Point vortex motion on the surface of a sphere with impenetrable boundaries

2006 ◽  
Vol 18 (3) ◽  
pp. 036602 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Vortex Motion ◽  
Point Vortex

Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part 1. Analysis of relative and absolute motions

2013 ◽  
Vol 717 ◽  
pp. 232-254 ◽  
Author(s):  
M. A. Sokolovskiy ◽  
K. V. Koshel ◽  
J. Verron
Keyword(s):  
Vortex Motion ◽  
Point Vortex ◽  
Bottom Layer ◽  
Phase Portraits ◽  
Fundamental Result ◽  
Barotropic Fluid ◽  
Oscillation Frequencies ◽  
Trilinear Coordinates ◽  
Analytical Expressions

AbstractThe results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

scholarly journals The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1175
Author(s):  
El Mostafa Kalmoun ◽  
Mohamed M. S. Nasser ◽  
Khalifa A. Hazaa
Keyword(s):  
Cross Sections ◽  
Single Point ◽  
Vortex Motion ◽  
Point Vortex ◽  
Multiply Connected Domains ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Contour Lines ◽  
Circular Domains ◽  
Prime Function

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

scholarly journals Vortex motion in shallow water with varying bottom topography and zero Froude number

2000 ◽  
Vol 411 ◽  
pp. 351-374 ◽  
Author(s):  
G. RICHARDSON
Keyword(s):  
Froude Number ◽  
Vortex Core ◽  
Bottom Topography ◽  
Vortex Motion ◽  
Point Vortex ◽  
Core Structure ◽  
Inviscid Fluid ◽  
Inviscid Flows ◽  
Matched Asymptotics ◽  
Vortex Core Structure

The methods of formal matched asymptotics are used to investigate the motion of a vortex in shallow inviscid fluid of varying depth and zero Froude number in the limit as the vortex core radius tends to zero. To leading order the vortex is driven by local gradients in the logarithm of the depth along an isobath (or depth contour). A further term in the vortex velocity is calculated in which effects arising from the global bottom topography, other vortices and the vortex core structure appear. The evolution of the vortex core structure is then calculated. A point-vortex model is formulated which describes the motion of a number of small vortices in terms of the bottom topography, the inviscid flows around the vortices and their evolving core structure. A numerical method for solving this model is discussed and finally some numerical simulations of the motion of vortex pairs over a varying bottom topography are presented.

scholarly journals Vortex motion and geometric function theory: the role of connections

2019 ◽  
Vol 377 (2158) ◽  
pp. 20180341
Author(s):  
Björn Gustafsson
Keyword(s):  
Vortex Dynamics ◽  
Affine Connection ◽  
Vortex Motion ◽  
Hamiltonian Formulation ◽  
Point Vortex ◽  
Hamiltonian Function ◽  
Riemannian Metric ◽  
Minimal Amount ◽  
Dynamical Equations ◽  
The Difference

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Latitudinal point vortex rings on the spheroid

2010 ◽  
Vol 466 (2118) ◽  
pp. 1749-1768 ◽  
Author(s):  
Sun-Chul Kim
Keyword(s):  
Vortex Ring ◽  
Stereographic Projection ◽  
Vortex Motion ◽  
Point Vortex ◽  
Vortex Rings ◽  
Dynamical Equations ◽  
Conformal Metric ◽  
The Stability ◽  
Pole Vortex

Point vortex motion on the surface of a spheroid is studied. Exact dynamical equations from the corresponding Hamiltonian are constructed by computing the conformal metric which induces a modified stereographic projection. As a concrete example, the motion of point vortices at the same latitude (called the point vortex ring ) is investigated as an extension of the sphere case. The role of eccentricity to the stability of the rotating motion is analysed. The influence of a pole vortex is also discussed.

scholarly journals Universal statistics of point vortex turbulence

2015 ◽  
Vol 779 ◽  
pp. 275-308 ◽  
Author(s):  
J. G. Esler ◽  
T. L. Ashbee
Keyword(s):  
Phase Transition ◽  
Probability Density ◽  
Vortex Motion ◽  
Point Vortex ◽  
Normal Modes ◽  
Bimodal Distribution ◽  
High Energy ◽  
Low Energy ◽  
Mean Flow ◽  
Energy States

A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container $\mathscr{D}$, as the number of vortices $N$ tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydrodynamic eigenvalue problem of the Laplacian in $\mathscr{D}$. The statistics of the projection of the vorticity distribution onto these eigenfunctions (‘vorticity projections’) are then investigated. The statistics are used first to obtain the density-of-states function and caloric curve for the system, generalising previous results to arbitrary (neutral) distributions of vortex circulations. Explicit expressions are then obtained for the microcanonical (i.e. fixed energy) probability density functions of the vorticity projections in a form that can be compared directly with direct numerical simulations of the dynamics. The energy spectra of the resulting flows are predicted analytically. Ensembles of simulations with $N=100$, in several conformal domains, are used to make a comprehensive validation of the theory, with good agreement found across a broad range of energies. The probability density function of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager–Kraichnan condensation in the literature, between low-energy states with no mean flow in the domain and high-energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of $\mathscr{D}$, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite $N$, are explored using a Markov chain phase-space sampling method.

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