Numerical analysis of anode boundary layer was carried out to make clear the physical basis of the anode fall in the atmospheric free-burning argon arc. The governing equations of the dominating processes took account of a physical state of non-local thermodynamic equilibrium, and they were solved by applying the finite difference method. The numerical results showed each anode fall, namely a positive anode fall and a negative one, and also showed the distributions of the temperature and electron number density in the case of each anode fall. We conclude that the gradient of electron number density played the most important role in driving an electron flux toward the anode and also determined the sign and magnitude of the anode fall. Furthermore, the electron density gradient was strongly affected by the relation between arc current density and convective flow in the anode boundary layer.