Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations
2021
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.
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