Magnetohydrodynamic
theory involves a further simplification of fluid theory, where the
proton and electron fluids are combined and assumed
to possess a common flow velocity *U*.
We also assume that
the relevant time scales are long in comparison
with microscopic particle motion time scales (
)
and that spatial scale lengths are long in comparison with the Debye length
and the thermal ion gyroradius.
The *equation of motion* for the MHD fluid is derived by
adding electron and proton forms of (), to give

(3.31) |

where

The fluid velocity is an

A further relation, linking
and the fields, is obtained
by multiplying the proton form of () by
- *e*/*m*_{e} and the electron form of () by
*e*/*m*_{p} (where *e* is the charge of an electron).
Terms quadratic in velocity are ignored ensuring that the
resulting expression will be linear in .
Adding the two
equations,

(3.33) |

The following simplifying approximations are made (given that ):

Further, we assume that the momentum exchanged between electrons and ions is proportional to the relative velocity of the two types of particles, with

(3.34) |

where is the conductivity coefficient, with . This gives the generalized form of

For low-frequency disturbances, with characteristic frequency , the second term on the left-hand side may be dropped. In situations where the electron cyclotron frequency , the third term on the left-hand side may also be dropped. If the pressure gradient term is also insignificant, then () reduces to

In the
perfectly conducting limit (
), ()
further simplifies to

This means that a highly conducting plasma with zero current must set up an electric field . This is a convection electric field that leads to drift of the plasma perpendicular to . See Lecture 2 for examples. Equation () also leads to the condition of frozen-in magnetic flux (not proven here) in which a plasma carries a magnetic field along with it. This condition may be stated formally as

(3.38) |

and using Gauss's law and Stoke's theorem.

We define a *magnetic flux tube* to be the surface generated by moving
any closed loop parallel to the magnetic field lines it intersects
at any given time. This surface encloses a constant amount of magnetic
flux. As a consequence of flux conservation, the same fluid elements
constitute a flux tube at different times. The fluid and magnetic field lines
move together. A further consequence of the frozen-in flux condition
is that all particles initially in a flux tube will remain in the
same flux tube at later times.

An equation of energy continuity is derived by taking the
second order moment of Boltzmann's equation, to give

where is the adiabatic index which takes the value 5/3 for a monatomic gas. Equation () assumes the adiabatic equation of state, for which there is no change in internal energy of a fluid element as it propagates, with