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.[1] 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.[2] 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=B0cos⁡(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]. The resulting current j=−enve 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.

## Contents

* 1 A Simple Example of Weibel Instability
* 2 See also
* 3 References
* 4 External Links

## A Simple Example of Weibel Instability

As a simple example of Weibel instability, consider an electron beam with density nb0 and initial velocity v0𝐳 propagating in a plasma of density np0=nb0 with velocity −v0𝐳. 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. 𝐁0=𝐄0=0. The perturbation will be taken as an electromagnetic wave propagating along x^, i.e., 𝐤=kx^. Assume the electric field has the form 𝐄1=Aei(kx−ωt)z^ With the assumed spatial and time dependence, we may use ∂∂t->−iω and ∇->ikx^. From Faraday's Law, we may obtain the perturbation magnetic field ∇×𝐄1=−∂𝐁1∂t⇒i𝐤×𝐄1=iω𝐁1⇒𝐁1=y^kωE1 Consider the electron beam. We assume small perturbations, and so linearize the velocity 𝐯b=𝐯b0+𝐯b1 and density nb=nb0+nb1. The goal is to find the perturbation electron beam current density 𝐉b1=−enb𝐯b=−enb0𝐯b1−enb1𝐯b0 where second-order terms have been neglected. To do that, we start with the fluid momentum equation for the electron beam m(∂𝐯b∂t+(𝐯b⋅∇)𝐯b)=−e(𝐄+𝐯b×𝐁) which can be simplified by noting that ∂𝐯b0∂t=∇⋅𝐯b0=0 and neglecting second-order terms. With the plane wave assumption for the derivatives, the momentum equation becomes −iωm𝐯b1=−e(𝐄1+𝐯b0×𝐁1) 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: vb1z=eE1miωvb1x=eE1miωkvb0ω

To find the perturbation density nb1, we use the fluid continuity equation for the electron beam ∂nb∂t+∇⋅(nb𝐯b)=0 which can again be simplified by noting that ∂nb0∂t=∇nb0=0 and neglecting second-order terms. The result is nb1=nb0kωvb1x

Using these results, we may use the equation for the beam perturbation current density given above to find Jb1x=−nb0e2E1kvb0imω2Jb1z=−nb0e2E11imω(1+k2vb02ω2)

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 v0, 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 v02, will add. The net current density perturbation is therefore 𝐉1=−2nb0e2E11imω(1+k2vb02ω2)z^

The dispersion relation can now be found from Maxwell's Equations: ∇×𝐄1=iω𝐁1∇×𝐁1=μ0𝐉1−iωε0μ0𝐄1 ⇒∇×∇×𝐄1=−∇2𝐄1+∇(∇⋅𝐄1)=k2𝐄1+i𝐤(i𝐤⋅𝐄1)=k2𝐄1=iω∇×𝐁1=iωc2ε0𝐉1+ω2c2𝐄1 where c=1ε0μ0 is the speed of light in free space. By defining the effective plasma frequency ωp2=2nb0e2ε0m, the equation above results in k2−ω2c2=−ωp2c2(1+k2v02ω2)⇒ω4−ω2(ωp2+k2c2)−ωp2k2v02=0 This bi-quadratic equation may be easily solved to give the dispersion relation ω2=12(ωp2+k2c2±(ωp2+k2c2)2+4ωp2k2v02) In the search for instabilities, we look for ℑ(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 non-relativistic assumption v0≪c to simplify the square root term, by noting that (ωp2+k2c2)2+4ωp2k2v02=(ωp2+k2c2)(1+4ωp2k2v02(ωp2+k2c2)2)1/2≈(ωp2+k2c2)(1+2ωp2k2v02(ωp2+k2c2)2) The resulting dispersion relation is then much simpler ω2=−ωp2k2v02ωp2+k2c2<0 ω is purely imaginary. Writing ω=iγ γ=ωpkv0(ωp2+k2c2)1/2=ωpv0c1(1+ωp2k2c2)1/2 we see that ℑ(ω)>0, indeed corresponding to an instability.

The electromagnetic fields then have the form 𝐄1=z^Aeγt+ikx𝐁1=y^kωE1=y^kiγAeγt+ikx Therefore, the electric and magnetic fields are 90∘ out of phase, and by noting that |B1||E1|=kγ∝cv0≫1 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 γ is on the order of the electron cyclotron frequency γ∼ωpv0c∼ωc⇒B∼meωpv0c

## See also

* Chromo–Weibel instability

## References

1. ↑ Weibel, Erich S. (1959-02-01). "Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution". Physical Review Letters (American Physical Society (APS)) 2 (3): 83–84. doi:10.1103/physrevlett.2.83. ISSN 0031-9007. Bibcode: 1959PhRvL...2...83W.
2. ↑ Fried, Burton D. (1959). "Mechanism for Instability of Transverse Plasma Waves". Physics of Fluids (AIP Publishing) 2 (3): 337. doi:10.1063/1.1705933. ISSN 0031-9171. Bibcode: 1959PhFl....2..337F.

## External Links

* IIT Delhi lecture on Weibel instability

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