MHD Equations

The MHD equations take the form:

$$\frac{\partial n}{\partial t} + \nabla\!\cdot\!(n\,{\bf V}_e) = 0,$$ (377)

$$m_e n\,\frac{\partial {\bf V}_e}{\partial t} + m_e n\,({\bf V}_e\!\cdot\!\nabla){\bf V}_e+ \nabla p_e$$ (378)

$$+ [\delta^{-1}]\,e n\, ({\bf E} + {\bf V}_e\times {\bf B}) = [\zeta]\,{\bf F}_U + {\bf F}_T,$$

$$\frac{3}{2}\frac{\partial p_e}{\partial t} + \frac{3}{2} \,({\bf V}_e\!\cdot\!\nabla)\, p_e + \frac{5}{2}\,p_e\,\nabla\!\cdot\!{\bf V}_e = -[\delta^{-1}\,\zeta\,\mu^2]\,W_i,$$ (379)

and

$$\frac{\partial n}{\partial t} + \nabla\!\cdot\!(n\,{\bf V}_i) = 0,$$ (380)

$$m_i n\,\frac{\partial {\bf V}_i}{\partial t} + m_i n\,({\bf V}_i\!\cdot\!\nabla) {\bf V}_i + \nabla p_i$$ (381)

$$- [\delta^{-1}]\, e n\, ({\bf E} + {\bf V}_i\times {\bf B}) = - [\zeta]\,{\bf F}_U -{\bf F}_T,$$

$$\frac{3}{2}\frac{\partial p_i}{\partial t} + \frac{3}{2} \,({\bf V}_i\!\cdot\!\nabla) \,p_i + \frac{5}{2}\,p_i\,\nabla\!\cdot\!{\bf V}_i = [\delta^{-1}\,\zeta\,\mu^2]\, W_i.$$ (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

$${\bf V}_i \simeq {\bf V} + O(m_e/m_i),$$ (383)

and

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

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (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:

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

$${\bf E} + {\bf V}\times {\bf B} \simeq 0.$$ (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:

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

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

$$\frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$$ (389)

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

$$W_i \simeq 0,$$ (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.