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.
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[clarification needed].
Weibel instability is also common in astrophysical plasmas, such as collisionless shock formation in supernova remnants and
As a simple example of Weibel instability, consider an electron beam with density
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 non-relativistic plasma for simplicity.
We assume there is no background electric or magnetic field i.e.
The perturbation will be taken as an electromagnetic wave propagating along
From Faraday's Law, we may obtain the perturbation magnetic field Consider the electron beam.
We assume small perturbations, and so linearize the velocity
The goal is to find the perturbation electron beam current density 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
With the plane wave assumption for the derivatives, the momentum equation becomes 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
, we use the fluid continuity equation for the electron beam which can again be simplified by noting that
The result is 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
, we see that with our assumptions for the beam and plasma unperturbed densities and velocities the x-component of the net current density will vanish, whereas the z-components, which are proportional to
The net current density perturbation is therefore The dispersion relation can now be found from Maxwell's Equations: where
is the speed of light in free space.
By defining the effective plasma frequency
, the equation above results in 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
to simplify the square root term, by noting that The resulting dispersion relation is then much simpler
The magnetic field growth results in the characteristic filamentation structure of Weibel instability.