## MHD equations - confused about derivation

I am confused about the usual derivation of the ideal magnetohydrodynamic equations, as given for example here:

http://theoretical-physics.net/dev/s...amics/mhd.html

The problem is that there are a few different "currents" to consider. For example, in the momentum equation, the current that feels the lorentz force and changes the dynamics of the plasma should only be the current in the plasma. In the induction equation, the current inducing the magnetic field should be the total current. But when we neglect the displacement current in Ampere's law and substitute the result into the Lorentz force, we're equating these two, which ignores the possibility of external current separate from the plasma generating magnetic fields which interact with the plasma.

Moreover, (and maybe I just don't understand Ohm's law), the current gotten from Ohm's law is just that induced in the plasma by external fields, and has nothing to do with the "innate" current in the plasma, which is due to whatever mechanism is inducing ionization.

so, i guess my question is, why are we allowed to conflate these three distinct notions of current into one quantity in the equations?
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 You can neglect the displacement current if the magnetic field and the current do not varry with big frequency. I guess one assumes that the total current obeys ohms law and maxwell-amperes law everywhere, regardless if whole or part of it is due to ionization mechanisms or due to fields accelerating charges. Maybe this assumption will not hold in some cases, i.e if the so called innate current in plasma is due to some mechanism that violates ohms law (maybe) or maxwell-amperes law (unlikely to happen). But seems to me that deep down whatever mechanism is inducing ionization would have to do with electromagnetic field accelerating charges.

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