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 counter-streaming beams. In this sense, it is like the two-stream 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 two-dimensional stream instabilities.
Consider an electron-ion plasma in which the ions are fixed and the electrons are hotter in the y-direction than in x or z-direction.
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 upward-moving-ev x B electrons congregate at B and downward-moving 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
A Simple Example of Weibel Instability
As a simple example of Weibel instability, consider an electron beam with density
We assume there is no background electric or magnetic field i.e.
With the assumed spatial and time dependence, we may use
Consider the electron beam. We assume small perturbations, and so linearize the velocity
where second-order terms have been neglected. To do that, we start with the fluid momentum equation for the electron beam
which can be simplified by noting that
We can decompose the above equations in components, paying attention to the cross product at the far right, and obtain the non-zero components of the beam velocity perturbation:
To find the perturbation density
which can again be simplified by noting that
Using these results, we may use the equation for the beam perturbation current density given above to find
Analogous expressions can be written for the perturbation current density of the left-moving plasma. By noting that the x-component of the perturbation current density is proportional to
The dispersion relation can now be found from Maxwell's Equations:
where
This bi-quadratic equation may be easily solved to give the dispersion relation
In the search for instabilities, we look for
To gain further insight on the instability, it is useful to harness our non-relativistic assumption
The resulting dispersion relation is then much simpler
we see that
The electromagnetic fields then have the form
Therefore, the electric and magnetic fields are
so we see this is a primarily magnetic perturbation although there is a non-zero electric perturbation. The magnetic field growth results in the characteristic filamentation structure of Weibel instability. Saturation will happen when the growth rate