We discuss a long Josephson junction formed by two s-wave superconductors on a surface of a 3D topological insulator. The stripe between the superconductors is either gapped by a ferromagnet or ungapped. We show that in both cases the Josephson current is carried by the bound states (1D Majorana modes running along the junction) as well as by the 2D scattering states. We investigate the relative strength of these contribution as a function of the junction’s width. We observe that the 1D Majorana modes (bound states) survive at energies above the gap.