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MHD Equations

The MHD equations take the form:
$\displaystyle \frac{\partial n}{\partial t} + \nabla\!\cdot\!(n\,{\bf V}_e)$ $\textstyle =$ $\displaystyle 0,$ (377)
$\displaystyle m_e n\,\frac{\partial {\bf V}_e}{\partial t} +
m_e n\,({\bf V}_e\!\cdot\!\nabla){\bf V}_e+
\nabla p_e$     (378)
$\displaystyle + [\delta^{-1}]\,e n\,
({\bf E} + {\bf V}_e\times {\bf B})$ $\textstyle =$ $\displaystyle [\zeta]\,{\bf F}_U + {\bf F}_T,$  
$\displaystyle \frac{3}{2}\frac{\partial p_e}{\partial t} + \frac{3}{2}
\,({\bf V}_e\!\cdot\!\nabla)\, p_e
+ \frac{5}{2}\,p_e\,\nabla\!\cdot\!{\bf V}_e$ $\textstyle =$ $\displaystyle -[\delta^{-1}\,\zeta\,\mu^2]\,W_i,$ (379)

and
$\displaystyle \frac{\partial n}{\partial t} + \nabla\!\cdot\!(n\,{\bf V}_i)$ $\textstyle =$ $\displaystyle 0,$ (380)
$\displaystyle m_i n\,\frac{\partial {\bf V}_i}{\partial t} +
m_i n\,({\bf V}_i\!\cdot\!\nabla) {\bf V}_i
+ \nabla p_i$     (381)
$\displaystyle - [\delta^{-1}]\, e n\,
({\bf E} + {\bf V}_i\times {\bf B})$ $\textstyle =$ $\displaystyle - [\zeta]\,{\bf F}_U
-{\bf F}_T,$  
$\displaystyle \frac{3}{2}\frac{\partial p_i}{\partial t} + \frac{3}{2}
\,({\bf V}_i\!\cdot\!\nabla) \,p_i +
\frac{5}{2}\,p_i\,\nabla\!\cdot\!{\bf V}_i$ $\textstyle =$ $\displaystyle [\delta^{-1}\,\zeta\,\mu^2]\, W_i.$ (382)

These equations can also be simplified by making use of the smallness of the mass ratio $m_e/m_i$. Now, according to the ordering adopted in Sect. 3.9, $U \sim \delta\,V_e\sim \delta\,V_i$ in the MHD limit. It follows from Eqs. (372) and (373) that
\begin{displaymath}
{\bf V}_i \simeq {\bf V} + O(m_e/m_i),
\end{displaymath} (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} (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:

\begin{displaymath}
\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,
\end{displaymath} (385)

where $d/dt \equiv \partial/\partial t + {\bf V}\!\cdot\!\nabla$.

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

\begin{displaymath}
m_i n\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times{\bf B} \simeq 0.
\end{displaymath} (386)

Here, $p=p_e+p_i$ is the total pressure. Note that all terms in the above equation are the same order in $\delta$.

The $O(\delta^{-1})$ components of Eqs. (378) and (381) yield the Ohm's law:

\begin{displaymath}
{\bf E} + {\bf V}\times {\bf B} \simeq 0.
\end{displaymath} (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:

\begin{displaymath}
\frac{3}{2} \frac{dp}{dt} + \frac{5}{2}\, p \,\nabla\!\cdot\!{\bf V} \simeq 0.
\end{displaymath} (388)

Equations (385) and (388) can be combined to give the more familiar adiabatic equation of state:
\begin{displaymath}
\frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.
\end{displaymath} (389)

Finally, the $O(\delta^{-1})$ components of Eqs. (379) and (382) yield

\begin{displaymath}
W_i \simeq 0,
\end{displaymath} (390)

or $T_e\simeq T_i$ [see Eq. (273)]. Thus, we expect equipartition of the thermal energy between electrons and ions in the MHD limit.


next up previous
Next: Drift Equations Up: Plasma Fluid Theory Previous: Cold-Plasma Equations
Richard Fitzpatrick 2011-03-31