VIDEOS 1 TO 50

Mod-01 Lec-18 Weibel Instability

Published: 2013/04/25

Channel: nptelhrd

2e The two-stream instability

Published: 2015/09/15

Channel: Plasma Physics and Applications

Quark-gluon plasma instability

Published: 2011/11/17

Channel: soundofthelittlebang

Quark-gluon plasma instability (with graphs)

Published: 2011/11/17

Channel: soundofthelittlebang

Mod-01 Lec-12 Two Stream Instability

Published: 2013/04/25

Channel: nptelhrd

Weibel Family Montage Video

Published: 2015/08/03

Channel: Colin Weibel

weibel

Published: 2013/01/01

Channel: Reinhard Beu

Double Weibel

Published: 2013/09/04

Channel: doubleweibel

Peter Weibel - Visionen | ZKM | @ trash-tv® film (english subtitles)

Published: 2011/01/10

Channel: trashtvfilm

Instability - Video Learning - WizScience.com

Published: 2015/09/10

Channel: Wiz Science™

Weibel School playground

Published: 2016/05/31

Channel: BusyKids TV

Story Connection: Weibel Family Winery

Published: 2015/09/25

Channel: Story Connection TV

Mod-01 Lec-09 Instability and Transition of Fluid Flows

Published: 2012/06/26

Channel: nptelhrd

Weibel SWISS QUALITY OIL

Published: 2017/04/05

Channel: Bernhard Hürzeler

18A Plasma Instabilities | Introduction to Plasma Physics by J D Callen

Published: 2015/12/24

Channel: Lucius Fox

thé-junior-weibel trailer

Published: 2012/08/20

Channel: milesdjw

19A Types Of Instabilities | Introduction to Plasma Physics by J D Callen

Published: 2015/12/24

Channel: Lucius Fox

Mod-01 Lec-25 Tokamak

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-11 Energy Flow with an Electrostatic Wave

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-10 Electrostatic Waves in Plasmas

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-19 Rayleigh Taylor Instability

Published: 2013/04/25

Channel: nptelhrd

Hene Weibel - Gottardo 16 EM

Published: 2016/03/23

Channel: Hene Weibel

Simulation of Quark-Gluon Plasma Instabilities

Published: 2011/09/22

Channel: TU Wien

Quark-gluon plasma instability (2D slice)

Published: 2011/11/17

Channel: soundofthelittlebang

Mod-01 Lec-39 Diffusion in magnetized plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-32 Electrostatic waves in magnetized plasma

Published: 2013/04/25

Channel: nptelhrd

Plasma Stability Assay

Published: 2015/06/22

Channel: The Edge

By evolution of 2D current sheets, drift-kink instability

Published: 2014/04/24

Channel: Pete Gingell

Weibel's Youtube Edit

Published: 2011/03/30

Channel: stephen weibel

Farley - Buneman

Published: 2008/04/27

Channel: Shahab Arabshahi

LINDSAY BUCHMAN : PROLIFERATION

Published: 2015/09/17

Channel: ERIC MINH SWENSON

Plasmas Torn Apart

Published: 2012/02/19

Channel: astropage.eu

Mod-01 Lec-01 Introduction to Plasmas

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-15 Free Electron Laser

Published: 2013/04/25

Channel: nptelhrd

Energetic PIC response to (1,1) internal kink mode

Published: 2011/09/01

Channel: charlsonification

Mod-01 Lec-38 Diffusion in plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-34 VIasov theory of plasma waves

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-31 Low frequency EM waves magnetized plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-27 Auxiliary heating and current drive in tokamak

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-41 Laser interaction with plasmas embedded with clusters

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-16 Free Electron Laser: Energy gain

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-03 DC Conductivity and Negative Differential Conductivity

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-09 Electromagnetic Wave Propagation Inhomogeneous Plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-37 Anomalous resistivity in a plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-35 Landau damping and growth of waves

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-30 Electromagnetic propagation at oblique angles to magnetic field in a plasma

Published: 2013/04/25

Channel: nptelhrd

Mod-01 Lec-14 Cerenkov Free Electron Laser

Published: 2013/04/25

Channel: nptelhrd

Demystifying Medicine 2017: Glycoproteins, Allergy, and Other Diseases

Published: 2017/02/02

Channel: nihvcast

Mod-01 Lec-13 Relativistic electron Beam- Plasma Interaction

Published: 2013/04/25

Channel: nptelhrd

Mod-05 Lec-08 Vanderwaal\'s equation of states-part03

Published: 2014/07/22

Channel: nptelhrd

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 -ray bursts.

As a simple example of Weibel instability, consider an electron beam with density and initial velocity propagating in a plasma of density with velocity . 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 i.e. . Assume the electric field has the form

With the assumed spatial and time dependence, we may use and . From Faraday's Law, we may obtain the perturbation magnetic field

Consider the electron beam. We assume small perturbations, and so linearize the velocity and density . 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 and neglecting second-order terms. 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 and neglecting second-order terms. 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 , will add. 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 ( 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 to simplify the square root term, by noting that

The resulting dispersion relation is then much simpler

is purely imaginary. Writing

we see that , indeed corresponding to an instability.

The electromagnetic fields then have the form

Therefore, the electric and magnetic fields are out of phase, and by noting that

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 is on the order of the electron cyclotron frequency

- E.S. Weibel, Phys. Rev. Lett. 2, 83 (1959); Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution
- B.D. Fried, Phys. Fluids 2, 337 (1959); Mechanism for Instability of Transverse Plasma Waves
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