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Mod-01 Lec-18 Weibel Instability
Mod-01 Lec-18 Weibel Instability
Published: 2013/04/25
Channel: nptelhrd
2e The two-stream instability
2e The two-stream instability
Published: 2015/09/15
Channel: Plasma Physics and Applications
Quark-gluon plasma instability
Quark-gluon plasma instability
Published: 2011/11/17
Channel: soundofthelittlebang
Quark-gluon plasma instability (with graphs)
Quark-gluon plasma instability (with graphs)
Published: 2011/11/17
Channel: soundofthelittlebang
Mod-01 Lec-12 Two Stream Instability
Mod-01 Lec-12 Two Stream Instability
Published: 2013/04/25
Channel: nptelhrd
Weibel Family Montage Video
Weibel Family Montage Video
Published: 2015/08/03
Channel: Colin Weibel
weibel
weibel
Published: 2013/01/01
Channel: Reinhard Beu
Double Weibel
Double Weibel
Published: 2013/09/04
Channel: doubleweibel
Peter Weibel - Visionen | ZKM | @ trash-tv® film (english subtitles)
Peter Weibel - Visionen | ZKM | @ trash-tv® film (english subtitles)
Published: 2011/01/10
Channel: trashtvfilm
Instability - Video Learning - WizScience.com
Instability - Video Learning - WizScience.com
Published: 2015/09/10
Channel: Wiz Science™
Weibel School playground
Weibel School playground
Published: 2016/05/31
Channel: BusyKids TV
Story Connection: Weibel Family Winery
Story Connection: Weibel Family Winery
Published: 2015/09/25
Channel: Story Connection TV
Mod-01 Lec-09 Instability and Transition of Fluid Flows
Mod-01 Lec-09 Instability and Transition of Fluid Flows
Published: 2012/06/26
Channel: nptelhrd
Weibel  SWISS QUALITY OIL
Weibel SWISS QUALITY OIL
Published: 2017/04/05
Channel: Bernhard Hürzeler
18A Plasma Instabilities | Introduction to Plasma Physics by J D Callen
18A Plasma Instabilities | Introduction to Plasma Physics by J D Callen
Published: 2015/12/24
Channel: Lucius Fox
thé-junior-weibel trailer
thé-junior-weibel trailer
Published: 2012/08/20
Channel: milesdjw
19A Types Of Instabilities | Introduction to Plasma Physics by J D Callen
19A Types Of Instabilities | Introduction to Plasma Physics by J D Callen
Published: 2015/12/24
Channel: Lucius Fox
Mod-01 Lec-25 Tokamak
Mod-01 Lec-25 Tokamak
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-11 Energy Flow with an Electrostatic Wave
Mod-01 Lec-11 Energy Flow with an Electrostatic Wave
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-10 Electrostatic Waves in Plasmas
Mod-01 Lec-10 Electrostatic Waves in Plasmas
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-19 Rayleigh Taylor Instability
Mod-01 Lec-19 Rayleigh Taylor Instability
Published: 2013/04/25
Channel: nptelhrd
Hene Weibel - Gottardo 16 EM
Hene Weibel - Gottardo 16 EM
Published: 2016/03/23
Channel: Hene Weibel
Simulation of Quark-Gluon Plasma Instabilities
Simulation of Quark-Gluon Plasma Instabilities
Published: 2011/09/22
Channel: TU Wien
Quark-gluon plasma instability (2D slice)
Quark-gluon plasma instability (2D slice)
Published: 2011/11/17
Channel: soundofthelittlebang
Mod-01 Lec-39 Diffusion in magnetized plasma
Mod-01 Lec-39 Diffusion in magnetized plasma
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-32 Electrostatic waves in magnetized plasma
Mod-01 Lec-32 Electrostatic waves in magnetized plasma
Published: 2013/04/25
Channel: nptelhrd
Plasma Stability Assay
Plasma Stability Assay
Published: 2015/06/22
Channel: The Edge
By evolution of 2D current sheets, drift-kink instability
By evolution of 2D current sheets, drift-kink instability
Published: 2014/04/24
Channel: Pete Gingell
Weibel
Weibel's Youtube Edit
Published: 2011/03/30
Channel: stephen weibel
Farley - Buneman
Farley - Buneman
Published: 2008/04/27
Channel: Shahab Arabshahi
LINDSAY BUCHMAN : PROLIFERATION
LINDSAY BUCHMAN : PROLIFERATION
Published: 2015/09/17
Channel: ERIC MINH SWENSON
Plasmas Torn Apart
Plasmas Torn Apart
Published: 2012/02/19
Channel: astropage.eu
Mod-01 Lec-01 Introduction to Plasmas
Mod-01 Lec-01 Introduction to Plasmas
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-15 Free Electron Laser
Mod-01 Lec-15 Free Electron Laser
Published: 2013/04/25
Channel: nptelhrd
Energetic PIC response to (1,1) internal kink mode
Energetic PIC response to (1,1) internal kink mode
Published: 2011/09/01
Channel: charlsonification
Mod-01 Lec-38 Diffusion in plasma
Mod-01 Lec-38 Diffusion in plasma
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-34 VIasov theory of plasma waves
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
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
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
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
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
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
Mod-01 Lec-09 Electromagnetic Wave Propagation Inhomogeneous Plasma
Published: 2013/04/25
Channel: nptelhrd
Mod-01 Lec-37 Anomalous resistivity in a plasma
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
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
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
Mod-01 Lec-14 Cerenkov Free Electron Laser
Published: 2013/04/25
Channel: nptelhrd
Demystifying Medicine 2017: Glycoproteins, Allergy, and Other Diseases
Demystifying Medicine 2017: Glycoproteins, Allergy, and Other Diseases
Published: 2017/02/02
Channel: nihvcast
Mod-01 Lec-13 Relativistic electron Beam- Plasma Interaction
Mod-01 Lec-13 Relativistic electron Beam- Plasma Interaction
Published: 2013/04/25
Channel: nptelhrd
Mod-05 Lec-08 Vanderwaal\
Mod-05 Lec-08 Vanderwaal\'s equation of states-part03
Published: 2014/07/22
Channel: nptelhrd
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WIKIPEDIA ARTICLE

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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.

A Simple Example of Weibel Instability[edit]

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

References[edit]

See also[edit]

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