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This article is cited in 2 scientific papers (total in 2 papers)
Special Issue: 200th birthday of Hermann von Helmholtz
Evolution of the Singularities of the Schwarz Function
Corresponding to the Motion of a Vortex Patch in the
Two-dimensional Euler Equations
Giorgio Riccardiab, David G. Dritschelc a Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”,
viale A. Lincoln 5, 8100 Caserta, Italy
b INM-CNR, Institute of Marine Engineering, National Research Council of Italy,
via di Vallerano 139, 00128 Rome, Italy
c Mathematical Institute, University of St Andrews,
St Andrews, Fife, KY16 9SS, UK
Abstract:
The paper deals with the calculation of the internal singularities of the Schwarz
function corresponding to the boundary of a planar vortex patch during its self-induced motion
in an inviscid, isochoric fluid. The vortex boundary is approximated by a simple, time-dependent
map onto the unit circle, whose coefficients are obtained by fitting to the boundary computed in
a contour dynamics numerical simulation of the motion. At any given time, the branch points of
the Schwarz function are calculated, and from them, the generally curved shape of the internal
branch cut, together with the jump of the Schwarz function across it. The knowledge of the
internal singularities enables the calculation of the Schwarz function at any point inside the
vortex, so that it is possible to check the validity of the map during the motion by comparing
left and right hand sides of the evolution equation of the Schwarz function. Our procedure
yields explicit functional forms of the analytic continuations of the velocity and its conjugate
on the vortex boundary. It also opens a new way to understand the relation between the time
evolution of the shape of a vortex patch during its motion, and the corresponding changes in
the singular set of its Schwarz function.
Keywords:
two-dimensional vortex dynamics, contour dynamics, Schwarz function, complex
analysis.
Received: 10.06.2021 Accepted: 09.09.2021
Citation:
Giorgio Riccardi, David G. Dritschel
Linking options:
https://www.mathnet.ru/eng/rcd1132
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