Magneto thermal free convection of air in a square enclosure under a non-uniform magnetic field provided by a permanent magnet is numerically studied. The results show that the natural convection of air in the square enclosure under magnetic field is quite different from that under the gravity field. The local value of Nusselt number under the magnetic field can reach to a much higher value than the maximum local value under the gravity field. Relatively uniform distribution of Nusselt number can be obtained along the cold wall of the enclosure under the magnetic field. A permanent magnet with high magnetic energy product with
reaches to 3.5 Tesla can plays a comparative role on the average Nusselt number compared with that under the gravity environment.