The Weibel instability is a plasma instability present in homogeneous or nearly homogeneous electromagnetic plasmas which possess an anisotropy in momentum (velocity) space. This anisotropy is most generally understood as two temperatures in different directions. Burton Fried showed that this instability can be understood more simply as the superposition of many counterstreaming beams. In this sense, it is like the twostream instability except that the perturbations are electromagnetic and result in filamentation as opposed to electrostatic perturbations which would result in charge bunching. In the linear limit the instability causes exponential growth of electromagnetic fields in the plasma which help restore momentum space isotropy. In very extreme cases, the Weibel instability is related to one or twodimensional stream instabilities.
Consider an electronion plasma in which the ions are fixed and the electrons are hotter in the ydirection than in x or zdirection.
To see how magnetic field perturbation would grow, suppose a field B = B cos kx spontaneously arises from noise. The Lorentz force then bends the electron trajectories with the result that upwardmovingev x B electrons congregate at B and downwardmoving ones at A. The resulting current j = en ve sheets generate magnetic field that enhances the original field and thus perturbation grows.
Weibel instability is also common in astrophysical plasmas, such as collisionless shock formation in supernova remnants and
γ
ray bursts.
As a simple example of Weibel instability, consider an electron beam with density
n
b
0
and initial velocity
v
0
z
propagating in a plasma of density
n
p
0
=
n
b
0
with velocity
−
v
0
z
. The analysis below will show how an electromagnetic perturbation in the form of a plane wave gives rise to a Weibel instability in this simple anisotropic plasma system. We assume a nonrelativistic plasma for simplicity.
We assume there is no background electric or magnetic field i.e.
B
0
=
E
0
=
0
. The perturbation will be taken as an electromagnetic wave propagating along
x
^
i.e.
k
=
k
x
^
. Assume the electric field has the form
E
1
=
A
e
i
(
k
x
−
ω
t
)
z
With the assumed spatial and time dependence, we may use
∂
∂
t
→
−
i
ω
and
∇
→
i
k
x
^
. From Faraday's Law, we may obtain the perturbation magnetic field
∇
×
E
1
=
−
∂
B
1
∂
t
⇒
i
k
×
E
1
=
i
ω
B
1
⇒
B
1
=
y
^
k
ω
E
1
Consider the electron beam. We assume small perturbations, and so linearize the velocity
v
b
=
v
b
0
+
v
b
1
and density
n
b
=
n
b
0
+
n
b
1
. The goal is to find the perturbation electron beam current density
J
b
1
=
−
e
n
b
v
b
=
−
e
n
b
0
v
b
1
+
−
e
n
b
1
v
b
0
where secondorder terms have been neglected. To do that, we start with the fluid momentum equation for the electron beam
m
(
∂
v
b
∂
t
+
(
v
b
⋅
∇
)
v
b
)
=
−
e
E
−
e
v
b
×
B
which can be simplified by noting that
∂
v
b
0
∂
t
=
∇
⋅
v
b
0
=
0
and neglecting secondorder terms. With the plane wave assumption for the derivatives, the momentum equation becomes
−
i
ω
m
v
b
1
=
−
e
E
1
−
e
v
b
0
×
B
1
We can decompose the above equations in components, paying attention to the cross product at the far right, and obtain the nonzero components of the beam velocity perturbation:
