Next: MHD Waves
Up: Magnetohydrodynamic Fluids
Previous: Magnetic Pressure
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 comoving
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 FaradayMaxwell 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 fieldlines must move with the
plasma. In other words, the fieldlines are frozen into the plasma.
A fluxtube is defined as a topologically cylindrical volume whose
sides are defined by magnetic fieldlines. Suppose that, at some initial
time, a fluxtube is embedded in the plasma. According to the fluxfreezing
constraint,

(701) 
the subsequent motion of the plasma and the magnetic field is always
such as to maintain the integrity of the fluxtube. Since magnetic
fieldlines can be regarded as infinitely thin fluxtubes, we conclude that
MHD plasma motion also maintains the integrity of fieldlines. In other words,
magnetic fieldlines embedded in an MHD plasma can never break and reconnect:
i.e., MHD forbids any change in topology of the fieldlines. 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
20110331