14. Magnetohydrodynamics
14.1 The equations of magnetohydrodynamics
In our preceding considerations we have treated gas independent of its state of ionization, yet
we know that an ionized medium can carry and be influenced by electromagnetic fields, that
should follow Maxwell’s equations.
vector
∇ ·
vector
E
= 4
π ρ
e
vector
∇ ×
vector
E
=
−
1
c
∂
vector
B
∂t
(14
.
1
−
2)
vector
∇ ·
vector
B
= 0
vector
∇ ×
vector
B
=
4
π
c
vector
j
e
+
1
c
∂
vector
E
∂t
(14
.
3
−
4)
where the charge density and current density are
ρ
e
=
Z e n
i
−
e n
e
(14
.
5)
vector
j
e
=
Z e n
i
vector
V
i
−
e n
e
vector
V
e
(14
.
6)
Implicit to Maxwell’s equations is the charge conservation, for
∂ρ
e
∂t
=
1
4
π
vector
∇ ·
∂
vector
E
∂t
=
c
4
π
vector
∇ ·
(
vector
∇ ×
vector
B
)
−
vector
∇ ·
vector
j
e
⇒
∂ρ
e
∂t
+
vector
∇ ·
vector
j
e
= 0
(14
.
7)
The effect of the electromagnetic fields is different depending on the spatial scale on which they
arise. Smallscale perturbations of the fields typically lead to electromagnetic waves which we
will discuss later. On large scales astrophysical plasma are usual neutral, as expressed in the
quasineutrality condition
vector
j
e
=
σ
vector
E
→
∂ρ
e
∂t
=
−
vector
∇ ·
vector
j
e
=
−
4
π σ ρ
e
⇒
ρ
e
≃
0
on large scales
(14
.
8)
This implies the absence of sources of a largescale electric field. The conductivity of astrophys
ical plasmas,
σ
, is high and consequently any largescale electric field would be quickly shorted
out. Currents can still exist, for only the bulk velocity of electrons and ions must be different,
and the Maxwell’s fourth equation can be simply written for large scales
vector
∇ ×
vector
B
=
4
π
c
vector
j
e
≃
4
π
c
e n
e
(
vector
V
i
−
vector
V
e
)
≃ −
4
π
c
e n
e
vector
V
e,rel
(14
.
9)
with the relative velocity of the electrons with respect to the ions,
vector
V
e,rel
=
vector
V
e
−
vector
V
i
. A largescale
velocity difference of the positive and negative charges, i.e. not the difference in the gyration
1
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motion around a largescale magnetic field, will likely be the result of a balance between an
electric field and collisional resistance with collision frequency
ν
C
, so in the rest frame of the
ions (upper index ”i”)
vector
F
= 0 =
−
e
vector
E
i
−
m
e
ν
C
vector
V
e,rel
⇒
vector
V
e,rel
=
−
e
m
e
ν
C
vector
E
i
vector
j
i
=
−
e n
e
vector
V
e,rel
=
n
e
e
2
m
e
ν
C
vector
E
i
=
σ
vector
E
i
(14
.
10)
Transforming back from the rest frame of the ions to the laboratory frame we note that the
current density involves only a velocity difference and hence is unchanged.
Then using the
Lorentz transformations for the fields we obtain
v
c
E
≪
E
≪
B
vector
j
e
=
vector
j
i
vector
B
=
vector
B
i
vector
E
i
=
vector
E
+
1
c
vector
V
i
×
vector
B
(14
.
11)
which carries two messages:
•
the notion of a vanishing electric field is a question of the frame of reference.
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 Fall '08
 RABE
 Physics, Electron, Plasma, group velocity, Langmuir waves

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