That's an interesting paper, I wish I could understand the math better. It seems to treat only the DC case, however. I wonder if there's an AC treatment around somewhere. It'll surely be much messier! One of the guiding assumptions is that the arc is a resistive fluid obeying Ohm's law. Arcs generally don't obey Ohm's law.
As the authors note, the paper is a "useful intermediate step".
Thanks for writing!
bob prohaska