Two-dimensional Vortex Dynamics Research Articles

Hugoniot-Maslov Chains for the System of Shallow-Water Equations Taking into Account Energy Exchange

Singular solutions of the system of shallow-water equations with structure of the “type of the square root” of a quadratic form were introduced in [1]. To such solutions one assigns a Hugoniot– Maslov chain, which is an infinite chain of ordinary differential equations essentially determining the solution dynamics. These chains have many interesting properties (see [2–7]). The aim of this note is to describe the Hugoniot–Maslov chain for a more complicated system of equations describing the dynamics of two-dimensional atmospheric vortices with the atmosphere–ocean energy exchange taken into account [8]. The dimensionless form of this system is as follows:

New Hall Resistivity Scaling Relations in the Presence of Competition between Point-like and Anisotropic Planar Pinning Potential

Explicit current- and temperature-dependent expressions for anisotropic longitudinal and transverse nonlinear magnetoresistivities are represented and analyzed on the basis of a Fokker-Planck approach for two-dimensional vortex dynamics in a washboard pinning potential in the presence of point-like disorder. It is shown that new scaling relations in discussed below longitudinal and transverse geometries of experiment can be presented in the form which is: a) similar formally to the well-known scaling relations for the point-like disorder, but contains instead of the bare Hall conductivity a renormalized one which depends on the pinning anisotropy b) just the same as for purely anisotropic scaling relations in the absence of point-like pins. Physical meaning of these results is discussed.

The point island approximation in vortex dynamics

The effect of approximating a circular island of non-zero radius by a point dipole – the point island approximation – on the dynamics of two-dimensional vortices is studied. The approximation enables the Hamiltonian for N point vortices moving among M point islands to be derived. Examples of exact (within the point island approximation) trajectories for a single point vortex in domains involving gaps and multiple islands are presented. In addition, vortex patch motion using the point island approximation is computed numerically, demonstrating that such patches, provided they stay close to a circular shape, are well-modelled by point vortices. The effect of making the point island approximation on the trajectories of a point vortex about, (i), two islands and, (ii), an island within a gap, is investigated by comparing them to the exact trajectories in each case. It is found that the point island approximation is an efficient and accurate way of determining the behaviour of the vortex provided that the vortex and islands all remain well-separated in comparison to the vortex and island radii.

On the effects of viscoelasticity on two-dimensional vortex dynamics in the cylinder wake

The present paper represents, as far as we are aware, the first linear stability analysis for inertial flows of a viscoelastic fluid around a bluff body. Our particular interest is the effect of polymer additives on the linear stability of two-dimensional viscous flow past a confined cylinder and our investigation involves both direct numerical simulation and the solution of a generalized eigenvalue problem arising from the linearized perturbation equations. For a constant viscosity FENE-type model polymer additive is shown to have a stabilizing effect upon two-dimensional flow past a confined cylinder. In particular, our results reveal that the greater the maximum extensibility of the FENE dumbbells, the larger the value of the critical Reynolds number marking the onset of vortex shedding. Thus the stabilization brought to bear by the presence of the FENE-type dumbbells would appear to depend strongly upon the extensional properties of the polymer solution. This is in agreement with the results of the recent experimental paper of Cressman et al. [J.R. Cressman, Q. Bailey, W.I. Goldburg, Modification of a vortex street by a polymer additive, Phys. Fluids 13 (2001) 867–871] which highlight the importance of the extensional properties of high molecular weight polymers (even at low concentrations) in reducing the kinetic energy contained in the velocity fluctuations behind a cylinder. The impact of elasticity on Strouhal number, time-averaged recirculation length, drag and lift in the vortex-shedding regime are also discussed at some length.

Two-dimensional vortex dynamics in decoupledYBa2Cu3O7/PrBa2Cu3O7superlattices

Two-dimensional (2D) vortex dynamics is studied in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}/{\mathrm{PrBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ superlattices by measuring the current-voltage $(I\ensuremath{-}V)$ characteristics. In the high current limit, the 2D collective creep is observed with an activation energy $U(j)\ensuremath{\propto}{j}^{\ensuremath{-}\ensuremath{\mu}},$ with j the current density. The exponent $\ensuremath{\mu}$ is 0.8 at low temperatures and $\ensuremath{\mu}=0.2$ at high temperatures. A dislocation-mediated vortex melting occurs when the temperature is increased. In the low current limit, the exponential growth of the energy barrier for elastic motion is prohibited by the plastic deformation of vortices. The observed plateau in the resistive transition is attributed to the possible quantum tunneling of vortices.

