account is taken of the dynamical penetration of the flux lines across the surface under the application of an external magnetic field. Our simulation produces flux penetration in the sample beyond ${H}_{c1}$ as expected, with the flux lines forming an Abrikosov lattice in the absence of the pinning centers. The pinning centers impede flux penetration into the bulk and the resulting critical current decreases with increasing vortex density roughly in agreement with the modified Bean's model.

Matrix formulation of hydrodynamics and extension of ptolemaic flows to three-dimensional motions

A new matrix formulation of Lagrange hydrodynamic equations is proposed. Exact solutions of those equations are obtained in matrix form. It is found that precession of vortex lines around some fixed axis in space is a general property of the flows described by those solutions. Two types of fluid motion are studied. Flows of the first type have straight vortex lines, and their particle trajectories are windings on toroidal surfaces. The other flows have plane particle trajectories, and their vortex lines are arbitrarily shaped plane curves. All these motions are shown to be three-dimensional generalizations of plane Ptolemaic flows [1,2].

Virial theorem for the anisotropic Ginzburg-Landau theory

The scalar virial theorem for the Ginzburg-Landau theory is generalized to a vector virial theorem and follows from similar scaling properties of the Gibbs free-energy density. All the components of the magnetic field H are determined in terms of average values of the kinetic and field tensor components of the Helmholtz free-energy density. We consider two frames, the crystal's and the magnetic induction's B, where the scaling properties yield useful relations due to anisotropy. In the last case the scaling relations do not completely determine H; instead, they provide useful identities that reflect collective properties of the vortex state. We compare both the scaling and the thermodynamic methods for the particular case of straight tilted parallel vortex lines in the London limit. \textcopyright{} 1996 The American Physical Society.

Phase singularities and Josephson coupling in layered superconductors

In the framework of the Lawrence-Doniach model, the properties of an Abrikosov vortex line, built by a stack of two-dimensional “pancake” vortices, is investigated. The role of phase singularities is discussed in detail, and a very compact expression for the free energy of arbitrary pancake configurations is derived. Linearization procedures for the nonlinear sine-Gordon type equations for the Josephson phase are discussed, and a suitable smallness parameter is found. Finally, explicit calculations for a titled straight vortex line and for a dislocated pancake are given.

Coexistence of orthogonal flux lines in uniaxial superconductors

We consider extremely type II uniaxial superconductors in the context of anistropic London theory and study the Helmholtz free energy when the magnetic induction makes an angle of 45° with the symmetry axis. For straight vortex lines we find that the free energy is lower when the straight lines are in multiple directions, namely at θ = 0° and at 90°, instead of all oriented at 45°.

Excitations bound to a vortex line in layered superconductors

Using Bogoliubov-de Gennes equation, we study quasiparticle excitations in layered superconductors in the presence of a straight vortex line which is parallel to the layers. The lowest bound state is shown to have energy eigenvalue of the order of magnitude Δ, the energy gap, in contrast to the corresponding value Δ2/EF when the line is perpendicular to the layers.

The energy and stability of a vortex line in anisotropic London theory

We show that the London theory expression for the energy of a system of vortex lines in anisotropic superconductors may be written in several different but equivalent forms because vortices form either closed loops or infinite lines. The divergence of the vortex lines' short distance interactions is handled by two procedures, the Gaussian and Pauli-Villars regularization methods. The stability of straight vortex lines perpendicular to the CuO 2 planes in high- T c superconductors is investigated from the point of view of anisotropic London theory. We show that the straight line is always an equilibrium configuration and its stability depends on the properties of the vortex core. The self-energy of spiral vortex lines whose axes coincide with the axis of anisotropy is computed and compared to that of the straight line parallel to the axis of anisotropy.

Flux pinning and creep in high- Tc superconductors

We report here calculations of intrinsic (lattice) as well as point defect induced pinning, using a realistic Ginzburg-Landau theory. The former mechanism is shown to be effective for a vortex line parallel to the CuO planes, and the latter for a vortex line perpendicular to them. We show that local kink-antikink pairs on otherwise straight vortex lines can be made metastable by pinning. We argue that these metastable defects cause flux noise and dissipative flux creep in the linear regime at low temperature and magnetic fields. These defects are shown to have the observed size and orientation dependence of the activation energy, and to rationalize a variety of properties of transport and magnetic behaviour.

