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For a field
and flow
, (4.2) becomes
 |
(4.10) |
Now consider the magnetic flux between two lines at
and
,
which are moving with the plasma. Thus,
and
such that
The flux is
Figure 4.4:
The flux between
and
.
|
Notice that the flux is a function of time only. The spatial
dependence has been integrated out. Flux changes in time due to
two effects, namely because
changes in time and because the end points move at speeds
and
reducing or increasing the range of integration. On
differentiating with respect to time we must remember to
differentiate not only the integrand but also the limits. Thus,
Hence, we may write
By (4.10) this is zero. In other words, the amount of
magnetic flux remains constant, and we say that it is frozen to the plasma.
Example 4. .1Consider the effect of a flow
on an initially
uniform field,
. Equation (4.10) becomes
Thus,
 |
(4.11) |
Look for a seperable solution to this linear equation of the form
so that (4.11) may be expressed as
Dividing by
and collecting all the functions of time onto one
side of the equation and all the functions of
onto the other we
get
Consider the
-dependence first of all.
Integrating we obtain
Similarly we may integrate the
dependent terms,
Hence, we obtain
In theory, we need to obtain the values for the separation constant
and add together all the possible solutions. However, this
example is particularly simple. Satisfying the initial condition gives
Therefore,
Thus, the flow carries in the field lines and makes the magnetic
field strength grow exponentially in time.
Next: MHD Equilibrium Structures
Up: MHD Equations
Previous: MHD Equations
Prof. Alan Hood
2000-11-06