New Hall voltages in a planar pinning potential

Two-dimensional vortex dynamics in a planar pinning potential (PPP) created by uniaxial or bianisotropic pinning planes in the presence of thermal fluctuations is considered on the basis of a Fokker–Planck equation. Explicit expressions for two new nonlinear anisotropic Hall voltages (longitudinal and transverse with respect to the current direction) are derived and analyzed. The physical origin of these odd (with respect to magnetic field reversal) voltages is caused by the subtle interplay between even effect of vortex guiding along the PPP and the odd Hall effect. Both new voltages are going to zero in the linear regimes of the vortex motion (i.e. in the thermoactivated flux flow (TAFF) and ohmic flux flow (FF) regimes) and have a bumplike current j or temperature θ dependence in the vicinity of highly nonlinear resistive transition from the TAFF to FF. As new odd voltages arise due to the Hall effect their characteristic scale is proportional to the Hall constant as for ordinary “steplike” odd Hall voltage, investigated earlier by Mawatari.

On the large-time behavior of two-dimensional vortex dynamics

In this paper we prove two results regarding the large-time behavior of vortex dynamics in the full plane. In the first result we show that the total integral of vorticity is confined in a region of diameter growing at most like the square root of time. In the second result we show that if a dynamic rescaling of the absolute value of vorticity with spatial scale growing linearly with time converges weakly, then it must converge to a discrete sum of Dirac masses. This last result extends in scope a previous result by the authors, valid for nonnegative initial vorticity on a half-plane.

Influence of Point-Like Disorder on the Hall Resistivity Behavior in an Anisotropic Planar Pinning Potential

Explicit current- and temperature-dependent expressions for anisotropic longitudinal and transverse nonlinear magnetoresistivities are derived and analyzed on the basis of a Fokker–Planck approach for two-dimensional vortex dynamics in a washboard pinning potential in the presence of point-like disorder. Gradually increasing the strength of the point-like pinning (in an experiment this is simply done by irradiation of the sample with different doses of high-energy electrons) this theory predicts a gradual decrease of the anisotropy of the magnetoresistivities. The physics of the transition from the recently discussed new scaling relations for anisotropic Hall resistivities in the absence of point-like pins to the well-known scaling relations for isotropic pinning is elucidated. This is discussed in terms of a gradual isotropization of the guided vortex motion responsible for the existence in a washboard pinning potential of new anisotropic Hall voltages which are odd with respect to magnetic field reversal.

Strong Effects of Background Vorticity Distribution on the Dynamics of Coherent Vortices in Electron Plasma

Two-dimensional vortex dynamics is characterized by motion of coherent vortices or clumps in a rugged distribution of background vortices. We study eminent features in this process by using a magnetized pure electron plasma starting from highly controlled initial density distribution. The topics included are (1) relaxation of many-clump dynamics states to crystallization, (2) single clump dynamics in a background vorticity distribution, (3) two-clump interaction in a low-level background vorticity distribution, and (4) accelerated crystallization of three clumps in a low-level background vorticity distribution.

Spiral vortices in compressible turbulent flows

We extend the spiral vortex solution of Lundgren [Phys. Fluids 25, 2193 (1982)] to compressible turbulent flows with a perfect gas. This model links the dynamical and the spectral properties of incompressible flows, providing a k−5/3 Kolmogorov energy spectrum. In so doing, a compressible spatiotemporal transformation is derived, reducing the dynamics of three-dimensional vortices, stretched by an axisymmetric incompressible strain, into a two-dimensional compressible vortex dynamics. It enables us to write the three-dimensional spectra of the incompressible and compressible square velocities in terms of, respectively, the two-dimensional spectra of the enstrophy and of the square velocity divergence, by the use of a temporal integration. Numerical results are presented from decaying direct simulations performed with 5122 grid points; initially, the rms Mach number is 0.23, with local values up to 0.9, the Reynolds number is 700, and the ratio between compressible and incompressible square velocities is 0.1. A k−5/3 inertial behavior is seen to result from the dynamical evolution for both the compressible and incompressible three-dimensional spectra.

