An appropriate coupling of an arc plasma column in the state of local thermodynamic equilibrium to a refractory cathode necessarily involves the non-equilibrium boundary layer between them. A model has been developed that combines a model of an equilibrium direct current arc plasma in atmospheric pressure argon with the assembly of a cathode made of tungsten and the boundary layer. A bidirectional coupling has been realized that allows us to consider a variable voltage drop across the boundary layer for different positions on the cathode. Results are obtained for arc currents between 10 and 150A in the cases of both a unidirectional and a bidirectional coupling. The results Show differences in the distributions of the temperature and the normal current density on the cathode surface, and the radial and axial distribution of the plasma temperature. Comparison with results of a fully non-equilibrium model of the arc plasma and experimental findings from optical emission spectroscopy show a fair agreement for currents, where the deviations from equilibrium in the arc column can be ignored. For arc currents beyond 100 A, the arc attachment on the cathode appears in two forms, which differ from each other in the distributions of the temperature and the normal current density on the cathode surface, whereas the values of the total arc voltage are close to each other.