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Wikipedia / Weibel instability

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.

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 sheets generate magnetic field that enhances the original field and thus perturbation grows.

-ray bursts.

A Simple Example of Weibel Instability

. 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.

. Assume the electric field We can decompose the above equations in components, paying attention to the cross product

Analogous expressions can be written for the perturbation current density of the left-moving plasma. By noting that the x, 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

The dispersion relation can now be found from Maxwell's Equations is the speed of light is assumed real). Therefore, we must take the dispersion relation/mode corresponding to the minus sign in the equation above.

, indeed corresponding to an instability.

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