v
b
1
z
=
e
E
1
m
i
ω
v
b
1
x
=
e
E
1
m
i
ω
k
v
b
0
ω
To find the perturbation density
n
b
1
, we use the fluid continuity equation for the electron beam
∂
n
b
∂
t
+
∇
⋅
(
n
b
v
b
)
=
0
which can again be simplified by noting that
∂
n
b
0
∂
t
=
∇
n
b
0
=
0
and neglecting secondorder terms. The result is
n
b
1
=
n
b
0
k
ω
v
b
1
x
Using these results, we may use the equation for the beam perturbation current density given above to find
J
b
1
x
=
−
n
b
0
e
2
E
1
k
v
b
0
i
m
ω
2
J
b
1
z
=
−
n
b
0
e
2
E
1
1
i
m
ω
(
1
+
k
2
v
b
0
2
ω
2
)
Analogous expressions can be written for the perturbation current density of the leftmoving plasma. By noting that the xcomponent of the perturbation current density is proportional to
v
0
, we see that with our assumptions for the beam and plasma unperturbed densities and velocities the xcomponent of the net current density will vanish, whereas the zcomponents, which are proportional to
v
0
2
, will add. The net current density perturbation is therefore
J
1
=
−
2
n
b
0
e
2
E
1
1
i
m
ω
(
1
+
k
2
v
b
0
2
ω
2
)
z
^
The dispersion relation can now be found from Maxwell's Equations:
∇
×
E
1
=
i
ω
B
1
∇
×
B
1
=
μ
0
J
1
−
i
ω
ϵ
0
μ
0
E
1
⇒
∇
×
∇
E
1
=
−
∇
2
E
1
+
∇
(
∇
⋅
E
1
)
=
k
2
E
1
+
i
k
(
i
k
⋅
E
1
)
=
k
2
E
1
=
i
ω
∇
×
B
1
=
i
ω
c
2
ϵ
0
J
1
+
ω
2
c
2
E
1
where
c
=
1
ϵ
0
μ
0
is the speed of light in free space. By defining the effective plasma frequency
ω
p
2
=
2
n
b
0
e
2
ϵ
0
m
, the equation above results in
k
2
−
ω
2
c
2
=
−
ω
p
2
c
2
(
1
+
k
2
v
0
2
ω
2
)
⇒
ω
4
−
ω
2
(
ω
p
2
+
k
2
c
2
)
−
ω
p
2
k
2
v
0
2
=
0
This biquadratic equation may be easily solved to give the dispersion relation
ω
2
=
1
2
(
ω
p
2
+
k
2
c
2
±
(
ω
p
2
+
k
2
c
2
)
2
+
4
ω
p
2
k
2
v
0
2
)
In the search for instabilities, we look for
I
m
(
ω
)
≠
0
(
k
is assumed real). Therefore, we must take the dispersion relation/mode corresponding to the minus sign in the equation above.
To gain further insight on the instability, it is useful to harness our nonrelativistic assumption
v
0
<<
c
to simplify the square root term, by noting that
(
ω
p
2
+
k
2
c
2
)
2
+
4
ω
p
2
k
2
v
0
2
=
(
ω
p
2
+
k
2
c
2
)
(
1
+
4
ω
p
2
k
2
v
0
2
(
ω
p
2
+
k
2
c
2
)
2
)
1
/
2
≈
(
ω
p
2
+
k
2
c
2
)
(
1
+
2
ω
p
2
k
2
v
0
2
(
ω
p
2
+
k
2
c
2
)
2
)
The resulting dispersion relation is then much simpler
ω
2
=
−
ω
p
2
k
2
v
0
2
ω
p
2
+
k
2
c
2
<
0
ω
is purely imaginary. Writing
ω
=
i
γ
γ
=
ω
p
k
v
0
(
ω
p
2
+
k
2
c
2
)
1
/
2
=
ω
p
v
0
c
1
(
1
+
ω
p
2
k
2
c
2
)
1
/
2
we see that
I
m
(
ω
)
>
0
, indeed corresponding to an instability.
The electromagnetic fields then have the form
E
1
=
A
z
^
e
γ
t
e
i
k
x
B
1
=
y
^
k
ω
E
1
=
y
^
k
i
γ
A
e
γ
t
e
i
k
x
Therefore, the electric and magnetic fields are
90
o
out of phase, and by noting that

B
1


E
1

=
k
γ
∝
c
v
0
>>
1
so we see this is a primarily magnetic perturbation although there is a nonzero electric perturbation. The magnetic field growth results in the characteristic filamentation structure of Weibel instability. Saturation will happen when the growth rate
γ
is on the order of the electron cyclotron frequency
γ
∼
ω
p
v
0
c
∼
ω
c
⇒
B
∼
m
e
ω
p
v
0
c