The supercurrents circulate around the vortex center which is situated inside the Josephson barrier, unlike Abrikosov vortices in type-II superconductors, which are located in the superconducting condensate.
The cores of Josephson vortices are more complex and depend on the physical nature of the barrier.
Similarly to Abrikosov vortices in superconductors, Josephson vortices in SNS Josephson junctions are characterized by cores in which the correlations are suppressed by destructive quantum interference and the normal state is recovered.
Rather, it depends on supercurrents circulating in superconducting electrodes, applied magnetic field etc.
In Superconductor-Insulator-Superconductor (SIS) Josephson tunnel junctions the cores are not expected to have a specific spectral signature; they were not observed.