The Interaction of Two Unsteady Point Vortex Sources in a Deformation Field in 2D Incompressible Flows

Taking into account the coupling of the ocean with the atmosphere is essential to properly describe vortex dynamics in the ocean. The forcing of a circular eddy with the relative wind stress curl leads to an Ekman pumping with a nonzero area integral. This in turn creates a source or a sink in the eddy. We revisit the two point vortex-source interaction, now coupled with an unsteady wind, leading to a time-varying circulation and source strength. Firstly, we recover the various fixed points of the two vortex-source system, and we calculate their stability. Then we show the effect of a weak amplitude, subharmonic, or harmonic time variation of the wind, leading to a similar variation of the circulation and the source strength of the vortex sources. We use a multiple time scale expansion of the variables to calculate the long time variation of these vortex trajectories around neutral fixed points. We study the amplitude equation and obtain its solution. We compute numerically the unstable evolution of the vortex sources when the source and circulation have a finite periodic variation. We also assess the influence of this time variation on the dispersion of a passive tracer near these vortex sources.