I've read that after a plasma channel is created in air, the resistance decreases significantly and the arc can be sustained with a much lower voltage, as long as the heat dissipated by the plasma is enough to sustain the plasma. I've been trying to calculate the minimum voltage required (if one exists) to sustain an arc with a given separation distance, air pressure, etc, assuming that the current depends only on the resistance of the plasma.
I have tried setting the rate of heat loss from the plasma into the surroundings (from Newton's law of cooling) equal to the power as given by Ohm's law (using Spitzer resistivity), but the unknown proportionality constant in the former means I cannot solve for any real numbers.
Am I approaching this completely wrong? What am I missing?