scholarly journals Equilibrium configurations of point vortices in doubly connected domains

1996 ◽  
Vol 1 ◽  
pp. 325-337
Author(s):  
Alan R. Elcrat ◽  
Chenglie Hu ◽  
Kenneth G. Miller
Keyword(s):  
Point Vortices ◽  
Equilibrium Configurations

Relative equilibrium configurations of point vortices on a sphere

2013 ◽  
Vol 18 (4) ◽  
pp. 344-355 ◽  
Author(s):  
Maria V. Demina ◽  
Nikolai A. Kudryashov
Keyword(s):  
Relative Equilibrium ◽  
Point Vortices ◽  
Equilibrium Configurations

scholarly journals Point vortex equilibria on the sphere via Brownian ratchets

2008 ◽  
Vol 465 (2102) ◽  
pp. 437-455 ◽  
Author(s):  
Paul K Newton ◽  
Takashi Sakajo
Keyword(s):  
Equilibrium Configuration ◽  
Null Space ◽  
Relative Equilibrium ◽  
Singular Values ◽  
Singular Value ◽  
Point Vortices ◽  
Basis Set ◽  
Frobenius Norm ◽  
Equilibrium Configurations ◽  
Smallest Singular Value

We describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, A Γ =0, where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ ∈ N is the (unit) vector of vortex strengths that must lie in the null space of A . Existence of an equilibrium is expressed by the condition det( A T A )=0, while uniqueness follows if Rank( A )= N −1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A , and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N =4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy.

Spectral gradient flow and equilibrium configurations of point vortices

2009 ◽  
Vol 466 (2118) ◽  
pp. 1687-1702 ◽  
Author(s):  
Andrea Barreiro ◽  
Jared Bronski ◽  
Paul K. Newton
Keyword(s):  
Symmetric Matrix ◽  
A Priori ◽  
Relative Equilibrium ◽  
Point Vortex ◽  
Gradient Flow ◽  
Rotational Frequency ◽  
Point Vortices ◽  
Equilibrium Configurations ◽  
The Matrix ◽  
Smallest Singular Value

We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices M T M . The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

scholarly journals Stationary states of identical point vortices and vortex foam on the sphere

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120622 ◽  
Author(s):  
Kevin A. O'Neil
Keyword(s):  
Initial Conditions ◽  
Point Vortex ◽  
Intermediate Energy ◽  
Point Vortices ◽  
Stationary States ◽  
Vortex Sheets ◽  
Identical Point ◽  
Equilibrium Configurations ◽  
Stationary Vortex ◽  
Point Vortex Equilibria

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.

scholarly journals Polynomial Method for Constructing Equilibrium Configurations of Point Vortices in the Plane

2015 ◽  
Vol 19 (5) ◽  
pp. 50-55 ◽  
Author(s):  
M. V. Demina ◽  
N. A. Kudryashov
Keyword(s):  
Point Vortices ◽  
Polynomial Method ◽  
Arbitrary Choice ◽  
Equilibrium Configurations ◽  
Vortex Systems

The problem of constructing and classifying stationary and translating configurations of point vortices with an arbitrary choice of circulations is studied. The polynomial method enabling one to find any such configuration is described in detail. Stationary configurations for vortex systems with circulations Γ, −µΓ are classified in the case of integer µ. New configurations are obtained.

Two Dimensional Bright and Dark Magnetoexcitons Interacting with Quantum Point Vortices

2019 ◽  
Vol 53 (16) ◽  
pp. 2055-2059
Author(s):  
S. A. Moskalenko ◽  
V. A. Moskalenko ◽  
I. V. Podlesny ◽  
I. A. Zubac
Keyword(s):  
Point Vortices ◽  
Two Dimensional ◽  
Quantum Point
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