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The MHD equations can be combined with Maxwell's equations,
to form a closed set. The displacement current is neglected in
Eq. (687) on the reasonable assumption that MHD motions are slow
compared to the velocity of light. Note that Eq. (688) guarantees
that
, provided that this relation is
presumed to hold initially. Similarly, the assumption of
quasineutrality renders the PoissonMaxwell equation,
, irrelevant.
Equations (682) and (687) can be combined to give the MHD equation
of motion:

(689) 
where

(690) 
Suppose that the magnetic field is approximately uniform, and
directed along the axis. In this case, the above equation of
motion reduces to

(691) 
where

(692) 
Note that the magnetic field increases the plasma pressure, by an
amount , in directions perpendicular to the magnetic field,
and decreases the plasma pressure, by the same amount, in the
parallel direction. Thus, the magnetic field gives rise to
a magnetic pressure, , acting perpendicular to fieldlines,
and a magnetic tension, , acting
along fieldlines.
Since, as we shall see presently, the plasma is tied to magnetic fieldlines,
it follows that magnetic fieldlines embedded in an
MHD plasma act rather like
mutually repulsive elastic bands.
Next: Flux Freezing
Up: Magnetohydrodynamic Fluids
Previous: Introduction
Richard Fitzpatrick
20110331