The formation of concentrated vortices like tornadoes and tropical cyclones in rotating fluids is of much interest in atmospheric flows. It is shown by direct numerical simulation that the selective decay of inviscid invariants leads to concentration of vorticity in a destabilized vortex. By selective decay we mean here that the circulation of the mean flow decays faster than the angular momentum or energy. Initially localized disturbances are superimposed onto the two-dimensional flattened Taylor Green vortices to trigger the elliptic instability. In the later stage of nonlinear evolution of the disturbance circulation decays faster than angular momentum and energy, giving rise to a sharp peak in the vorticity distribution of the mean flow. During the selective decay vortex pairs reconnect and eventually annihilate at the cell boundaries of the Taylor Green vortices. By evaluating the weight function of the inviscid invariants it is shown that the loss of angular momentum is much smaller than that of circulation when vorticity is lost at the cell boundary by reconnection or annihilation. Thus the reconnection and subsequent annihilation of vortex pairs is responsible for the selective decay and concentration of vorticity. The relevance of the mechanism to previous experiments and general cases is also discussed.