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trapped in the earth's magnetic field forever. In practice, some species with par-
ticular energies (e.g., protons with energies
100 MeV) are trapped for
100
10 10 cm
10 9 s
10 19 cm. Since
years. These particles travel
1
×
/
s
×
3
×
=
3
×
10 9 cm, such particles traverse the system
10 10
the scale size of the system is
times before escaping.
We now raise the interesting question “Can the current in some complicated
plasma problem be computed using only the guiding center equations?” As
pointed out by Spitzer (1962), the answer is no, since the guiding center current
is only part of the total current. The particle gyromotion produces an additional
current if there is a population gradient, exactly analogous to the magnetization
current, that arises from edge effects in solids. When this contribution is calcu-
lated exactly, it contains a term that can even cancel the gradient and curvature
drift terms. The actual currents have nothing to do with the values of B and
B ,
and we must use the macroscopic equations given in Section 2.5.2. The sign of
the current is generally given correctly by the sign of the gradient and curvature
drifts, but the magnitudes must be derived from other equations.
2.5.2 Magnetohydrodynamics
We now take up a second viewpoint on the dynamics of a collisionless magnetized
plasma. In magnetohydrodynamics it is conjectured that the electrical conduc-
tivity is so high parallel to B that in a reference frame moving with velocity
B 2
V
=
E
×
B
/
the electric field vanishes both parallel and perpendicular to B . Since this is the
plasma reference frame, the plasma can be considered as a single fluid having
infinite conductivity. Study of such a one-fluid model is referred to as magneto-
hydrodynamics. Its range of applicability is particularly great in astrophysics due
to the large distance scales involved. Thus, plasmas in the solar wind, solar flares,
sunspots, the earth's magnetosphere, and interstellar regions can all be treated
to first order with a magnetohydrodynamic approach.
A close analogy exists with hydrodynamics, inwhich, as we have noted, a single
fluid is subject to a variety of forces: pressure, viscosity, gravity, and so forth.
The presence of an electrical conductivity requires the inclusion of Maxwell's
equations and a volume force on the fluid given by J
×
B . We thus have the
continuity equation
∂ρ/∂
t
+∇·
V
) =
0
(2.55a)
the equation of motion,
ρ d V dt =−∇
p
+ ρ
g
+
J
×
B
(2.55b)
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