Two-Dimensional Vortex Dynamics in a Potential-Confined Electron Plasma

We study dynamics of discrete vortex strings interacting in a background vorticity distribution by using a magnetized electron plasma confined by a potential distribution in a cylindrical trap, and demonstrate relevant 2-dimensional features characterizing the incompressible dissipation-free guiding-center fluid. Some specific elementary processes are examined and compared with a recently proposed theoretical model that describes a motion of a point vortex moving up a hill of background vorticity distribution. Radial transport of particles is observed to be very limited and associated with several conserved parameters.

Vortex dynamics and zonal flows

Two-dimensional vortex dynamics have been studied in plasmas by exploiting the analogy between fluid velocity and the E×B drift velocity. The analogy extends to geophysical flows by including physics that mimic zonal flows, dissipation and the β-effect due to the variation in the Coriolis parameter. Vortices with the same sign as the ambient zonal shear are stable, while opposite-signed vortices fragment. Rules for vortex merger derived by maximizing entropy or minimizing enstrophy do not work for vortices embedded in zonal flows. New rules based on the minimization of energy hold. When zonal flows are not imposed, and the flow is forced at small scales, large, coherent jet streams or eddies form that co-exist with turbulence. Their sizes are determined by an energy balance, not the length scales of the forcing or boundaries. The motivation for this work is to understand atmospheric and ocean vortices: Gulf stream meanders and eddies, the Antarctic ozone hole, the jet streams of Earth and Jupiter, and the Jovian Great Red Spot and White Ovals.

Vortices in rotating systems: Centrifugal, elliptic and hyperbolic type instabilities

This paper is devoted to the effects of rotation on the linear dynamics of two-dimensional vortices. The asymmetric behavior of cyclones and anticyclones, a basic problem with respect to the dynamics of rotating flows, is particularly addressed. This problem is investigated by means of linear stability analyses of flattened Taylor–Green vortices in a rotating system. This flow constitutes an infinite array of contra-rotating one-signed nonaxisymmetric vorticity structures. We address the stability of this flow with respect to three-dimensional short-wave perturbations via both the geometrical optics method and via a classical normal mode analysis, based on a matrix eigenvalue method. From a physical point of view, we show that vortices are affected by elliptic, hyperbolic and centrifugal instabilities. A complete picture of the short-wave stability properties of the flow is given for various levels of the background rotation. For Taylor–Green cells with aspect ratio E=2, we show that anticyclones undergo centrifugal instability if the Rossby number verifies Ro>1, elliptic instability for all values of Ro except 0.75<Ro<1.25 and hyperbolic instability. The Rossby number is here defined as the ratio of the maximum amplitude of vorticity to twice the background rotation. On the other hand, cyclones bear elliptic and hyperbolic instabilities whatever the Rossby number. Besides, depending on the Rossby number, rotation can either strengthen (anticyclonic vortices) or weaken elliptic instability. From a technical point of view, in this article we bring an assessment of the links between the short-wave asymptotics and the normal mode analysis. Normal modes are exhibited which are in complete agreement with the short-wave asymptotics both with respect to the amplification rate and with respect to the structure of the eigenmode. For example, we show centrifugal eigenmodes which are localized in the vicinity of closed streamlines in the anticyclones; elliptical eigenmodes which are concentrated in the center of the cyclones or anticyclones; hyperbolic eigenmodes which are localized in the neighborhood of closed streamlines in cyclones.

Solitons in (2 + 0) dimensions and their applications in vortex dynamics

Globally smooth, exact solutions of inviscid, two-dimensional vortex dynamics are derived by exploiting techniques from soliton theory. The Stuart and Mallier-Maslowe vortices are rederived using the Hirota method and a 2-soliton solution. A 3-soliton expansion yields a complex flow pattern. Doubly periodic arrays of vortices are expressed in terms of elliptic and theta functions. The implications and interpretations in the dynamics of shear flows are discussed.

