MHD Equations

The MHD equations take the form:

$\displaystyle \frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_e)$ $\displaystyle =0,$ (4.186)
$\displaystyle m_e \,n\,\frac{\partial {\bf V}_e}{\partial t} +
m_e \,n\,({\bf V...
...bf V}_e+
\nabla p_e+ [\delta^{-1}]\,e\, n\,
({\bf E} + {\bf V}_e\times {\bf B})$ $\displaystyle = [\zeta]\,{\bf F}_U + {\bf F}_T,$ (4.187)
$\displaystyle \frac{3}{2}\frac{\partial p_e}{\partial t} + \frac{3}{2}
\,({\bf V}_e\cdot\nabla)\, p_e
+ \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e$ $\displaystyle = -[\delta^{-1}\,\zeta\,\mu^2]\,w_i,$ (4.188)

and

$\displaystyle \frac{\partial n}{\partial t} + \nabla\cdot(n\,{\bf V}_i)$ $\displaystyle =0,$ (4.189)
$\displaystyle m_i \,n\,\frac{\partial {\bf V}_i}{\partial t} +
m_i\, n\,({\bf V...
...onumber\\ [0.5ex]
- [\delta^{-1}]\, e\, n\,
({\bf E} + {\bf V}_i\times {\bf B})$ $\displaystyle =- [\zeta]\,{\bf F}_U
-{\bf F}_T,$ (4.190)
$\displaystyle \frac{3}{2}\frac{\partial p_i}{\partial t} + \frac{3}{2}
\,({\bf V}_i\cdot\nabla) \,p_i +
\frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i$ $\displaystyle =[\delta^{-1}\,\zeta\,\mu^2]\, w_i.$ (4.191)

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 Section 4.12, $U \sim \delta\,V_e\sim \delta\,V_i$ in the MHD limit. It follows from Equations (4.181) and (4.182) that

$\displaystyle {\bf V}_i \simeq {\bf V} + {\cal O}\left(\frac{m_e}{m_i}\right),$ (4.192)

and

$\displaystyle {\bf V}_e \simeq {\bf V} - [\delta]\,\frac{{\bf j}}{n\,e} + {\cal O}\left(\frac{m_e}
{m_i}\right).$ (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:

$\displaystyle \frac{dn}{dt} + n\,\nabla\cdot{\bf V} = 0,$ (4.194)

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

Equations (4.187) and (4.190) can be summed to give the equation of motion:

$\displaystyle m_i \,n\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times{\bf B} \simeq 0.$ (4.195)

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

The ${\cal O}(\delta^{-1})$ components of Equations (4.187) and (4.190) yield the Ohm's law:

$\displaystyle {\bf E} + {\bf V}\times {\bf B} \simeq {\bf0}.$ (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:

$\displaystyle \frac{3}{2} \frac{dp}{dt} + \frac{5}{2}\, p \,\nabla\cdot{\bf V} \simeq 0.$ (4.197)

Equations (4.194) and (4.197) can be combined to give the more familiar adiabatic equation of state:

$\displaystyle \frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$ (4.198)

Finally, the ${\cal O}(\delta^{-1})$ components of Equations (4.188) and (4.191) yield

$\displaystyle w_i \simeq 0,$ (4.199)

or $T_e\simeq T_i$ [see Equation (4.82)]. Thus, we expect equipartition of the thermal energy between electrons and ions in the MHD limit.