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2
B
2
B
yields finally
But using
∇×
(
V
×
B
)
=∇
(
∇·
B
)
−∇
=−∇
2
B
∂
B
/∂
t
=∇×
(
V
×
B
)
+
(
1
/σμ
0
)
∇
(2.59)
The MHD equations can now be written
+∇·
(
ρ
)
=
∂ρ/∂
t
V
0
(2.60a)
∇·
B
=
0
=∇·
E
(2.60b)
p
μ
0
ρ
d
V
dt
=−∇
B
2
+
(
·∇
)
/
+
/
2
B
B
/μ
0
(2.60c)
2
B
∂
B
/∂
t
=∇×
(
V
×
B
)
+
(
1
/σμ
0
)
∇
(2.60d)
One of the equations of state completes the system.
The concepts of magnetic pressure and tension are derivable from (2.60c). One
should remember that the terms involving the magnetic field
B
all stem from the
J
B
force but that
J
and
B
are inextricably related through Maxwell's equa-
tions. Thus, any deviation of
B
from a force-free configuration must be balanced
by fluid pressure when the fluid is in equilibrium. Fluid acceleration occurs in
the case of a nonequilibrium condition. For example, it is straightforward to
show that the dipole magnetic field in the upper portion of Fig. 2.12 is such that
the magnetic terms in (2.60c) cancel everywhere outside the core of the earth.
The distorted dipole field shown following, however, has forces that will cause
the plasma to flow back toward the earth. Since
×
0 in this distorted
field, there must be a current in the region, as indicated. Such a current is asso-
ciated with a
J
∇×
B
=
B
force on the plasma in the direction toward the earth, since
there is a small component of
B
upward while
J
is out of the page. This
J
×
B
force is equivalent to the magnetic pressure and magnetic tension forces. In a
steady state, then, there must be a particle pressure gradient pointing toward the
earth.
The meaning of the magnetic terms can be understood in a local coordinate
system (see Fig. 2.13) defined by a unit vector
×
ˆ
s
parallel to
B
, a unit vector
n
ˆ
Force-free dipole
(no plasma)
3
B
5
0
Distorted dipole
(plasma sheet)
B
J
3
B
J
3
B
5
0
J
Figure 2.12
Illustration of the difference between a force-free dipole and a distorted field
configuration. The magnetic tension force is related to the
J
×
B
force shown.
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