One can consider the plasma flow to have currents because of the motion of the particles, but one has to think of it as both positive and negative currents, with the net current tending towards zero, since there is negligible accumulating charge.
In the Wiki article I cited earlier Birkeland is cited as a pioneer:
My reading of it is that Birkeland is an early worker in the field, with a very useful but crude understanding of the solar wind (the proton is generally considered to have been discovered in 1919, and plasmas still elude full understanding).In 1916, Birkeland was probably the first person to successfully predict that, "From a physical point of view it is most probable that solar rays are neither exclusively negative nor positive rays, but of both kinds". In other words, the solar wind consists of both negative electrons and positive ions.
I found the thunderbolts.info article a bit confusing. The notion of complicated electrical and magnetic effects in the solar system, especially effects due to the interaction with the earth's magnetic field, has been a subject of intense investigation for years. That an informal term like "ropes" is used by NASA to describe some of the observed effects does not seem to be a useful point of criticism.

