Dynamics of vortex clusters is essential for understanding diverse superfluidphenomena. In this paper, we examine the dynamics of vortex quadrupoles in atrapped two-dimensional (2D) Bose-Einstein condensate. We find that themovement of these vortex-clusters fall into three distinct regimes which arefully described by the radial positions of the vortices in a 2D isotropicharmonic trap, or by the major radius (minor radius) of the ellipticalequipotential lines decided by the vortex positions in a 2D anisotropicharmonic trap. In the "recombination" and "exchange" regimes the quadrupolestructure maintains, while the vortices annihilate each other permanently inthe "annihilation" regime. We find that the mechanism of the charge flipping inthe "exchange" regime and the disappearance of the quadrupole structure in the"annihilation" regime are both through an intermediate state where two vortexdipoles connected through a soliton ring. We give the parameter ranges forthese three regimes in coordinate space for a specific initial configurationand phase diagram of the vortex positions with respect to the Thomas-Fermiradius of the condensate. We show that the results are also applicable tosystems with quantum fluctuations for the short-time evolution.
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