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The MHD equations take the form:
and
These equations can also be simplified by making use of the smallness
of the mass ratio . Now, according to the ordering adopted in Sect. 3.9,
in the MHD limit. It follows from Eqs. (372) and (373)
that

(383) 
and

(384) 
The main point, here, is that in the MHD limit the velocity difference between
the electron and ion fluids is relatively small.
Equations (377) and (380) yield the continuity equation:

(385) 
where
.
Equations (378) and (381) can be summed to give the
equation of motion:

(386) 
Here, is the total pressure.
Note that all terms in the above equation are the same order in .
The
components of Eqs. (378) and (381) yield
the Ohm's law:

(387) 
This is sometimes called the perfect conductivity equation, since
it is identical to the Ohm's law in a perfectly conducting liquid.
Equations (379) and (382) can be summed to give the
energy evolution equation:

(388) 
Equations (385) and (388) can be combined to give the more familiar
adiabatic equation of state:

(389) 
Finally, the
components of Eqs. (379) and (382)
yield

(390) 
or [see Eq. (273)]. Thus, we expect equipartition of the
thermal energy between electrons and ions in the MHD limit.
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Richard Fitzpatrick
20110331