Abstract:
We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.
Keywords:
ideal flow, vortex dynamics, point vortices.
Funding agency
V.S.K. acknowledges financial support from CAPES/Brazil through a Science Without Borders postdoctoral program during his stay at the Federal University of Pernambuco, and the ESI Junior Research Fellowship Program during his stay at the Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna.
Citation:
Vikas S. Krishnamurthy, Mark A. Stremler, “Finite-time Collapse of Three Point Vortices in the Plane”, Regul. Chaotic Dyn., 23:5 (2018), 530–550
\Bibitem{KriStr18}
\by Vikas S. Krishnamurthy, Mark A. Stremler
\paper Finite-time Collapse of Three Point Vortices in the Plane
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 5
\pages 530--550
\mathnet{http://mi.mathnet.ru/rcd343}
\crossref{https://doi.org/10.1134/S1560354718050040}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000447268600004}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85054605889}
Linking options:
https://www.mathnet.ru/eng/rcd343
https://www.mathnet.ru/eng/rcd/v23/i5/p530
This publication is cited in the following 15 articles:
Francesco Grotto, Marco Romito, Milo Viviani, “Zero-noise dynamics after collapse for three point vortices”, Physica D: Nonlinear Phenomena, 457 (2024), 133947
Jiahe Chen, Qihuai Liu, “Sufficient and necessary conditions for self-similar motions of three point vortices in generalized fluid systems”, Physica D: Nonlinear Phenomena, 2024, 134392
Martin Donati, Ludovic Godard-Cadillac, Dragoş Iftimie, “On the dynamics of point vortices with positive intensities collapsing with the boundary”, Physica D: Nonlinear Phenomena, 470 (2024), 134402
Martin Donati, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains”, SIAM Rev., 65:1 (2023), 227
Ludovic Godard-Cadillac, “Hölder estimate for the 3 point-vortex problem with alpha-models”, Comptes Rendus. Mathématique, 361:G1 (2023), 355
Martin Donati, Ludovic Godard-Cadillac, “Hölder regularity for collapses of point-vortices”, Nonlinearity, 36:11 (2023), 5773
Sergey G. Chefranov, Igor I. Mokhov, Alexander G. Chefranov, “Investigating the dynamics of point helical vortices on a rotating sphere to model tropical cyclones”, Physics of Fluids, 35:4 (2023)
Habin Yim, Sun-Chul Kim, Sung-Ik Sohn, “Motion of three geostrophic Bessel vortices”, Physica D: Nonlinear Phenomena, 441 (2022), 133509
Francesco Grotto, Umberto Pappalettera, “Burst of Point Vortices and Non-uniqueness of 2D Euler Equations”, Arch Rational Mech Anal, 245:1 (2022), 89
Qian Luo, Yufei Chen, Qihuai Liu, “Global Phase Diagrams of Three Point Vortices”, Int. J. Bifurcation Chaos, 32:02 (2022)
Jean N. Reinaud, David G. Dritschel, Richard K. Scott, “Self-similar collapse of three vortices in the generalised Euler and quasi-geostrophic equations”, Physica D: Nonlinear Phenomena, 434 (2022), 133226
M. A. Stremler, “Something Old, Something New:
Three Point Vortices on the Plane”, Regul. Chaotic Dyn., 26:5 (2021), 482–504
S. S. Kallyadan, P. Shukla, “Dynamics of two moving vortices in the presence of a fixed vortex”, Eur. J. Mech. B-Fluids, 89 (2021), 458–472
H. Kudela, “Collapse of N point vortices, formation of the vortex sheets and transport of passive markers”, Energies, 14:4 (2021), 943
Citation in format AMSBIB
Contact us: