Ginzburg‐Landau vortex dynamics driven by an applied boundary current

In this paper we study the time‐dependent Ginzburg‐Landau equations on a smooth, bounded domain Ω ? ?², subject to an electrical current applied on the boundary. The dynamics with an applied current are nondissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit ε → 0. We first consider the original time scale in which the vortices do not move and the solutions undergo a “phase relaxation.” Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current. © 2010 Wiley Periodicals, Inc.     

Geomagnetic responses in high latitudes during the storm of july 15—16, 2000

The ionospheric equivalent currents in the high latitudes and the auroral electrojet system during the geomagnetic storm on July 15–16, 2000 are analyzed by using geomagnetic data from IMAGE chain. The large-scale vortices of equivalent currents are observed in the storm. The vortices on the dusk side of ionosphere correspond to four-celled pattern of plasma convection associated with NBZ, region I and region II field-aligned currents. Only one vortex can be found on the dawn side of ionosphere after interplanetary magnetic field (IMF) turns southward. In the initial phase of the storm, the center of eastward electrojets tends to shift equatorward. It arrives at 58.62° latitude of corrected geomagnetic coordinates (CGM). The westward electrojets are strong in the main phase. The center of westward electrojets in this period moves equatorward and may be beyond the southernmost station (56.45°) of the chain.

     

Stationary corner vortex configurations

Given a stable configuration of point vortices for steady two dimensional inviscid, incompressible fluid flow in a domainD, it is shown that there is another stable configuration of stationary point vortices inD with vortices near the original vortices plus additional vortices near any of the convex corners ofD. It follows that there are steady flows which have a finite sequence, of arbitrary length, of vortices of alternating sign descending into any convex corner ofD. Several computed examples are given.     

Existence of Optical Vortex Solitons in Pair Plasmas

Acta Mathematicae Applicatae Sinica, English Series - Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical physics. In this study, we employ the...     

Geomagnetic responses in high latitudes during the storm of July 15-16, 2000

The ionospheric equivalent currents in the high latitudes and the auroral electrojet system during the geomagnetic storm on July 15-16, 2000 are analyzed by using geomagnetic data from IMAGE chain. The large-scale vortices of equivalent currents are observed in the storm. The vortices on the dusk side of ionosphere correspond to four-celled pattern of plasma convection associated with NBZ, region I and region II field-aligned currents. Only one vortex can be found on the dawn side of ionosphere after interplanetary magnetic field (IMF) turns southward. In the initial phase of the storm, the center of eastward electrojets tends to shift equatorward. It arrives at 58.62o latitude of corrected geomagnetic coordinates (CGM). The westward electrojets are strong in the main phase. The center of westward electrojets in this period moves equatorward and may be beyond the southernmost station (56.45°) of the chain.     

Numerical computation of end plate effect on Taylor vortices between rotating conical cylinders

End plate effect on Taylor vortices between rotating conical cylinders is studied by numerical method in this paper. We suppose that the inner cone rotates together with the end plate at the top and the outer one as well as the end plate at the bottom remains at rest. It is found that the instability sets in at a critical Reynolds number about Re = 80. Increase Re to about Re = 200 the first single clockwise vortex is formed near the top of the flow system. Further increase Re to about Re = 440 another clockwise vortex is formed under the first one. At about Re = 448 the third vortex is formed which rotates in counterclockwise direction between the first two vortices. With increasing of Re the process continues. Finally, a configuration is obtained with an odd number of vortices in the annulus at about Re = 700, which confirms the experimental observation. Moreover, the local extreme values of pressure and velocity are achieved at the adjacent lines between neighboring vortices or at the medium lines of vortices. The effect of gap size on vortices is also considered. It is shown that the number of vortices increases with decreasing of the gap size and the end plates play an important role in the parity of the number of the vortices.     

A Ginzburg–Landau model for the phase transition in Helium II

The paper deals with the second-order phase transition in Helium II by a Ginzburg–Landau model, in which any particle has simultaneously a normal and superfluid velocity. This pattern is able to describe the classical effects of Helium II as the phase diagram, the vortices, the second sound and the thermomechanical effect. Finally, the vorticities and turbulence are described by an extension of the model in which the material time derivative is used.     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

Two-component Bose-Einstein condensates

Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe Bose-Einstein condensates. In this paper, we study the structure of vortices of the coupled nonlinear equations for two-component Bose-Einstein condensates (BEC) in a three-dimensional space. We show that vortices is 1-rectifiable set, and give its mean curvature. In particular, we show that large interspecies scattering length causes vortices for two-component BEC.     