Vortices in layered high-Tc superconductors

We investigate the properties of a point-like vortex, vortex-antivortex pairs and vortex lines in layered superconductors. The formation energy of a single point-like vortex turns out to diverge logarithmically with the size of the sample. The spontaneous creation of vortex-antivortex pairs leads to a Kosterlitz-Thouless transition (at a temperature T KT ) that is similar to the transition in 2D superconductors. The interaction between pairs in different planes is small. A single straight vortex line and a vortex line pair have the same energy as in continuous superconductors. However, the stiffness of a single vortex line in a layered superconductor is considerably smaller than in conventional systems, and the line “melts” in the absence of pinning at T KT .

Spur-type instability observed on numerically simulated vortex filaments

An instability observed on vortex filaments during numerical simulations of the three-dimensional, time-dependent dynamics of vortex wakes is studied to determine when and why it occurs. It is concluded that the observed instability is a consequence of the use of straight-line vortex segments of finite length to model continuously curving vortex filaments. The instability appears to occur only when the link length is a sizable fraction of the vortex span and, therefore, is not expected in an experiment. Guidelines are then given that help avoid numerical instabilities when vortex filaments are used in flow simulations.

Experiments on the trajectory and circulation of the starting vortex

Various reasons have been given as to why the trajectories and circulations of vortices generated at sharp edges do not follow classical similarity-theory predictions for at least an initial short time. Amongst these are the effect of the particular flow geometry (e.g. duct with wedge, nozzle) distant from the salient edge (for rectilinear vortices); axisymmetry (for ring vortices); end effects (for rectilinear vortices); viscous diffusion; finite thickness of the detaching shear layer, as well as secondary vorticity caused by the interaction of the primary vortex with the edge at which it was generated. A further process that may be active is that of viscous entrainment. Experiments, in which essentially straight-line vortices were generated, indicate that of the seven possibilities mentioned, the first five do not play a significant part. All models consist of a basic flow onto which the modelled vortex is superposed. Thus either the basic flow or the vortex model are at fault. The basic flow onto which the vortex is superposed may well not be a pure edge flow, but one that is already taking on the character of an entraining jet flow. On the other hand, the vortex model fails to incorporate secondary vorticity which, particularly when rolled up, might be expected to be dynamically important.

The Estimation of the Drag of Circular Cylinders at Subcritical Reynolds Numbers and Subsonic Speeds

In an investigation, on the characteristics of the wake from slender cone-cylinders at large angles of attack, the strength and position of the vortices in the wake were determined for a large range of flow conditions. Two types of experiments were conducted, namely schlieren studies and wake traverses.Schlieren photographs (such as Fig. 1) clearly reveal straight parallel vortex lines in the wake for subcritical values of the cross-flow Reynolds number component and for cross-flow components of the free stream Mach number up to at least 1·4. In ref. 1 numerous such schlieren photographs were analysed using the impulse flow analogy (refs. 2 and 3 for example). According to the analogy the progressive development of the wake along the body when viewed in cross-flow planes is similar to the growth with time of the flow behind a two-dimensional cylinder started impulsively from rest with the same cross-flow conditions.

Extraction of electrons from quantized vortex lines

If electrons are trapped in vortex lines of rotating He II, they can emerge into the vapor phase without overcoming any detectable energy barrier. This observation of the extraction of electrons from straight vortex lines above 1.1°K agrees very well with the evaporation of electrons from vortex rings studied by Surko and Reif below 0.7°K

A Direct Aerodynamic Proof of the Biot-Savart Law

The expression for the induced velocity of a vortex element,in Glauert's notation (where K is the strength of the vortex, h the length of the normal, and r the distance from the field point to the element, of length ds) which is often known as the Biot-Savart law, and is of fundamental importance in aerodynamics, is frequently left unproved in textbooks. Sometimes reference is made to the electro-magnetic analogy, or to a standard treatise such as Lamb. One well-known work implies that the law is the solution of an integral equation derived from the known induced velocity of an infinite straight-line vortex, but this is misleading, since there is an infinite number of solutions, of which only one is correct. Another textbook gives a lengthy argument starting from the replacement of a line vortex by a sheet of sources and sinks, which is probably not too convincing to the average student. Lamb himself gives a long and rather involved proof based on Helmholtz and Stokes, with appeals to the Theory of Attractions, with which few are likely to be familiar. A simple and direct proof from fluid mechanics seems therefore worth while.