MHD Equations

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 4.11, in the MHD limit. It follows from Equations (4.200) and (4.201) that

(4.211) |

and

(4.212) |

The main point, here, is that in the MHD limit the velocity difference between the electron and ion fluids is relatively small.

Equations (4.205) and (4.208) yield the *continuity equation*:

where .

Equations (4.206) and (4.209) can be summed to give the
*equation of motion*:

(4.214) |

Here, is the total pressure. Note that all terms in the previous equation are the same order in .

The
components of Equations (4.206) and (4.209) yield
the *Ohm's law*:

This is sometimes called the

Equations (4.207) and (4.210) can be summed to give the
*energy evolution equation*:

Equations (4.213) and (4.216) can be combined to give the more familiar

(4.217) |

Finally, the components of Equations (4.207) and (4.210) yield

(4.218) |

or [see Equation (4.101)]. Thus, we expect equipartition of the thermal energy between electrons and ions in the MHD limit.