Characteristics of two-dimensional vortex dynamics from XY-type models with Ginzburg-Landau dynamics

The characteristic features of vortex dynamics corresponding to two-dimensional XY-type models with Ginzburg-Landau dynamics are extracted from simulations. The cases covered are with and without frustration, as well as above and below the Kosterlitz-Thouless transition. Most of the results are very well described by a phenomenological response function. The dependence of the characteristic frequency for this response function on the vortex density, frustration, correlation length, and temperature is obtained. A critical behavior for vortex dynamics at the Kosterlitz-Thouless transition is suggested by the simulations. The agreements with experiments and other simulations are discussed. {copyright} {ital 1997} {ital The American Physical Society}

Three-dimensional representation of two-dimensional vortex dynamics in lasers

We observe that the dynamics of vortices in paraxial optics, including vortex pair-creation and pair-annihilation, can be represented in a simplified manner by the motion of a locus obtained from the complex optical field, for which the real and imaginary parts of the field are equal. When the locus lies within the transverse plane a dark line is present in the optical field with a phase change of π across. When the locus intersects the transverse plane, a vortex exists at the intersection point. Thus the locus line appears like a three-dimensional vortex line. Examples are given ranging from emission in low order laser mode families to patterns calculated for plane mirror resonators.

Two-dimensional vortex dynamics in a stratified barotropic fluid

We show that two-dimensional ‘point’ vortex dynamics in both a polytropic fluid of γ = 3/2 and an isothermal fluid stratified by a constant gravitational field can be written in Hamiltonian form. We find that the formulation admits only one constant of the motion in addition to the Hamiltonian, so that two vortices are the most for which the motion is generally integrable. We study in detail the two-vortex problem and find a rich collection of behaviour: closed trajectories analogous to the circular orbits of the uniform-fluid two-vortex problem, open trajectories for which the self-propelled vortices scatter off each other, and both unstable and stable steadily translating pairs of vortices. Comparison is made to the case of two vortices in a uniform-density fluid bounded by a wall.

Dynamics of two-dimensional pancake vortices in layered superconductors.

The dynamics of 2D pancake vortices in Josephson-coupled superconducting/normal - metal multilayers is considered within the time-dependent Ginzburg-Landau theory. For temperatures close to $T_{c}$ a viscous drag force acting on a moving 2D vortex is shown to depend strongly on the conductivity of normal metal layers. For a tilted vortex line consisting of 2D vortices the equation of viscous motion in the presence of a transport current parallel to the layers is obtained. The specific structure of the vortex line core leads to a new dynamic behavior and to substantial deviations from the Bardeen-Stephen theory. The viscosity coefficient is found to depend essentially on the angle $\gamma$ between the magnetic field ${\bf B}$ and the ${\bf c}$ axis normal to the layers. For field orientations close to the layers the nonlinear effects in the vortex motion appear even for slowly moving vortex lines (when the in-plane transport current is much smaller than the Ginzburg-Landau critical current). In this nonlinear regime the viscosity coefficient depends logarithmically on the vortex velocity $V$.

Effects of vortex-pair creation on the vortex dynamics with transport current

The dynamics of two-dimensional magnetic vortices with logarithmic interaction is studied numerically on the basis of a langevin-type stochastic model, recently developed by us. Especially, the I– V curves are examined by taking into account the effects of vortex-pair creation. From large-scale computer simulations, we find that the I– V curves exhibit a non-ohmic behavior as V ∼ I a( T) below a certain temperature depending on a magnetic induction. It is also found that for a weak magnetic induction the jump in the power law exponent a( T) as a function of temperature T is observed as is expected by the Kosterlitz and Thouless model, while for a strong induction the jump in a( T) is absent.

Free vortex calculations using analytical/numerical matching with solution pyramiding

The motion of a free vortex system is predicted using a novel method called analytical/numerical matching (ANM). Analytical/numerical matching is a hybrid mathematical and numerical technique that results in a new problem formulation with high accuracy and increased computational efficiency. The approach capitalizer on the disparity of characteristic length scaler that occur in many fluid dynamics problems. Accordingly, many problems can be decomposed into two distinct problems involving a local high-resolution near field, often solvable analytically, end a global, low-resolution far field, solved numerically. The ANM technique breaks the free vortex problem into these local and global problems each of which are easier end more efficient to solve than the traditional problem formulation. In addition, the ANM decomposition can be repeated to form a hierarchy of nested problems of increasing resolution using a method called pyramiding. The ANM method with pyramiding, called ANM/P, is studied to assess its accuracy and computational efficiency for two-dimensional and three-dimensional vortex dynamics problems. In calculations of vortex sheet rollup end vortex ring motion and interactions, this method war found to be relatively easy to implement end exhibited reductions in computational time from a factor of three to more than an order of magnitude