MHD Equations
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 Section 4.12,
in the MHD limit. It follows from Equations (4.181) and (4.182)
that
![$\displaystyle {\bf V}_i \simeq {\bf V} + {\cal O}\left(\frac{m_e}{m_i}\right),$](img1476.png) |
(4.192) |
and
![$\displaystyle {\bf V}_e \simeq {\bf V} - [\delta]\,\frac{{\bf j}}{n\,e} + {\cal O}\left(\frac{m_e}
{m_i}\right).$](img1489.png) |
(4.193) |
The main point, here, is that in the MHD limit the velocity difference between
the electron and ion fluids is relatively small.
Equations (4.186) and (4.189) yield the continuity equation:
![$\displaystyle \frac{dn}{dt} + n\,\nabla\cdot{\bf V} = 0,$](img1478.png) |
(4.194) |
where
.
Equations (4.187) and (4.190) can be summed to give the
equation of motion:
![$\displaystyle m_i \,n\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times{\bf B} \simeq 0.$](img1490.png) |
(4.195) |
Here,
is the total pressure.
Note that all terms in the previous equation are the same order in
.
The
components of Equations (4.187) and (4.190) yield
the Ohm's law:
![$\displaystyle {\bf E} + {\bf V}\times {\bf B} \simeq {\bf0}.$](img1493.png) |
(4.196) |
This is sometimes called the perfect conductivity equation, because
it is identical to the Ohm's law in a perfectly conducting liquid.
Equations (4.188) and (4.191) can be summed to give the
energy evolution equation:
![$\displaystyle \frac{3}{2} \frac{dp}{dt} + \frac{5}{2}\, p \,\nabla\cdot{\bf V} \simeq 0.$](img1494.png) |
(4.197) |
Equations (4.194) and (4.197) can be combined to give the more familiar
adiabatic equation of state:
![$\displaystyle \frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$](img1495.png) |
(4.198) |
Finally, the
components of Equations (4.188) and (4.191)
yield
![$\displaystyle w_i \simeq 0,$](img1496.png) |
(4.199) |
or
[see Equation (4.82)]. Thus, we expect equipartition of the
thermal energy between electrons and ions in the MHD limit.