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A magnetic arcade in the corona, in equilibrium with no
flow, may be modelled by solving,
which satisfies (4.1) when
and
. Thus,
is a solution to
. This is
equivalent to
 |
(4.12) |
In addition,
and so, taking the curl of
(4.12) we obtain
Using a vector identity we obtain
Thus, as the first term in the middle expression is zero, we have
 |
(4.13) |
For
, (4.12)
and (4.13) become
 |
(4.14) |
and
 |
(4.15) |
respectively.
To solve these equations we note that they are linear equations and
that the coefficients are constants and so we look for seperable solutions
of the form
Substituting into (4.15) gives
Dividing by
and rearranging gives
Taking the constant as
we obtain equations for
and
as
These are easily solved in terms of trigonometric functions and
expontential functions respectively to give one possible solution as
and
The field lines are shown in Figure 4.5.
Figure 4.5:
The field lines for the magnetic arcade model.
|
Subsections
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Up: Magnetohydrodynamics MHD
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Prof. Alan Hood
2000-11-06