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The ground state of a two-dimensional square superconducting TT ring array has been investigated. The circulating currents of the π ring array will spontaneously magnetize to the ’antiferromagnetic’ arrangement with directions of the nearest-neighbouring currents circulating oppositely in the absence of an extal magnetic field. It is found that the extal magnetic field could destroy the anti-parallel configuration effectively. The extal magnetic field needed to destroy the anti-parallel configuration is related to the superconducting π ring’s inductance parameter β= 2πLIc/φ0.For a small β the anti-parallel configuration, which is the lowest-energy ground state of the system, will be fully destroyed and changed to the configuration that the circulating currents have the same direction and parallel to the extal magnetic field when the magnetic flux reachesφ0/4 in each ring. Moreover, the magnetic field needed to destroy the anti-parallel configuration will be very small when βis large enough.