[throwawayx134ax]

Question: Do you mean current as in movement of material, or current as in movement of electric fields?

Or is that not a valid distinction in this case?

[pfdietz]

Both meanings were used here. There are electrical currents, and there are flows of material through the magnetic field, producing voltages that sustain currents. The JxB forces also affect the flows. It's a highly nonlinear phenomenon (magnetohydronamics, or MHD).

[pfdietz]

Magnetohydrodynamics (spelling)