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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
![\begin{displaymath}
{\bf V}_e \simeq {\bf V} - [\delta]\,\frac{{\bf j}}{ne} + O\!\left(\frac{m_e}
{m_i}\right).
\end{displaymath}](img945.png) |
(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.
Next: Drift Equations
Up: Plasma Fluid Theory
Previous: Cold-Plasma Equations
Richard Fitzpatrick
2011-03-31