A pulsed slit discharge nozzle (PDN) has been developed in our laboratory to generate molecular ions in an astrophysically relevant environment. The free cold molecular ions are formed through soft (Penning) ionization of the neutral precursor molecules seeded in a supersonic free jet and are probed with cavity ringdown spectroscopy. An attempt is made to characterize the nature and the structure of the plasma that is generated in these experiments to optimize the yield of formation of ions in the jet. The experimental conditions are characterized by a strong pressure gradient in a short discharge zone. We find that the plasma generated in the PDN source is best characterized as an intense abnormal glow discharge and that its structure is reduced to a negative glow and to dark zones near the electrodes. We have calculated the parameters (length, thickness and cathode voltage fall) that are associated with the Crookes dark space and the negative glow in the plasma. We have also estimated the electron temperature (Te) and density (ne) in the plasma. All these parameters are required to optimize the yield of formation of ions and radicals in the jet expansions, a key requirement in our experiments.