Occurrence of Taylor vortices in the flow between two rotating conical cylinders

The behavior of the flow between two coaxial conical cylinders with the inner one rotating and the outer one stationary is studied numerically. Special attention is paid to the occurrence of Taylor vortices in basic flow and unsteady helical vortices. It is found that, in basic flow, the vortices occur in the direction toward smaller radius, while toward bigger radius in unsteady helical vortices; moreover, the unsteady helical vortices can coexist with unstable steady Taylor vortices. The results suggest that the behavior of conical flow is dominated by a competition between the meridional flow and radial flow. The effect of meridional flow is most significant at small apex angle or in basic flow and helical vortices, while the radial flow dominates the structure at larger apex angle or in steady vortical flow. In order to get better understanding the competition and the transition of Taylor–Couette flow to conical flow, a velocity angle related to velocity components is defined, and the pattern evolution of velocity, streamlines and the velocity angle are examined with respect to apex angle, as well as Reynolds number. Finally, the statistical properties of turbulent conical flow are investigated.     

Zero Modes of the Dirac Operator in Vortex-Type Gauge Fields

We consider the D4 Euclidean Dirac equation in a SU(2) gauge field with the form of intersecting central vortices and obtain its solutions corresponding to normalizable zero modes. In the context of the problem of constructing the mechanism of chiral symmetry breaking based on the model with randomly distributed central vortices, we study a possibility for splitting fermionic zero modes in the case of a superposition of many vortices with Pontryagin indices of different signs.     

Equilibrium configurations of point vortices on a sphere

Configurations of point vortices on the sphere are considered in which all vortex velocities are zero. A sharp upper bound for the number of equilibria lying on a great circle is found, valid for generic circulations, and some unusual equilibrium configurations with a free real parameter are described. Equilibria of rings (vortices evenly spaced along circles of latitude) are also discussed. All equilibrium configurations of four vortices are determined.      

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.     

Homostrophic vortex interaction under external strain,in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.     

On the motion of two-dimensional vortices with mass

Summary The Helmholtz-Kirchhoff ODEs governing the planar motion ofN point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two “massive” vortices totwo degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently by Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the Helmholtz-Kirchhoff equations.     

Computational fluid dynamics simulation of fluid particle fragmentation in turbulent flows

A simulation methodology is presented that allows detailed studies of the breakup mechanism of fluid particles in turbulent flows. The simulations, based on large eddy and volume of fluid simulations, agree very well with high-speed measurements of the breakup dynamics with respect to deformation time and length scales, and also the resulting size of the daughter fragments. The simulations reveal the size of the turbulent vortices that contribute to the breakup and how fast the interaction and energy transfer occurs. It is concluded that the axis of the deformed particle and the vortex core axis are aligned perpendicular to each other, and that breakup sometimes occurs due to interaction with two vortices at the same time. Analysis of the energy transfer from the continuous phase turbulence to the fluid particles reveals that the deformed particle attains it maximum in interfacial energy before the breakup is finalized. Similar to transition state theory in chemistry this implies that an activation barrier exists. Consequently, by considering the dynamics of the phenomenon, more energy than required at the final stage needs to be transferred from the turbulent vortices for breakup to occur. This knowledge helps developing new, more physical sound models for the breakup phenomenon required to solve scale separation problems in computational fluid dynamics simulations.     

Dynamics of planar vortex clusters with binaries

We study an N -vortex problem having J of them forming a cluster, which means the distances between the vortices in the cluster is much smaller by O (ε) than the distances, O (ℓ), to the N – J vortices outside of the cluster. With the strengths of N vortices being of the same order, the velocity and time scales for the motion of the J vortices relative to those of the N – J vortices are O (ε^–1) and O (ε²) respectively. We show that this two-time and two-length scale problem can be converted to a standard two-time scale problem and then the leading order solution of the N -vortex problem can be uncoupled to two problems, one for the motion of J vortices in the cluster relative to the center of the cluster and one for the motion of the N – J vortex plus the center of the cluster. For N = 3 and J = 2, the 3-vortex problem is uncoupled to two binary vortices problems in the length scales ℓ and ℓε respectively. When perturbed in the scale ℓ, say by a fourth vortex even of finite strength, the binary problem becomes a 3-vortex problem, admitting periodic solutions. Since 3-vortex problems are solvable, the uncoupling enables us to solve 3-cluster problems having at most three vortices in each cluster. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Film flow over strongly undulated bottom profiles at low Reynolds numbers – Experimental study

We study gravity driven films of a Newtonian fluid flowing down sinusoidal bottom profiles. We give experimental evidence of vortices in the film flow under creeping flow conditions down to Reynolds numbers of the order of 10^–5. The vortices are created in the valleys of the undulated bottom profile. They are visualized employing a particle image velocimeter with fluorescent tracers. It turns out that the vortices are generated beyond a critical film thickness.