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Flux Freezing
The MHD Ohm's law,
 |
(693) |
is sometimes referred to as the perfect conductivity equation, for
obvious reasons, and sometimes as the flux freezing equation.
The latter nomenclature comes about because Eq. (693) implies that the magnetic
flux through any closed contour in the plasma, each element of
which moves with the local plasma velocity, is a conserved quantity.
In order to verify the above assertion, let us consider the
magnetic flux,
, through a contour,
, which is co-moving
with the plasma:
 |
(694) |
Here,
is some surface which spans
. The time rate of
change of
is made up of two parts. Firstly, there
is the part due to the time variation of
over the
surface
. This can be written
 |
(695) |
Using the Faraday-Maxwell equation, this reduces to
 |
(696) |
Secondly, there is the part due to the motion of
. If
is an element of
then
is the area swept out
by
per unit time. Hence, the flux crossing this area is
.
It follows that
 |
(697) |
Using Stokes's theorem, we obtain
 |
(698) |
Hence, the total time rate of change of
is given by
 |
(699) |
The condition
 |
(700) |
clearly implies that
remains constant in time
for any arbitrary contour.
This, in turn, implies that magnetic field-lines must move with the
plasma. In other words, the field-lines are frozen into the plasma.
A flux-tube is defined as a topologically cylindrical volume whose
sides are defined by magnetic field-lines. Suppose that, at some initial
time, a flux-tube is embedded in the plasma. According to the flux-freezing
constraint,
 |
(701) |
the subsequent motion of the plasma and the magnetic field is always
such as to maintain the integrity of the flux-tube. Since magnetic
field-lines can be regarded as infinitely thin flux-tubes, we conclude that
MHD plasma motion also maintains the integrity of field-lines. In other words,
magnetic field-lines embedded in an MHD plasma can never break and reconnect:
i.e., MHD forbids any change in topology of the field-lines. It turns
out that this is an extremely restrictive constraint. Later on, we shall discuss
situations in which this constraint is relaxed.
Next: MHD Waves
Up: Magnetohydrodynamic Fluids
Previous: Magnetic Pressure
Richard Fitzpatrick
2011-03-31