  Dynamics of vortex clusters is essential for understanding diverse superfluid
phenomena. In this paper, we examine the dynamics of vortex quadrupoles in a
trapped two-dimensional (2D) Bose-Einstein condensate. We find that the
movement of these vortex-clusters fall into three distinct regimes which are
fully described by the radial positions of the vortices in a 2D isotropic
harmonic trap, or by the major radius (minor radius) of the elliptical
equipotential lines decided by the vortex positions in a 2D anisotropic
harmonic trap. In the "recombination" and "exchange" regimes the quadrupole
structure maintains, while the vortices annihilate each other permanently in
the "annihilation" regime. We find that the mechanism of the charge flipping in
the "exchange" regime and the disappearance of the quadrupole structure in the
"annihilation" regime are both through an intermediate state where two vortex
dipoles connected through a soliton ring. We give the parameter ranges for
these three regimes in coordinate space for a specific initial configuration
and phase diagram of the vortex positions with respect to the Thomas-Fermi
radius of the condensate. We show that the results are also applicable to
systems with quantum fluctuations for the short-time evolution.
