Spectral gradient flow and equilibrium configurations of point vortices

Barreiro, Andrea K.; Bronski, Jared C.; Newton, Paul K.

doi:10.1098/rspa.2009.0419

Cited by 6 publications

(8 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By using the conditions required for a point vortex equilibrium, differential equations are derived for these polynomials which are then used to study the equilibria and establish connections to various polynomial systems. For an alternative approach to point vortex equilibria that is based on matrix methods, see [35,36].…”

Section: Introductionmentioning

confidence: 99%

A transformation between stationary point vortex equilibria

et al. 2020

View full text Add to dashboard Cite

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.

show abstract

Section: Introductionmentioning

confidence: 99%

A transformation between stationary point vortex equilibria

et al. 2020

View full text Add to dashboard Cite

show abstract

“…We remark that since the integrand expression in (25) is purely real and the integral is over the segment [0, 1] of the real axis, the quantity defined by (25) is also purely real. The circulation density γ(s) appearing in the Birkhoff-Rott equation (3) can then be deduced from (25) using relation (5). Noting that since on the contour [0, 1] we have t = s, we can conclude that γ p (s) = ϕ p (s), i.e., expression (25) can be interpreted as the circulation densities for different values of p. Consequently, we obtain the following main theorem.…”

Section: Rotating Equilibriamentioning

confidence: 99%

“…The circulation density γ(s) appearing in the Birkhoff-Rott equation (3) can then be deduced from (25) using relation (5). Noting that since on the contour [0, 1] we have t = s, we can conclude that γ p (s) = ϕ p (s), i.e., expression (25) can be interpreted as the circulation densities for different values of p. Consequently, we obtain the following main theorem.…”

Section: Rotating Equilibriamentioning

confidence: 99%

“…In this formulation the problems of determining the shapes L of the sheets and the corresponding circulation densities γ decouple: first, one needs to find the sheets L satisfying certain conditions after which obtaining the circulation density γ reduces to evaluating an integral formula. In this sense, there an analogy between the proposed approach and the "Brownian ratchet" method developed by Newton to find relative equilibria of point vortices [19,5]. Using the proposed approach we then construct a family of rotating equilibrium configurations consisting of a different number of sheets each with one endpoint at the center of rotation and the other at a vertex of a polygon.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rotating equilibria of vortex sheets

Protas

Sakajo

2020

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.

show abstract

“…Fluid velocity is high over relatively large areas on the sphere, with correspondingly larger interaction energies. These configurations may be compared with relative equilibrium states of vortex sheets on the plane [9,10].…”

Section: Introductionmentioning

confidence: 99%

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

O’Neil

2013

Proc. R. Soc. A.

View full text Add to dashboard Cite

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.

show abstract

Spectral gradient flow and equilibrium configurations of point vortices

Cited by 6 publications

References 27 publications

A transformation between stationary point vortex equilibria

A transformation between stationary point vortex equilibria

Rotating equilibria of vortex sheets

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

Contact Info

Product

Resources

About